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

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%------------------------------------------------------------------------------
% File     : Leo-III---1.8.0
% Problem  : SCT105+1 : TPTP v9.3.1. Released v5.2.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39

% Computer : n026.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8046.5625MB
% OS       : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Sun Sep 27 08:27:16 AM UTC 2026

% Result   : Theorem 141.70s 33.17s
% Output   : Refutation 136.72s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :   16
%            Number of leaves      :  527
% Syntax   : Number of formulae    : 1164 ( 301 unt;   0 typ;   7 def)
%            Number of atoms       : 3346 ( 718 equ;   0 cnn)
%            Maximal formula atoms :   40 (   2 avg)
%            Number of connectives : 21141 ( 503   ~; 287   |; 284   &;18511   @)
%                                         ( 101 <=>;1455  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   28 (  10 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   94 (  91 usr;  22 con; 0-6 aty)
%            Number of variables   : 3830 (   2   ^;3765   !;  56   ?;3830   :)
%                                         (   0  !>;   0  ?*;   0  @-;   7  @+)

% Comments : 
%------------------------------------------------------------------------------
thf(c_Relation_Otrans_decl,type,
    c_Relation_Otrans: $i > $i > $o ).

thf(tc_Arrow__Order__Mirabelle_Oalt_decl,type,
    tc_Arrow__Order__Mirabelle_Oalt: $i ).

thf(v_L_decl,type,
    v_L: $i ).

thf(c_member_decl,type,
    c_member: $i > $i > $i > $o ).

thf(tc_prod_decl,type,
    tc_prod: $i > $i > $i ).

thf(c_Product__Type_OPair_decl,type,
    c_Product__Type_OPair: $i > $i > $i > $i > $i ).

thf(class_Lattices_Osemilattice__sup_decl,type,
    class_Lattices_Osemilattice__sup: $i > $o ).

thf(c_Lattices_Osemilattice__sup__class_Osup_decl,type,
    c_Lattices_Osemilattice__sup__class_Osup: $i > $i > $i > $i ).

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

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

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

thf(tc_HOL_Obool_decl,type,
    tc_HOL_Obool: $i ).

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

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

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

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

thf(class_Lattices_Olattice_decl,type,
    class_Lattices_Olattice: $i > $o ).

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

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

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

thf(class_Lattices_Obounded__lattice__bot_decl,type,
    class_Lattices_Obounded__lattice__bot: $i > $o ).

thf(c_Set_Oinsert_decl,type,
    c_Set_Oinsert: $i > $i > $i > $i ).

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

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

thf(c_Transitive__Closure_Ortrancl_decl,type,
    c_Transitive__Closure_Ortrancl: $i > $i > $i ).

thf(c_Relation_OId_decl,type,
    c_Relation_OId: $i > $i ).

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

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

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

thf(c_Relation_ORange_decl,type,
    c_Relation_ORange: $i > $i > $i > $i ).

thf(c_Transitive__Closure_Otrancl_decl,type,
    c_Transitive__Closure_Otrancl: $i > $i > $i ).

thf(c_Wellfounded_Owf_decl,type,
    c_Wellfounded_Owf: $i > $i > $o ).

thf(c_Wellfounded_Omlex__prod_decl,type,
    c_Wellfounded_Omlex__prod: $i > $i > $i > $i ).

thf(c_Relation_Oantisym_decl,type,
    c_Relation_Oantisym: $i > $i > $o ).

thf(c_Relation_OId__on_decl,type,
    c_Relation_OId__on: $i > $i > $i ).

thf(c_Wellfounded_Omin__ext_decl,type,
    c_Wellfounded_Omin__ext: $i > $i > $i ).

thf(c_Wellfounded_Omeasure_decl,type,
    c_Wellfounded_Omeasure: $i > $i ).

thf(c_Relation_Oinv__image_decl,type,
    c_Relation_Oinv__image: $i > $i > $i > $i ).

thf(c_Wellfounded_Oless__than_decl,type,
    c_Wellfounded_Oless__than: $i ).

thf(c_Relation_ODomain_decl,type,
    c_Relation_ODomain: $i > $i > $i > $i ).

thf(c_Set_Othe__elem_decl,type,
    c_Set_Othe__elem: $i > $i > $i ).

thf(c_Wellfounded_Opred__nat_decl,type,
    c_Wellfounded_Opred__nat: $i ).

thf(c_Nitpick_Ofold__graph_H_decl,type,
    c_Nitpick_Ofold__graph_H: $i > $i > $i > $i > $i > $i > $o ).

thf(c_Relation_OImage_decl,type,
    c_Relation_OImage: $i > $i > $i > $i > $i ).

thf(c_Relation_Osingle__valued_decl,type,
    c_Relation_Osingle__valued: $i > $i > $i > $o ).

thf(c_Recdef_Owfrec_decl,type,
    c_Recdef_Owfrec: $i > $i > $i > $i > $i ).

thf(c_Recdef_Ocut_decl,type,
    c_Recdef_Ocut: $i > $i > $i > $i > $i > $i ).

thf(c_Wellfounded_Oacyclic_decl,type,
    c_Wellfounded_Oacyclic: $i > $i > $o ).

thf(c_Order__Relation_Opartial__order__on_decl,type,
    c_Order__Relation_Opartial__order__on: $i > $i > $i > $o ).

thf(c_Order__Relation_Opreorder__on_decl,type,
    c_Order__Relation_Opreorder__on: $i > $i > $i > $o ).

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

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

thf(c_Relation_Ototal__on_decl,type,
    c_Relation_Ototal__on: $i > $i > $i > $o ).

thf(c_Relation_OField_decl,type,
    c_Relation_OField: $i > $i > $i ).

thf(c_Wellfounded_Omax__ext_decl,type,
    c_Wellfounded_Omax__ext: $i > $i > $i ).

thf(c_Nitpick_Owf__wfrec_H_decl,type,
    c_Nitpick_Owf__wfrec_H: $i > $i > $i > $i > $i > $i ).

thf(c_Nitpick_Owf__wfrec_decl,type,
    c_Nitpick_Owf__wfrec: $i > $i > $i > $i > $i ).

thf(c_FunDef_Opair__less_decl,type,
    c_FunDef_Opair__less: $i ).

thf(c_Wellfounded_Olex__prod_decl,type,
    c_Wellfounded_Olex__prod: $i > $i > $i > $i > $i ).

thf(c_FunDef_Oreduction__pair_decl,type,
    c_FunDef_Oreduction__pair: $i > $i > $o ).

thf(c_FunDef_Orp__inv__image_decl,type,
    c_FunDef_Orp__inv__image: $i > $i > $i > $i > $i ).

thf(c_Wellfounded_OwfP_decl,type,
    c_Wellfounded_OwfP: $i > $i > $o ).

thf(c_FunDef_Oin__rel_decl,type,
    c_FunDef_Oin__rel: $i > $i > $i > $i ).

thf(c_Order__Relation_Olinear__order__on_decl,type,
    c_Order__Relation_Olinear__order__on: $i > $i > $i > $o ).

thf(c_Relation_Orefl__on_decl,type,
    c_Relation_Orefl__on: $i > $i > $i > $o ).

thf(c_Recdef_Osame__fst_decl,type,
    c_Recdef_Osame__fst: $i > $i > $i > $i > $i ).

thf(c_Equiv__Relations_Oequiv_decl,type,
    c_Equiv__Relations_Oequiv: $i > $i > $i > $o ).

thf(c_Relation_Orel__comp_decl,type,
    c_Relation_Orel__comp: $i > $i > $i > $i > $i > $i ).

thf(c_Relation_Oirrefl_decl,type,
    c_Relation_Oirrefl: $i > $i > $o ).

thf(c_Recdef_Owfrec__rel_decl,type,
    c_Recdef_Owfrec__rel: $i > $i > $i > $i > $i > $i > $o ).

thf(c_Wellfounded_Oacc_decl,type,
    c_Wellfounded_Oacc: $i > $i > $i ).

thf(c_FunDef_Opair__leq_decl,type,
    c_FunDef_Opair__leq: $i ).

thf(c_Product__Type_Ocurry_decl,type,
    c_Product__Type_Ocurry: $i > $i > $i > $i > $i > $i > $i ).

thf(c_Order__Relation_Owell__order__on_decl,type,
    c_Order__Relation_Owell__order__on: $i > $i > $i > $o ).

thf(c_Product__Type_Oprod_Oprod__rec_decl,type,
    c_Product__Type_Oprod_Oprod__rec: $i > $i > $i > $i > $i > $i ).

thf(c_FunDef_Omin__weak_decl,type,
    c_FunDef_Omin__weak: $i ).

thf(c_FunDef_Omax__weak_decl,type,
    c_FunDef_Omax__weak: $i ).

thf(c_FunDef_Omax__strict_decl,type,
    c_FunDef_Omax__strict: $i ).

thf(c_FunDef_Omin__strict_decl,type,
    c_FunDef_Omin__strict: $i ).

thf(c_Order__Relation_Ostrict__linear__order__on_decl,type,
    c_Order__Relation_Ostrict__linear__order__on: $i > $i > $i > $o ).

thf(c_Nitpick_Orefl_H_decl,type,
    c_Nitpick_Orefl_H: $i > $i > $o ).

thf(c_Product__Type_Ointernal__split_decl,type,
    c_Product__Type_Ointernal__split: $i > $i > $i > $i > $i > $i ).

thf(c_Arrow__Order__Mirabelle_OLin_decl,type,
    c_Arrow__Order__Mirabelle_OLin: $i ).

thf(sk1_decl,type,
    sk1: $o ).

thf(sk2_decl,type,
    sk2: $i ).

thf(sk3_decl,type,
    sk3: $i ).

thf(sk4_decl,type,
    sk4: $i ).

thf(sk5_decl,type,
    sk5: $i ).

thf(sk6_decl,type,
    sk6: $i ).

thf(sk7_decl,type,
    sk7: $i ).

thf(sk26_decl,type,
    sk26: $i > $i > $i ).

thf(sk2_def,definition,
    ( sk2
    = ( @+[A: $i] : ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) ) ).

thf(sk3_def,definition,
    ( sk3
    = ( @+[A: $i] :
          ~ ! [B: $i] :
              ( ( A != B )
             => ( ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
                | ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) ) ) ) ).

thf(sk4_def,definition,
    ( sk4
    = ( @+[A: $i] :
          ~ ( ( sk3 != A )
           => ( ( v_L @ ( sk3 @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
              | ( v_L @ ( A @ ( sk3 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) ) ) ) ).

thf(sk5_def,definition,
    ( sk5
    = ( @+[A: $i] : ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) ) ).

thf(sk6_def,definition,
    ( sk6
    = ( @+[A: $i] :
          ~ ! [B: $i] :
              ( ( A != B )
             => ( ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
                | ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) ) ) ) ).

thf(sk7_def,definition,
    ( sk7
    = ( @+[A: $i] :
          ~ ( ( sk6 != A )
           => ( ( v_L @ ( A @ ( sk6 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
              | ( v_L @ ( sk6 @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) ) ) ) ).

thf(sk26_def,definition,
    ( sk26
    = ( ^ [A: $i,B: $i] :
        @+[C: $i] : ( B @ ( C @ ( A @ c_member ) ) ) ) ) ).

thf(155,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
          & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__le__not__le) ).

thf(1036,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( ( ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
         => ( ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(36,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( C = B )
       => ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I1_J) ).

thf(629,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( C = B )
       => ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(408,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] : ( D @ ( C @ hAPP ) @ hBOOL )
    <=> ! [D: $i,E: $i] : ( E @ ( D @ ( A @ ( B @ c_Product__Type_OPair ) ) ) @ ( C @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__paired__All) ).

thf(2031,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i,E: $i] : ( E @ ( D @ ( A @ ( B @ c_Product__Type_OPair ) ) ) @ ( C @ hAPP ) @ hBOOL )
       => ! [D: $i] : ( D @ ( C @ hAPP ) @ hBOOL ) )
      & ( ! [D: $i] : ( D @ ( C @ hAPP ) @ hBOOL )
       => ! [D: $i,E: $i] : ( E @ ( D @ ( A @ ( B @ c_Product__Type_OPair ) ) ) @ ( C @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[408]) ).

thf(463,axiom,
    ( c_FunDef_Omin__strict
    = ( c_FunDef_Opair__less @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Wellfounded_Omin__ext ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_min__strict__def) ).

thf(2263,plain,
    ( c_FunDef_Omin__strict
    = ( c_FunDef_Opair__less @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Wellfounded_Omin__ext ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).

thf(117,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Wellfounded_Owf ) )
     => ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( B @ c_Wellfounded_Owf ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__trancl) ).

thf(893,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Wellfounded_Owf ) )
     => ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( B @ c_Wellfounded_Owf ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(503,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( C @ c_Wellfounded_Oacc ) @ ( A @ ( C @ c_Wellfounded_Oacc ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acc__subset) ).

thf(2427,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( C @ c_Wellfounded_Oacc ) @ ( A @ ( C @ c_Wellfounded_Oacc ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[503]) ).

thf(130,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__leE) ).

thf(941,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(6,axiom,
    tc_HOL_Obool @ class_Lattices_Osemilattice__sup,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL__Obool__Lattices_Osemilattice__sup) ).

thf(555,plain,
    tc_HOL_Obool @ class_Lattices_Osemilattice__sup,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(224,axiom,
    ( c_FunDef_Opair__less
    = ( c_Wellfounded_Oless__than @ ( c_Wellfounded_Oless__than @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Wellfounded_Olex__prod ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__less__def) ).

thf(1295,plain,
    ( c_FunDef_Opair__less
    = ( c_Wellfounded_Oless__than @ ( c_Wellfounded_Oless__than @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Wellfounded_Olex__prod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(387,axiom,
    ! [A: $i,B: $i] :
      ~ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ c_member ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_emptyE) ).

thf(1950,plain,
    ! [A: $i,B: $i] :
      ~ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ c_member ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).

thf(442,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Opred__nat @ ( tc_Nat_Onat @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pred__nat__trancl__eq__le) ).

thf(2163,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( c_Wellfounded_Opred__nat @ ( tc_Nat_Onat @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) ) )
      & ( ( c_Wellfounded_Opred__nat @ ( tc_Nat_Onat @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[442]) ).

thf(42,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__trans) ).

thf(645,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(299,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( D @ ( E @ c_Relation_Orefl__on ) ) )
     => ( ( A @ ( B @ ( E @ c_Relation_Orefl__on ) ) )
       => ( A @ ( C @ ( tc_HOL_Obool @ ( E @ ( E @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( D @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( E @ c_Relation_Orefl__on ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__on__Un) ).

thf(1586,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( D @ ( E @ c_Relation_Orefl__on ) ) )
     => ( ( A @ ( B @ ( E @ c_Relation_Orefl__on ) ) )
       => ( A @ ( C @ ( tc_HOL_Obool @ ( E @ ( E @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( D @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( E @ c_Relation_Orefl__on ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).

thf(510,axiom,
    ( c_FunDef_Omin__weak
    = ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_Set_Oinsert ) ) @ ( c_FunDef_Opair__leq @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Wellfounded_Omin__ext ) @ ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_min__weak__def) ).

thf(2452,plain,
    ( c_FunDef_Omin__weak
    = ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_Set_Oinsert ) ) @ ( c_FunDef_Opair__leq @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Wellfounded_Omin__ext ) @ ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[510]) ).

thf(93,axiom,
    ! [A: $i,B: $i] : ( A @ ( B @ c_Relation_OId__on ) @ ( B @ c_Relation_Oantisym ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_antisym__Id__on) ).

thf(808,plain,
    ! [A: $i,B: $i] : ( A @ ( B @ c_Relation_OId__on ) @ ( B @ c_Relation_Oantisym ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(809,plain,
    ! [B: $i,A: $i] : ( A @ ( B @ c_Relation_OId__on ) @ ( B @ c_Relation_Oantisym ) ),
    inference(cnf,[status(esa)],[808]) ).

thf(487,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( D @ ( E @ c_Equiv__Relations_Oequiv ) ) )
     => ( ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( E @ c_Set_Oinsert ) ) @ ( C @ ( E @ ( E @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( E @ c_Set_Oinsert ) ) @ ( C @ ( E @ ( E @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( D @ ( B @ ( E @ c_member ) ) )
         => ( C @ ( B @ ( A @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__equiv__class) ).

thf(2363,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( D @ ( E @ c_Equiv__Relations_Oequiv ) ) )
     => ( ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( E @ c_Set_Oinsert ) ) @ ( C @ ( E @ ( E @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( E @ c_Set_Oinsert ) ) @ ( C @ ( E @ ( E @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( D @ ( B @ ( E @ c_member ) ) )
         => ( C @ ( B @ ( A @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[487]) ).

thf(20,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__left__commute) ).

thf(582,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(69,axiom,
    tc_Nat_Onat @ class_Lattices_Osemilattice__sup,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Lattices_Osemilattice__sup) ).

thf(730,plain,
    tc_Nat_Onat @ class_Lattices_Osemilattice__sup,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(416,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ( C @ ( B @ ( F @ ( E @ ( A @ ( D @ c_Recdef_Ocut ) ) ) ) @ hAPP ) )
        = ( C @ ( E @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_tfl__cut__apply) ).

thf(2062,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ( C @ ( B @ ( F @ ( E @ ( A @ ( D @ c_Recdef_Ocut ) ) ) ) @ hAPP ) )
        = ( C @ ( E @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[416]) ).

thf(350,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ c_FunDef_Oin__rel ) ) @ hAPP ) @ hAPP ) @ hBOOL )
    <=> ( C @ ( A @ ( B @ ( D @ ( E @ c_Product__Type_OPair ) ) ) @ ( D @ ( E @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__rel__def) ).

thf(1808,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( C @ ( A @ ( B @ ( D @ ( E @ c_Product__Type_OPair ) ) ) @ ( D @ ( E @ tc_prod ) @ c_member ) ) )
       => ( A @ ( B @ ( C @ ( D @ ( E @ c_FunDef_Oin__rel ) ) @ hAPP ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( C @ ( D @ ( E @ c_FunDef_Oin__rel ) ) @ hAPP ) @ hAPP ) @ hBOOL )
       => ( C @ ( A @ ( B @ ( D @ ( E @ c_Product__Type_OPair ) ) ) @ ( D @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[350]) ).

thf(293,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__empty) ).

thf(1569,plain,
    ! [A: $i,B: $i] :
      ( ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).

thf(165,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ ( C @ ( D @ c_Nitpick_Ofold__graph_H ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__graph_H_Oequations_I1_J) ).

thf(1073,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ ( C @ ( D @ c_Nitpick_Ofold__graph_H ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(469,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Oless__than @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__than__iff) ).

thf(2282,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( c_Wellfounded_Oless__than @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) ) )
      & ( ( c_Wellfounded_Oless__than @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[469]) ).

thf(111,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( E @ ( F @ ( G @ c_member ) ) )
     => ( ( A @ ( tc_HOL_Obool @ ( G @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( F @ ( G @ c_Set_Oinsert ) ) @ ( E @ ( tc_HOL_Obool @ ( G @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( C @ ( D @ ( G @ c_Nitpick_Ofold__graph_H ) ) ) ) ) )
       => ( A @ ( F @ ( C @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( C @ ( D @ ( G @ c_Nitpick_Ofold__graph_H ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__graph_H_Ointros_I2_J) ).

thf(878,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( E @ ( F @ ( G @ c_member ) ) )
     => ( ( A @ ( tc_HOL_Obool @ ( G @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( F @ ( G @ c_Set_Oinsert ) ) @ ( E @ ( tc_HOL_Obool @ ( G @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( C @ ( D @ ( G @ c_Nitpick_Ofold__graph_H ) ) ) ) ) )
       => ( A @ ( F @ ( C @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( C @ ( D @ ( G @ c_Nitpick_Ofold__graph_H ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(346,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__into__trancl2) ).

thf(1793,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).

thf(228,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_FunDef_Oreduction__pair ) )
     => ( A @ ( C @ ( B @ ( D @ c_FunDef_Orp__inv__image ) ) ) @ ( B @ c_FunDef_Oreduction__pair ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rp__inv__image__rp) ).

thf(1305,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_FunDef_Oreduction__pair ) )
     => ( A @ ( C @ ( B @ ( D @ c_FunDef_Orp__inv__image ) ) ) @ ( B @ c_FunDef_Oreduction__pair ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(156,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( B @ ( D @ c_member ) ) )
       => ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__single__insert) ).

thf(1041,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( B @ ( D @ c_member ) ) )
       => ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(271,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ c_Set_Oinsert ) ) )
      = ( A @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( B @ ( D @ c_Set_Oinsert ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__commute) ).

thf(1486,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ c_Set_Oinsert ) ) )
      = ( A @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( B @ ( D @ c_Set_Oinsert ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(378,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ c_fequal @ ( B @ ( C @ c_member ) ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__mem) ).

thf(1917,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B = A )
       => ( A @ c_fequal @ ( B @ ( C @ c_member ) ) ) )
      & ( ( A @ c_fequal @ ( B @ ( C @ c_member ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[378]) ).

thf(1918,plain,
    ( ! [A: $i,B: $i] :
        ( ( B = A )
       => ! [C: $i] : ( A @ c_fequal @ ( B @ ( C @ c_member ) ) ) )
    & ! [A: $i,B: $i] :
        ( ? [C: $i] : ( A @ c_fequal @ ( B @ ( C @ c_member ) ) )
       => ( B = A ) ) ),
    inference(miniscope,[status(thm)],[1917]) ).

thf(1919,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ c_fequal @ ( E @ ( F @ c_member ) ) )
        | ( E != D ) )
      & ( ( B = A )
        | ~ ( A @ c_fequal @ ( B @ ( C @ c_member ) ) ) ) ),
    inference(cnf,[status(esa)],[1918]) ).

thf(1921,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ c_fequal @ ( B @ ( C @ c_member ) ) )
      | ( B != A ) ),
    inference(cnfConj,[status(thm)],[1919]) ).

thf(1923,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ c_fequal @ ( B @ ( C @ c_member ) ) )
      | ( B != A ) ),
    inference(lifteq,[status(thm)],[1921]) ).

thf(1924,plain,
    ! [B: $i,A: $i] : ( A @ c_fequal @ ( A @ ( B @ c_member ) ) ),
    inference(simp,[status(thm)],[1923]) ).

thf(147,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__supI1) ).

thf(1005,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(519,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ c_Relation_OField ) )
      = ( A @ ( C @ c_Relation_OField ) @ ( B @ ( C @ c_Relation_OField ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Field__Un) ).

thf(2495,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ c_Relation_OField ) )
      = ( A @ ( C @ c_Relation_OField ) @ ( B @ ( C @ c_Relation_OField ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[519]) ).

thf(493,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] : ( A @ ( B @ c_Wellfounded_Oacc ) @ ( C @ ( B @ c_member ) ) )
     => ( A @ ( B @ c_Wellfounded_Owf ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acc__wfI) ).

thf(2381,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] : ( A @ ( B @ c_Wellfounded_Oacc ) @ ( C @ ( B @ c_member ) ) )
     => ( A @ ( B @ c_Wellfounded_Owf ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[493]) ).

thf(109,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( B != A )
       => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I12_J) ).

thf(873,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( B != A )
       => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(203,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__not__le) ).

thf(1205,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) )
         => ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(275,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B = A )
          | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__le__imp__less__or__eq) ).

thf(1497,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B = A )
          | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(167,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( B != A )
       => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__neq__le__trans) ).

thf(1078,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( B != A )
       => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(316,axiom,
    ! [A: $i] :
      ( ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Transitive__Closure_Otrancl ) )
      = ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__empty) ).

thf(1655,plain,
    ! [A: $i] :
      ( ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Transitive__Closure_Otrancl ) )
      = ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).

thf(230,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_OwfP ) )
     => ( ( B @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ c_Wellfounded_OwfP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wfP__subset) ).

thf(1310,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_OwfP ) )
     => ( ( B @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ c_Wellfounded_OwfP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(363,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Transitive__Closure_Ortrancl ) )
      = ( A @ ( A @ ( B @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( B @ ( B @ c_Relation_Orel__comp ) ) ) ) @ ( B @ c_Relation_OId @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__unfold) ).

thf(1867,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Transitive__Closure_Ortrancl ) )
      = ( A @ ( A @ ( B @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( B @ ( B @ c_Relation_Orel__comp ) ) ) ) @ ( B @ c_Relation_OId @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).

thf(437,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_member ) ) )
     => ( B @ ( C @ ( D @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DiffD1) ).

thf(2141,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_member ) ) )
     => ( B @ ( C @ ( D @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[437]) ).

thf(65,axiom,
    tc_HOL_Obool @ class_Orderings_Opreorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL__Obool__Orderings_Opreorder) ).

thf(710,plain,
    tc_HOL_Obool @ class_Orderings_Opreorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(5,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( B = A )
       => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__eq__refl) ).

thf(551,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( B = A )
       => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(552,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
      | ( B != A )
      | ~ ( C @ class_Orderings_Opreorder ) ),
    inference(cnf,[status(esa)],[551]) ).

thf(553,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
      | ~ ( C @ class_Orderings_Opreorder )
      | ( B != A ) ),
    inference(lifteq,[status(thm)],[552]) ).

thf(554,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
      | ~ ( B @ class_Orderings_Opreorder ) ),
    inference(simp,[status(thm)],[553]) ).

thf(5144,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_HOL_Obool @ class_Orderings_Opreorder )
       != ( B @ class_Orderings_Opreorder ) )
      | ( A @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
      | ~ $true ),
    inference(paramod_ordered,[status(thm)],[710,554]) ).

thf(5145,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_HOL_Obool @ class_Orderings_Opreorder )
       != ( B @ class_Orderings_Opreorder ) )
      | ( A @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(simp,[status(thm)],[5144]) ).

thf(5146,plain,
    ! [A: $i] : ( A @ ( A @ ( tc_HOL_Obool @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(pattern_uni,[status(thm)],[5145:[bind(A,$thf( A )),bind(B,$thf( tc_HOL_Obool ))]]) ).

thf(235,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__empty__left) ).

thf(1327,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(75,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( C @ c_Set_Oinsert ) ) @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( C @ c_Set_Oinsert ) ) )
      = ( A @ ( B @ ( C @ c_Set_Oinsert ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__Diff__single) ).

thf(747,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( C @ c_Set_Oinsert ) ) @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( C @ c_Set_Oinsert ) ) )
      = ( A @ ( B @ ( C @ c_Set_Oinsert ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(1,conjecture,
    ( ( ! [A: $i,B: $i] :
          ( ( A != B )
         => ( ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
            | ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) )
      & ! [A: $i] :
          ~ ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      & ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) )
  <=> ( ! [A: $i,B: $i] :
          ( ( A != B )
         => ( ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
            | ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) )
      & ! [A: $i] :
          ~ ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      & ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).

thf(2,negated_conjecture,
    ~ ( ( ! [A: $i,B: $i] :
            ( ( A != B )
           => ( ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
              | ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) )
        & ! [A: $i] :
            ~ ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        & ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) )
    <=> ( ! [A: $i,B: $i] :
            ( ( A != B )
           => ( ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
              | ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) )
        & ! [A: $i] :
            ~ ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        & ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(522,plain,
    ~ ( ( ( ! [A: $i,B: $i] :
              ( ( A != B )
             => ( ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
                | ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) )
          & ! [A: $i] :
              ~ ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
          & ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) )
       => ( ! [A: $i,B: $i] :
              ( ( A != B )
             => ( ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
                | ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) )
          & ! [A: $i] :
              ~ ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
          & ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) ) )
      & ( ( ! [A: $i,B: $i] :
              ( ( A != B )
             => ( ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
                | ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) )
          & ! [A: $i] :
              ~ ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
          & ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) )
       => ( ! [A: $i,B: $i] :
              ( ( A != B )
             => ( ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
                | ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) )
          & ! [A: $i] :
              ~ ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
          & ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(523,plain,
    ~ ( ( ( ! [A: $i,B: $i] :
              ( ( A != B )
             => ( ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
                | ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) )
          & ~ ? [A: $i] : ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
          & ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) )
       => ( ! [A: $i,B: $i] :
              ( ( A != B )
             => ( ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
                | ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) )
          & ~ ? [A: $i] : ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
          & ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) ) )
      & ( ( ! [A: $i,B: $i] :
              ( ( A != B )
             => ( ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
                | ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) )
          & ~ ? [A: $i] : ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
          & ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) )
       => ( ! [A: $i,B: $i] :
              ( ( A != B )
             => ( ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
                | ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) )
          & ~ ? [A: $i] : ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
          & ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) ) ) ),
    inference(miniscope,[status(thm)],[522]) ).

thf(524,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ~ ( v_L @ ( sk7 @ ( sk6 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        | ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
        | sk1 )
      & ( ~ ( v_L @ ( sk6 @ ( sk7 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        | ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
        | sk1 )
      & ( ( sk6 != sk7 )
        | ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
        | sk1 )
      & ( ( v_L @ ( E @ ( F @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        | ( v_L @ ( F @ ( E @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        | ( E = F )
        | sk1 )
      & ( ~ ( v_L @ ( D @ ( D @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        | sk1 )
      & ( ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
        | sk1 )
      & ( ~ sk1
        | ~ ( v_L @ ( sk3 @ ( sk4 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        | ( v_L @ ( sk2 @ ( sk2 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) )
      & ( ~ sk1
        | ~ ( v_L @ ( sk4 @ ( sk3 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        | ( v_L @ ( sk2 @ ( sk2 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) )
      & ( ~ sk1
        | ( sk3 != sk4 )
        | ( v_L @ ( sk2 @ ( sk2 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) )
      & ( ~ sk1
        | ( v_L @ ( C @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        | ( v_L @ ( B @ ( C @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        | ( B = C ) )
      & ( ~ sk1
        | ~ ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) )
      & ( ~ sk1
        | ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) ) ),
    inference(cnf,[status(esa)],[523]) ).

thf(533,plain,
    ! [B: $i,A: $i] :
      ( ~ sk1
      | ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | ( A = B ) ),
    inference(cnfConj,[status(thm)],[524]) ).

thf(537,plain,
    ! [B: $i,A: $i] :
      ( ~ sk1
      | ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | ( A = B ) ),
    inference(lifteq,[status(thm)],[533]) ).

thf(538,plain,
    ! [B: $i,A: $i] :
      ( ~ sk1
      | ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | ( A = B ) ),
    inference(simp,[status(thm)],[537]) ).

thf(279,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__less__mono2) ).

thf(1507,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(263,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ ( E @ c_Relation_OImage ) ) ) )
      = ( A @ ( C @ ( D @ ( E @ c_Relation_OImage ) ) ) @ ( B @ ( C @ ( D @ ( E @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__Un) ).

thf(1458,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ ( E @ c_Relation_OImage ) ) ) )
      = ( A @ ( C @ ( D @ ( E @ c_Relation_OImage ) ) ) @ ( B @ ( C @ ( D @ ( E @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(401,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_Otrans ) )
     => ( B @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ c_Relation_Otrans ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__reflclI) ).

thf(1999,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_Otrans ) )
     => ( B @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ c_Relation_Otrans ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[401]) ).

thf(475,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( A @ ( E @ ( D @ ( B @ ( C @ c_Product__Type_OPair ) ) ) @ ( B @ ( C @ tc_prod ) @ c_member ) ) )
         => ! [F: $i] :
              ( ( A @ ( F @ ( D @ ( B @ ( C @ c_Product__Type_OPair ) ) ) @ ( B @ ( C @ tc_prod ) @ c_member ) ) )
             => ( E = F ) ) )
     => ( A @ ( B @ ( C @ c_Relation_Osingle__valued ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_single__valuedI) ).

thf(2316,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( A @ ( E @ ( D @ ( B @ ( C @ c_Product__Type_OPair ) ) ) @ ( B @ ( C @ tc_prod ) @ c_member ) ) )
         => ! [F: $i] :
              ( ( A @ ( F @ ( D @ ( B @ ( C @ c_Product__Type_OPair ) ) ) @ ( B @ ( C @ tc_prod ) @ c_member ) ) )
             => ( E = F ) ) )
     => ( A @ ( B @ ( C @ c_Relation_Osingle__valued ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[475]) ).

thf(97,axiom,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__not__refl) ).

thf(827,plain,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(828,plain,
    ~ ? [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(miniscope,[status(thm)],[827]) ).

thf(829,plain,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(cnf,[status(esa)],[828]) ).

thf(199,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( D @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( D @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__mono) ).

thf(1196,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( D @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( D @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(377,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( E @ ( F @ c_Product__Type_OPair ) ) ) @ ( C @ ( D @ ( E @ ( F @ c_Product__Type_Oprod_Oprod__rec ) ) ) ) )
      = ( A @ ( B @ ( C @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_prod_Orecs) ).

thf(1914,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( E @ ( F @ c_Product__Type_OPair ) ) ) @ ( C @ ( D @ ( E @ ( F @ c_Product__Type_Oprod_Oprod__rec ) ) ) ) )
      = ( A @ ( B @ ( C @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[377]) ).

thf(176,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__mono) ).

thf(1127,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(458,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_HOL_Obool @ ( A @ ( B @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ c_Relation_ODomain ) ) )
      = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__empty) ).

thf(2227,plain,
    ! [A: $i,B: $i] :
      ( ( tc_HOL_Obool @ ( A @ ( B @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ c_Relation_ODomain ) ) )
      = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[458]) ).

thf(102,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ~ ( ( B @ ( C @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ~ ( C @ ( B @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equalityE) ).

thf(841,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ~ ( ( B @ ( C @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ~ ( C @ ( B @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(306,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_r__r__into__trancl) ).

thf(1613,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).

thf(312,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( E @ c_Relation_Osingle__valued ) ) )
     => ( ( D @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( ( D @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
         => ( ( D @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( A @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
            | ( D @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( B @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_single__valued__confluent) ).

thf(1645,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( E @ c_Relation_Osingle__valued ) ) )
     => ( ( D @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( ( D @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
         => ( ( D @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( A @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
            | ( D @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( B @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).

thf(418,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_member ) ) )
     => ( ( A @ ( B @ ( C @ c_Set_Oinsert ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__absorb) ).

thf(2067,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_member ) ) )
     => ( ( A @ ( B @ ( C @ c_Set_Oinsert ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).

thf(372,axiom,
    ( c_FunDef_Opair__leq
    = ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Relation_OId @ ( c_FunDef_Opair__less @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__leq__def) ).

thf(1903,plain,
    ( c_FunDef_Opair__leq
    = ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Relation_OId @ ( c_FunDef_Opair__less @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).

thf(1904,plain,
    ( ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Relation_OId @ ( c_FunDef_Opair__less @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
    = c_FunDef_Opair__leq ),
    inference(lifteq,[status(thm)],[1903]) ).

thf(270,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ~ ( ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ~ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__supE) ).

thf(1482,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ~ ( ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ~ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(44,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Opreorder )
     => ~ ( A @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__irrefl) ).

thf(651,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Opreorder )
     => ~ ( A @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(142,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__antisym) ).

thf(973,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(114,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_psubset__imp__subset) ).

thf(885,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(158,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_psubset__subset__trans) ).

thf(1046,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(445,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) )
     => ( ( c_FunDef_Opair__less @ ( B @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) )
       => ( ( c_FunDef_Omin__strict @ ( A @ ( C @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) )
         => ( c_FunDef_Omin__strict @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_smin__insertI) ).

thf(2173,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) )
     => ( ( c_FunDef_Opair__less @ ( B @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) )
       => ( ( c_FunDef_Omin__strict @ ( A @ ( C @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) )
         => ( c_FunDef_Omin__strict @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[445]) ).

thf(13,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B != A )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__cases) ).

thf(565,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B != A )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(330,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Relation_ODomain ) ) @ ( C @ ( D @ c_member ) ) )
     => ~ ! [E: $i] :
            ~ ( A @ ( E @ ( C @ ( B @ ( D @ c_Product__Type_OPair ) ) ) @ ( B @ ( D @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DomainE) ).

thf(1709,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Relation_ODomain ) ) @ ( C @ ( D @ c_member ) ) )
     => ~ ! [E: $i] :
            ~ ( A @ ( E @ ( C @ ( B @ ( D @ c_Product__Type_OPair ) ) ) @ ( B @ ( D @ tc_prod ) @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).

thf(410,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Wellfounded_Omeasure @ hAPP ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
    <=> ( B @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__measure) ).

thf(2039,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( B @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( D @ c_Wellfounded_Omeasure @ hAPP ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) )
      & ( ( A @ ( D @ c_Wellfounded_Omeasure @ hAPP ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ( B @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[410]) ).

thf(374,axiom,
    ! [A: $i] :
      ( ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Relation_OField ) )
      = ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Field__empty) ).

thf(1907,plain,
    ! [A: $i] :
      ( ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Relation_OField ) )
      = ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).

thf(124,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__le__less__trans) ).

thf(917,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(220,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( B @ ( B @ c_Relation_ODomain ) ) )
      = ( A @ ( B @ ( B @ c_Relation_ODomain ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__domain) ).

thf(1279,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( B @ ( B @ c_Relation_ODomain ) ) )
      = ( A @ ( B @ ( B @ c_Relation_ODomain ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(262,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( C @ c_Transitive__Closure_Ortrancl ) )
          = ( B @ ( C @ c_Transitive__Closure_Ortrancl ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__subset) ).

thf(1455,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( C @ c_Transitive__Closure_Ortrancl ) )
          = ( B @ ( C @ c_Transitive__Closure_Ortrancl ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(197,axiom,
    ! [A: $i,B: $i] : ( A @ ( B @ c_Relation_OId__on ) @ ( B @ ( B @ c_Relation_Osingle__valued ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_single__valued__Id__on) ).

thf(1191,plain,
    ! [A: $i,B: $i] : ( A @ ( B @ c_Relation_OId__on ) @ ( B @ ( B @ c_Relation_Osingle__valued ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(528,plain,
    ! [B: $i,A: $i] :
      ( ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | ( A = B )
      | sk1 ),
    inference(cnfConj,[status(thm)],[524]) ).

thf(541,plain,
    ! [B: $i,A: $i] :
      ( ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | sk1
      | ( A = B ) ),
    inference(lifteq,[status(thm)],[528]) ).

thf(542,plain,
    ! [B: $i,A: $i] :
      ( ( v_L @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | ( v_L @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | sk1
      | ( A = B ) ),
    inference(simp,[status(thm)],[541]) ).

thf(340,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_member ) ) )
     => ( ~ ( A @ ( C @ ( D @ c_member ) ) )
       => ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DiffI) ).

thf(1760,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_member ) ) )
     => ( ~ ( A @ ( C @ ( D @ c_member ) ) )
       => ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[340]) ).

thf(318,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( E @ ( C @ c_Relation_OImage ) ) ) @ ( D @ ( E @ c_member ) ) )
    <=> ? [F: $i] :
          ( ( B @ ( D @ ( F @ ( E @ ( C @ c_Product__Type_OPair ) ) ) @ ( E @ ( C @ tc_prod ) @ c_member ) ) )
          & ( A @ ( F @ ( C @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__iff) ).

thf(1660,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ? [F: $i] :
            ( ( B @ ( D @ ( F @ ( E @ ( C @ c_Product__Type_OPair ) ) ) @ ( E @ ( C @ tc_prod ) @ c_member ) ) )
            & ( A @ ( F @ ( C @ c_member ) ) ) )
       => ( A @ ( B @ ( E @ ( C @ c_Relation_OImage ) ) ) @ ( D @ ( E @ c_member ) ) ) )
      & ( ( A @ ( B @ ( E @ ( C @ c_Relation_OImage ) ) ) @ ( D @ ( E @ c_member ) ) )
       => ? [F: $i] :
            ( ( B @ ( D @ ( F @ ( E @ ( C @ c_Product__Type_OPair ) ) ) @ ( E @ ( C @ tc_prod ) @ c_member ) ) )
            & ( A @ ( F @ ( C @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).

thf(236,axiom,
    ! [A: $i,B: $i,C: $i] : ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__Un__subset) ).

thf(1330,plain,
    ! [A: $i,B: $i,C: $i] : ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(508,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_member ) ) )
     => ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UnI1) ).

thf(2444,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_member ) ) )
     => ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[508]) ).

thf(361,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_Relation_Oantisym ) )
     => ( ( C @ ( A @ ( B @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ( ( C @ ( B @ ( A @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
         => ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_antisymD) ).

thf(1860,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_Relation_Oantisym ) )
     => ( ( C @ ( A @ ( B @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ( ( C @ ( B @ ( A @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).

thf(22,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Olattice )
     => ( B @ ( A @ tc_fun ) @ class_Lattices_Osemilattice__sup ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Lattices_Osemilattice__sup) ).

thf(587,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Olattice )
     => ( B @ ( A @ tc_fun ) @ class_Lattices_Osemilattice__sup ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(101,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_OId__on ) @ ( B @ ( B @ c_Relation_ODomain ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__Id__on) ).

thf(838,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_OId__on ) @ ( B @ ( B @ c_Relation_ODomain ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(163,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ c_Transitive__Closure_Ortrancl ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__r__diff__Id) ).

thf(1059,plain,
    ! [A: $i,B: $i] :
      ( ( B @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ c_Transitive__Closure_Ortrancl ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(177,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_fequal @ hAPP ) @ hBOOL )
      | ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fequal__2) ).

thf(1129,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_fequal @ hAPP ) @ hBOOL )
      | ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(1130,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ c_fequal @ hAPP ) @ hBOOL )
      | ( B != A ) ),
    inference(cnf,[status(esa)],[1129]) ).

thf(1131,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ c_fequal @ hAPP ) @ hBOOL )
      | ( B != A ) ),
    inference(lifteq,[status(thm)],[1130]) ).

thf(1132,plain,
    ! [A: $i] : ( A @ ( A @ c_fequal @ hAPP ) @ hBOOL ),
    inference(simp,[status(thm)],[1131]) ).

thf(500,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( D @ ( E @ ( F @ c_member ) ) )
     => ( ( A @ ( B @ ( E @ ( C @ ( F @ c_Product__Type_OPair ) ) ) @ ( C @ ( F @ tc_prod ) @ c_member ) ) )
       => ( D @ ( A @ ( C @ ( F @ c_Relation_OImage ) ) ) @ ( B @ ( C @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rev__ImageI) ).

thf(2420,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( D @ ( E @ ( F @ c_member ) ) )
     => ( ( A @ ( B @ ( E @ ( C @ ( F @ c_Product__Type_OPair ) ) ) @ ( C @ ( F @ tc_prod ) @ c_member ) ) )
       => ( D @ ( A @ ( C @ ( F @ c_Relation_OImage ) ) ) @ ( B @ ( C @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[500]) ).

thf(121,axiom,
    c_Wellfounded_Oless__than @ ( tc_Nat_Onat @ c_Wellfounded_Owf ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__less__than) ).

thf(902,plain,
    c_Wellfounded_Oless__than @ ( tc_Nat_Onat @ c_Wellfounded_Owf ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(9,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( A @ ( B @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup_Oidem) ).

thf(559,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( A @ ( B @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(247,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Set_Oinsert ) ) )
      = ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( C @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__is__Un) ).

thf(1379,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Set_Oinsert ) ) )
      = ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( C @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(182,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I7_J) ).

thf(1147,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(294,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__absorb2) ).

thf(1572,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).

thf(83,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_leI) ).

thf(770,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(336,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( B @ ( A @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ ( E @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) )
    <=> ? [G: $i] :
          ( ! [H: $i] :
              ( ( D @ ( B @ ( H @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) )
             => ( H @ ( G @ hAPP ) @ ( H @ ( C @ ( D @ ( E @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) ) )
          & ( ( B @ ( A @ ( C @ hAPP ) @ hAPP ) )
            = ( B @ ( G @ ( C @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wfrec__rel_Oequations) ).

thf(1741,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ? [G: $i] :
            ( ! [H: $i] :
                ( ( D @ ( B @ ( H @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) )
               => ( H @ ( G @ hAPP ) @ ( H @ ( C @ ( D @ ( E @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) ) )
            & ( ( B @ ( A @ ( C @ hAPP ) @ hAPP ) )
              = ( B @ ( G @ ( C @ hAPP ) @ hAPP ) ) ) )
       => ( B @ ( A @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ ( E @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) ) )
      & ( ( B @ ( A @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ ( E @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) )
       => ? [G: $i] :
            ( ! [H: $i] :
                ( ( D @ ( B @ ( H @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) )
               => ( H @ ( G @ hAPP ) @ ( H @ ( C @ ( D @ ( E @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) ) )
            & ( ( B @ ( A @ ( C @ hAPP ) @ hAPP ) )
              = ( B @ ( G @ ( C @ hAPP ) @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).

thf(382,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( D @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( C @ ( D @ c_Relation_ODomain ) ) @ ( B @ ( C @ ( D @ c_Relation_ODomain ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__mono) ).

thf(1931,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( D @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( C @ ( D @ c_Relation_ODomain ) ) @ ( B @ ( C @ ( D @ c_Relation_ODomain ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[382]) ).

thf(210,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__ge1) ).

thf(1232,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(311,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ~ ( A @ ( C @ ( B @ c_member ) ) )
    <=> ( A
        = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_all__not__in__conv) ).

thf(1637,plain,
    ! [A: $i,B: $i] :
      ( ( ( A
          = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
       => ! [C: $i] :
            ~ ( A @ ( C @ ( B @ c_member ) ) ) )
      & ( ! [C: $i] :
            ~ ( A @ ( C @ ( B @ c_member ) ) )
       => ( A
          = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).

thf(1638,plain,
    ( ! [A: $i,B: $i] :
        ( ( A
          = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
       => ~ ? [C: $i] : ( A @ ( C @ ( B @ c_member ) ) ) )
    & ! [A: $i,B: $i] :
        ( ~ ? [C: $i] : ( A @ ( C @ ( B @ c_member ) ) )
       => ( A
          = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ),
    inference(miniscope,[status(thm)],[1637]) ).

thf(1639,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ~ ( C @ ( E @ ( D @ c_member ) ) )
        | ( C
         != ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) )
      & ( ( A
          = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
        | ( A @ ( A @ ( B @ sk26 ) @ ( B @ c_member ) ) ) ) ),
    inference(cnf,[status(esa)],[1638]) ).

thf(1641,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_member ) ) )
      | ( A
       != ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ),
    inference(cnfConj,[status(thm)],[1639]) ).

thf(1643,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_member ) ) )
      | ( A
       != ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ),
    inference(lifteq,[status(thm)],[1641]) ).

thf(1644,plain,
    ! [B: $i,A: $i] :
      ~ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( A @ c_member ) ) ),
    inference(simp,[status(thm)],[1643]) ).

thf(4731,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ c_fequal @ ( A @ ( B @ c_member ) ) )
       != ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( D @ ( C @ c_member ) ) ) )
      | ~ $true ),
    inference(paramod_ordered,[status(thm)],[1924,1644]) ).

thf(4732,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ c_fequal @ ( A @ ( B @ c_member ) ) )
     != ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( D @ ( C @ c_member ) ) ) ),
    inference(simp,[status(thm)],[4731]) ).

thf(4745,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ c_fequal )
       != ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
      | ( A != D )
      | ( B != C ) ),
    inference(simp,[status(thm)],[4732]) ).

thf(4756,plain,
    ! [B: $i,A: $i] :
      ( ( B @ c_fequal )
     != ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    inference(simp,[status(thm)],[4745]) ).

thf(443,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( D @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( D @ c_Relation_Osingle__valued ) ) )
       => ( B @ ( C @ ( D @ c_Relation_Osingle__valued ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_single__valued__subset) ).

thf(2169,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( D @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( D @ c_Relation_Osingle__valued ) ) )
       => ( B @ ( C @ ( D @ c_Relation_Osingle__valued ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[443]) ).

thf(498,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( B @ ( C @ ( D @ c_member ) ) )
       => ( C = A ) )
     => ( B @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insertCI) ).

thf(2403,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( B @ ( C @ ( D @ c_member ) ) )
       => ( C = A ) )
     => ( B @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[498]) ).

thf(272,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__imp__diff__less) ).

thf(1489,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(266,axiom,
    ! [A: $i,B: $i,C: $i] : ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__upper1) ).

thf(1465,plain,
    ! [A: $i,B: $i,C: $i] : ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(513,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Oacyclic ) )
     => ( ( B @ ( A @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ c_Wellfounded_Oacyclic ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acyclic__subset) ).

thf(2469,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Oacyclic ) )
     => ( ( B @ ( A @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ c_Wellfounded_Oacyclic ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[513]) ).

thf(99,axiom,
    ! [A: $i] :
      ( ( A @ c_Wellfounded_Omeasure )
      = ( c_Wellfounded_Oless__than @ ( A @ ( tc_Nat_Onat @ c_Relation_Oinv__image ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_measure__def) ).

thf(832,plain,
    ! [A: $i] :
      ( ( A @ c_Wellfounded_Omeasure )
      = ( c_Wellfounded_Oless__than @ ( A @ ( tc_Nat_Onat @ c_Relation_Oinv__image ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(833,plain,
    ! [A: $i] :
      ( ( A @ c_Wellfounded_Omeasure )
      = ( c_Wellfounded_Oless__than @ ( A @ ( tc_Nat_Onat @ c_Relation_Oinv__image ) ) ) ),
    inference(cnf,[status(esa)],[832]) ).

thf(834,plain,
    ! [A: $i] :
      ( ( c_Wellfounded_Oless__than @ ( A @ ( tc_Nat_Onat @ c_Relation_Oinv__image ) ) )
      = ( A @ c_Wellfounded_Omeasure ) ),
    inference(lifteq,[status(thm)],[833]) ).

thf(184,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( C @ c_Set_Oinsert ) ) )
        = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( C @ c_Set_Oinsert ) ) ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_singleton__inject) ).

thf(1151,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( C @ c_Set_Oinsert ) ) )
        = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( C @ c_Set_Oinsert ) ) ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(16,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__le__cases) ).

thf(574,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(297,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Obot )
     => ( ( A @ ( C @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ hAPP ) )
        = ( C @ c_Orderings_Obot__class_Obot ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bot__apply) ).

thf(1581,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Obot )
     => ( ( A @ ( C @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ hAPP ) )
        = ( C @ c_Orderings_Obot__class_Obot ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).

thf(160,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( A @ ( B @ ( C @ c_member ) ) )
     => ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( C @ c_Set_Oinsert ) ) @ ( A @ ( B @ ( C @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__insert__absorb) ).

thf(1051,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( A @ ( B @ ( C @ c_member ) ) )
     => ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( C @ c_Set_Oinsert ) ) @ ( A @ ( B @ ( C @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(404,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Opred__nat @ ( tc_Nat_Onat @ c_Transitive__Closure_Otrancl ) @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__eq) ).

thf(2017,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( c_Wellfounded_Opred__nat @ ( tc_Nat_Onat @ c_Transitive__Closure_Otrancl ) @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) ) )
      & ( ( c_Wellfounded_Opred__nat @ ( tc_Nat_Onat @ c_Transitive__Closure_Otrancl ) @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[404]) ).

thf(227,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ c_Relation_OId @ ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_reflcl__trancl) ).

thf(1302,plain,
    ! [A: $i,B: $i] :
      ( ( B @ c_Relation_OId @ ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(78,axiom,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__neqE__nat) ).

thf(756,plain,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(477,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( D @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Relation_ODomain ) ) )
      = ( A @ ( C @ ( D @ c_Relation_ODomain ) ) @ ( B @ ( C @ ( D @ c_Relation_ODomain ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__Un__eq) ).

thf(2324,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( D @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Relation_ODomain ) ) )
      = ( A @ ( C @ ( D @ c_Relation_ODomain ) ) @ ( B @ ( C @ ( D @ c_Relation_ODomain ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[477]) ).

thf(406,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Product__Type_OPair ) ) ) @ ( D @ ( E @ tc_prod ) @ c_Set_Oinsert ) ) @ ( D @ ( E @ c_Relation_ORange ) ) )
      = ( A @ ( D @ ( E @ c_Relation_ORange ) ) @ ( B @ ( D @ c_Set_Oinsert ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__insert) ).

thf(2027,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Product__Type_OPair ) ) ) @ ( D @ ( E @ tc_prod ) @ c_Set_Oinsert ) ) @ ( D @ ( E @ c_Relation_ORange ) ) )
      = ( A @ ( D @ ( E @ c_Relation_ORange ) ) @ ( B @ ( D @ c_Set_Oinsert ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[406]) ).

thf(424,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ c_Relation_Otrans ) )
     => ( ( D @ ( B @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( ( D @ ( A @ ( B @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
         => ( D @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_transD) ).

thf(2090,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ c_Relation_Otrans ) )
     => ( ( D @ ( B @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( ( D @ ( A @ ( B @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
         => ( D @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[424]) ).

thf(24,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Olattice )
     => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( B @ ( A @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf__sup__aci_I5_J) ).

thf(591,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Olattice )
     => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( B @ ( A @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(360,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( C @ c_Relation_OField ) @ ( B @ ( C @ c_Relation_OField ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mono__Field) ).

thf(1858,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( C @ c_Relation_OField ) @ ( B @ ( C @ c_Relation_OField ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).

thf(434,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( A @ ( F @ ( G @ tc_prod ) @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( F @ ( G @ c_Product__Type_OPair ) ) ) @ ( D @ ( E @ ( F @ ( G @ c_Product__Type_OPair ) ) ) @ ( F @ ( G @ tc_prod ) @ ( F @ ( G @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( F @ ( G @ tc_prod ) @ ( F @ ( G @ tc_prod ) @ tc_prod ) @ c_member ) ) )
     => ( ( ( D @ ( E @ ( F @ ( G @ c_Product__Type_OPair ) ) ) )
         != ( B @ ( C @ ( F @ ( G @ c_Product__Type_OPair ) ) ) ) )
       => ~ ! [H: $i,I: $i] :
              ( ( A @ ( I @ ( H @ ( F @ ( G @ c_Product__Type_OPair ) ) ) @ ( D @ ( E @ ( F @ ( G @ c_Product__Type_OPair ) ) ) @ ( F @ ( G @ tc_prod ) @ ( F @ ( G @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( F @ ( G @ tc_prod ) @ ( F @ ( G @ tc_prod ) @ tc_prod ) @ c_member ) ) )
             => ~ ( A @ ( F @ ( G @ tc_prod ) @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( F @ ( G @ c_Product__Type_OPair ) ) ) @ ( I @ ( H @ ( F @ ( G @ c_Product__Type_OPair ) ) ) @ ( F @ ( G @ tc_prod ) @ ( F @ ( G @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( F @ ( G @ tc_prod ) @ ( F @ ( G @ tc_prod ) @ tc_prod ) @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__rtranclE2) ).

thf(2122,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( A @ ( F @ ( G @ tc_prod ) @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( F @ ( G @ c_Product__Type_OPair ) ) ) @ ( D @ ( E @ ( F @ ( G @ c_Product__Type_OPair ) ) ) @ ( F @ ( G @ tc_prod ) @ ( F @ ( G @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( F @ ( G @ tc_prod ) @ ( F @ ( G @ tc_prod ) @ tc_prod ) @ c_member ) ) )
     => ( ( ( D @ ( E @ ( F @ ( G @ c_Product__Type_OPair ) ) ) )
         != ( B @ ( C @ ( F @ ( G @ c_Product__Type_OPair ) ) ) ) )
       => ~ ! [H: $i,I: $i] :
              ( ( A @ ( I @ ( H @ ( F @ ( G @ c_Product__Type_OPair ) ) ) @ ( D @ ( E @ ( F @ ( G @ c_Product__Type_OPair ) ) ) @ ( F @ ( G @ tc_prod ) @ ( F @ ( G @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( F @ ( G @ tc_prod ) @ ( F @ ( G @ tc_prod ) @ tc_prod ) @ c_member ) ) )
             => ~ ( A @ ( F @ ( G @ tc_prod ) @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( F @ ( G @ c_Product__Type_OPair ) ) ) @ ( I @ ( H @ ( F @ ( G @ c_Product__Type_OPair ) ) ) @ ( F @ ( G @ tc_prod ) @ ( F @ ( G @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( F @ ( G @ tc_prod ) @ ( F @ ( G @ tc_prod ) @ tc_prod ) @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[434]) ).

thf(390,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_Wellfounded_Owf ) )
     => ( ( C @ ( A @ ( B @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ~ ( C @ ( B @ ( A @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__not__sym) ).

thf(1960,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_Wellfounded_Owf ) )
     => ( ( C @ ( A @ ( B @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ~ ( C @ ( B @ ( A @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).

thf(92,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Owf ) )
     => ( B @ ( A @ ( C @ c_Wellfounded_Omlex__prod ) ) @ ( C @ c_Wellfounded_Owf ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__mlex) ).

thf(806,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Owf ) )
     => ( B @ ( A @ ( C @ c_Wellfounded_Omlex__prod ) ) @ ( C @ c_Wellfounded_Owf ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(29,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( B = A )
      <=> ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
          & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__eq__iff) ).

thf(602,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(414,axiom,
    ! [A: $i,B: $i] : ( A @ ( B @ c_Relation_OId__on ) @ ( B @ c_Relation_Otrans ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__Id__on) ).

thf(2058,plain,
    ! [A: $i,B: $i] : ( A @ ( B @ c_Relation_OId__on ) @ ( B @ c_Relation_Otrans ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[414]) ).

thf(57,axiom,
    tc_HOL_Obool @ class_Lattices_Obounded__lattice__bot,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL__Obool__Lattices_Obounded__lattice__bot) ).

thf(692,plain,
    tc_HOL_Obool @ class_Lattices_Obounded__lattice__bot,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(368,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_member ) ) )
    <=> ( ~ ( A @ ( C @ ( D @ c_member ) ) )
        & ( B @ ( C @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__iff) ).

thf(1886,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ~ ( A @ ( C @ ( D @ c_member ) ) )
          & ( B @ ( C @ ( D @ c_member ) ) ) )
       => ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_member ) ) ) )
      & ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_member ) ) )
       => ( ~ ( A @ ( C @ ( D @ c_member ) ) )
          & ( B @ ( C @ ( D @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[368]) ).

thf(242,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__antisym__conv1) ).

thf(1351,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( ( B = A )
           => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) )
          & ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(430,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_Otrans ) )
     => ( ( A @ ( B @ c_Transitive__Closure_Otrancl ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__id) ).

thf(2103,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_Otrans ) )
     => ( ( A @ ( B @ c_Transitive__Closure_Otrancl ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[430]) ).

thf(452,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( D @ c_Transitive__Closure_Otrancl ) @ ( C @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ ( D @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( C @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__mono) ).

thf(2205,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( D @ c_Transitive__Closure_Otrancl ) @ ( C @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ ( D @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( C @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[452]) ).

thf(189,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Lattices_Olattice )
     => ( ( A @ ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ hAPP ) )
        = ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__apply) ).

thf(1169,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Lattices_Olattice )
     => ( ( A @ ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ hAPP ) )
        = ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(54,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__trans) ).

thf(687,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(384,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_member ) ) )
     => ( B @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insertI2) ).

thf(1943,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_member ) ) )
     => ( B @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[384]) ).

thf(201,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__supI2) ).

thf(1201,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(171,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Obounded__lattice__bot )
     => ( ( B @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__bot__right) ).

thf(1104,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Obounded__lattice__bot )
     => ( ( B @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(103,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__diff__cancel) ).

thf(849,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(362,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ? [E: $i] :
          ( ( A @ ( B @ ( E @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
          & ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( E @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_tranclD2) ).

thf(1863,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ? [E: $i] :
          ( ( A @ ( B @ ( E @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
          & ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( E @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).

thf(314,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_Equiv__Relations_Oequiv ) ) )
     => ( ( C @ ( A @ ( D @ c_member ) ) )
       => ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( B @ ( D @ ( D @ c_Relation_OImage ) ) ) @ ( A @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equiv__class__self) ).

thf(1651,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_Equiv__Relations_Oequiv ) ) )
     => ( ( C @ ( A @ ( D @ c_member ) ) )
       => ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( B @ ( D @ ( D @ c_Relation_OImage ) ) ) @ ( A @ ( D @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).

thf(304,axiom,
    ! [A: $i,B: $i] : ( B @ c_Relation_OId @ ( A @ ( A @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_IdI) ).

thf(1604,plain,
    ! [A: $i,B: $i] : ( B @ c_Relation_OId @ ( A @ ( A @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).

thf(3,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup_Oleft__idem) ).

thf(544,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(545,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) )
      | ~ ( C @ class_Lattices_Osemilattice__sup ) ),
    inference(cnf,[status(esa)],[544]) ).

thf(546,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ class_Lattices_Osemilattice__sup )
      | ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    inference(lifteq,[status(thm)],[545]) ).

thf(80,axiom,
    ! [A: $i,B: $i,C: $i] : ( B @ ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__subset) ).

thf(761,plain,
    ! [A: $i,B: $i,C: $i] : ( B @ ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(94,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__left__absorb) ).

thf(810,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(471,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_member ) ) )
     => ~ ( ( B @ ( C @ ( D @ c_member ) ) )
         => ( A @ ( C @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DiffE) ).

thf(2296,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_member ) ) )
     => ~ ( ( B @ ( C @ ( D @ c_member ) ) )
         => ( A @ ( C @ ( D @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[471]) ).

thf(162,axiom,
    ! [A: $i] : ( A @ c_Relation_OId @ ( A @ c_Relation_Oantisym ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_antisym__Id) ).

thf(1057,plain,
    ! [A: $i] : ( A @ c_Relation_OId @ ( A @ c_Relation_Oantisym ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(52,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Opreorder )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Orderings_Opreorder) ).

thf(678,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Opreorder )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(515,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( C @ ( D @ ( E @ ( F @ c_Product__Type_OPair ) ) ) )
        = ( A @ ( B @ ( E @ ( F @ c_Product__Type_OPair ) ) ) ) )
     => ~ ( ( D = B )
         => ( C != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Pair__inject) ).

thf(2473,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( C @ ( D @ ( E @ ( F @ c_Product__Type_OPair ) ) ) )
        = ( A @ ( B @ ( E @ ( F @ c_Product__Type_OPair ) ) ) ) )
     => ~ ( ( D = B )
         => ( C != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[515]) ).

thf(529,plain,
    ( ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
    | sk1 ),
    inference(cnfConj,[status(thm)],[524]) ).

thf(326,axiom,
    ! [A: $i] : ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Order__Relation_Olinear__order__on ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lnear__order__on__empty) ).

thf(1691,plain,
    ! [A: $i] : ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Order__Relation_Olinear__order__on ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).

thf(412,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( C @ ( D @ c_Relation_OField ) @ ( D @ c_Relation_Orefl__on ) ) )
     => ( ( C @ ( D @ c_Relation_Oantisym ) )
       => ( ( C @ ( D @ c_Relation_OField ) @ ( B @ ( D @ c_member ) ) )
         => ( ( C @ ( D @ c_Relation_OField ) @ ( A @ ( D @ c_member ) ) )
           => ( ( ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) )
                = ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) ) )
            <=> ( B = A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Refl__antisym__eq__Image1__Image1__iff) ).

thf(2047,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( C @ ( D @ c_Relation_OField ) @ ( D @ c_Relation_Orefl__on ) ) )
     => ( ( C @ ( D @ c_Relation_Oantisym ) )
       => ( ( C @ ( D @ c_Relation_OField ) @ ( B @ ( D @ c_member ) ) )
         => ( ( C @ ( D @ c_Relation_OField ) @ ( A @ ( D @ c_member ) ) )
           => ( ( ( B = A )
               => ( ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) )
                  = ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) ) ) )
              & ( ( ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) )
                  = ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) ) )
               => ( B = A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[412]) ).

thf(525,plain,
    ( ~ ( v_L @ ( sk6 @ ( sk7 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
    | sk1 ),
    inference(cnfConj,[status(thm)],[524]) ).

thf(2504,plain,
    ( ( ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
     != ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) )
    | ~ ( v_L @ ( sk6 @ ( sk7 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ~ $true
    | sk1
    | sk1 ),
    inference(paramod_ordered,[status(thm)],[529,525]) ).

thf(2505,plain,
    ( ( ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
     != ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) )
    | ~ ( v_L @ ( sk6 @ ( sk7 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | sk1 ),
    inference(simp,[status(thm)],[2504]) ).

thf(2506,plain,
    ( ~ ( v_L @ ( sk6 @ ( sk7 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | sk1 ),
    inference(pattern_uni,[status(thm)],[2505:[]]) ).

thf(4735,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_fequal @ ( A @ ( B @ c_member ) ) )
       != ( v_L @ ( sk6 @ ( sk7 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) )
      | ~ $true
      | ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | sk1 ),
    inference(paramod_ordered,[status(thm)],[1924,2506]) ).

thf(4736,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_fequal @ ( A @ ( B @ c_member ) ) )
       != ( v_L @ ( sk6 @ ( sk7 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) )
      | ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | sk1 ),
    inference(simp,[status(thm)],[4735]) ).

thf(4742,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_fequal )
       != v_L )
      | ( A
       != ( sk6 @ ( sk7 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) ) )
      | ( B
       != ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) ) )
      | ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | sk1 ),
    inference(simp,[status(thm)],[4736]) ).

thf(4752,plain,
    ( ( ( sk6 @ ( sk7 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ c_fequal )
     != v_L )
    | ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | sk1 ),
    inference(simp,[status(thm)],[4742]) ).

thf(527,plain,
    ! [A: $i] :
      ( ~ ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | sk1 ),
    inference(cnfConj,[status(thm)],[524]) ).

thf(539,plain,
    ! [A: $i] :
      ( ~ ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
      | sk1 ),
    inference(simp,[status(thm)],[527]) ).

thf(5115,plain,
    ! [A: $i] :
      ( ( ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
       != ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) )
      | ~ $true
      | sk1
      | ( ( sk6 @ ( sk7 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ c_fequal )
       != v_L )
      | sk1 ),
    inference(paramod_ordered,[status(thm)],[4752,539]) ).

thf(5116,plain,
    ! [A: $i] :
      ( ( ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
       != ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) )
      | ( ( sk6 @ ( sk7 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ c_fequal )
       != v_L )
      | sk1 ),
    inference(simp,[status(thm)],[5115]) ).

thf(5117,plain,
    ( ( ( sk6 @ ( sk7 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ c_fequal )
     != v_L )
    | sk1 ),
    inference(pattern_uni,[status(thm)],[5116:[bind(A,$thf( sk5 ))]]) ).

thf(96,axiom,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__refl) ).

thf(825,plain,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(134,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__insertI2) ).

thf(952,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(174,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_Wellfounded_Owf ) )
     => ( A @ ( C @ ( B @ ( D @ c_Relation_Oinv__image ) ) @ hAPP ) @ ( B @ c_Wellfounded_Owf ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__inv__image) ).

thf(1123,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_Wellfounded_Owf ) )
     => ( A @ ( C @ ( B @ ( D @ c_Relation_Oinv__image ) ) @ hAPP ) @ ( B @ c_Wellfounded_Owf ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(191,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ hAPP ) @ hBOOL )
       => ( A @ ( B @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_predicate1D) ).

thf(1174,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ hAPP ) @ hBOOL )
       => ( A @ ( B @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(95,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
    <=> ( ( C @ ( B @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
        & ( B @ ( C @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__eq__subset) ).

thf(813,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( C @ ( B @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
          & ( B @ ( C @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
       => ( C = B ) )
      & ( ( C = B )
       => ( ( C @ ( B @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
          & ( B @ ( C @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(814,plain,
    ( ! [A: $i,B: $i,C: $i] :
        ( ( ( C @ ( B @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
          & ( B @ ( C @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
       => ( C = B ) )
    & ! [A: $i,B: $i,C: $i] :
        ( ( C = B )
       => ( ( C @ ( B @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
          & ( B @ ( C @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(miniscope,[status(thm)],[813]) ).

thf(815,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( F = E )
        | ~ ( F @ ( E @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
        | ~ ( E @ ( F @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( C @ ( B @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
        | ( C != B ) )
      & ( ( B @ ( C @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
        | ( C != B ) ) ),
    inference(cnf,[status(esa)],[814]) ).

thf(816,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
      | ( C != B ) ),
    inference(cnfConj,[status(thm)],[815]) ).

thf(819,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
      | ( C != B ) ),
    inference(lifteq,[status(thm)],[816]) ).

thf(820,plain,
    ! [B: $i,A: $i] : ( B @ ( B @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(simp,[status(thm)],[819]) ).

thf(58,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Orderings_Oorder) ).

thf(693,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(218,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__least) ).

thf(1274,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(507,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ~ ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( C @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
     => ( A @ ( B @ c_Wellfounded_Oacyclic ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acyclicI) ).

thf(2441,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ~ ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( C @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
     => ( A @ ( B @ c_Wellfounded_Oacyclic ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[507]) ).

thf(2264,plain,
    ( ( c_FunDef_Opair__less @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Wellfounded_Omin__ext ) )
    = c_FunDef_Omin__strict ),
    inference(lifteq,[status(thm)],[2263]) ).

thf(499,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( A @ ( B @ ( G @ ( H @ c_Wellfounded_Olex__prod ) ) ) @ ( C @ ( D @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( E @ ( F @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ tc_prod ) @ c_member ) ) )
    <=> ( ( ( A @ ( C @ ( E @ ( G @ ( G @ c_Product__Type_OPair ) ) ) @ ( G @ ( G @ tc_prod ) @ c_member ) ) )
          & ( F = D ) )
        | ( B @ ( D @ ( F @ ( H @ ( H @ c_Product__Type_OPair ) ) ) @ ( H @ ( H @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__lex__prod) ).

thf(2409,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( ( ( A @ ( C @ ( E @ ( G @ ( G @ c_Product__Type_OPair ) ) ) @ ( G @ ( G @ tc_prod ) @ c_member ) ) )
            & ( F = D ) )
          | ( B @ ( D @ ( F @ ( H @ ( H @ c_Product__Type_OPair ) ) ) @ ( H @ ( H @ tc_prod ) @ c_member ) ) ) )
       => ( A @ ( B @ ( G @ ( H @ c_Wellfounded_Olex__prod ) ) ) @ ( C @ ( D @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( E @ ( F @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ tc_prod ) @ c_member ) ) ) )
      & ( ( A @ ( B @ ( G @ ( H @ c_Wellfounded_Olex__prod ) ) ) @ ( C @ ( D @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( E @ ( F @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ tc_prod ) @ c_member ) ) )
       => ( ( ( A @ ( C @ ( E @ ( G @ ( G @ c_Product__Type_OPair ) ) ) @ ( G @ ( G @ tc_prod ) @ c_member ) ) )
            & ( F = D ) )
          | ( B @ ( D @ ( F @ ( H @ ( H @ c_Product__Type_OPair ) ) ) @ ( H @ ( H @ tc_prod ) @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[499]) ).

thf(371,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( D @ ( E @ ( F @ c_Relation_Osingle__valued ) ) )
     => ( ( D @ ( B @ ( C @ ( E @ ( F @ c_Product__Type_OPair ) ) ) @ ( E @ ( F @ tc_prod ) @ c_member ) ) )
       => ( ( D @ ( A @ ( C @ ( E @ ( F @ c_Product__Type_OPair ) ) ) @ ( E @ ( F @ tc_prod ) @ c_member ) ) )
         => ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_single__valuedD) ).

thf(1900,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( D @ ( E @ ( F @ c_Relation_Osingle__valued ) ) )
     => ( ( D @ ( B @ ( C @ ( E @ ( F @ c_Product__Type_OPair ) ) ) @ ( E @ ( F @ tc_prod ) @ c_member ) ) )
       => ( ( D @ ( A @ ( C @ ( E @ ( F @ c_Product__Type_OPair ) ) ) @ ( E @ ( F @ tc_prod ) @ c_member ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[371]) ).

thf(333,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ? [E: $i] :
          ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( E @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
          & ( A @ ( E @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_tranclD) ).

thf(1723,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ? [E: $i] :
          ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( E @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
          & ( A @ ( E @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).

thf(136,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__subset__rtrancl) ).

thf(957,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(207,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( C @ c_Set_Oinsert ) ) )
      = ( A @ ( B @ ( C @ c_Set_Oinsert ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__absorb2) ).

thf(1223,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( C @ c_Set_Oinsert ) ) )
      = ( A @ ( B @ ( C @ c_Set_Oinsert ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(462,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
    <=> ( ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
          & ( C != B ) )
        | ( C = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__eq__or__trancl) ).

thf(2249,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
            & ( C != B ) )
          | ( C = B ) )
       => ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) )
      & ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ( ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
            & ( C != B ) )
          | ( C = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[462]) ).

thf(86,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( ( B = A )
        | ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__iff__psubset__eq) ).

thf(778,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( B = A )
          | ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) )
       => ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B = A )
          | ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(27,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__not__sym) ).

thf(598,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(440,axiom,
    ! [A: $i] : ( A @ c_Relation_OId @ ( A @ c_Relation_Otrans ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__Id) ).

thf(2147,plain,
    ! [A: $i] : ( A @ c_Relation_OId @ ( A @ c_Relation_Otrans ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[440]) ).

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

thf(788,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B != A )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(309,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ ( B @ c_Relation_ORange ) ) @ ( C @ ( D @ c_member ) ) )
    <=> ? [E: $i] : ( A @ ( C @ ( E @ ( D @ ( B @ c_Product__Type_OPair ) ) ) @ ( D @ ( B @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__iff) ).

thf(1619,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ? [E: $i] : ( A @ ( C @ ( E @ ( D @ ( B @ c_Product__Type_OPair ) ) ) @ ( D @ ( B @ tc_prod ) @ c_member ) ) )
       => ( A @ ( D @ ( B @ c_Relation_ORange ) ) @ ( C @ ( D @ c_member ) ) ) )
      & ( ( A @ ( D @ ( B @ c_Relation_ORange ) ) @ ( C @ ( D @ c_member ) ) )
       => ? [E: $i] : ( A @ ( C @ ( E @ ( D @ ( B @ c_Product__Type_OPair ) ) ) @ ( D @ ( B @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).

thf(256,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Obounded__lattice__bot )
     => ( ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
          = ( C @ c_Orderings_Obot__class_Obot ) )
      <=> ( ( A
            = ( C @ c_Orderings_Obot__class_Obot ) )
          & ( B
            = ( C @ c_Orderings_Obot__class_Obot ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__eq__bot__iff) ).

thf(1423,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Obounded__lattice__bot )
     => ( ( ( ( A
              = ( C @ c_Orderings_Obot__class_Obot ) )
            & ( B
              = ( C @ c_Orderings_Obot__class_Obot ) ) )
         => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
            = ( C @ c_Orderings_Obot__class_Obot ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
            = ( C @ c_Orderings_Obot__class_Obot ) )
         => ( ( A
              = ( C @ c_Orderings_Obot__class_Obot ) )
            & ( B
              = ( C @ c_Orderings_Obot__class_Obot ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(4737,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_fequal @ ( A @ ( B @ c_member ) ) )
       != ( v_L @ ( C @ ( C @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) )
      | ~ $true
      | sk1 ),
    inference(paramod_ordered,[status(thm)],[1924,539]) ).

thf(4738,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_fequal @ ( A @ ( B @ c_member ) ) )
       != ( v_L @ ( C @ ( C @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) )
      | sk1 ),
    inference(simp,[status(thm)],[4737]) ).

thf(4746,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_fequal )
       != v_L )
      | ( A
       != ( C @ ( C @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) ) )
      | ( B
       != ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) ) )
      | sk1 ),
    inference(simp,[status(thm)],[4738]) ).

thf(4757,plain,
    ! [A: $i] :
      ( ( ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ c_fequal )
       != v_L )
      | sk1 ),
    inference(simp,[status(thm)],[4746]) ).

thf(324,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( C @ ( A @ ( D @ c_member ) ) )
       => ( B @ ( A @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_psubsetD) ).

thf(1687,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( C @ ( A @ ( D @ c_member ) ) )
       => ( B @ ( A @ ( D @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[324]) ).

thf(353,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Wellfounded_Oacyclic ) )
    <=> ! [C: $i] :
          ~ ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( C @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acyclic__def) ).

thf(1819,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ~ ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( C @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
       => ( A @ ( B @ c_Wellfounded_Oacyclic ) ) )
      & ( ( A @ ( B @ c_Wellfounded_Oacyclic ) )
       => ! [C: $i] :
            ~ ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( C @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[353]) ).

thf(187,axiom,
    ! [A: $i,B: $i] : ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ c_Relation_Ototal__on ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_total__on__empty) ).

thf(1159,plain,
    ! [A: $i,B: $i] : ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ c_Relation_Ototal__on ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(447,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_Otrans ) )
    <=> ! [C: $i,D: $i] :
          ( ( A @ ( D @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
         => ! [E: $i] :
              ( ( A @ ( E @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
             => ( A @ ( E @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__def) ).

thf(2181,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i,D: $i] :
            ( ( A @ ( D @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
           => ! [E: $i] :
                ( ( A @ ( E @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
               => ( A @ ( E @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ) )
       => ( A @ ( B @ c_Relation_Otrans ) ) )
      & ( ( A @ ( B @ c_Relation_Otrans ) )
       => ! [C: $i,D: $i] :
            ( ( A @ ( D @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
           => ! [E: $i] :
                ( ( A @ ( E @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
               => ( A @ ( E @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[447]) ).

thf(252,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__absorb) ).

thf(1407,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(375,axiom,
    ! [A: $i,B: $i,C: $i] : ( A @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( B @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl_Ortrancl__refl) ).

thf(1910,plain,
    ! [A: $i,B: $i,C: $i] : ( A @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( B @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).

thf(154,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( B @ ( C @ ( D @ c_member ) ) )
     => ( ( A @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__insert) ).

thf(1032,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( B @ ( C @ ( D @ c_member ) ) )
     => ( ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( A @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(356,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( C @ ( D @ ( E @ ( F @ c_Product__Type_OPair ) ) ) )
        = ( A @ ( B @ ( E @ ( F @ c_Product__Type_OPair ) ) ) ) )
    <=> ( ( C = A )
        & ( D = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Pair__eq) ).

thf(1837,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( C = A )
          & ( D = B ) )
       => ( ( C @ ( D @ ( E @ ( F @ c_Product__Type_OPair ) ) ) )
          = ( A @ ( B @ ( E @ ( F @ c_Product__Type_OPair ) ) ) ) ) )
      & ( ( ( C @ ( D @ ( E @ ( F @ c_Product__Type_OPair ) ) ) )
          = ( A @ ( B @ ( E @ ( F @ c_Product__Type_OPair ) ) ) ) )
       => ( ( C = A )
          & ( D = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).

thf(35,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B = A )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord__less__eq__trans) ).

thf(625,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B = A )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(505,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ c_Relation_Oantisym ) )
    <=> ( A @ ( B @ c_Relation_Oantisym ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_antisym__reflcl) ).

thf(2433,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ c_Relation_Oantisym ) )
       => ( B @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ c_Relation_Oantisym ) ) )
      & ( ( B @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ c_Relation_Oantisym ) )
       => ( A @ ( B @ c_Relation_Oantisym ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[505]) ).

thf(104,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I5_J) ).

thf(852,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(480,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( E @ ( E @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ! [F: $i] :
            ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( F @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
           => ! [G: $i] :
                ( ( A @ ( G @ ( F @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
               => ( F = G ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__Un__separatorE) ).

thf(2332,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( E @ ( E @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ! [F: $i] :
            ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( F @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
           => ! [G: $i] :
                ( ( A @ ( G @ ( F @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
               => ( F = G ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).

thf(1296,plain,
    ( ( c_Wellfounded_Oless__than @ ( c_Wellfounded_Oless__than @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Wellfounded_Olex__prod ) ) ) )
    = c_FunDef_Opair__less ),
    inference(lifteq,[status(thm)],[1295]) ).

thf(47,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B = A )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord__le__eq__trans) ).

thf(665,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B = A )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(98,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Wellfounded_Owf ) )
     => ( A @ ( B @ c_Wellfounded_Omin__ext ) @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Wellfounded_Owf ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_min__ext__wf) ).

thf(830,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Wellfounded_Owf ) )
     => ( A @ ( B @ c_Wellfounded_Omin__ext ) @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Wellfounded_Owf ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(152,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( B @ ( A @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__ge2) ).

thf(1027,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( B @ ( A @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(478,axiom,
    ! [A: $i] : ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Order__Relation_Opartial__order__on ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_partial__order__on__empty) ).

thf(2327,plain,
    ! [A: $i] : ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Order__Relation_Opartial__order__on ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[478]) ).

thf(482,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( B @ ( G @ ( H @ tc_prod ) @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( E @ ( F @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ tc_prod ) @ c_member ) ) )
     => ( ( C @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
       => ( ! [I: $i,J: $i,K: $i,L: $i] :
              ( ( B @ ( L @ ( K @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( J @ ( I @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ tc_prod ) @ c_member ) ) )
             => ( ( B @ ( G @ ( H @ tc_prod ) @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( L @ ( K @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ tc_prod ) @ c_member ) ) )
               => ( ( L @ ( K @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
                 => ( J @ ( I @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ) ) )
         => ( E @ ( F @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__rtrancl__induct2) ).

thf(2344,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( B @ ( G @ ( H @ tc_prod ) @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( E @ ( F @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ tc_prod ) @ c_member ) ) )
     => ( ( C @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
       => ( ! [I: $i,J: $i,K: $i,L: $i] :
              ( ( B @ ( L @ ( K @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( J @ ( I @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ tc_prod ) @ c_member ) ) )
             => ( ( B @ ( G @ ( H @ tc_prod ) @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( L @ ( K @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ tc_prod ) @ c_member ) ) )
               => ( ( L @ ( K @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
                 => ( J @ ( I @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ) ) )
         => ( E @ ( F @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[482]) ).

thf(518,axiom,
    ! [A: $i,B: $i] :
      ~ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ c_member ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_empty__iff) ).

thf(2492,plain,
    ! [A: $i,B: $i] :
      ~ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ c_member ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[518]) ).

thf(357,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( D @ ( E @ c_Relation_Orefl__on ) ) )
     => ( ( C @ ( A @ ( B @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( D @ ( B @ ( E @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__onD1) ).

thf(1851,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( D @ ( E @ c_Relation_Orefl__on ) ) )
     => ( ( C @ ( A @ ( B @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( D @ ( B @ ( E @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).

thf(393,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Transitive__Closure_Otrancl__into__trancl) ).

thf(1970,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[393]) ).

thf(359,axiom,
    ! [A: $i] : ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Order__Relation_Owell__order__on ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_well__order__on__empty) ).

thf(1856,plain,
    ! [A: $i] : ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Order__Relation_Owell__order__on ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[359]) ).

thf(145,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( ( B != A )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__less__le) ).

thf(980,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B != A )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B != A )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(535,plain,
    ( ~ sk1
    | ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) ),
    inference(cnfConj,[status(thm)],[524]) ).

thf(536,plain,
    ( ~ sk1
    | ( sk3 != sk4 )
    | ( v_L @ ( sk2 @ ( sk2 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) ),
    inference(cnfConj,[status(thm)],[524]) ).

thf(540,plain,
    ( ~ sk1
    | ( v_L @ ( sk2 @ ( sk2 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
    | ( sk4 != sk3 ) ),
    inference(lifteq,[status(thm)],[536]) ).

thf(531,plain,
    ! [A: $i] :
      ( ~ sk1
      | ~ ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ),
    inference(cnfConj,[status(thm)],[524]) ).

thf(2521,plain,
    ! [A: $i] :
      ( ( ( v_L @ ( sk2 @ ( sk2 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
       != ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) )
      | ~ sk1
      | ~ $true
      | ~ sk1
      | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
      | ( sk4 != sk3 ) ),
    inference(paramod_ordered,[status(thm)],[540,531]) ).

thf(2522,plain,
    ! [A: $i] :
      ( ( ( v_L @ ( sk2 @ ( sk2 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
       != ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) )
      | ~ sk1
      | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
      | ( sk4 != sk3 ) ),
    inference(simp,[status(thm)],[2521]) ).

thf(2523,plain,
    ( ~ sk1
    | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
    | ( sk4 != sk3 ) ),
    inference(pattern_uni,[status(thm)],[2522:[bind(A,$thf( sk2 ))]]) ).

thf(2535,plain,
    ( ( ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
     != ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) )
    | ~ sk1
    | ~ $true
    | ( sk4 != sk3 )
    | ~ sk1 ),
    inference(paramod_ordered,[status(thm)],[535,2523]) ).

thf(2536,plain,
    ( ( ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
     != ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) )
    | ( sk4 != sk3 )
    | ~ sk1 ),
    inference(simp,[status(thm)],[2535]) ).

thf(2537,plain,
    ( ( sk4 != sk3 )
    | ~ sk1 ),
    inference(pattern_uni,[status(thm)],[2536:[]]) ).

thf(338,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ ( B @ c_Relation_ORange ) ) @ ( C @ ( D @ c_member ) ) )
     => ~ ! [E: $i] :
            ~ ( A @ ( C @ ( E @ ( D @ ( B @ c_Product__Type_OPair ) ) ) @ ( D @ ( B @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_RangeE) ).

thf(1755,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ ( B @ c_Relation_ORange ) ) @ ( C @ ( D @ c_member ) ) )
     => ~ ! [E: $i] :
            ~ ( A @ ( C @ ( E @ ( D @ ( B @ c_Product__Type_OPair ) ) ) @ ( D @ ( B @ tc_prod ) @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).

thf(106,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__less__mono) ).

thf(857,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(302,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ( C != B )
       => ~ ! [E: $i] :
              ( ( A @ ( E @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
             => ~ ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( E @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__rtranclE) ).

thf(1592,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ( C != B )
       => ~ ! [E: $i] :
              ( ( A @ ( E @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
             => ~ ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( E @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).

thf(472,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) )
     => ( ( c_FunDef_Opair__less @ ( D @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) )
       => ( ( c_FunDef_Omax__strict @ ( C @ ( A @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) )
         => ( c_FunDef_Omax__strict @ ( C @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_smax__insertI) ).

thf(2300,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) )
     => ( ( c_FunDef_Opair__less @ ( D @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) )
       => ( ( c_FunDef_Omax__strict @ ( C @ ( A @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) )
         => ( c_FunDef_Omax__strict @ ( C @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[472]) ).

thf(60,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__antisym) ).

thf(697,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(520,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( A @ ( B @ ( D @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
         => ( A @ ( C @ c_Wellfounded_Oacc ) @ ( D @ ( C @ c_member ) ) ) )
     => ( A @ ( C @ c_Wellfounded_Oacc ) @ ( B @ ( C @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acc_OaccI) ).

thf(2498,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( A @ ( B @ ( D @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
         => ( A @ ( C @ c_Wellfounded_Oacc ) @ ( D @ ( C @ c_member ) ) ) )
     => ( A @ ( C @ c_Wellfounded_Oacc ) @ ( B @ ( C @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[520]) ).

thf(211,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ ( C @ ( D @ c_Nitpick_Ofold__graph_H ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__graph_H_Ointros_I1_J) ).

thf(1234,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ ( C @ ( D @ c_Nitpick_Ofold__graph_H ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(285,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__le__mono) ).

thf(1541,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(180,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Olattice )
     => ( B @ ( A @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf__sup__ord_I4_J) ).

thf(1143,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Olattice )
     => ( B @ ( A @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(433,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( B @ ( C @ ( D @ c_member ) ) )
     => ( ~ ( A @ ( C @ ( D @ c_member ) ) )
       => ( ( ( B @ ( C @ ( D @ c_Set_Oinsert ) ) )
            = ( A @ ( C @ ( D @ c_Set_Oinsert ) ) ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__ident) ).

thf(2115,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( B @ ( C @ ( D @ c_member ) ) )
     => ( ~ ( A @ ( C @ ( D @ c_member ) ) )
       => ( ( ( B = A )
           => ( ( B @ ( C @ ( D @ c_Set_Oinsert ) ) )
              = ( A @ ( C @ ( D @ c_Set_Oinsert ) ) ) ) )
          & ( ( ( B @ ( C @ ( D @ c_Set_Oinsert ) ) )
              = ( A @ ( C @ ( D @ c_Set_Oinsert ) ) ) )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[433]) ).

thf(341,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Owf ) )
     => ( ( B @ ( A @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ c_Wellfounded_Owf ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__subset) ).

thf(1762,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Owf ) )
     => ( ( B @ ( A @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ c_Wellfounded_Owf ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).

thf(428,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__into__trancl2) ).

thf(2100,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).

thf(491,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( C @ ( F @ c_Wellfounded_Omax__ext ) @ ( D @ ( E @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ tc_prod ) @ c_member ) ) )
     => ( ( C @ ( F @ c_Wellfounded_Omax__ext ) @ ( A @ ( B @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ tc_prod ) @ c_member ) ) )
       => ( C @ ( F @ c_Wellfounded_Omax__ext ) @ ( A @ ( D @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( E @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_max__ext__additive) ).

thf(2377,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( C @ ( F @ c_Wellfounded_Omax__ext ) @ ( D @ ( E @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ tc_prod ) @ c_member ) ) )
     => ( ( C @ ( F @ c_Wellfounded_Omax__ext ) @ ( A @ ( B @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ tc_prod ) @ c_member ) ) )
       => ( C @ ( F @ c_Wellfounded_Omax__ext ) @ ( A @ ( D @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( E @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[491]) ).

thf(133,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
        = ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
     => ( A @ ( C @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__by__empty) ).

thf(948,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
        = ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
     => ( A @ ( C @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(150,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( A
        = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__empty) ).

thf(1017,plain,
    ! [A: $i,B: $i] :
      ( ( ( A
          = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
       => ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A
          = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(388,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
     => ( A @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( C @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_r__into__rtrancl) ).

thf(1953,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
     => ( A @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( C @ tc_prod ) @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).

thf(405,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ~ ( A @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ~ ! [E: $i] :
              ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( E @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
             => ~ ( A @ ( B @ ( E @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_tranclE) ).

thf(2023,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ~ ( A @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ~ ! [E: $i] :
              ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( E @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
             => ~ ( A @ ( B @ ( E @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[405]) ).

thf(244,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Wellfounded_Owf ) )
     => ( A @ ( B @ ( B @ c_FunDef_Oin__rel ) ) @ ( B @ c_Wellfounded_OwfP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__in__rel) ).

thf(1361,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Wellfounded_Owf ) )
     => ( A @ ( B @ ( B @ c_FunDef_Oin__rel ) ) @ ( B @ c_Wellfounded_OwfP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(392,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ( C != B )
       => ~ ! [E: $i] :
              ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( E @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
             => ~ ( A @ ( B @ ( E @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtranclE) ).

thf(1964,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ( C != B )
       => ~ ! [E: $i] :
              ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( E @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
             => ~ ( A @ ( B @ ( E @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).

thf(194,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__trans) ).

thf(1182,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(48,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Olattice )
     => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf__sup__aci_I8_J) ).

thf(669,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Olattice )
     => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(166,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( C @ ( D @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ ( F @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ hAPP ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_predicate2D) ).

thf(1075,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( C @ ( D @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ ( F @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ hAPP ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(4723,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_fequal @ ( A @ ( B @ c_member ) ) )
       != ( v_L @ ( C @ ( C @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) )
      | ~ sk1
      | ~ $true ),
    inference(paramod_ordered,[status(thm)],[1924,531]) ).

thf(4724,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_fequal @ ( A @ ( B @ c_member ) ) )
       != ( v_L @ ( C @ ( C @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) )
      | ~ sk1 ),
    inference(simp,[status(thm)],[4723]) ).

thf(4741,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_fequal )
       != v_L )
      | ( A
       != ( C @ ( C @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) ) )
      | ( B
       != ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) ) )
      | ~ sk1 ),
    inference(simp,[status(thm)],[4724]) ).

thf(4750,plain,
    ! [A: $i] :
      ( ( ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ c_fequal )
       != v_L )
      | ~ sk1 ),
    inference(simp,[status(thm)],[4741]) ).

thf(526,plain,
    ( ~ ( v_L @ ( sk7 @ ( sk6 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
    | sk1 ),
    inference(cnfConj,[status(thm)],[524]) ).

thf(274,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__not__less) ).

thf(1493,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(322,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( D @ ( E @ ( F @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ c_member ) ) )
     => ( ( A @ ( B @ ( E @ ( C @ ( G @ c_Product__Type_OPair ) ) ) @ ( C @ ( G @ tc_prod ) @ c_member ) ) )
       => ( A @ ( D @ ( C @ ( G @ ( H @ c_Relation_Orel__comp ) ) ) ) @ ( B @ ( F @ ( C @ ( H @ c_Product__Type_OPair ) ) ) @ ( C @ ( H @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__compI) ).

thf(1683,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( D @ ( E @ ( F @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ c_member ) ) )
     => ( ( A @ ( B @ ( E @ ( C @ ( G @ c_Product__Type_OPair ) ) ) @ ( C @ ( G @ tc_prod ) @ c_member ) ) )
       => ( A @ ( D @ ( C @ ( G @ ( H @ c_Relation_Orel__comp ) ) ) ) @ ( B @ ( F @ ( C @ ( H @ c_Product__Type_OPair ) ) ) @ ( C @ ( H @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[322]) ).

thf(77,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
      | ~ ( A @ ( B @ c_fequal @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fequal__1) ).

thf(753,plain,
    ! [A: $i,B: $i] :
      ( ( B = A )
      | ~ ( A @ ( B @ c_fequal @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(335,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ( A @ ( D @ c_Wellfounded_Oacc ) @ ( B @ ( D @ c_member ) ) )
       => ( A @ ( D @ c_Wellfounded_Oacc ) @ ( C @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acc__downwards__aux) ).

thf(1739,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ( A @ ( D @ c_Wellfounded_Oacc ) @ ( B @ ( D @ c_member ) ) )
       => ( A @ ( D @ c_Wellfounded_Oacc ) @ ( C @ ( D @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).

thf(264,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Lattices_Osemilattice__sup )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( E @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( D @ ( E @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__mono) ).

thf(1461,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Lattices_Osemilattice__sup )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( E @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( D @ ( E @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(38,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup_Oleft__commute) ).

thf(637,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(298,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__into__trancl1) ).

thf(1584,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).

thf(432,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Order__Relation_Opreorder__on ) ) )
    <=> ( ( A @ ( C @ c_Relation_Otrans ) )
        & ( A @ ( B @ ( C @ c_Relation_Orefl__on ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_preorder__on__def) ).

thf(2108,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( A @ ( C @ c_Relation_Otrans ) )
          & ( A @ ( B @ ( C @ c_Relation_Orefl__on ) ) ) )
       => ( A @ ( B @ ( C @ c_Order__Relation_Opreorder__on ) ) ) )
      & ( ( A @ ( B @ ( C @ c_Order__Relation_Opreorder__on ) ) )
       => ( ( A @ ( C @ c_Relation_Otrans ) )
          & ( A @ ( B @ ( C @ c_Relation_Orefl__on ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[432]) ).

thf(260,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B @ ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__least) ).

thf(1446,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B @ ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(185,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B @ ( A @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__supI2) ).

thf(1155,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B @ ( A @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(4,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C = A )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I4_J) ).

thf(547,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C = A )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(34,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Olattice )
     => ( ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf__sup__aci_I7_J) ).

thf(622,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Olattice )
     => ( ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(354,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( A @ ( D @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
         => ( ( A @ ( C @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
           => ( C = D ) ) )
     => ( A @ ( B @ c_Relation_Oantisym ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_antisymI) ).

thf(1825,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( A @ ( D @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
         => ( ( A @ ( C @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
           => ( C = D ) ) )
     => ( A @ ( B @ c_Relation_Oantisym ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).

thf(43,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( C = B )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord__eq__less__trans) ).

thf(647,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( C = B )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(300,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( A @ ( D @ c_member ) ) )
       => ( B @ ( A @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subsetD) ).

thf(1588,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( A @ ( D @ c_member ) ) )
       => ( B @ ( A @ ( D @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).

thf(183,axiom,
    ( c_Wellfounded_Oless__than
    = ( c_Wellfounded_Opred__nat @ ( tc_Nat_Onat @ c_Transitive__Closure_Otrancl ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__than__def) ).

thf(1149,plain,
    ( c_Wellfounded_Oless__than
    = ( c_Wellfounded_Opred__nat @ ( tc_Nat_Onat @ c_Transitive__Closure_Otrancl ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(1150,plain,
    ( ( c_Wellfounded_Opred__nat @ ( tc_Nat_Onat @ c_Transitive__Closure_Otrancl ) )
    = c_Wellfounded_Oless__than ),
    inference(lifteq,[status(thm)],[1149]) ).

thf(225,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Ominus )
     => ( ( A @ ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ hAPP ) )
        = ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_minus__apply) ).

thf(1297,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Ominus )
     => ( ( A @ ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ hAPP ) )
        = ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(248,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Obounded__lattice__bot )
     => ( ( A @ ( B @ c_Orderings_Obot__class_Obot @ ( B @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__bot__left) ).

thf(1382,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Obounded__lattice__bot )
     => ( ( A @ ( B @ c_Orderings_Obot__class_Obot @ ( B @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(355,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Relation_ODomain ) ) @ ( C @ ( D @ c_member ) ) )
    <=> ? [E: $i] : ( A @ ( E @ ( C @ ( B @ ( D @ c_Product__Type_OPair ) ) ) @ ( B @ ( D @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__iff) ).

thf(1831,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ? [E: $i] : ( A @ ( E @ ( C @ ( B @ ( D @ c_Product__Type_OPair ) ) ) @ ( B @ ( D @ tc_prod ) @ c_member ) ) )
       => ( A @ ( B @ ( D @ c_Relation_ODomain ) ) @ ( C @ ( D @ c_member ) ) ) )
      & ( ( A @ ( B @ ( D @ c_Relation_ODomain ) ) @ ( C @ ( D @ c_member ) ) )
       => ? [E: $i] : ( A @ ( E @ ( C @ ( B @ ( D @ c_Product__Type_OPair ) ) ) @ ( B @ ( D @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).

thf(461,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( E @ ( C @ c_Relation_OImage ) ) ) @ ( D @ ( E @ c_member ) ) )
    <=> ( B @ ( D @ ( A @ ( E @ ( C @ c_Product__Type_OPair ) ) ) @ ( E @ ( C @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__singleton__iff) ).

thf(2243,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( B @ ( D @ ( A @ ( E @ ( C @ c_Product__Type_OPair ) ) ) @ ( E @ ( C @ tc_prod ) @ c_member ) ) )
       => ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( E @ ( C @ c_Relation_OImage ) ) ) @ ( D @ ( E @ c_member ) ) ) )
      & ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( E @ ( C @ c_Relation_OImage ) ) ) @ ( D @ ( E @ c_member ) ) )
       => ( B @ ( D @ ( A @ ( E @ ( C @ c_Product__Type_OPair ) ) ) @ ( E @ ( C @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[461]) ).

thf(485,axiom,
    ! [A: $i] :
      ( ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Relation_OId__on ) )
      = ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Id__on__empty) ).

thf(2355,plain,
    ! [A: $i] :
      ( ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Relation_OId__on ) )
      = ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[485]) ).

thf(348,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( C @ ( D @ c_Relation_OField ) @ ( D @ c_Order__Relation_Opartial__order__on ) ) )
     => ( ( C @ ( D @ c_Relation_OField ) @ ( B @ ( D @ c_member ) ) )
       => ( ( C @ ( D @ c_Relation_OField ) @ ( A @ ( D @ c_member ) ) )
         => ( ( ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) )
              = ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) ) )
          <=> ( B = A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Partial__order__eq__Image1__Image1__iff) ).

thf(1799,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( C @ ( D @ c_Relation_OField ) @ ( D @ c_Order__Relation_Opartial__order__on ) ) )
     => ( ( C @ ( D @ c_Relation_OField ) @ ( B @ ( D @ c_member ) ) )
       => ( ( C @ ( D @ c_Relation_OField ) @ ( A @ ( D @ c_member ) ) )
         => ( ( ( B = A )
             => ( ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) )
                = ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) ) ) )
            & ( ( ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) )
                = ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) ) )
             => ( B = A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).

thf(365,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( tc_HOL_Obool @ ( D @ ( E @ c_Product__Type_Ocurry ) ) ) ) ) @ hBOOL )
     => ( A @ ( B @ ( D @ ( E @ c_Product__Type_OPair ) ) ) @ ( C @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_curryD) ).

thf(1874,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( tc_HOL_Obool @ ( D @ ( E @ c_Product__Type_Ocurry ) ) ) ) ) @ hBOOL )
     => ( A @ ( B @ ( D @ ( E @ c_Product__Type_OPair ) ) ) @ ( C @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[365]) ).

thf(467,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( C @ ( D @ c_Relation_OField ) @ ( D @ c_Order__Relation_Opreorder__on ) ) )
     => ( ( C @ ( D @ c_Relation_OField ) @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( D @ c_Relation_OField ) @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( A @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) @ ( B @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
          <=> ! [E: $i] :
                ( ( B @ ( E @ ( D @ c_member ) ) )
               => ? [F: $i] :
                    ( ( C @ ( E @ ( F @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
                    & ( A @ ( F @ ( D @ c_member ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__Image__Image__iff) ).

thf(2272,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( C @ ( D @ c_Relation_OField ) @ ( D @ c_Order__Relation_Opreorder__on ) ) )
     => ( ( C @ ( D @ c_Relation_OField ) @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( D @ c_Relation_OField ) @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( ! [E: $i] :
                  ( ( B @ ( E @ ( D @ c_member ) ) )
                 => ? [F: $i] :
                      ( ( C @ ( E @ ( F @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
                      & ( A @ ( F @ ( D @ c_member ) ) ) ) )
             => ( A @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) @ ( B @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
            & ( ( A @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) @ ( B @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
             => ! [E: $i] :
                  ( ( B @ ( E @ ( D @ c_member ) ) )
                 => ? [F: $i] :
                      ( ( C @ ( E @ ( F @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
                      & ( A @ ( F @ ( D @ c_member ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).

thf(398,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( c_FunDef_Opair__less @ ( A @ ( C @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( B @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__lessI1) ).

thf(1991,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( c_FunDef_Opair__less @ ( A @ ( C @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( B @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[398]) ).

thf(451,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Product__Type_OPair ) ) ) @ ( D @ ( E @ tc_prod ) @ c_member ) ) )
     => ( A @ ( D @ ( E @ c_Relation_ORange ) ) @ ( B @ ( D @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_RangeI) ).

thf(2203,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Product__Type_OPair ) ) ) @ ( D @ ( E @ tc_prod ) @ c_member ) ) )
     => ( A @ ( D @ ( E @ c_Relation_ORange ) ) @ ( B @ ( D @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).

thf(351,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( D @ ( E @ c_Relation_Orefl__on ) ) )
     => ( ( C @ ( A @ ( B @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( D @ ( A @ ( E @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__onD2) ).

thf(1814,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( D @ ( E @ c_Relation_Orefl__on ) ) )
     => ( ( C @ ( A @ ( B @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( D @ ( A @ ( E @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).

thf(37,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C = A )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I2_J) ).

thf(633,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C = A )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(453,axiom,
    ( c_FunDef_Omax__strict
    = ( c_FunDef_Opair__less @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Wellfounded_Omax__ext ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_max__strict__def) ).

thf(2207,plain,
    ( c_FunDef_Omax__strict
    = ( c_FunDef_Opair__less @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Wellfounded_Omax__ext ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[453]) ).

thf(2208,plain,
    ( ( c_FunDef_Opair__less @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Wellfounded_Omax__ext ) )
    = c_FunDef_Omax__strict ),
    inference(lifteq,[status(thm)],[2207]) ).

thf(328,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_Oantisym ) )
    <=> ! [C: $i,D: $i] :
          ( ( A @ ( D @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
         => ( ( A @ ( C @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
           => ( C = D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_antisym__def) ).

thf(1695,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i,D: $i] :
            ( ( A @ ( D @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
           => ( ( A @ ( C @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
             => ( C = D ) ) )
       => ( A @ ( B @ c_Relation_Oantisym ) ) )
      & ( ( A @ ( B @ c_Relation_Oantisym ) )
       => ! [C: $i,D: $i] :
            ( ( A @ ( D @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
           => ( ( A @ ( C @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
             => ( C = D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).

thf(222,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Olattice )
     => ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf__sup__ord_I3_J) ).

thf(1286,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Olattice )
     => ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(8,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__imp__not__less) ).

thf(557,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(558,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
      | ~ ( C @ class_Orderings_Opreorder ) ),
    inference(cnf,[status(esa)],[557]) ).

thf(307,axiom,
    ! [A: $i,B: $i] : ( A @ ( B @ c_Transitive__Closure_Ortrancl ) @ ( B @ c_Relation_Otrans ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__rtrancl) ).

thf(1615,plain,
    ! [A: $i,B: $i] : ( A @ ( B @ c_Transitive__Closure_Ortrancl ) @ ( B @ c_Relation_Otrans ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).

thf(1616,plain,
    ! [B: $i,A: $i] : ( A @ ( B @ c_Transitive__Closure_Ortrancl ) @ ( B @ c_Relation_Otrans ) ),
    inference(cnf,[status(esa)],[1615]) ).

thf(530,plain,
    ( ~ sk1
    | ~ ( v_L @ ( sk4 @ ( sk3 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ( v_L @ ( sk2 @ ( sk2 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) ),
    inference(cnfConj,[status(thm)],[524]) ).

thf(67,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B != A )
      <=> ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) )
          | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__neq__iff) ).

thf(718,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) )
            | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) )
         => ( B != A ) )
        & ( ( B != A )
         => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) )
            | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(291,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( C @ ( B @ hAPP ) )
          = ( C @ ( A @ hAPP ) ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ext) ).

thf(1563,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( C @ ( B @ hAPP ) )
          = ( C @ ( A @ hAPP ) ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).

thf(46,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( ( B = A )
          | ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__less__iff__gr__or__eq) ).

thf(657,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( ( ( B = A )
            | ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) )
         => ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
         => ( ( B = A )
            | ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(331,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] : ( A @ ( C @ ( B @ c_member ) ) )
    <=> ( A
       != ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ex__in__conv) ).

thf(1712,plain,
    ! [A: $i,B: $i] :
      ( ( ( A
         != ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
       => ? [C: $i] : ( A @ ( C @ ( B @ c_member ) ) ) )
      & ( ? [C: $i] : ( A @ ( C @ ( B @ c_member ) ) )
       => ( A
         != ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).

thf(161,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__insert) ).

thf(1054,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(196,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Set_Oinsert ) ) )
     != ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__not__empty) ).

thf(1187,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Set_Oinsert ) ) )
     != ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(51,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__imp__not__eq) ).

thf(674,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(282,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( ( ~ ( A @ ( B @ ( D @ c_member ) ) )
         => ( ( ~ ( C @ ( B @ ( D @ c_member ) ) )
             => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
            & ( ( C @ ( B @ ( D @ c_member ) ) )
             => ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) ) )
        & ( ( A @ ( B @ ( D @ c_member ) ) )
         => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_psubset__insert__iff) ).

thf(1515,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( ~ ( A @ ( B @ ( D @ c_member ) ) )
           => ( ( ~ ( C @ ( B @ ( D @ c_member ) ) )
               => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
              & ( ( C @ ( B @ ( D @ c_member ) ) )
               => ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) ) )
          & ( ( A @ ( B @ ( D @ c_member ) ) )
           => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) )
       => ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( ~ ( A @ ( B @ ( D @ c_member ) ) )
           => ( ( ~ ( C @ ( B @ ( D @ c_member ) ) )
               => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
              & ( ( C @ ( B @ ( D @ c_member ) ) )
               => ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) ) )
          & ( ( A @ ( B @ ( D @ c_member ) ) )
           => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(457,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_member ) ) )
    <=> ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mem__def) ).

thf(2221,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( A @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ c_member ) ) ) )
      & ( ( A @ ( B @ ( C @ c_member ) ) )
       => ( B @ ( A @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[457]) ).

thf(89,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( ~ ( B @ ( A @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
          & ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__fun__def) ).

thf(794,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( ( ~ ( B @ ( A @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
         => ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
         => ( ~ ( B @ ( A @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(481,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nitpick_Orefl_H ) )
    <=> ! [C: $i] : ( A @ ( C @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nitpick_Orefl_H__def) ).

thf(2338,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] : ( A @ ( C @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
       => ( A @ ( B @ c_Nitpick_Orefl_H ) ) )
      & ( ( A @ ( B @ c_Nitpick_Orefl_H ) )
       => ! [C: $i] : ( A @ ( C @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[481]) ).

thf(219,axiom,
    ! [A: $i,B: $i] :
      ~ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__psubset__empty) ).

thf(1276,plain,
    ! [A: $i,B: $i] :
      ~ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(1277,plain,
    ~ ? [A: $i,B: $i] : ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ),
    inference(miniscope,[status(thm)],[1276]) ).

thf(1278,plain,
    ! [B: $i,A: $i] :
      ~ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ),
    inference(cnf,[status(esa)],[1277]) ).

thf(402,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( A @ ( B @ ( C @ c_Relation_ORange ) ) )
        = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
    <=> ( A
        = ( tc_HOL_Obool @ ( B @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__empty__iff) ).

thf(2001,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( A
          = ( tc_HOL_Obool @ ( B @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
       => ( ( A @ ( B @ ( C @ c_Relation_ORange ) ) )
          = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) )
      & ( ( ( A @ ( B @ ( C @ c_Relation_ORange ) ) )
          = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
       => ( A
          = ( tc_HOL_Obool @ ( B @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[402]) ).

thf(495,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_member ) ) )
     => ~ ( A @ ( C @ ( D @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DiffD2) ).

thf(2391,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_member ) ) )
     => ~ ( A @ ( C @ ( D @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[495]) ).

thf(441,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( ( D @ ( A @ ( B @ c_member ) ) )
         => ~ ( C @ ( A @ ( B @ c_member ) ) ) )
       => ~ ( ~ ( D @ ( A @ ( B @ c_member ) ) )
           => ( C @ ( A @ ( B @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equalityCE) ).

thf(2149,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( ( D @ ( A @ ( B @ c_member ) ) )
         => ~ ( C @ ( A @ ( B @ c_member ) ) ) )
       => ~ ( ~ ( D @ ( A @ ( B @ c_member ) ) )
           => ( C @ ( A @ ( B @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[441]) ).

thf(380,axiom,
    ( c_FunDef_Omax__weak
    = ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_Set_Oinsert ) ) @ ( c_FunDef_Opair__leq @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Wellfounded_Omax__ext ) @ ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_max__weak__def) ).

thf(1927,plain,
    ( c_FunDef_Omax__weak
    = ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_Set_Oinsert ) ) @ ( c_FunDef_Opair__leq @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Wellfounded_Omax__ext ) @ ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[380]) ).

thf(55,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__asym_H) ).

thf(689,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(71,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( B @ ( A @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__commute) ).

thf(733,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( B @ ( A @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(140,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B = A )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__or__eq__imp__le) ).

thf(965,plain,
    ! [A: $i,B: $i] :
      ( ( ( B = A )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(76,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ c_Relation_OId @ ( A @ ( B @ c_Transitive__Closure_Ortrancl ) @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__reflcl__absorb) ).

thf(750,plain,
    ! [A: $i,B: $i] :
      ( ( B @ c_Relation_OId @ ( A @ ( B @ c_Transitive__Closure_Ortrancl ) @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(334,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) )
    <=> ? [G: $i] :
          ( ! [H: $i] :
              ( ( D @ ( B @ ( H @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) )
             => ( H @ ( G @ hAPP ) @ ( H @ ( C @ ( D @ ( E @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) ) )
          & ( A
            = ( B @ ( G @ ( C @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wfrec__rel_Osimps) ).

thf(1727,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ? [G: $i] :
            ( ! [H: $i] :
                ( ( D @ ( B @ ( H @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) )
               => ( H @ ( G @ hAPP ) @ ( H @ ( C @ ( D @ ( E @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) ) )
            & ( A
              = ( B @ ( G @ ( C @ hAPP ) @ hAPP ) ) ) )
       => ( A @ ( B @ ( C @ ( D @ ( E @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) ) )
      & ( ( A @ ( B @ ( C @ ( D @ ( E @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) )
       => ? [G: $i] :
            ( ! [H: $i] :
                ( ( D @ ( B @ ( H @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) )
               => ( H @ ( G @ hAPP ) @ ( H @ ( C @ ( D @ ( E @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) ) )
            & ( A
              = ( B @ ( G @ ( C @ hAPP ) @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).

thf(215,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( A @ ( B @ ( E @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( F @ ( G @ c_Nitpick_Ofold__graph_H ) ) ) ) ) )
    <=> ( ? [H: $i,I: $i] :
            ( ( I @ ( tc_HOL_Obool @ ( G @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( H @ ( G @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( G @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ ( E @ ( F @ ( G @ c_Nitpick_Ofold__graph_H ) ) ) ) ) )
            & ( C @ ( H @ ( G @ c_member ) ) )
            & ( ( A @ ( B @ ( E @ hAPP ) @ hAPP ) )
              = ( I @ ( H @ ( E @ hAPP ) @ hAPP ) ) ) )
        | ( ( ( A @ ( B @ ( E @ hAPP ) @ hAPP ) )
            = D )
          & ( C
            = ( tc_HOL_Obool @ ( G @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__graph_H_Oequations_I2_J) ).

thf(1251,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( ? [H: $i,I: $i] :
              ( ( I @ ( tc_HOL_Obool @ ( G @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( H @ ( G @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( G @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ ( E @ ( F @ ( G @ c_Nitpick_Ofold__graph_H ) ) ) ) ) )
              & ( C @ ( H @ ( G @ c_member ) ) )
              & ( ( A @ ( B @ ( E @ hAPP ) @ hAPP ) )
                = ( I @ ( H @ ( E @ hAPP ) @ hAPP ) ) ) )
          | ( ( ( A @ ( B @ ( E @ hAPP ) @ hAPP ) )
              = D )
            & ( C
              = ( tc_HOL_Obool @ ( G @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) )
       => ( A @ ( B @ ( E @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( F @ ( G @ c_Nitpick_Ofold__graph_H ) ) ) ) ) ) )
      & ( ( A @ ( B @ ( E @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( F @ ( G @ c_Nitpick_Ofold__graph_H ) ) ) ) ) )
       => ( ? [H: $i,I: $i] :
              ( ( I @ ( tc_HOL_Obool @ ( G @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( H @ ( G @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( G @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ ( E @ ( F @ ( G @ c_Nitpick_Ofold__graph_H ) ) ) ) ) )
              & ( C @ ( H @ ( G @ c_member ) ) )
              & ( ( A @ ( B @ ( E @ hAPP ) @ hAPP ) )
                = ( I @ ( H @ ( E @ hAPP ) @ hAPP ) ) ) )
          | ( ( ( A @ ( B @ ( E @ hAPP ) @ hAPP ) )
              = D )
            & ( C
              = ( tc_HOL_Obool @ ( G @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(33,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( C = B )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord__eq__le__trans) ).

thf(618,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( C = B )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(342,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( D @ c_Wellfounded_Oacc ) @ ( C @ ( D @ c_member ) ) )
     => ( ( B @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( A @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ( B @ ( D @ c_Wellfounded_Oacc ) @ ( A @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acc__downwards) ).

thf(1764,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( D @ c_Wellfounded_Oacc ) @ ( C @ ( D @ c_member ) ) )
     => ( ( B @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( A @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ( B @ ( D @ c_Wellfounded_Oacc ) @ ( A @ ( D @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).

thf(10,axiom,
    tc_Nat_Onat @ class_Groups_Ominus,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ominus) ).

thf(562,plain,
    tc_Nat_Onat @ class_Groups_Ominus,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(148,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ ( B @ c_Wellfounded_Owf ) )
     => ! [E: $i] :
          ( ( E @ ( C @ ( D @ ( A @ ( B @ c_Recdef_Owfrec ) ) ) @ hAPP ) )
          = ( E @ ( E @ ( D @ ( C @ ( D @ ( A @ ( B @ c_Recdef_Owfrec ) ) ) @ ( A @ ( B @ c_Recdef_Ocut ) ) ) ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_tfl__wfrec) ).

thf(1007,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ ( B @ c_Wellfounded_Owf ) )
     => ! [E: $i] :
          ( ( E @ ( C @ ( D @ ( A @ ( B @ c_Recdef_Owfrec ) ) ) @ hAPP ) )
          = ( E @ ( E @ ( D @ ( C @ ( D @ ( A @ ( B @ c_Recdef_Owfrec ) ) ) @ ( A @ ( B @ c_Recdef_Ocut ) ) ) ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(367,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Owf ) )
     => ~ ( B @ ( A @ ( A @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__irrefl) ).

thf(1884,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Owf ) )
     => ~ ( B @ ( A @ ( A @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).

thf(175,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__mono) ).

thf(1125,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(84,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B != A )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__le__neq__trans) ).

thf(772,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B != A )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(496,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( A @ ( D @ ( B @ c_member ) ) )
       => ( A @ ( B @ c_Relation_OId__on ) @ ( C @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Id__on__eqI) ).

thf(2393,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( A @ ( D @ ( B @ c_member ) ) )
       => ( A @ ( B @ c_Relation_OId__on ) @ ( C @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[496]) ).

thf(381,axiom,
    ! [A: $i,B: $i] : ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( B @ c_Relation_Otrans ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__trancl) ).

thf(1929,plain,
    ! [A: $i,B: $i] : ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( B @ c_Relation_Otrans ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[381]) ).

thf(250,axiom,
    ! [A: $i,B: $i] : ( A @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__refl) ).

thf(1388,plain,
    ! [A: $i,B: $i] : ( A @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(19,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( B @ ( A @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup_Ocommute) ).

thf(579,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( B @ ( A @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(283,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__assoc) ).

thf(1535,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).

thf(347,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i] :
          ( ( D @ ( E @ ( G @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) )
         => ( G @ ( A @ hAPP ) @ ( G @ ( B @ ( D @ ( C @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) ) )
     => ( E @ ( A @ ( B @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( D @ ( C @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wfrec__rel_Ointros) ).

thf(1795,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i] :
          ( ( D @ ( E @ ( G @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) )
         => ( G @ ( A @ hAPP ) @ ( G @ ( B @ ( D @ ( C @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) ) )
     => ( E @ ( A @ ( B @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( D @ ( C @ ( F @ c_Recdef_Owfrec__rel ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[347]) ).

thf(208,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Nitpick_Owf__wfrec_H ) ) ) ) )
      = ( A @ ( A @ ( C @ ( B @ ( C @ ( D @ ( E @ c_Nitpick_Owf__wfrec ) ) ) @ ( D @ ( E @ c_Recdef_Ocut ) ) ) ) @ ( B @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nitpick_Owf__wfrec_H__def) ).

thf(1226,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Nitpick_Owf__wfrec_H ) ) ) ) )
      = ( A @ ( A @ ( C @ ( B @ ( C @ ( D @ ( E @ c_Nitpick_Owf__wfrec ) ) ) @ ( D @ ( E @ c_Recdef_Ocut ) ) ) ) @ ( B @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(826,plain,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(cnf,[status(esa)],[825]) ).

thf(18,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oord )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Oord ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Orderings_Oord) ).

thf(577,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oord )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Oord ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(66,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) )
        <=> ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__antisym__conv3) ).

thf(711,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( ( A = B )
           => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) )
          & ( ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) )
           => ( A = B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(277,axiom,
    ! [A: $i,B: $i] : ( A @ ( B @ c_Relation_OId__on ) @ ( A @ ( B @ c_Relation_Orefl__on ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__on__Id__on) ).

thf(1503,plain,
    ! [A: $i,B: $i] : ( A @ ( B @ c_Relation_OId__on ) @ ( A @ ( B @ c_Relation_Orefl__on ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(128,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_empty__Diff) ).

thf(936,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(329,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_member ) ) )
     => ( ~ ( B @ ( C @ ( D @ c_member ) ) )
       => ( A @ ( C @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UnE) ).

thf(1707,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_member ) ) )
     => ( ~ ( B @ ( C @ ( D @ c_member ) ) )
       => ( A @ ( C @ ( D @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).

thf(231,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_psubset__trans) ).

thf(1312,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(779,plain,
    ( ! [A: $i,B: $i,C: $i] :
        ( ( ( B = A )
          | ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) )
       => ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
    & ! [A: $i,B: $i,C: $i] :
        ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B = A )
          | ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(miniscope,[status(thm)],[778]) ).

thf(780,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( E @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
        | ( E != D ) )
      & ( ( D @ ( E @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
        | ~ ( D @ ( E @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( B = A )
        | ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
        | ~ ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(cnf,[status(esa)],[779]) ).

thf(783,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
      | ( B != A ) ),
    inference(cnfConj,[status(thm)],[780]) ).

thf(786,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
      | ( B != A ) ),
    inference(lifteq,[status(thm)],[783]) ).

thf(787,plain,
    ! [B: $i,A: $i] : ( A @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(simp,[status(thm)],[786]) ).

thf(400,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Owf ) )
     => ~ ( B @ ( A @ ( A @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__not__refl) ).

thf(1997,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Owf ) )
     => ~ ( B @ ( A @ ( A @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).

thf(237,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
        <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__diff__iff) ).

thf(1332,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
           => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
          & ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
           => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(502,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__rtrancl__trancl) ).

thf(2425,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[502]) ).

thf(49,axiom,
    tc_HOL_Obool @ class_Orderings_Obot,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL__Obool__Orderings_Obot) ).

thf(672,plain,
    tc_HOL_Obool @ class_Orderings_Obot,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(217,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__empty__right) ).

thf(1271,plain,
    ! [A: $i,B: $i] :
      ( ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(246,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( ( B != A )
          & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__le) ).

thf(1371,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( ( ( B != A )
            & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
         => ( ( B != A )
            & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(1058,plain,
    ! [A: $i] : ( A @ c_Relation_OId @ ( A @ c_Relation_Oantisym ) ),
    inference(cnf,[status(esa)],[1057]) ).

thf(172,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__ab__group__add )
     => ( ( ( C @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) )
       => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) ) )
        <=> ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__eq__diff__less) ).

thf(1107,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__ab__group__add )
     => ( ( ( C @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) )
       => ( ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) ) )
           => ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) ) ) )
          & ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) ) )
           => ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(125,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( D @ c_Set_Oinsert ) ) @ hAPP ) @ hBOOL )
    <=> ( ( A @ ( B @ hAPP ) @ hBOOL )
        | ( C = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__code) ).

thf(919,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( A @ ( B @ hAPP ) @ hBOOL )
          | ( C = A ) )
       => ( A @ ( B @ ( C @ ( D @ c_Set_Oinsert ) ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( C @ ( D @ c_Set_Oinsert ) ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ hAPP ) @ hBOOL )
          | ( C = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(91,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__le__trans) ).

thf(804,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(2148,plain,
    ! [A: $i] : ( A @ c_Relation_OId @ ( A @ c_Relation_Otrans ) ),
    inference(cnf,[status(esa)],[2147]) ).

thf(486,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ c_Relation_OId__on ) @ ( B @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
     => ~ ! [D: $i] :
            ( ( A @ ( D @ ( C @ c_member ) ) )
           => ( B
             != ( D @ ( D @ ( C @ ( C @ c_Product__Type_OPair ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Id__onE) ).

thf(2358,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ c_Relation_OId__on ) @ ( B @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
     => ~ ! [D: $i] :
            ( ( A @ ( D @ ( C @ c_member ) ) )
           => ( B
             != ( D @ ( D @ ( C @ ( C @ c_Product__Type_OPair ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[486]) ).

thf(421,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Order__Relation_Owell__order__on ) ) )
    <=> ( ( C @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Wellfounded_Owf ) )
        & ( A @ ( B @ ( C @ c_Order__Relation_Olinear__order__on ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_well__order__on__def) ).

thf(2079,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( C @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Wellfounded_Owf ) )
          & ( A @ ( B @ ( C @ c_Order__Relation_Olinear__order__on ) ) ) )
       => ( A @ ( B @ ( C @ c_Order__Relation_Owell__order__on ) ) ) )
      & ( ( A @ ( B @ ( C @ c_Order__Relation_Owell__order__on ) ) )
       => ( ( C @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Wellfounded_Owf ) )
          & ( A @ ( B @ ( C @ c_Order__Relation_Olinear__order__on ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).

thf(204,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( B @ ( C @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equalityD1) ).

thf(1209,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( B @ ( C @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(120,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__funE) ).

thf(900,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(358,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ ( F @ c_Product__Type_Ocurry ) ) ) ) ) )
      = ( A @ ( B @ ( E @ ( F @ c_Product__Type_OPair ) ) ) @ ( C @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_curry__conv) ).

thf(1853,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ ( F @ c_Product__Type_Ocurry ) ) ) ) ) )
      = ( A @ ( B @ ( E @ ( F @ c_Product__Type_OPair ) ) ) @ ( C @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).

thf(221,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
        <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__diff__iff) ).

thf(1282,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
          & ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(494,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Order__Relation_Ostrict__linear__order__on ) ) )
    <=> ( ( A @ ( B @ ( C @ c_Relation_Ototal__on ) ) )
        & ( A @ ( C @ c_Relation_Oirrefl ) )
        & ( A @ ( C @ c_Relation_Otrans ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_strict__linear__order__on__def) ).

thf(2383,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( A @ ( B @ ( C @ c_Relation_Ototal__on ) ) )
          & ( A @ ( C @ c_Relation_Oirrefl ) )
          & ( A @ ( C @ c_Relation_Otrans ) ) )
       => ( A @ ( B @ ( C @ c_Order__Relation_Ostrict__linear__order__on ) ) ) )
      & ( ( A @ ( B @ ( C @ c_Order__Relation_Ostrict__linear__order__on ) ) )
       => ( ( A @ ( B @ ( C @ c_Relation_Ototal__on ) ) )
          & ( A @ ( C @ c_Relation_Oirrefl ) )
          & ( A @ ( C @ c_Relation_Otrans ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[494]) ).

thf(501,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( D @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Relation_ORange ) ) )
      = ( A @ ( C @ ( D @ c_Relation_ORange ) ) @ ( B @ ( C @ ( D @ c_Relation_ORange ) ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__Un__eq) ).

thf(2422,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( D @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Relation_ORange ) ) )
      = ( A @ ( C @ ( D @ c_Relation_ORange ) ) @ ( B @ ( C @ ( D @ c_Relation_ORange ) ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[501]) ).

thf(349,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( c_FunDef_Opair__less @ ( A @ ( C @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( B @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__lessI2) ).

thf(1806,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( c_FunDef_Opair__less @ ( A @ ( C @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( B @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[349]) ).

thf(32,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I10_J) ).

thf(616,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(417,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__rtrancl__into__rtrancl) ).

thf(2065,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[417]) ).

thf(159,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A != B )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I11_J) ).

thf(1048,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A != B )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(72,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( A != B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__imp__not__eq2) ).

thf(736,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(146,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ? [E: $i] :
          ( ( E @ ( A @ hAPP ) @ hBOOL )
          & ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( E @ ( D @ c_member ) ) ) )
    <=> ( ? [E: $i] :
            ( ( E @ ( A @ hAPP ) @ hBOOL )
            & ( B @ ( E @ ( D @ c_member ) ) ) )
        | ? [E: $i] :
            ( ( E @ ( A @ hAPP ) @ hBOOL )
            & ( C @ ( E @ ( D @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bex__Un) ).

thf(990,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ? [E: $i] :
              ( ( E @ ( A @ hAPP ) @ hBOOL )
              & ( B @ ( E @ ( D @ c_member ) ) ) )
          | ? [E: $i] :
              ( ( E @ ( A @ hAPP ) @ hBOOL )
              & ( C @ ( E @ ( D @ c_member ) ) ) ) )
       => ? [E: $i] :
            ( ( E @ ( A @ hAPP ) @ hBOOL )
            & ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( E @ ( D @ c_member ) ) ) ) )
      & ( ? [E: $i] :
            ( ( E @ ( A @ hAPP ) @ hBOOL )
            & ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( E @ ( D @ c_member ) ) ) )
       => ( ? [E: $i] :
              ( ( E @ ( A @ hAPP ) @ hBOOL )
              & ( B @ ( E @ ( D @ c_member ) ) ) )
          | ? [E: $i] :
              ( ( E @ ( A @ hAPP ) @ hBOOL )
              & ( C @ ( E @ ( D @ c_member ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(383,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ c_member ) ) )
    <=> ( ( A @ ( C @ ( D @ c_member ) ) )
        | ( C = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__iff) ).

thf(1933,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( A @ ( C @ ( D @ c_member ) ) )
          | ( C = B ) )
       => ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ c_member ) ) ) )
      & ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ c_member ) ) )
       => ( ( A @ ( C @ ( D @ c_member ) ) )
          | ( C = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).

thf(112,axiom,
    ! [A: $i,B: $i] : ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_empty__subsetI) ).

thf(880,plain,
    ! [A: $i,B: $i] : ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(157,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equalityI) ).

thf(1043,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(449,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
        | ( B @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) )
     => ( A @ ( B @ ( tc_HOL_Obool @ ( D @ ( D @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__rtrancl__UnI) ).

thf(2197,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
        | ( B @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) )
     => ( A @ ( B @ ( tc_HOL_Obool @ ( D @ ( D @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[449]) ).

thf(229,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__absorb1) ).

thf(1307,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(399,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( B @ ( C @ ( D @ c_member ) ) )
       => ( A @ ( C @ ( D @ c_member ) ) ) )
     => ( B @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UnCI) ).

thf(1993,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( B @ ( C @ ( D @ c_member ) ) )
       => ( A @ ( C @ ( D @ c_member ) ) ) )
     => ( B @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[399]) ).

thf(25,axiom,
    tc_Nat_Onat @ class_Orderings_Opreorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Opreorder) ).

thf(594,plain,
    tc_Nat_Onat @ class_Orderings_Opreorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(90,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
          & ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__sup__iff) ).

thf(799,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) )
         => ( A @ ( B @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( A @ ( B @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(509,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( E @ ( E @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ! [F: $i] :
            ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( F @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
           => ! [G: $i] :
                ( ( A @ ( F @ ( G @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
               => ( G = F ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__Un__separator__converseE) ).

thf(2446,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( E @ ( E @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ! [F: $i] :
            ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( F @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
           => ! [G: $i] :
                ( ( A @ ( F @ ( G @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
               => ( G = F ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[509]) ).

thf(61,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B != A )
       => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__neqE) ).

thf(700,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B != A )
       => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(288,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I8_J) ).

thf(1556,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).

thf(131,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( B @ c_Transitive__Closure_Ortrancl ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__rtrancl__absorb) ).

thf(943,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( B @ c_Transitive__Closure_Ortrancl ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(192,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_OField ) )
      = ( A @ ( B @ ( B @ c_Relation_ORange ) ) @ ( A @ ( B @ ( B @ c_Relation_ODomain ) ) @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Field__def) ).

thf(1177,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_OField ) )
      = ( A @ ( B @ ( B @ c_Relation_ORange ) ) @ ( A @ ( B @ ( B @ c_Relation_ODomain ) ) @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(512,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ c_Wellfounded_Oacc ) @ ( B @ ( C @ c_member ) ) )
    <=> ! [D: $i] :
          ( ( A @ ( B @ ( D @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
         => ( A @ ( C @ c_Wellfounded_Oacc ) @ ( D @ ( C @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acc_Osimps) ).

thf(2461,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( A @ ( B @ ( D @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
           => ( A @ ( C @ c_Wellfounded_Oacc ) @ ( D @ ( C @ c_member ) ) ) )
       => ( A @ ( C @ c_Wellfounded_Oacc ) @ ( B @ ( C @ c_member ) ) ) )
      & ( ( A @ ( C @ c_Wellfounded_Oacc ) @ ( B @ ( C @ c_member ) ) )
       => ! [D: $i] :
            ( ( A @ ( B @ ( D @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
           => ( A @ ( C @ c_Wellfounded_Oacc ) @ ( D @ ( C @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).

thf(116,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_OId @ ( B @ ( B @ c_Relation_OImage ) ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__Id) ).

thf(890,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_OId @ ( B @ ( B @ c_Relation_OImage ) ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(319,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( C @ ( A @ hAPP ) @ hBOOL )
       => ( ! [F: $i,G: $i] :
              ( ( B @ ( G @ ( F @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
             => ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( G @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
               => ( ( G @ ( A @ hAPP ) @ hBOOL )
                 => ( F @ ( A @ hAPP ) @ hBOOL ) ) ) )
         => ( D @ ( A @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__rtrancl__induct) ).

thf(1667,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( C @ ( A @ hAPP ) @ hBOOL )
       => ( ! [F: $i,G: $i] :
              ( ( B @ ( G @ ( F @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
             => ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( G @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
               => ( ( G @ ( A @ hAPP ) @ hBOOL )
                 => ( F @ ( A @ hAPP ) @ hBOOL ) ) ) )
         => ( D @ ( A @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).

thf(153,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( D @ c_Set_Oinsert ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__insert__right) ).

thf(1029,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( D @ c_Set_Oinsert ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(118,axiom,
    ! [A: $i] : ( A @ c_Relation_OId @ ( A @ ( A @ c_Relation_Osingle__valued ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_single__valued__Id) ).

thf(895,plain,
    ! [A: $i] : ( A @ c_Relation_OId @ ( A @ ( A @ c_Relation_Osingle__valued ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(479,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ c_member ) ) )
     => ( ( C != B )
       => ( A @ ( C @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insertE) ).

thf(2329,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ c_member ) ) )
     => ( ( C != B )
       => ( A @ ( C @ ( D @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[479]) ).

thf(127,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F
        = ( B @ ( C @ ( D @ ( E @ c_Recdef_Owfrec ) ) ) ) )
     => ( ( C @ ( E @ c_Wellfounded_Owf ) )
       => ( ( A @ ( F @ hAPP ) )
          = ( A @ ( A @ ( C @ ( F @ ( D @ ( E @ c_Recdef_Ocut ) ) ) ) @ ( B @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_def__wfrec) ).

thf(932,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F
        = ( B @ ( C @ ( D @ ( E @ c_Recdef_Owfrec ) ) ) ) )
     => ( ( C @ ( E @ c_Wellfounded_Owf ) )
       => ( ( A @ ( F @ hAPP ) )
          = ( A @ ( A @ ( C @ ( F @ ( D @ ( E @ c_Recdef_Ocut ) ) ) ) @ ( B @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(488,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_Set_Oinsert ) ) @ ( D @ c_Wellfounded_Owf ) )
    <=> ( ~ ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( B @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
        & ( A @ ( D @ c_Wellfounded_Owf ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__insert) ).

thf(2365,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ~ ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( B @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
          & ( A @ ( D @ c_Wellfounded_Owf ) ) )
       => ( A @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_Set_Oinsert ) ) @ ( D @ c_Wellfounded_Owf ) ) )
      & ( ( A @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_Set_Oinsert ) ) @ ( D @ c_Wellfounded_Owf ) )
       => ( ~ ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( B @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
          & ( A @ ( D @ c_Wellfounded_Owf ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[488]) ).

thf(483,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( E @ ( F @ c_Product__Type_OPair ) ) ) @ ( C @ ( D @ ( E @ ( F @ c_Product__Type_Ointernal__split ) ) ) ) )
      = ( A @ ( B @ ( C @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_internal__split__conv) ).

thf(2350,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( E @ ( F @ c_Product__Type_OPair ) ) ) @ ( C @ ( D @ ( E @ ( F @ c_Product__Type_Ointernal__split ) ) ) ) )
      = ( A @ ( B @ ( C @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).

thf(59,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I6_J) ).

thf(695,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(422,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ c_Relation_Oantisym ) )
       => ( B @ ( C @ c_Relation_Oantisym ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_antisym__subset) ).

thf(2086,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ c_Relation_Oantisym ) )
       => ( B @ ( C @ c_Relation_Oantisym ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).

thf(436,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( c_FunDef_Opair__leq @ ( A @ ( C @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( B @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__leqI1) ).

thf(2139,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( c_FunDef_Opair__leq @ ( A @ ( C @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( B @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[436]) ).

thf(470,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ c_Relation_OId @ ( A @ ( B @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__in__Id__conv) ).

thf(2288,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B = A )
       => ( C @ c_Relation_OId @ ( A @ ( B @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) ) )
      & ( ( C @ c_Relation_OId @ ( A @ ( B @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[470]) ).

thf(258,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Order__Relation_Olinear__order__on ) ) )
    <=> ( ( A @ ( B @ ( C @ c_Relation_Ototal__on ) ) )
        & ( A @ ( B @ ( C @ c_Order__Relation_Opartial__order__on ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linear__order__on__def) ).

thf(1435,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( A @ ( B @ ( C @ c_Relation_Ototal__on ) ) )
          & ( A @ ( B @ ( C @ c_Order__Relation_Opartial__order__on ) ) ) )
       => ( A @ ( B @ ( C @ c_Order__Relation_Olinear__order__on ) ) ) )
      & ( ( A @ ( B @ ( C @ c_Order__Relation_Olinear__order__on ) ) )
       => ( ( A @ ( B @ ( C @ c_Relation_Ototal__on ) ) )
          & ( A @ ( B @ ( C @ c_Order__Relation_Opartial__order__on ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(268,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B @ ( A @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__absorb1) ).

thf(1477,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B @ ( A @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(170,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) )
        | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__le__less__linear) ).

thf(1102,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) )
        | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(45,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__imp__neq) ).

thf(653,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(200,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ c_Transitive__Closure_Otrancl ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__reflcl) ).

thf(1198,plain,
    ! [A: $i,B: $i] :
      ( ( B @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ c_Transitive__Closure_Otrancl ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(506,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__into__rtrancl) ).

thf(2439,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[506]) ).

thf(137,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__funD) ).

thf(959,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(444,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( A @ ( D @ c_member ) ) )
       => ( B @ ( A @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__mono) ).

thf(2171,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( A @ ( D @ c_member ) ) )
       => ( B @ ( A @ ( D @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[444]) ).

thf(205,axiom,
    ! [A: $i] :
      ( ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Transitive__Closure_Ortrancl ) )
      = ( A @ c_Relation_OId ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__empty) ).

thf(1213,plain,
    ! [A: $i] :
      ( ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Transitive__Closure_Ortrancl ) )
      = ( A @ c_Relation_OId ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(173,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( ( B = A )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__eq__less__or__eq) ).

thf(1113,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B = A )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B = A )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(11,axiom,
    tc_HOL_Obool @ class_Lattices_Olattice,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL__Obool__Lattices_Olattice) ).

thf(563,plain,
    tc_HOL_Obool @ class_Lattices_Olattice,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(339,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
     => ( A @ ( C @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( C @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_r__into__trancl_H) ).

thf(1758,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
     => ( A @ ( C @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( C @ tc_prod ) @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).

thf(267,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
    <=> ( ( A
          = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
        & ( B
          = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__empty) ).

thf(1467,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( A
            = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
          & ( B
            = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) )
       => ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
          = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) )
      & ( ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
          = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
       => ( ( A
            = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
          & ( B
            = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(53,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
        <=> ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__antisym__conv) ).

thf(680,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( ( A = B )
           => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) )
          & ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( A = B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(514,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_Otrans ) )
     => ( ( A @ ( B @ c_Relation_Oantisym ) )
       => ( B @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ c_Relation_Otrans ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__diff__Id) ).

thf(2471,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_Otrans ) )
     => ( ( A @ ( B @ c_Relation_Oantisym ) )
       => ( B @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ c_Relation_Otrans ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[514]) ).

thf(409,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) )
     => ( ( c_FunDef_Opair__leq @ ( D @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) )
       => ( ( c_FunDef_Omax__weak @ ( C @ ( A @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) )
         => ( c_FunDef_Omax__weak @ ( C @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wmax__insertI) ).

thf(2037,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) )
     => ( ( c_FunDef_Opair__leq @ ( D @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) )
       => ( ( c_FunDef_Omax__weak @ ( C @ ( A @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) )
         => ( c_FunDef_Omax__weak @ ( C @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[409]) ).

thf(460,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Relation_OId__on ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
    <=> ( ( A @ ( C @ ( D @ c_member ) ) )
        & ( C = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Id__on__iff) ).

thf(2234,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( A @ ( C @ ( D @ c_member ) ) )
          & ( C = B ) )
       => ( A @ ( D @ c_Relation_OId__on ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) )
      & ( ( A @ ( D @ c_Relation_OId__on ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ( ( A @ ( C @ ( D @ c_member ) ) )
          & ( C = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[460]) ).

thf(280,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__not__refl3) ).

thf(1509,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(168,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( E @ ( D @ c_member ) ) )
         => ( E @ ( A @ hAPP ) @ hBOOL ) )
    <=> ( ! [E: $i] :
            ( ( B @ ( E @ ( D @ c_member ) ) )
           => ( E @ ( A @ hAPP ) @ hBOOL ) )
        & ! [E: $i] :
            ( ( C @ ( E @ ( D @ c_member ) ) )
           => ( E @ ( A @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ball__Un) ).

thf(1081,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ! [E: $i] :
              ( ( B @ ( E @ ( D @ c_member ) ) )
             => ( E @ ( A @ hAPP ) @ hBOOL ) )
          & ! [E: $i] :
              ( ( C @ ( E @ ( D @ c_member ) ) )
             => ( E @ ( A @ hAPP ) @ hBOOL ) ) )
       => ! [E: $i] :
            ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( E @ ( D @ c_member ) ) )
           => ( E @ ( A @ hAPP ) @ hBOOL ) ) )
      & ( ! [E: $i] :
            ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( E @ ( D @ c_member ) ) )
           => ( E @ ( A @ hAPP ) @ hBOOL ) )
       => ( ! [E: $i] :
              ( ( B @ ( E @ ( D @ c_member ) ) )
             => ( E @ ( A @ hAPP ) @ hBOOL ) )
          & ! [E: $i] :
              ( ( C @ ( E @ ( D @ c_member ) ) )
             => ( E @ ( A @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(181,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__le__mono2) ).

thf(1145,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(413,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( A @ ( C @ c_Wellfounded_Oacc ) @ ( B @ ( C @ c_member ) ) )
     => ~ ! [D: $i] :
            ( ( A @ ( B @ ( D @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
           => ( A @ ( C @ c_Wellfounded_Oacc ) @ ( D @ ( C @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__acc__down) ).

thf(2054,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( A @ ( C @ c_Wellfounded_Oacc ) @ ( B @ ( C @ c_member ) ) )
     => ~ ! [D: $i] :
            ( ( A @ ( B @ ( D @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
           => ( A @ ( C @ c_Wellfounded_Oacc ) @ ( D @ ( C @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).

thf(386,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__trancl__trancl) ).

thf(1948,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).

thf(202,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( D @ ( E @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( F @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rev__predicate2D) ).

thf(1203,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( D @ ( E @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( F @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(23,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
        | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__linear) ).

thf(589,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
        | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(373,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( E @ hAPP ) @ ( D @ ( E @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( E @ ( B @ c_Wellfounded_Omlex__prod ) ) @ ( C @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mlex__less) ).

thf(1905,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( E @ hAPP ) @ ( D @ ( E @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( E @ ( B @ c_Wellfounded_Omlex__prod ) ) @ ( C @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[373]) ).

thf(255,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( A @ ( B @ ( C @ ( C @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( C @ c_Relation_OImage ) ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__closed__trancl) ).

thf(1420,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( A @ ( B @ ( C @ ( C @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( C @ c_Relation_OImage ) ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(129,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__supI) ).

thf(939,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(239,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__subset__conv) ).

thf(1340,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(303,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( C @ c_Relation_Ototal__on ) ) )
    <=> ( A @ ( B @ ( C @ c_Relation_Ototal__on ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_total__on__diff__Id) ).

thf(1598,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( A @ ( B @ ( C @ c_Relation_Ototal__on ) ) )
       => ( C @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( C @ c_Relation_Ototal__on ) ) ) )
      & ( ( C @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( C @ c_Relation_Ototal__on ) ) )
       => ( A @ ( B @ ( C @ c_Relation_Ototal__on ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).

thf(39,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__assoc) ).

thf(640,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(63,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Opreorder )
     => ( A @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__refl) ).

thf(705,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Opreorder )
     => ( A @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(115,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Set_Oinsert ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__insert__left) ).

thf(887,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Set_Oinsert ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(141,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( D @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Relation_ODomain ) ) @ ( A @ ( C @ ( D @ c_Relation_ODomain ) ) @ ( B @ ( C @ ( D @ c_Relation_ODomain ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__Diff__subset) ).

thf(971,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( D @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Relation_ODomain ) ) @ ( A @ ( C @ ( D @ c_Relation_ODomain ) ) @ ( B @ ( C @ ( D @ c_Relation_ODomain ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(313,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( E @ ( D @ hAPP ) @ hBOOL )
         => ( E @ ( B @ hAPP ) @ ( C @ c_Wellfounded_Owf ) ) )
     => ( B @ ( D @ ( C @ ( A @ c_Recdef_Osame__fst ) ) ) @ ( C @ ( A @ tc_prod ) @ c_Wellfounded_Owf ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__same__fst) ).

thf(1647,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( E @ ( D @ hAPP ) @ hBOOL )
         => ( E @ ( B @ hAPP ) @ ( C @ c_Wellfounded_Owf ) ) )
     => ( B @ ( D @ ( C @ ( A @ c_Recdef_Osame__fst ) ) ) @ ( C @ ( A @ tc_prod ) @ c_Wellfounded_Owf ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).

thf(135,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ c_Wellfounded_Owf ) )
     => ( ( A @ ( B @ ( D @ ( C @ ( E @ c_Recdef_Owfrec ) ) ) @ hAPP ) )
        = ( A @ ( A @ ( D @ ( B @ ( D @ ( C @ ( E @ c_Recdef_Owfrec ) ) ) @ ( C @ ( E @ c_Recdef_Ocut ) ) ) ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wfrec) ).

thf(954,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ c_Wellfounded_Owf ) )
     => ( ( A @ ( B @ ( D @ ( C @ ( E @ c_Recdef_Owfrec ) ) ) @ hAPP ) )
        = ( A @ ( A @ ( D @ ( B @ ( D @ ( C @ ( E @ c_Recdef_Owfrec ) ) ) @ ( C @ ( E @ c_Recdef_Ocut ) ) ) ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(446,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( B @ ( G @ ( H @ tc_prod ) @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( E @ ( F @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ tc_prod ) @ c_member ) ) )
     => ( ( E @ ( F @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
       => ( ! [I: $i,J: $i] :
              ( ( B @ ( G @ ( H @ tc_prod ) @ c_Transitive__Closure_Ortrancl ) @ ( J @ ( I @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( E @ ( F @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ tc_prod ) @ c_member ) ) )
             => ! [K: $i,L: $i] :
                  ( ( B @ ( L @ ( K @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( J @ ( I @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ tc_prod ) @ c_member ) ) )
                 => ( ( J @ ( I @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
                   => ( L @ ( K @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ) ) )
         => ( C @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__induct2) ).

thf(2175,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( B @ ( G @ ( H @ tc_prod ) @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( E @ ( F @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ tc_prod ) @ c_member ) ) )
     => ( ( E @ ( F @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
       => ( ! [I: $i,J: $i] :
              ( ( B @ ( G @ ( H @ tc_prod ) @ c_Transitive__Closure_Ortrancl ) @ ( J @ ( I @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( E @ ( F @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ tc_prod ) @ c_member ) ) )
             => ! [K: $i,L: $i] :
                  ( ( B @ ( L @ ( K @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( J @ ( I @ ( G @ ( H @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( G @ ( H @ tc_prod ) @ ( G @ ( H @ tc_prod ) @ tc_prod ) @ c_member ) ) )
                 => ( ( J @ ( I @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
                   => ( L @ ( K @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ) ) )
         => ( C @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[446]) ).

thf(337,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( c_FunDef_Opair__leq @ ( A @ ( C @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( B @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__leqI2) ).

thf(1753,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( c_FunDef_Opair__leq @ ( A @ ( C @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( B @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).

thf(169,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( C @ c_member ) ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_singletonE) ).

thf(1098,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( C @ c_member ) ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(344,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( B @ ( D @ ( D @ c_Relation_ODomain ) ) @ ( C @ ( D @ c_member ) ) )
     => ( ( B @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
      <=> ( C = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Not__Domain__rtrancl) ).

thf(1770,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( B @ ( D @ ( D @ c_Relation_ODomain ) ) @ ( C @ ( D @ c_member ) ) )
     => ( ( ( C = A )
         => ( B @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) )
        & ( ( B @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
         => ( C = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).

thf(497,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Arrow__Order__Mirabelle_OLin @ ( C @ ( tc_HOL_Obool @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ tc_fun ) @ c_member ) ) )
     => ( ( B != A )
       => ( ~ ( C @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
        <=> ( C @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_notin__Lin__iff) ).

thf(2397,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Arrow__Order__Mirabelle_OLin @ ( C @ ( tc_HOL_Obool @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ tc_fun ) @ c_member ) ) )
     => ( ( B != A )
       => ( ( ( C @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
           => ~ ( C @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) )
          & ( ~ ( C @ ( A @ ( B @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
           => ( C @ ( B @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[497]) ).

thf(151,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ hAPP ) @ hBOOL )
     => ( C @ ( D @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup1I2) ).

thf(1025,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ hAPP ) @ hBOOL )
     => ( C @ ( D @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(415,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( tc_HOL_Obool @ ( D @ ( E @ c_Product__Type_Ocurry ) ) ) ) ) @ hBOOL )
     => ( A @ ( B @ ( D @ ( E @ c_Product__Type_OPair ) ) ) @ ( C @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_curryE) ).

thf(2060,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( tc_HOL_Obool @ ( D @ ( E @ c_Product__Type_Ocurry ) ) ) ) ) @ hBOOL )
     => ( A @ ( B @ ( D @ ( E @ c_Product__Type_OPair ) ) ) @ ( C @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[415]) ).

thf(64,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) )
        | ( B = A )
        | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__less__linear) ).

thf(707,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) )
        | ( B = A )
        | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(292,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_member ) ) )
     => ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( C @ c_Set_Oinsert ) ) @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( C @ c_Set_Oinsert ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__Diff) ).

thf(1566,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_member ) ) )
     => ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( C @ c_Set_Oinsert ) ) @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( C @ c_Set_Oinsert ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(438,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_Relation_Orefl__on ) ) )
     => ( ( C @ ( A @ ( D @ c_member ) ) )
       => ( B @ ( A @ ( A @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__onD) ).

thf(2143,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_Relation_Orefl__on ) ) )
     => ( ( C @ ( A @ ( D @ c_member ) ) )
       => ( B @ ( A @ ( A @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[438]) ).

thf(511,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_Set_Oinsert ) ) @ ( D @ c_Wellfounded_Oacyclic ) )
    <=> ( ~ ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( B @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
        & ( A @ ( D @ c_Wellfounded_Oacyclic ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acyclic__insert) ).

thf(2454,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ~ ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( B @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
          & ( A @ ( D @ c_Wellfounded_Oacyclic ) ) )
       => ( A @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_Set_Oinsert ) ) @ ( D @ c_Wellfounded_Oacyclic ) ) )
      & ( ( A @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_Set_Oinsert ) ) @ ( D @ c_Wellfounded_Oacyclic ) )
       => ( ~ ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( B @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
          & ( A @ ( D @ c_Wellfounded_Oacyclic ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[511]) ).

thf(257,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ hAPP ) @ hBOOL )
     => ( ~ ( A @ ( C @ hAPP ) @ hBOOL )
       => ( A @ ( B @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup1E) ).

thf(1432,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ hAPP ) @ hBOOL )
     => ( ~ ( A @ ( C @ hAPP ) @ hBOOL )
       => ( A @ ( B @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(88,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__insert2) ).

thf(791,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(21,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Olattice )
     => ( B @ ( A @ tc_fun ) @ class_Lattices_Olattice ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Lattices_Olattice) ).

thf(585,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Olattice )
     => ( B @ ( A @ tc_fun ) @ class_Lattices_Olattice ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(504,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( C @ ( D @ c_Relation_OField ) @ ( D @ c_Order__Relation_Opreorder__on ) ) )
     => ( ( C @ ( D @ c_Relation_OField ) @ ( B @ ( D @ c_member ) ) )
       => ( ( C @ ( D @ c_Relation_OField ) @ ( A @ ( D @ c_member ) ) )
         => ( ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
          <=> ( C @ ( B @ ( A @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__Image1__Image1__iff) ).

thf(2429,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( C @ ( D @ c_Relation_OField ) @ ( D @ c_Order__Relation_Opreorder__on ) ) )
     => ( ( C @ ( D @ c_Relation_OField ) @ ( B @ ( D @ c_member ) ) )
       => ( ( C @ ( D @ c_Relation_OField ) @ ( A @ ( D @ c_member ) ) )
         => ( ( ( C @ ( B @ ( A @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
             => ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
            & ( ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ ( D @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
             => ( C @ ( B @ ( A @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[504]) ).

thf(233,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( C @ ( D @ hAPP ) @ hBOOL )
       => ( C @ ( B @ hAPP ) @ hBOOL ) )
     => ( C @ ( D @ ( B @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup1CI) ).

thf(1317,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( C @ ( D @ hAPP ) @ hBOOL )
       => ( C @ ( B @ hAPP ) @ hBOOL ) )
     => ( C @ ( D @ ( B @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(50,axiom,
    tc_HOL_Obool @ class_Orderings_Oorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL__Obool__Orderings_Oorder) ).

thf(673,plain,
    tc_HOL_Obool @ class_Orderings_Oorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(425,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( tc_HOL_Obool @ ( D @ ( E @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ ( E @ c_Relation_OImage ) ) ) )
      = ( A @ ( B @ ( D @ ( E @ c_Relation_OImage ) ) ) @ ( A @ ( C @ ( D @ ( E @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__Image) ).

thf(2092,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( tc_HOL_Obool @ ( D @ ( E @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ ( E @ c_Relation_OImage ) ) ) )
      = ( A @ ( B @ ( D @ ( E @ c_Relation_OImage ) ) ) @ ( A @ ( C @ ( D @ ( E @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[425]) ).

thf(226,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__psubset__trans) ).

thf(1300,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(456,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Order__Relation_Olinear__order__on ) ) )
     => ( C @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( C @ c_Order__Relation_Ostrict__linear__order__on ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_strict__linear__order__on__diff__Id) ).

thf(2219,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Order__Relation_Olinear__order__on ) ) )
     => ( C @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( C @ c_Order__Relation_Ostrict__linear__order__on ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[456]) ).

thf(395,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_Oirrefl ) )
    <=> ! [C: $i] :
          ~ ( A @ ( C @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_irrefl__def) ).

thf(1980,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ~ ( A @ ( C @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
       => ( A @ ( B @ c_Relation_Oirrefl ) ) )
      & ( ( A @ ( B @ c_Relation_Oirrefl ) )
       => ! [C: $i] :
            ~ ( A @ ( C @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).

thf(435,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( B @ ( C @ ( D @ ( E @ ( F @ c_Recdef_Ocut ) ) ) ) )
        = ( B @ ( C @ ( A @ ( E @ ( F @ c_Recdef_Ocut ) ) ) ) ) )
    <=> ! [G: $i] :
          ( ( C @ ( B @ ( G @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) )
         => ( ( G @ ( D @ hAPP ) )
            = ( G @ ( A @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cuts__eq) ).

thf(2128,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ! [G: $i] :
            ( ( C @ ( B @ ( G @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) )
           => ( ( G @ ( D @ hAPP ) )
              = ( G @ ( A @ hAPP ) ) ) )
       => ( ( B @ ( C @ ( D @ ( E @ ( F @ c_Recdef_Ocut ) ) ) ) )
          = ( B @ ( C @ ( A @ ( E @ ( F @ c_Recdef_Ocut ) ) ) ) ) ) )
      & ( ( ( B @ ( C @ ( D @ ( E @ ( F @ c_Recdef_Ocut ) ) ) ) )
          = ( B @ ( C @ ( A @ ( E @ ( F @ c_Recdef_Ocut ) ) ) ) ) )
       => ! [G: $i] :
            ( ( C @ ( B @ ( G @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) )
           => ( ( G @ ( D @ hAPP ) )
              = ( G @ ( A @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[435]) ).

thf(17,axiom,
    tc_HOL_Obool @ class_Groups_Ominus,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL__Obool__Groups_Ominus) ).

thf(576,plain,
    tc_HOL_Obool @ class_Groups_Ominus,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(317,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( E @ hAPP ) @ ( D @ ( E @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
       => ( A @ ( E @ ( B @ c_Wellfounded_Omlex__prod ) ) @ ( C @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mlex__leq) ).

thf(1658,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( E @ hAPP ) @ ( D @ ( E @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
       => ( A @ ( E @ ( B @ c_Wellfounded_Omlex__prod ) ) @ ( C @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).

thf(476,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__trans) ).

thf(2322,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Otrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).

thf(284,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Transitive__Closure_Ortrancl ) @ ( B @ c_Transitive__Closure_Otrancl ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__trancl__absorb) ).

thf(1538,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Transitive__Closure_Ortrancl ) @ ( B @ c_Transitive__Closure_Otrancl ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(517,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Wellfounded_Owf ) )
    <=> ! [C: $i] : ( A @ ( B @ c_Wellfounded_Oacc ) @ ( C @ ( B @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__acc__iff) ).

thf(2486,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] : ( A @ ( B @ c_Wellfounded_Oacc ) @ ( C @ ( B @ c_member ) ) )
       => ( A @ ( B @ c_Wellfounded_Owf ) ) )
      & ( ( A @ ( B @ c_Wellfounded_Owf ) )
       => ! [C: $i] : ( A @ ( B @ c_Wellfounded_Oacc ) @ ( C @ ( B @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[517]) ).

thf(238,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__imp__le) ).

thf(1336,plain,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(40,axiom,
    tc_Nat_Onat @ class_Lattices_Olattice,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Lattices_Olattice) ).

thf(643,plain,
    tc_Nat_Onat @ class_Lattices_Olattice,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(243,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__idemp) ).

thf(1358,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(74,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__left__idem) ).

thf(744,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(370,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl_Ortrancl__into__rtrancl) ).

thf(1898,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).

thf(62,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__asym) ).

thf(703,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(179,axiom,
    ! [A: $i,B: $i,C: $i] : ( B @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__upper2) ).

thf(1141,plain,
    ! [A: $i,B: $i,C: $i] : ( B @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(492,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Owf ) )
     => ( ( B @ ( A @ ( B @ ( C @ ( C @ ( C @ c_Relation_Orel__comp ) ) ) ) @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( C @ c_FunDef_Oreduction__pair ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_reduction__pairI) ).

thf(2379,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Owf ) )
     => ( ( B @ ( A @ ( B @ ( C @ ( C @ ( C @ c_Relation_Orel__comp ) ) ) ) @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( C @ c_FunDef_Oreduction__pair ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).

thf(31,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( A @ ( B @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__idem) ).

thf(613,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( A @ ( B @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(394,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( B @ ( C @ ( D @ c_member ) ) )
       => ( ( B @ ( A @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
          = ( B @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Set_Oinsert ) ) ) ) )
      & ( ( B @ ( C @ ( D @ c_member ) ) )
       => ( ( B @ ( A @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
          = ( B @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__Diff__if) ).

thf(1972,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( B @ ( C @ ( D @ c_member ) ) )
       => ( ( B @ ( A @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
          = ( B @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Set_Oinsert ) ) ) ) )
      & ( ( B @ ( C @ ( D @ c_member ) ) )
       => ( ( B @ ( A @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
          = ( B @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).

thf(144,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__imp__le__nat) ).

thf(978,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(407,axiom,
    c_FunDef_Omax__weak @ ( c_FunDef_Omax__strict @ ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_FunDef_Oreduction__pair ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_max__rpair__set) ).

thf(2030,plain,
    c_FunDef_Omax__weak @ ( c_FunDef_Omax__strict @ ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_FunDef_Oreduction__pair ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[407]) ).

thf(15,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Olattice )
     => ( ( A @ ( B @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf__sup__aci_I6_J) ).

thf(571,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Olattice )
     => ( ( A @ ( B @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(26,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I5_J) ).

thf(595,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(30,axiom,
    tc_Nat_Onat @ class_Orderings_Oorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Oorder) ).

thf(612,plain,
    tc_Nat_Onat @ class_Orderings_Oorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(107,axiom,
    c_Wellfounded_Opred__nat @ ( tc_Nat_Onat @ c_Wellfounded_Owf ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__pred__nat) ).

thf(859,plain,
    c_Wellfounded_Opred__nat @ ( tc_Nat_Onat @ c_Wellfounded_Owf ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(286,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( ( B != A )
        & ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_psubset__eq) ).

thf(1543,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( B != A )
          & ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
       => ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B != A )
          & ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).

thf(343,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( E @ ( C @ c_Relation_OImage ) ) ) @ ( D @ ( E @ c_member ) ) )
     => ~ ! [F: $i] :
            ( ( B @ ( D @ ( F @ ( E @ ( C @ c_Product__Type_OPair ) ) ) @ ( E @ ( C @ tc_prod ) @ c_member ) ) )
           => ~ ( A @ ( F @ ( C @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ImageE) ).

thf(1766,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( E @ ( C @ c_Relation_OImage ) ) ) @ ( D @ ( E @ c_member ) ) )
     => ~ ! [F: $i] :
            ( ( B @ ( D @ ( F @ ( E @ ( C @ c_Product__Type_OPair ) ) ) @ ( E @ ( C @ tc_prod ) @ c_member ) ) )
           => ~ ( A @ ( F @ ( C @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[343]) ).

thf(468,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Product__Type_OPair ) ) ) @ ( D @ ( E @ tc_prod ) @ c_Set_Oinsert ) ) @ ( D @ ( E @ c_Relation_ODomain ) ) )
      = ( A @ ( D @ ( E @ c_Relation_ODomain ) ) @ ( C @ ( E @ c_Set_Oinsert ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__insert) ).

thf(2279,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Product__Type_OPair ) ) ) @ ( D @ ( E @ tc_prod ) @ c_Set_Oinsert ) ) @ ( D @ ( E @ c_Relation_ODomain ) ) )
      = ( A @ ( D @ ( E @ c_Relation_ODomain ) ) @ ( C @ ( E @ c_Set_Oinsert ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[468]) ).

thf(28,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I9_J) ).

thf(600,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(389,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ~ ( C @ ( D @ c_Transitive__Closure_Otrancl ) @ ( E @ ( E @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ( C @ ( A @ ( B @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_irrefl__trancl__rD) ).

thf(1955,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ~ ( C @ ( D @ c_Transitive__Closure_Otrancl ) @ ( E @ ( E @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ( C @ ( A @ ( B @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[389]) ).

thf(439,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_member ) ) )
     => ( B @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UnI2) ).

thf(2145,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_member ) ) )
     => ( B @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[439]) ).

thf(465,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
        = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
     => ~ ( C @ ( A @ ( B @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equals0D) ).

thf(2266,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
        = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
     => ~ ( C @ ( A @ ( B @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[465]) ).

thf(110,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( A @ ( B @ ( tc_HOL_Obool @ ( D @ ( C @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ ( C @ c_Relation_ORange ) ) @ ( A @ ( D @ ( C @ c_Relation_ORange ) ) @ ( B @ ( D @ ( C @ c_Relation_ORange ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__Diff__subset) ).

thf(876,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( A @ ( B @ ( tc_HOL_Obool @ ( D @ ( C @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ ( C @ c_Relation_ORange ) ) @ ( A @ ( D @ ( C @ c_Relation_ORange ) ) @ ( B @ ( D @ ( C @ c_Relation_ORange ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(213,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ( ( B = A )
          | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__le__less) ).

thf(1240,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( ( ( B = A )
            | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( B = A )
            | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(308,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( C @ ( D @ ( tc_HOL_Obool @ ( E @ ( F @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( E @ ( F @ c_Relation_OImage ) ) ) @ ( B @ ( D @ ( E @ ( F @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__mono) ).

thf(1617,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( C @ ( D @ ( tc_HOL_Obool @ ( E @ ( F @ tc_prod ) @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( E @ ( F @ c_Relation_OImage ) ) ) @ ( B @ ( D @ ( E @ ( F @ c_Relation_OImage ) ) ) @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).

thf(423,axiom,
    ! [A: $i] : ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Relation_Orefl__on ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__on__empty) ).

thf(2088,plain,
    ! [A: $i] : ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Relation_Orefl__on ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[423]) ).

thf(411,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_Relation_Otrans ) )
     => ( ( A @ ( B @ c_Relation_Otrans ) )
       => ( A @ ( C @ ( B @ ( D @ c_Wellfounded_Olex__prod ) ) ) @ ( B @ ( D @ tc_prod ) @ c_Relation_Otrans ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__lex__prod) ).

thf(2045,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_Relation_Otrans ) )
     => ( ( A @ ( B @ c_Relation_Otrans ) )
       => ( A @ ( C @ ( B @ ( D @ c_Wellfounded_Olex__prod ) ) ) @ ( B @ ( D @ tc_prod ) @ c_Relation_Otrans ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[411]) ).

thf(193,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Wellfounded_Owf ) )
     => ( A @ ( B @ c_Wellfounded_Omax__ext ) @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Wellfounded_Owf ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_max__ext__wf) ).

thf(1180,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Wellfounded_Owf ) )
     => ( A @ ( B @ c_Wellfounded_Omax__ext ) @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Wellfounded_Owf ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(7,axiom,
    tc_Nat_Onat @ class_Orderings_Oord,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Oord) ).

thf(556,plain,
    tc_Nat_Onat @ class_Orderings_Oord,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(287,axiom,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__irrefl__nat) ).

thf(1553,plain,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(455,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ c_Relation_OId @ ( A @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
     => ~ ! [C: $i] :
            ( A
           != ( C @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_IdE) ).

thf(2215,plain,
    ! [A: $i,B: $i] :
      ( ( B @ c_Relation_OId @ ( A @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
     => ~ ! [C: $i] :
            ( A
           != ( C @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[455]) ).

thf(79,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__supI1) ).

thf(759,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(261,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
            = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__diff__iff) ).

thf(1448,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( ( B = A )
           => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
              = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) )
          & ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
              = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(420,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( D @ ( A @ ( B @ c_member ) ) )
      <=> ( C @ ( A @ ( B @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eqset__imp__iff) ).

thf(2071,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( ( C @ ( A @ ( B @ c_member ) ) )
         => ( D @ ( A @ ( B @ c_member ) ) ) )
        & ( ( D @ ( A @ ( B @ c_member ) ) )
         => ( C @ ( A @ ( B @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[420]) ).

thf(323,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__trans) ).

thf(1685,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( C @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( A @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).

thf(186,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ hAPP ) @ hBOOL )
     => ( C @ ( A @ ( D @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup1I1) ).

thf(1157,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ hAPP ) @ hBOOL )
     => ( C @ ( A @ ( D @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(234,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__iff__sup) ).

thf(1321,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( ( ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
            = A )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(143,axiom,
    ! [A: $i,B: $i] : ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__le__self) ).

thf(976,plain,
    ! [A: $i,B: $i] : ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

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

thf(903,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) )
     => ( ( ( C = B )
         => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) )
       => ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
           => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) )
         => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(206,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
        & ( A @ ( C @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__subset) ).

thf(1216,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
          & ( A @ ( C @ ( D @ c_member ) ) ) )
       => ( A @ ( B @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ ( B @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
          & ( A @ ( C @ ( D @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(454,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( F @ ( C @ c_Relation_Oinv__image ) ) @ hAPP ) @ ( D @ ( E @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) )
    <=> ( B @ ( D @ ( A @ hAPP ) @ ( E @ ( A @ hAPP ) @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__inv__image) ).

thf(2209,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( B @ ( D @ ( A @ hAPP ) @ ( E @ ( A @ hAPP ) @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
       => ( A @ ( B @ ( F @ ( C @ c_Relation_Oinv__image ) ) @ hAPP ) @ ( D @ ( E @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) ) )
      & ( ( A @ ( B @ ( F @ ( C @ c_Relation_Oinv__image ) ) @ hAPP ) @ ( D @ ( E @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) )
       => ( B @ ( D @ ( A @ hAPP ) @ ( E @ ( A @ hAPP ) @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).

thf(149,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Order__Relation_Opartial__order__on ) ) )
    <=> ( ( A @ ( C @ c_Relation_Oantisym ) )
        & ( A @ ( B @ ( C @ c_Order__Relation_Opreorder__on ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_partial__order__on__def) ).

thf(1010,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( A @ ( C @ c_Relation_Oantisym ) )
          & ( A @ ( B @ ( C @ c_Order__Relation_Opreorder__on ) ) ) )
       => ( A @ ( B @ ( C @ c_Order__Relation_Opartial__order__on ) ) ) )
      & ( ( A @ ( B @ ( C @ c_Order__Relation_Opartial__order__on ) ) )
       => ( ( A @ ( C @ c_Relation_Oantisym ) )
          & ( A @ ( B @ ( C @ c_Order__Relation_Opreorder__on ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(251,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( C @ ( E @ c_Set_Oinsert ) ) @ ( D @ ( E @ c_Set_Oinsert ) ) )
        = ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( E @ c_Set_Oinsert ) ) @ ( B @ ( E @ c_Set_Oinsert ) ) ) )
    <=> ( ( ( C = B )
          & ( D = A ) )
        | ( ( C = A )
          & ( D = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_doubleton__eq__iff) ).

thf(1390,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( ( C = B )
            & ( D = A ) )
          | ( ( C = A )
            & ( D = B ) ) )
       => ( ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( C @ ( E @ c_Set_Oinsert ) ) @ ( D @ ( E @ c_Set_Oinsert ) ) )
          = ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( E @ c_Set_Oinsert ) ) @ ( B @ ( E @ c_Set_Oinsert ) ) ) ) )
      & ( ( ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( C @ ( E @ c_Set_Oinsert ) ) @ ( D @ ( E @ c_Set_Oinsert ) ) )
          = ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( E @ c_Set_Oinsert ) ) @ ( B @ ( E @ c_Set_Oinsert ) ) ) )
       => ( ( ( C = B )
            & ( D = A ) )
          | ( ( C = A )
            & ( D = B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(474,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Wellfounded_Owf ) )
    <=> ! [C: $i] :
          ( ? [D: $i] : ( C @ ( D @ ( B @ c_member ) ) )
         => ? [D: $i] :
              ( ! [E: $i] :
                  ( ( A @ ( D @ ( E @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
                 => ~ ( C @ ( E @ ( B @ c_member ) ) ) )
              & ( C @ ( D @ ( B @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__eq__minimal) ).

thf(2305,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ? [D: $i] : ( C @ ( D @ ( B @ c_member ) ) )
           => ? [D: $i] :
                ( ! [E: $i] :
                    ( ( A @ ( D @ ( E @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
                   => ~ ( C @ ( E @ ( B @ c_member ) ) ) )
                & ( C @ ( D @ ( B @ c_member ) ) ) ) )
       => ( A @ ( B @ c_Wellfounded_Owf ) ) )
      & ( ( A @ ( B @ c_Wellfounded_Owf ) )
       => ! [C: $i] :
            ( ? [D: $i] : ( C @ ( D @ ( B @ c_member ) ) )
           => ? [D: $i] :
                ( ! [E: $i] :
                    ( ( A @ ( D @ ( E @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
                   => ~ ( C @ ( E @ ( B @ c_member ) ) ) )
                & ( C @ ( D @ ( B @ c_member ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[474]) ).

thf(295,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ c_Transitive__Closure_Ortrancl ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ c_Transitive__Closure_Ortrancl ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__Un__rtrancl) ).

thf(1575,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ c_Transitive__Closure_Ortrancl ) @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ c_Transitive__Closure_Ortrancl ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( C @ ( C @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ c_Transitive__Closure_Ortrancl ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).

thf(139,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_leD) ).

thf(963,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(652,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) )
      | ~ ( B @ class_Orderings_Opreorder ) ),
    inference(cnf,[status(esa)],[651]) ).

thf(5270,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_HOL_Obool @ class_Orderings_Opreorder )
       != ( B @ class_Orderings_Opreorder ) )
      | ~ ( A @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) )
      | ~ $true ),
    inference(paramod_ordered,[status(thm)],[710,652]) ).

thf(5271,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_HOL_Obool @ class_Orderings_Opreorder )
       != ( B @ class_Orderings_Opreorder ) )
      | ~ ( A @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(simp,[status(thm)],[5270]) ).

thf(5272,plain,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_HOL_Obool @ c_Orderings_Oord__class_Oless ) ) ),
    inference(pattern_uni,[status(thm)],[5271:[bind(A,$thf( A )),bind(B,$thf( tc_HOL_Obool ))]]) ).

thf(164,axiom,
    ! [A: $i,B: $i] :
      ( ( B != A )
    <=> ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__neq__iff) ).

thf(1062,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
       => ( B != A ) )
      & ( ( B != A )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(132,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Wellfounded_Owf ) )
     => ( A @ ( B @ c_Wellfounded_Oacyclic ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__acyclic) ).

thf(946,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Wellfounded_Owf ) )
     => ( A @ ( B @ c_Wellfounded_Oacyclic ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(70,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ominus )
     => ( B @ ( A @ tc_fun ) @ class_Groups_Ominus ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Groups_Ominus) ).

thf(731,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ominus )
     => ( B @ ( A @ tc_fun ) @ class_Groups_Ominus ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(289,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__Diff) ).

thf(1558,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).

thf(301,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_Wellfounded_Owf ) )
     => ( ( C @ ( A @ ( B @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ~ ( C @ ( B @ ( A @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__asym) ).

thf(1590,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_Wellfounded_Owf ) )
     => ( ( C @ ( A @ ( B @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ~ ( C @ ( B @ ( A @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).

thf(138,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
      | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__le__linear) ).

thf(961,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
      | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(249,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__commute) ).

thf(1385,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(534,plain,
    ( ( sk6 != sk7 )
    | ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
    | sk1 ),
    inference(cnfConj,[status(thm)],[524]) ).

thf(543,plain,
    ( ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
    | sk1
    | ( sk7 != sk6 ) ),
    inference(lifteq,[status(thm)],[534]) ).

thf(2571,plain,
    ! [A: $i] :
      ( ( ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
       != ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) )
      | ~ $true
      | sk1
      | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
      | sk1
      | ( sk7 != sk6 ) ),
    inference(paramod_ordered,[status(thm)],[543,539]) ).

thf(2572,plain,
    ! [A: $i] :
      ( ( ( v_L @ ( sk5 @ ( sk5 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
       != ( v_L @ ( A @ ( A @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) ) )
      | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
      | sk1
      | ( sk7 != sk6 ) ),
    inference(simp,[status(thm)],[2571]) ).

thf(2573,plain,
    ( ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
    | sk1
    | ( sk7 != sk6 ) ),
    inference(pattern_uni,[status(thm)],[2572:[bind(A,$thf( sk5 ))]]) ).

thf(2592,plain,
    ( ( ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
     != ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) )
    | ~ $true
    | sk1
    | ( sk7 != sk6 )
    | sk1 ),
    inference(paramod_ordered,[status(thm)],[529,2573]) ).

thf(2593,plain,
    ( ( ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) )
     != ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) )
    | ( sk7 != sk6 )
    | sk1 ),
    inference(simp,[status(thm)],[2592]) ).

thf(2594,plain,
    ( ( sk7 != sk6 )
    | sk1 ),
    inference(pattern_uni,[status(thm)],[2593:[]]) ).

thf(332,axiom,
    ! [A: $i,B: $i,C: $i] : ( A @ ( B @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( C @ c_member ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insertI1) ).

thf(1721,plain,
    ! [A: $i,B: $i,C: $i] : ( A @ ( B @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( C @ c_member ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).

thf(896,plain,
    ! [A: $i] : ( A @ c_Relation_OId @ ( A @ ( A @ c_Relation_Osingle__valued ) ) ),
    inference(cnf,[status(esa)],[895]) ).

thf(427,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Product__Type_OPair ) ) ) @ ( D @ ( E @ tc_prod ) @ c_member ) ) )
     => ( A @ ( D @ ( E @ c_Relation_ODomain ) ) @ ( C @ ( E @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DomainI) ).

thf(2098,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Product__Type_OPair ) ) ) @ ( D @ ( E @ tc_prod ) @ c_member ) ) )
     => ( A @ ( D @ ( E @ c_Relation_ODomain ) ) @ ( C @ ( E @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[427]) ).

thf(403,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_member ) ) )
    <=> ( ( A @ ( C @ ( D @ c_member ) ) )
        | ( B @ ( C @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__iff) ).

thf(2009,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( A @ ( C @ ( D @ c_member ) ) )
          | ( B @ ( C @ ( D @ c_member ) ) ) )
       => ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_member ) ) ) )
      & ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_member ) ) )
       => ( ( A @ ( C @ ( D @ c_member ) ) )
          | ( B @ ( C @ ( D @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[403]) ).

thf(273,axiom,
    ! [A: $i,B: $i,C: $i] : ( B @ ( A @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__insertI) ).

thf(1491,plain,
    ! [A: $i,B: $i,C: $i] : ( B @ ( A @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(325,axiom,
    ! [A: $i] : ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Relation_Oantisym ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_antisym__empty) ).

thf(1689,plain,
    ! [A: $i] : ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Relation_Oantisym ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).

thf(14,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup_Oassoc) ).

thf(568,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = ( A @ ( B @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(123,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat_Odiff__diff__eq) ).

thf(914,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(345,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Relation_Ototal__on ) ) )
    <=> ! [D: $i] :
          ( ( B @ ( D @ ( C @ c_member ) ) )
         => ! [E: $i] :
              ( ( B @ ( E @ ( C @ c_member ) ) )
             => ( ( D != E )
               => ( ( A @ ( D @ ( E @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
                  | ( A @ ( E @ ( D @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_total__on__def) ).

thf(1777,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( B @ ( D @ ( C @ c_member ) ) )
           => ! [E: $i] :
                ( ( B @ ( E @ ( C @ c_member ) ) )
               => ( ( D != E )
                 => ( ( A @ ( D @ ( E @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
                    | ( A @ ( E @ ( D @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) ) ) ) ) )
       => ( A @ ( B @ ( C @ c_Relation_Ototal__on ) ) ) )
      & ( ( A @ ( B @ ( C @ c_Relation_Ototal__on ) ) )
       => ! [D: $i] :
            ( ( B @ ( D @ ( C @ c_member ) ) )
           => ! [E: $i] :
                ( ( B @ ( E @ ( C @ c_member ) ) )
               => ( ( D != E )
                 => ( ( A @ ( D @ ( E @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) )
                    | ( A @ ( E @ ( D @ ( C @ ( C @ c_Product__Type_OPair ) ) ) @ ( C @ ( C @ tc_prod ) @ c_member ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[345]) ).

thf(126,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__left__commute) ).

thf(929,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(198,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__Diff__cancel) ).

thf(1193,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(81,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ! [E: $i] : ( E @ ( A @ hAPP ) @ ( E @ ( B @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__fun__def) ).

thf(763,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( ! [E: $i] : ( E @ ( A @ hAPP ) @ ( E @ ( B @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ! [E: $i] : ( E @ ( A @ hAPP ) @ ( E @ ( B @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(281,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__imp__le) ).

thf(1513,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(259,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot )
     != ( A @ ( B @ ( C @ c_Set_Oinsert ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_empty__not__insert) ).

thf(1442,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot )
     != ( A @ ( B @ ( C @ c_Set_Oinsert ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(188,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( C @ c_member ) ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_singleton__iff) ).

thf(1161,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B = A )
       => ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( C @ c_member ) ) ) )
      & ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( C @ c_member ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(426,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_Set_Oinsert ) ) @ ( D @ c_Relation_OField ) )
      = ( A @ ( D @ c_Relation_OField ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Field__insert) ).

thf(2095,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_Set_Oinsert ) ) @ ( D @ c_Relation_OField ) )
      = ( A @ ( D @ c_Relation_OField ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[426]) ).

thf(459,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ~ ( A @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ~ ! [E: $i] :
              ( ( A @ ( E @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
             => ~ ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( E @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__tranclE) ).

thf(2230,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ~ ( A @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ~ ! [E: $i] :
              ( ( A @ ( E @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
             => ~ ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( E @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[459]) ).

thf(369,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( A @ ( D @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
         => ! [E: $i] :
              ( ( A @ ( E @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
             => ( A @ ( E @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ) )
     => ( A @ ( B @ c_Relation_Otrans ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_transI) ).

thf(1893,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( A @ ( D @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
         => ! [E: $i] :
              ( ( A @ ( E @ ( D @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) )
             => ( A @ ( E @ ( C @ ( B @ ( B @ c_Product__Type_OPair ) ) ) @ ( B @ ( B @ tc_prod ) @ c_member ) ) ) ) )
     => ( A @ ( B @ c_Relation_Otrans ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).

thf(178,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oab__group__add )
     => ( ( ( C @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) )
       => ( ( D = C )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__eq__diff__eq) ).

thf(1133,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oab__group__add )
     => ( ( ( C @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) )
       => ( ( ( B = A )
           => ( D = C ) )
          & ( ( D = C )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(429,axiom,
    c_FunDef_Omin__weak @ ( c_FunDef_Omin__strict @ ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_FunDef_Oreduction__pair ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_min__rpair__set) ).

thf(2102,plain,
    c_FunDef_Omin__weak @ ( c_FunDef_Omin__strict @ ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_FunDef_Oreduction__pair ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).

thf(473,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( B @ ( C @ ( D @ ( D @ ( D @ c_Relation_Orel__comp ) ) ) ) )
        = ( tc_HOL_Obool @ ( D @ ( D @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
     => ( ( ( A @ ( C @ ( D @ ( D @ ( D @ c_Relation_Orel__comp ) ) ) ) )
          = ( tc_HOL_Obool @ ( D @ ( D @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
       => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ ( D @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ ( D @ ( D @ c_Relation_Orel__comp ) ) ) ) )
          = ( tc_HOL_Obool @ ( D @ ( D @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_union__comp__emptyR) ).

thf(2302,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( B @ ( C @ ( D @ ( D @ ( D @ c_Relation_Orel__comp ) ) ) ) )
        = ( tc_HOL_Obool @ ( D @ ( D @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
     => ( ( ( A @ ( C @ ( D @ ( D @ ( D @ c_Relation_Orel__comp ) ) ) ) )
          = ( tc_HOL_Obool @ ( D @ ( D @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
       => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ ( D @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( C @ ( D @ ( D @ ( D @ c_Relation_Orel__comp ) ) ) ) )
          = ( tc_HOL_Obool @ ( D @ ( D @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[473]) ).

thf(296,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ ( C @ c_Relation_OImage ) ) ) )
      = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__empty) ).

thf(1578,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ ( C @ c_Relation_OImage ) ) ) )
      = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).

thf(100,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__absorb2) ).

thf(835,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Osemilattice__sup )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( C @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(209,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B
          = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( C @ c_Set_Oinsert ) ) ) )
        | ( B
          = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__singletonD) ).

thf(1229,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( C @ c_Set_Oinsert ) ) @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B
          = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( C @ c_Set_Oinsert ) ) ) )
        | ( B
          = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(310,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Relation_Osingle__valued ) ) )
    <=> ! [D: $i,E: $i] :
          ( ( A @ ( E @ ( D @ ( B @ ( C @ c_Product__Type_OPair ) ) ) @ ( B @ ( C @ tc_prod ) @ c_member ) ) )
         => ! [F: $i] :
              ( ( A @ ( F @ ( D @ ( B @ ( C @ c_Product__Type_OPair ) ) ) @ ( B @ ( C @ tc_prod ) @ c_member ) ) )
             => ( E = F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_single__valued__def) ).

thf(1625,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i,E: $i] :
            ( ( A @ ( E @ ( D @ ( B @ ( C @ c_Product__Type_OPair ) ) ) @ ( B @ ( C @ tc_prod ) @ c_member ) ) )
           => ! [F: $i] :
                ( ( A @ ( F @ ( D @ ( B @ ( C @ c_Product__Type_OPair ) ) ) @ ( B @ ( C @ tc_prod ) @ c_member ) ) )
               => ( E = F ) ) )
       => ( A @ ( B @ ( C @ c_Relation_Osingle__valued ) ) ) )
      & ( ( A @ ( B @ ( C @ c_Relation_Osingle__valued ) ) )
       => ! [D: $i,E: $i] :
            ( ( A @ ( E @ ( D @ ( B @ ( C @ c_Product__Type_OPair ) ) ) @ ( B @ ( C @ tc_prod ) @ c_member ) ) )
           => ! [F: $i] :
                ( ( A @ ( F @ ( D @ ( B @ ( C @ c_Product__Type_OPair ) ) ) @ ( B @ ( C @ tc_prod ) @ c_member ) ) )
               => ( E = F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).

thf(366,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( A @ ( B @ ( C @ c_Relation_ODomain ) ) )
        = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
    <=> ( A
        = ( tc_HOL_Obool @ ( B @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__empty__iff) ).

thf(1876,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( A
          = ( tc_HOL_Obool @ ( B @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
       => ( ( A @ ( B @ ( C @ c_Relation_ODomain ) ) )
          = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) )
      & ( ( ( A @ ( B @ ( C @ c_Relation_ODomain ) ) )
          = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
       => ( A
          = ( tc_HOL_Obool @ ( B @ ( C @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).

thf(320,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( A @ ( D @ ( B @ c_member ) ) )
      <=> ( A @ ( C @ ( B @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eqelem__imp__iff) ).

thf(1673,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( ( A @ ( C @ ( B @ c_member ) ) )
         => ( A @ ( D @ ( B @ c_member ) ) ) )
        & ( ( A @ ( D @ ( B @ c_member ) ) )
         => ( A @ ( C @ ( B @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).

thf(214,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ c_Transitive__Closure_Ortrancl ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__reflcl) ).

thf(1248,plain,
    ! [A: $i,B: $i] :
      ( ( B @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) @ ( B @ c_Transitive__Closure_Ortrancl ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(269,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Obot )
     => ( A @ ( B @ c_Orderings_Obot__class_Obot @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bot__least) ).

thf(1480,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Obot )
     => ( A @ ( B @ c_Orderings_Obot__class_Obot @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(490,axiom,
    ! [A: $i] :
      ( ( A
       != ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
     => ( c_FunDef_Omin__strict @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_smin__emptyI) ).

thf(2374,plain,
    ! [A: $i] :
      ( ( A
       != ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot ) )
     => ( c_FunDef_Omin__strict @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[490]) ).

thf(241,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Owf ) )
     => ( ( A @ ( B @ ( C @ ( C @ c_Relation_OImage ) ) ) @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A
          = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wfE__pf) ).

thf(1348,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Owf ) )
     => ( ( A @ ( B @ ( C @ ( C @ c_Relation_OImage ) ) ) @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A
          = ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(105,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ c_Set_Oinsert ) ) @ ( B @ c_Set_Othe__elem ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_the__elem__eq) ).

thf(854,plain,
    ! [A: $i,B: $i] :
      ( ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ c_Set_Oinsert ) ) @ ( B @ c_Set_Othe__elem ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(396,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_HOL_Obool @ ( A @ ( B @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ c_Relation_ORange ) ) )
      = ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__empty) ).

thf(1986,plain,
    ! [A: $i,B: $i] :
      ( ( tc_HOL_Obool @ ( A @ ( B @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( B @ c_Relation_ORange ) ) )
      = ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).

thf(12,axiom,
    tc_HOL_Obool @ class_Orderings_Oord,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL__Obool__Orderings_Oord) ).

thf(564,plain,
    tc_HOL_Obool @ class_Orderings_Oord,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(352,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( C @ ( D @ ( E @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) )
     => ( ( E @ ( D @ ( C @ ( A @ ( B @ ( F @ c_Recdef_Ocut ) ) ) ) @ hAPP ) )
        = ( E @ ( A @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cut__apply) ).

thf(1816,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( C @ ( D @ ( E @ ( F @ ( F @ c_Product__Type_OPair ) ) ) @ ( F @ ( F @ tc_prod ) @ c_member ) ) )
     => ( ( E @ ( D @ ( C @ ( A @ ( B @ ( F @ c_Recdef_Ocut ) ) ) ) @ hAPP ) )
        = ( E @ ( A @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).

thf(290,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Wellfounded_Oacyclic ) )
     => ( A @ ( B @ c_Transitive__Closure_Ortrancl ) @ ( B @ c_Relation_Oantisym ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acyclic__impl__antisym__rtrancl) ).

thf(1561,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Wellfounded_Oacyclic ) )
     => ( A @ ( B @ c_Transitive__Closure_Ortrancl ) @ ( B @ c_Relation_Oantisym ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).

thf(223,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__antisym__conv2) ).

thf(1288,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( ( B = A )
           => ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) ) )
          & ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(532,plain,
    ( ~ sk1
    | ~ ( v_L @ ( sk3 @ ( sk4 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ( v_L @ ( sk2 @ ( sk2 @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Product__Type_OPair ) ) ) @ ( tc_Arrow__Order__Mirabelle_Oalt @ ( tc_Arrow__Order__Mirabelle_Oalt @ tc_prod ) @ c_member ) ) )
    | ~ ( v_L @ ( tc_Arrow__Order__Mirabelle_Oalt @ c_Relation_Otrans ) ) ),
    inference(cnfConj,[status(thm)],[524]) ).

thf(240,axiom,
    ! [A: $i,B: $i] : ( A @ ( B @ c_Wellfounded_Omeasure @ hAPP ) @ ( B @ c_Wellfounded_Owf ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__measure) ).

thf(1346,plain,
    ! [A: $i,B: $i] : ( A @ ( B @ c_Wellfounded_Omeasure @ hAPP ) @ ( B @ c_Wellfounded_Owf ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(450,axiom,
    ! [A: $i] : ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Wellfounded_Owf ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__empty) ).

thf(2201,plain,
    ! [A: $i] : ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Wellfounded_Owf ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).

thf(397,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( D @ c_Product__Type_OPair ) ) ) @ ( E @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( E @ ( tc_HOL_Obool @ ( C @ ( D @ c_Product__Type_Ocurry ) ) ) ) ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_curryI) ).

thf(1989,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( C @ ( D @ c_Product__Type_OPair ) ) ) @ ( E @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( E @ ( tc_HOL_Obool @ ( C @ ( D @ c_Product__Type_Ocurry ) ) ) ) ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[397]) ).

thf(245,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__Un__eq) ).

thf(1363,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
          = A )
       => ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(232,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_OId__on ) @ ( B @ ( B @ c_Relation_ORange ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__Id__on) ).

thf(1314,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Relation_OId__on ) @ ( B @ ( B @ c_Relation_ORange ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(327,axiom,
    ! [A: $i,B: $i] : ( B @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ c_Relation_Oirrefl ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_irrefl__diff__Id) ).

thf(1693,plain,
    ! [A: $i,B: $i] : ( B @ c_Relation_OId @ ( A @ ( tc_HOL_Obool @ ( B @ ( B @ tc_prod ) @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ c_Relation_Oirrefl ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).

thf(419,axiom,
    c_Wellfounded_Oless__than @ ( tc_Nat_Onat @ c_Relation_Otrans ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__less__than) ).

thf(2070,plain,
    c_Wellfounded_Oless__than @ ( tc_Nat_Onat @ c_Relation_Otrans ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[419]) ).

thf(190,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ hAPP ) @ hBOOL )
     => ( ( A @ ( D @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( C @ ( A @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rev__predicate1D) ).

thf(1172,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ hAPP ) @ hBOOL )
     => ( ( A @ ( D @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( C @ ( A @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(119,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( B @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__commute) ).

thf(897,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( B @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(253,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__ab__group__add )
     => ( ( ( C @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) )
       => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
        <=> ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__eq__diff__less__eq) ).

thf(1410,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__ab__group__add )
     => ( ( ( C @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) )
       => ( ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) )
          & ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(82,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__cancel) ).

thf(767,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Obot__class_Obot ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(216,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
          = C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_double__diff) ).

thf(1268,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
          = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(108,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( ( ~ ( C @ ( B @ ( D @ c_member ) ) )
         => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( C @ ( B @ ( D @ c_member ) ) )
         => ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__insert__iff) ).

thf(860,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( ~ ( C @ ( B @ ( D @ c_member ) ) )
           => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
          & ( ( C @ ( B @ ( D @ c_member ) ) )
           => ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) )
       => ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( ~ ( C @ ( B @ ( D @ c_member ) ) )
           => ( A @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
          & ( ( C @ ( B @ ( D @ c_member ) ) )
           => ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( B @ ( D @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(464,axiom,
    c_FunDef_Opair__less @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Wellfounded_Owf ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__pair__less) ).

thf(2265,plain,
    c_FunDef_Opair__less @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Wellfounded_Owf ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[464]) ).

thf(276,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Transitive__Closure_Ortrancl ) @ ( B @ c_Transitive__Closure_Ortrancl ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__idemp) ).

thf(1500,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Transitive__Closure_Ortrancl ) @ ( B @ c_Transitive__Closure_Ortrancl ) )
      = ( A @ ( B @ c_Transitive__Closure_Ortrancl ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(379,axiom,
    ! [A: $i] : ( c_FunDef_Omin__weak @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wmin__emptyI) ).

thf(1925,plain,
    ! [A: $i] : ( c_FunDef_Omin__weak @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).

thf(56,axiom,
    tc_Nat_Onat @ class_Orderings_Olinorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Olinorder) ).

thf(691,plain,
    tc_Nat_Onat @ class_Orderings_Olinorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(516,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Owf ) )
     => ( B @ ( C @ c_Wellfounded_Oacc ) @ ( A @ ( C @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acc__wfD) ).

thf(2484,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ c_Wellfounded_Owf ) )
     => ( B @ ( C @ c_Wellfounded_Oacc ) @ ( A @ ( C @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[516]) ).

thf(489,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( A @ ( D @ c_member ) ) )
       => ( B @ ( A @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__mp) ).

thf(2372,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( A @ ( D @ c_member ) ) )
       => ( B @ ( A @ ( D @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[489]) ).

thf(1930,plain,
    ! [B: $i,A: $i] : ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( B @ c_Relation_Otrans ) ),
    inference(cnf,[status(esa)],[1929]) ).

thf(484,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl_Or__into__trancl) ).

thf(2353,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[484]) ).

thf(68,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Obot )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Obot ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Orderings_Obot) ).

thf(728,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Obot )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Obot ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(73,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( C = B )
       => ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I3_J) ).

thf(740,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( C = B )
       => ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(212,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( C @ ( B @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equalityD2) ).

thf(1236,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( C @ ( B @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(521,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( D @ c_Wellfounded_Oacc ) @ ( C @ ( D @ c_member ) ) )
     => ( ( B @ ( C @ ( A @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ( B @ ( D @ c_Wellfounded_Oacc ) @ ( A @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acc__downward) ).

thf(2502,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( D @ c_Wellfounded_Oacc ) @ ( C @ ( D @ c_member ) ) )
     => ( ( B @ ( C @ ( A @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
       => ( B @ ( D @ c_Wellfounded_Oacc ) @ ( A @ ( D @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[521]) ).

thf(41,axiom,
    tc_Nat_Onat @ class_Orderings_Obot,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Obot) ).

thf(644,plain,
    tc_Nat_Onat @ class_Orderings_Obot,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(1544,plain,
    ( ! [A: $i,B: $i,C: $i] :
        ( ( ( B != A )
          & ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
       => ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) )
    & ! [A: $i,B: $i,C: $i] :
        ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B != A )
          & ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(miniscope,[status(thm)],[1543]) ).

thf(1545,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( E @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
        | ( E = D )
        | ~ ( D @ ( E @ ( tc_HOL_Obool @ ( F @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( B != A )
        | ~ ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
        | ~ ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(cnf,[status(esa)],[1544]) ).

thf(1547,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B != A )
      | ~ ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(cnfConj,[status(thm)],[1545]) ).

thf(1549,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
      | ( B != A ) ),
    inference(lifteq,[status(thm)],[1547]) ).

thf(1550,plain,
    ! [B: $i,A: $i] :
      ~ ( A @ ( A @ ( tc_HOL_Obool @ ( B @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ),
    inference(simp,[status(thm)],[1549]) ).

thf(278,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__trans) ).

thf(1505,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(466,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) )
     => ( ( c_FunDef_Opair__leq @ ( B @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) )
       => ( ( c_FunDef_Omin__weak @ ( A @ ( C @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) )
         => ( c_FunDef_Omin__weak @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wmin__insertI) ).

thf(2270,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_member ) ) )
     => ( ( c_FunDef_Opair__leq @ ( B @ ( D @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_prod ) @ c_member ) ) )
       => ( ( c_FunDef_Omin__weak @ ( A @ ( C @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) )
         => ( c_FunDef_Omin__weak @ ( A @ ( B @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ c_Set_Oinsert ) ) @ ( C @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ c_Product__Type_OPair ) ) ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ ( tc_HOL_Obool @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ tc_prod ) @ tc_fun ) @ tc_prod ) @ c_member ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[466]) ).

thf(305,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
          & ( C != B ) )
        | ( C = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtranclD) ).

thf(1606,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( D @ c_Transitive__Closure_Ortrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
     => ( ( ( A @ ( D @ c_Transitive__Closure_Otrancl ) @ ( B @ ( C @ ( D @ ( D @ c_Product__Type_OPair ) ) ) @ ( D @ ( D @ tc_prod ) @ c_member ) ) )
          & ( C != B ) )
        | ( C = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).

thf(315,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( F @ ( G @ hAPP ) @ hBOOL )
     => ( ( F @ ( B @ hAPP ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( G @ ( E @ ( A @ c_Recdef_Osame__fst ) ) ) @ ( C @ ( F @ ( E @ ( A @ c_Product__Type_OPair ) ) ) @ ( D @ ( F @ ( E @ ( A @ c_Product__Type_OPair ) ) ) @ ( E @ ( A @ tc_prod ) @ ( E @ ( A @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( E @ ( A @ tc_prod ) @ ( E @ ( A @ tc_prod ) @ tc_prod ) @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_same__fstI) ).

thf(1653,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( F @ ( G @ hAPP ) @ hBOOL )
     => ( ( F @ ( B @ hAPP ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
       => ( B @ ( G @ ( E @ ( A @ c_Recdef_Osame__fst ) ) ) @ ( C @ ( F @ ( E @ ( A @ c_Product__Type_OPair ) ) ) @ ( D @ ( F @ ( E @ ( A @ c_Product__Type_OPair ) ) ) @ ( E @ ( A @ tc_prod ) @ ( E @ ( A @ tc_prod ) @ c_Product__Type_OPair ) ) ) @ ( E @ ( A @ tc_prod ) @ ( E @ ( A @ tc_prod ) @ tc_prod ) @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).

thf(391,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_Wellfounded_Owf ) )
     => ( ( A @ ( B @ c_Wellfounded_Owf ) )
       => ( A @ ( C @ ( B @ ( D @ c_Wellfounded_Olex__prod ) ) ) @ ( B @ ( D @ tc_prod ) @ c_Wellfounded_Owf ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__lex__prod) ).

thf(1962,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_Wellfounded_Owf ) )
     => ( ( A @ ( B @ c_Wellfounded_Owf ) )
       => ( A @ ( C @ ( B @ ( D @ c_Wellfounded_Olex__prod ) ) ) @ ( B @ ( D @ tc_prod ) @ c_Wellfounded_Owf ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).

thf(385,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_member ) ) )
     => ( ( B @ ( A @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__Diff1) ).

thf(1945,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_member ) ) )
     => ( ( B @ ( A @ ( C @ ( D @ c_Set_Oinsert ) ) @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ ( A @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).

thf(265,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( D @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B @ ( C @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( D @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__mono) ).

thf(1463,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( C @ ( D @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B @ ( C @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( D @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( E @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(2059,plain,
    ! [B: $i,A: $i] : ( A @ ( B @ c_Relation_OId__on ) @ ( B @ c_Relation_Otrans ) ),
    inference(cnf,[status(esa)],[2058]) ).

thf(85,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( B @ ( B @ c_Relation_ORange ) ) )
      = ( A @ ( B @ ( B @ c_Relation_ORange ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__range) ).

thf(775,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Transitive__Closure_Otrancl ) @ ( B @ ( B @ c_Relation_ORange ) ) )
      = ( A @ ( B @ ( B @ c_Relation_ORange ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(254,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__not__refl2) ).

thf(1416,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(448,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( D @ ( A @ hAPP ) @ hBOOL )
       => ( ! [F: $i] :
              ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( F @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
             => ! [G: $i] :
                  ( ( B @ ( G @ ( F @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
                 => ( ( F @ ( A @ hAPP ) @ hBOOL )
                   => ( G @ ( A @ hAPP ) @ hBOOL ) ) ) )
         => ( C @ ( A @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__induct) ).

thf(2191,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( C @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
     => ( ( D @ ( A @ hAPP ) @ hBOOL )
       => ( ! [F: $i] :
              ( ( B @ ( E @ c_Transitive__Closure_Ortrancl ) @ ( F @ ( D @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
             => ! [G: $i] :
                  ( ( B @ ( G @ ( F @ ( E @ ( E @ c_Product__Type_OPair ) ) ) @ ( E @ ( E @ tc_prod ) @ c_member ) ) )
                 => ( ( F @ ( A @ hAPP ) @ hBOOL )
                   => ( G @ ( A @ hAPP ) @ hBOOL ) ) ) )
         => ( C @ ( A @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).

thf(431,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_member ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( D @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__rev__mp) ).

thf(2106,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B @ ( C @ ( D @ c_member ) ) )
     => ( ( A @ ( B @ ( tc_HOL_Obool @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( D @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[431]) ).

thf(376,axiom,
    ! [A: $i] : ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Order__Relation_Opreorder__on ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_preorder__on__empty) ).

thf(1912,plain,
    ! [A: $i] : ( tc_HOL_Obool @ ( A @ ( A @ tc_prod ) @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( tc_HOL_Obool @ ( A @ tc_fun ) @ c_Orderings_Obot__class_Obot @ ( A @ c_Order__Relation_Opreorder__on ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).

thf(113,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__Diff__cancel2) ).

thf(882,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
      = ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(364,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( A @ ( C @ ( B @ c_member ) ) )
       => ( A @ ( D @ ( B @ c_member ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__mem__trans) ).

thf(1870,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( A @ ( C @ ( B @ c_member ) ) )
       => ( A @ ( D @ ( B @ c_member ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).

thf(321,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_Relation_Otrans ) )
     => ( A @ ( C @ ( B @ ( D @ c_Relation_Oinv__image ) ) @ hAPP ) @ ( B @ c_Relation_Otrans ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__inv__image) ).

thf(1681,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ c_Relation_Otrans ) )
     => ( A @ ( C @ ( B @ ( D @ c_Relation_Oinv__image ) ) @ hAPP ) @ ( B @ c_Relation_Otrans ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).

thf(195,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__partition) ).

thf(1184,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_HOL_Obool @ ( C @ tc_fun ) @ c_Lattices_Osemilattice__sup__class_Osup ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(28037,plain,
    $false,
    inference(e,[status(thm)],[1036,629,2031,2263,893,2427,941,555,1295,1950,2163,645,1586,2452,809,2363,582,730,2062,1808,1569,1073,2282,878,1793,1305,1041,1486,1924,1005,2495,2381,873,1205,1497,1078,1655,1310,1867,2141,5146,1327,747,538,1507,1458,1999,2316,829,1196,1914,1127,2227,841,1613,1645,2067,1904,1482,651,973,885,1046,2173,565,1709,2039,1907,917,1279,1455,1191,542,1760,1660,1330,2444,1860,587,838,1059,1132,2420,902,559,1379,1147,1572,770,1741,1931,1232,4756,2169,2403,1489,1465,2469,834,1151,574,1581,1051,2017,1302,756,2324,2027,2090,591,1858,2122,1960,806,602,2058,692,1886,1351,2103,2205,1169,687,1943,1201,1104,849,1863,1651,1604,546,761,810,2296,1057,678,2473,529,1691,2047,5117,825,952,1123,1174,820,693,1274,2441,2264,2409,1900,1723,957,1223,2249,778,598,2147,788,1619,1423,4757,1687,1819,1159,2181,1407,710,1910,1032,1837,625,2433,852,2332,1296,665,522,830,1027,2327,2344,551,2492,1851,1970,1856,980,2537,1755,857,1592,2300,697,2498,1234,1541,1143,2115,1762,2100,2377,948,1017,1953,2023,1361,1964,1182,669,1075,4750,526,1644,1493,1683,554,753,1739,1461,637,1584,2108,1446,1155,547,622,1825,647,1588,1150,1297,1382,1831,2243,2355,1799,1874,2272,1991,2203,1814,633,2208,1695,1286,558,1616,530,718,1563,657,1712,1054,1187,674,1515,2221,794,2338,1278,2001,2391,2149,1927,689,733,965,750,1727,1251,618,1764,562,1007,1884,1615,1125,772,2393,1929,1388,579,1535,1795,1226,826,577,711,1503,936,1707,1312,787,1997,1332,2425,672,1271,1371,1058,1107,919,804,2148,2358,2079,1209,900,1853,1282,2383,2422,1806,616,2065,1048,736,990,1933,880,1043,2197,1307,535,1993,594,799,2446,700,1149,1556,943,1177,2461,890,1667,1029,895,2329,932,2365,2350,695,2086,2139,2288,1435,1477,1102,2207,653,2506,1198,2439,959,2171,1213,827,1113,563,1758,531,1467,680,2471,2037,2234,1509,1081,1145,2054,1948,1203,589,1905,1420,939,1340,1598,640,705,887,971,1647,954,2175,1753,1098,557,1770,2397,1025,2060,707,1566,2143,2454,1432,791,585,2429,1317,544,673,2092,1300,1637,2219,1980,2128,576,1658,2322,1538,2486,808,1336,643,1358,744,1898,703,1141,2379,613,1972,978,2030,571,595,612,859,1543,1766,2279,600,1955,2145,2266,876,1240,1617,2088,2045,1180,556,1553,2215,759,1448,2071,1685,539,1157,1321,976,903,1216,2209,1010,1390,2305,1575,963,5272,1903,1062,946,652,731,1558,1276,1590,961,813,1385,2594,1721,896,2098,2009,1491,1689,568,914,1777,929,1193,763,1513,1442,1161,2095,832,2230,1893,1133,2102,2302,1578,835,1229,1625,1876,1673,1248,1480,2374,1348,854,1986,564,1816,1561,1288,532,1346,2201,1989,1363,1314,1693,2070,1172,897,1410,767,1129,1268,860,2265,1500,1925,691,2484,2372,1930,2353,728,740,1236,2502,644,1550,1505,2270,1606,1653,1962,1945,1463,2059,775,1416,1917,2191,2106,1912,882,1870,1681,1184]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03  % Problem  : SCT105+1 : TPTP v9.3.1. Released v5.2.0.
% 0.00/0.08  % Command  : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.15/0.42  % Computer : n026.cluster.edu
% 0.15/0.42  % Model    : x86_64 x86_64
% 0.15/0.42  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.42  % Memory   : 8046.5625MB
% 0.15/0.42  % OS       : Linux 6.8.0-71-generic
% 0.15/0.42  % CPULimit : 300
% 0.15/0.43  % WCLimit  : 300
% 0.15/0.43  % DateTime : Sat Sep 26 06:30:55 UTC 2026
% 0.15/0.43  % CPUTime  : 
% 0.15/0.43  Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 1.15/1.10  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 3.28/1.71  % [INFO] 	 Parsing done (608ms). 
% 3.28/1.74  % [INFO] 	 Running in sequential loop mode. 
% 4.66/2.31  % [INFO] 	 eprover registered as external prover. 
% 4.66/2.32  % [INFO] 	 Scanning for conjecture ... 
% 5.50/2.62  % [INFO] 	 Found a conjecture (or negated_conjecture) and 522 axioms. Running axiom selection ... 
% 6.69/2.98  % [INFO] 	 Axiom selection finished. Selected 519 axioms (removed 3 axioms). 
% 8.80/3.65  % [INFO] 	 Problem is first-order (TPTP FOF). 
% 9.18/3.72  % [INFO] 	 Type checking passed. 
% 9.18/3.72  % [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 ... 
% 141.70/33.16  % External prover 'e' found a proof!
% 141.70/33.16  % [INFO] 	 Killing All external provers ... 
% 141.70/33.16  % Time passed: 32527ms (effective reasoning time: 31415ms)
% 141.70/33.16  % 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)>
% 141.70/33.17  % Axioms used in derivation (519): fact_wf__not__refl, fact_order__le__neq__trans, fact_wmax__insertI, fact_transI, fact_wfrec__rel_Ointros, fact_single__valued__Id__on, fact_insertCI, fact_ImageE, fact_wf__not__sym, fact_refl__onD2, fact_wf__inv__image, fact_acc__wfI, fact_Field__empty, fact_Un__absorb1, fact_xt1_I1_J, fact_converse__rtrancl__induct2, fact_singleton__iff, fact_wmin__emptyI, fact_Diff__insert__absorb, arity_fun__Orderings_Oorder, fact_cut__apply, fact_sup1I1, fact_order__le__less__trans, fact_pair__lessI2, fact_smax__insertI, fact_insert__iff, arity_Nat__Onat__Lattices_Olattice, fact_le__eq__less__or__eq, fact_nat__less__cases, fact_diff__le__mono2, fact_xt1_I4_J, fact_UnI1, fact_termination__basic__simps_I5_J, fact_insert__subset, fact_rtranclD, arity_Nat__Onat__Orderings_Oorder, fact_pred__nat__trancl__eq__le, fact_sup__apply, fact_Un__mono, fact_Transitive__Closure_Otrancl__into__trancl, fact_wmin__insertI, fact_curry__conv, fact_IdI, fact_insertI1, fact_order__less__asym_H, fact_Un__empty__left, fact_less__imp__le__nat, fact_less__supI1, fact_bot__apply, fact_prod_Orecs, fact_union__comp__emptyR, fact_trancl__id, fact_mono__Field, fact_min__ext__wf, fact_linorder__not__le, fact_linorder__antisym__conv2, fact_acc_Osimps, fact_diff__diff__cancel, arity_HOL__Obool__Orderings_Opreorder, fact_linorder__neqE, fact_sup__assoc, fact_curryE, fact_leI, fact_empty__not__insert, fact_trans__Id__on, fact_rtrancl__empty, fact_order__eq__refl, fact_Un__insert__left, arity_fun__Lattices_Olattice, fact_max__ext__additive, fact_rtrancl_Ortrancl__refl, fact_r__into__trancl_H, fact_rel__compI, help_c__fequal__2, fact_converse__rtrancl__induct, fact_min__strict__def, fact_acyclic__impl__antisym__rtrancl, fact_le__neq__implies__less, fact_tranclD2, fact_linorder__antisym__conv3, fact_order__less__imp__not__eq, fact_max__rpair__set, fact_pair__leq__def, fact_rtrancl__mono, fact_rtrancl__subset__rtrancl, fact_insert__not__empty, fact_converse__rtranclE2, fact_split__paired__All, fact_Diff__subset, fact_sup__absorb1, fact_equalityD2, fact_antisymI, fact_Field__Un, fact_less__eq, fact_Range__empty, fact_rev__ImageI, fact_linorder__antisym__conv1, fact_tranclE, fact_acc_OaccI, fact_le__supI, fact_le__supI2, fact_single__valued__Id, fact_predicate2D, fact_Diff__idemp, fact_diff__le__self, fact_singleton__inject, fact_le__funD, fact_emptyE, fact_le__fun__def, fact_acc__subset, fact_DiffD2, fact_reduction__pairI, fact_sup__ge2, fact_wf__irrefl, fact_linorder__less__linear, fact_Image__Un, fact_order__less__imp__not__less, fact_lnear__order__on__empty, fact_DiffE, fact_Domain__empty__iff, fact_ord__less__eq__trans, fact_in__mono, fact_Domain__Un__eq, fact_wf__lex__prod, fact_wf__measure, fact_diff__commute, fact_xt1_I9_J, fact_Diff__subset__conv, fact_sup__least, fact_rtrancl__unfold, fact_in__measure, fact_minus__apply, fact_rtrancl__subset, fact_max__weak__def, fact_subset__singletonD, fact_sup__eq__bot__iff, fact_single__valued__subset, fact_insert__is__Un, fact_less__or__eq__imp__le, arity_HOL__Obool__Lattices_Olattice, fact_wfE__pf, fact_subset__iff__psubset__eq, fact_Id__on__iff, fact_Diff__insert, fact_tfl__cut__apply, fact_fold__graph_H_Oequations_I1_J, fact_wf__subset, fact_rtrancl__Un__separator__converseE, fact_Domain__insert, fact_rtrancl__reflcl, arity_Nat__Onat__Orderings_Opreorder, fact_antisym__empty, fact_order__less__imp__not__eq2, fact_order__antisym, fact_xt1_I10_J, fact_less__irrefl__nat, fact_refl__onD, fact_sup__bot__left, fact_order__refl, arity_fun__Lattices_Osemilattice__sup, fact_acc__downwards, fact_Un__left__commute, fact_sup1I2, fact_Un__empty, fact_Domain__mono, fact_preorder__on__def, arity_HOL__Obool__Lattices_Osemilattice__sup, fact_xt1_I5_J, fact_insert__code, fact_Domain__empty, fact_wf__empty, fact_bot__least, fact_subset__Image1__Image1__iff, fact_insert__Diff__single, fact_Domain__Diff__subset, fact_refl__on__empty, arity_Nat__Onat__Groups_Ominus, fact_trans__inv__image, fact_rtrancl__Un__separatorE, fact_rtrancl__Un__rtrancl, fact_RangeI, arity_HOL__Obool__Groups_Ominus, fact_nat__neq__iff, fact_rtrancl__reflcl__absorb, fact_DomainI, fact_Diff__empty, fact_ex__in__conv, fact_sup__left__commute, arity_Nat__Onat__Lattices_Osemilattice__sup, fact_subset__insertI, fact_notin__Lin__iff, fact_xt1_I3_J, fact_wf__pred__nat, fact_less__imp__neq, fact_xt1_I8_J, fact_predicate1D, fact_less__not__refl3, fact_eq__mem, fact_trans__diff__Id, fact_smin__insertI, arity_HOL__Obool__Orderings_Oord, fact_Diff__partition, fact_Id__onE, fact_RangeE, fact_sup__left__idem, fact_insert__absorb2, fact_le__supI1, fact_converse__rtranclE, fact_equals0D, fact_insert__absorb, fact_subset__insert, fact_Un__Image, fact_pair__in__Id__conv, fact_max__ext__wf, fact_insertI2, fact_Diff__cancel, fact_same__fstI, fact_trans__Id, fact_trancl__trans, fact_equalityD1, fact_trans__def, fact_Refl__antisym__eq__Image1__Image1__iff, fact_diff__single__insert, fact_le__funE, fact_wfrec, fact_eqset__imp__iff, fact_trancl__empty, fact_in__rtrancl__UnI, fact_order__neq__le__trans, fact_converse__rtrancl__into__rtrancl, fact_Un__insert__right, arity_fun__Orderings_Opreorder, fact_rtrancl__trans, fact_less__by__empty, fact_less__imp__diff__less, help_c__fequal__1, fact_refl__on__Un, fact_linorder__neq__iff, fact_DiffD1, arity_Nat__Onat__Orderings_Oord, fact_irrefl__def, fact_UnI2, fact_rp__inv__image__rp, fact_insert__commute, fact_subset__Un__eq, fact_le__iff__sup, fact_sup_Oassoc, fact_ord__le__eq__trans, fact_antisym__subset, fact_all__not__in__conv, fact_mlex__leq, fact_set__mp, fact_less__than__iff, fact_linorder__neqE__nat, fact_Image__singleton__iff, fact_not__less__iff__gr__or__eq, fact_Un__Diff__cancel, fact_wfrec__rel_Oequations, fact_r__r__into__trancl, fact_def__wfrec, fact_subset__equiv__class, fact_total__on__def, fact_irrefl__trancl__rD, fact_Diff__mono, fact_psubsetD, fact_fold__graph_H_Ointros_I1_J, fact_bex__Un, fact_partial__order__on__def, fact_Range__Id__on, fact_reflcl__trancl, fact_inf__sup__aci_I8_J, fact_antisym__def, fact_Range__iff, fact_sup1CI, fact_Image__empty, fact_rtrancl__into__trancl2, fact_Nitpick_Orefl_H__def, fact_trans__reflclI, fact_psubset__trans, fact_double__diff, fact_order__less__le, fact_sup__ge1, fact_wf__acc__iff, arity_Nat__Onat__Orderings_Obot, fact_set__eq__subset, fact_acyclic__insert, fact_partial__order__on__empty, fact_Pair__inject, fact_xt1_I12_J, fact_linear__order__on__def, fact_subset__refl, fact_fold__graph_H_Ointros_I2_J, fact_internal__split__conv, fact_single__valued__confluent, arity_fun__Orderings_Oord, fact_equalityI, fact_ext, fact_order__less__not__sym, fact_le__supE, fact_smin__emptyI, fact_acc__wfD, fact_eq__diff__iff, arity_HOL__Obool__Orderings_Oorder, fact_linorder__le__less__linear, arity_fun__Orderings_Obot, fact_empty__iff, fact_wf__in__rel, fact_Image__closed__trancl, fact_sup__idem, fact_less__le__not__le, arity_Nat__Onat__Orderings_Olinorder, fact_rtrancl__induct2, fact_rtrancl__trancl__absorb, fact_nat__le__linear, fact_linorder__le__cases, fact_Id__on__eqI, fact_trancl__range, fact_acyclic__def, fact_order__less__le__trans, fact_max__strict__def, fact_rtrancl__eq__or__trancl, fact_refl__onD1, fact_not__leE, fact_leD, fact_total__on__empty, fact_insert__ident, fact_subset__Image__Image__iff, fact_acyclic__subset, fact_curryD, fact_subset__empty, fact_singletonE, fact_diff__less__mono2, fact_subset__insert__iff, fact_trancl__into__rtrancl, fact_rtrancl__Un__subset, fact_diff__eq__diff__less, fact_sup__mono, fact_Range__Diff__subset, fact_converse__tranclE, fact_rtrancl__r__diff__Id, fact_sup_Oidem, fact_preorder__on__empty, fact_wf__eq__minimal, arity_HOL__Obool__Lattices_Obounded__lattice__bot, fact_order__less__asym, fact_mlex__less, fact_psubset__imp__subset, fact_less__than__def, fact_Un__absorb, fact_trancl_Or__into__trancl, fact_min__rpair__set, fact_set__rev__mp, fact_tranclD, fact_Domain__iff, fact_DiffI, fact_pair__less__def, fact_eq__mem__trans, fact_antisymD, fact_trans__lex__prod, fact_acc__downwards__aux, fact_Un__empty__right, fact_wf__insert, fact_diff__less__mono, fact_rtrancl__induct, fact_sup__commute, fact_trancl__rtrancl__absorb, fact_rev__predicate2D, fact_linorder__linear, fact_total__on__diff__Id, fact_transD, fact_trancl__into__trancl2, fact_single__valuedI, fact_subset__insertI2, fact_Un__absorb2, fact_less__not__refl2, fact_single__valued__def, fact_DomainE, fact_xt1_I6_J, fact_Nat_Odiff__diff__eq, fact_le__refl, fact_antisym__reflcl, fact_strict__linear__order__on__diff__Id, fact_in__lex__prod, fact_strict__linear__order__on__def, fact_inf__sup__aci_I5_J, fact_ball__Un, fact_Un__upper1, fact_xt1_I2_J, fact_UnCI, fact_linorder__not__less, fact_less__fun__def, fact_rev__predicate1D, fact_insert__Diff, fact_trans__rtrancl, fact_Domain__Id__on, fact_cuts__eq, fact_wf__trancl, fact_pair__leqI1, fact_equalityE, fact_Pair__eq, fact_tfl__wfrec, fact_in__rel__def, fact_inf__sup__ord_I4_J, fact_le__sup__iff, fact_subset__psubset__trans, fact_irrefl__diff__Id, fact_IdE, fact_measure__def, fact_order__eq__iff, fact_rtrancl_Ortrancl__into__rtrancl, fact_rtranclE, fact_empty__subsetI, fact_insert__Diff__if, fact_less__not__refl, fact_insert__mono, fact_ord__eq__less__trans, fact_sup_Ocommute, fact_Un__Diff__cancel2, fact_Range__Un__eq, fact_wf__mlex, arity_fun__Groups_Ominus, fact_Nitpick_Owf__wfrec_H__def, fact_le__trans, fact_less__supI2, fact_equalityCE, fact_r__into__rtrancl, fact_psubset__insert__iff, fact_rtrancl__into__trancl1, fact_Un__iff, fact_Not__Domain__rtrancl, fact_rtrancl__idemp, fact_Diff__insert2, fact_subsetD, fact_Un__Diff, fact_psubset__eq, fact_trancl__rtrancl__trancl, fact_sup1E, fact_inf__sup__aci_I7_J, fact_inf__sup__ord_I3_J, fact_wf__same__fst, fact_diff__eq__diff__less__eq, fact_Id__on__empty, fact_Range__insert, fact_in__inv__image, fact_the__elem__eq, fact_insertE, fact_empty__Diff, fact_trans__less__than, fact_acyclicI, fact_acc__downward, fact_Range__empty__iff, fact_inf__sup__aci_I6_J, fact_eq__imp__le, fact_diff__eq__diff__eq, fact_order__le__imp__less__or__eq, fact_antisym__Id__on, fact_order__antisym__conv, fact_le__antisym, fact_Un__commute, fact_eqelem__imp__iff, arity_HOL__Obool__Orderings_Obot, fact_Image__iff, fact_diff__le__mono, fact_sup__bot__right, fact_pair__lessI1, fact_psubset__subset__trans, fact_Un__least, fact_curryI, fact_well__order__on__def, fact_wf__acyclic, fact_subset__trans, fact_sup__absorb2, fact_min__weak__def, fact_Diff__iff, fact_order__less__imp__le, fact_order__less__trans, fact_pair__leqI2, fact_trans__trancl, fact_Un__upper2, fact_antisym__Id, fact_order__less__irrefl, fact_Partial__order__eq__Image1__Image1__iff, fact_Un__left__absorb, fact_order__le__less, fact_sup_Oleft__idem, fact_le__diff__iff, fact_wf__asym, fact_not__psubset__empty, fact_mem__def, fact_refl__on__Id__on, fact_wfrec__rel_Osimps, fact_nat__less__le, fact_wf__pair__less, fact_trancl__mono, fact_ord__eq__le__trans, fact_fold__graph_H_Oequations_I2_J, fact_Field__insert, fact_less__diff__iff, fact_sup_Oleft__commute, fact_linorder__cases, fact_xt1_I11_J, fact_UnE, fact_equiv__class__self, fact_wf__less__than, fact_Image__mono, fact_wfP__subset, fact_rtrancl__trancl__trancl, fact_not__acc__down, fact_doubleton__eq__iff, fact_order__trans, fact_trancl__reflcl, fact_Field__def, fact_Image__Id, fact_trancl__domain, fact_Un__assoc, fact_single__valuedD, fact_well__order__on__empty, fact_insert__Diff1, fact_xt1_I7_J
% 141.70/33.17  % No. of inferences in proof: 1157
% 141.70/33.17  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 32527 ms resp. 31415 ms w/o parsing
% 136.72/33.87  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 136.72/33.89  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------