TSTP Solution File: ITP153^2 by Leo-III---1.7.12

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : ITP153^2 : TPTP v8.2.0. Released v7.5.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n008.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Mon May 20 22:23:26 EDT 2024

% Result   : Theorem 175.48s 152.87s
% Output   : Refutation 176.05s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :  294
% Syntax   : Number of formulae    :  550 ( 173 unt;  40 typ;   0 def)
%            Number of atoms       : 2133 ( 396 equ;   4 cnn)
%            Maximal formula atoms :   16 (   4 avg)
%            Number of connectives : 8813 ( 178   ~;  30   |; 147   &;7886   @)
%                                         (   0 <=>; 572  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   28 (   9 avg)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :  164 ( 164   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   44 (  41 usr;   8 con; 0-5 aty)
%            Number of variables   : 1943 (  60   ^1816   !;  22   ?;1943   :)
%                                         (  45  !>;   0  ?*;   0  @-;   0  @+)

% Comments : 
%------------------------------------------------------------------------------
thf(a_type,type,
    a: $tType ).

thf(bot_type,type,
    bot: 
      !>[TA: $tType] : $o ).

thf(ord_type,type,
    ord: 
      !>[TA: $tType] : $o ).

thf(linorder_type,type,
    linorder: 
      !>[TA: $tType] : $o ).

thf(preorder_type,type,
    preorder: 
      !>[TA: $tType] : $o ).

thf(order_bot_type,type,
    order_bot: 
      !>[TA: $tType] : $o ).

thf(bNF_Ca1785829860lChain_type,type,
    bNF_Ca1785829860lChain: 
      !>[TA: $tType,TB: $tType] : ( ( set @ ( product_prod @ TB @ TB ) ) > ( TB > TA ) > $o ) ).

thf(uminus_uminus_type,type,
    uminus_uminus: 
      !>[TA: $tType] : ( TA > TA ) ).

thf(order_1409979114der_on_type,type,
    order_1409979114der_on: 
      !>[TA: $tType] : ( ( set @ TA ) > ( set @ ( product_prod @ TA @ TA ) ) > $o ) ).

thf(order_preorder_on_type,type,
    order_preorder_on: 
      !>[TA: $tType] : ( ( set @ TA ) > ( set @ ( product_prod @ TA @ TA ) ) > $o ) ).

thf(order_under_type,type,
    order_under: 
      !>[TA: $tType] : ( ( set @ ( product_prod @ TA @ TA ) ) > TA > ( set @ TA ) ) ).

thf(bot_bot_type,type,
    bot_bot: 
      !>[TA: $tType] : TA ).

thf(ord_less_eq_type,type,
    ord_less_eq: 
      !>[TA: $tType] : ( TA > TA > $o ) ).

thf(prefer951318096ood_as_type,type,
    prefer951318096ood_as: 
      !>[TA: $tType] : ( TA > ( set @ TA ) > ( set @ ( product_prod @ TA @ TA ) ) > ( set @ TA ) ) ).

thf(prefer310429814s_good_type,type,
    prefer310429814s_good: 
      !>[TA: $tType] : ( TA > ( set @ TA ) > ( set @ ( product_prod @ TA @ TA ) ) > ( set @ TA ) ) ).

thf(prefer1532642881r_than_type,type,
    prefer1532642881r_than: 
      !>[TA: $tType] : ( TA > ( set @ TA ) > ( set @ ( product_prod @ TA @ TA ) ) > ( set @ TA ) ) ).

thf(prefer199794634erence_type,type,
    prefer199794634erence: 
      !>[TA: $tType] : ( ( set @ TA ) > ( set @ ( product_prod @ TA @ TA ) ) > $o ) ).

thf(prefer1997167224erence_type,type,
    prefer1997167224erence: 
      !>[TA: $tType] : ( ( set @ TA ) > ( set @ ( product_prod @ TA @ TA ) ) > $o ) ).

thf(prefer1801827867axioms_type,type,
    prefer1801827867axioms: 
      !>[TA: $tType] : ( ( set @ TA ) > ( set @ ( product_prod @ TA @ TA ) ) > $o ) ).

thf(product_Pair_type,type,
    product_Pair: 
      !>[TA: $tType,TB: $tType] : ( TB > TA > ( product_prod @ TB @ TA ) ) ).

thf(produc2004651681e_prod_type,type,
    produc2004651681e_prod: 
      !>[TA: $tType,TB: $tType,TC: $tType] : ( ( TC > TB > TA ) > ( product_prod @ TC @ TB ) > TA ) ).

thf(product_rec_prod_type,type,
    product_rec_prod: 
      !>[TA: $tType,TB: $tType,TC: $tType] : ( ( TC > TB > TA ) > ( product_prod @ TC @ TB ) > TA ) ).

thf(product_fst_type,type,
    product_fst: 
      !>[TA: $tType,TB: $tType] : ( ( product_prod @ TB @ TA ) > TB ) ).

thf(product_snd_type,type,
    product_snd: 
      !>[TA: $tType,TB: $tType] : ( ( product_prod @ TB @ TA ) > TA ) ).

thf(domain_type,type,
    domain: 
      !>[TA: $tType,TB: $tType] : ( ( set @ ( product_prod @ TB @ TA ) ) > ( set @ TB ) ) ).

thf(id_on_type,type,
    id_on: 
      !>[TA: $tType] : ( ( set @ TA ) > ( set @ ( product_prod @ TA @ TA ) ) ) ).

thf(range_type,type,
    range: 
      !>[TA: $tType,TB: $tType] : ( ( set @ ( product_prod @ TB @ TA ) ) > ( set @ TA ) ) ).

thf(refl_on_type,type,
    refl_on: 
      !>[TA: $tType] : ( ( set @ TA ) > ( set @ ( product_prod @ TA @ TA ) ) > $o ) ).

thf(total_on_type,type,
    total_on: 
      !>[TA: $tType] : ( ( set @ TA ) > ( set @ ( product_prod @ TA @ TA ) ) > $o ) ).

thf(trans_type,type,
    trans: 
      !>[TA: $tType] : ( ( set @ ( product_prod @ TA @ TA ) ) > $o ) ).

thf(collect_type,type,
    collect: 
      !>[TA: $tType] : ( ( TA > $o ) > ( set @ TA ) ) ).

thf(insert_type,type,
    insert: 
      !>[TA: $tType] : ( TA > ( set @ TA ) > ( set @ TA ) ) ).

thf(is_empty_type,type,
    is_empty: 
      !>[TA: $tType] : ( ( set @ TA ) > $o ) ).

thf(is_singleton_type,type,
    is_singleton: 
      !>[TA: $tType] : ( ( set @ TA ) > $o ) ).

thf(the_elem_type,type,
    the_elem: 
      !>[TA: $tType] : ( ( set @ TA ) > TA ) ).

thf(member_type,type,
    member: 
      !>[TA: $tType] : ( TA > ( set @ TA ) > $o ) ).

thf(carrier_type,type,
    carrier: set @ a ).

thf(relation_type,type,
    relation: set @ ( product_prod @ a @ a ) ).

thf(y_type,type,
    y: a ).

thf(z_type,type,
    z: a ).

thf(226,axiom,
    ! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
      ( ~ ( member @ TA @ A @ B )
     => ( ~ ( member @ TA @ A @ C )
       => ( ( ( insert @ TA @ A @ B )
            = ( insert @ TA @ A @ C ) )
          = ( B = C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195_insert__ident) ).

thf(892,plain,
    ! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
      ( ~ ( member @ TA @ A @ B )
     => ( ~ ( member @ TA @ A @ C )
       => ( ( ( insert @ TA @ A @ B )
            = ( insert @ TA @ A @ C ) )
          = ( B = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(136,axiom,
    ! [TA: $tType] : ( trans @ TA @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_154_trans__empty) ).

thf(629,plain,
    ! [TA: $tType] : ( trans @ TA @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(181,axiom,
    ! [TA: $tType,A: product_prod @ TA @ TA,B: set @ TA] :
      ( ( member @ ( product_prod @ TA @ TA ) @ A @ ( id_on @ TA @ B ) )
     => ~ ! [C: TA] :
            ( ( member @ TA @ C @ B )
           => ( A
             != ( product_Pair @ TA @ TA @ C @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_Id__onE) ).

thf(760,plain,
    ! [TA: $tType,A: product_prod @ TA @ TA,B: set @ TA] :
      ( ( member @ ( product_prod @ TA @ TA ) @ A @ ( id_on @ TA @ B ) )
     => ~ ! [C: TA] :
            ( ( member @ TA @ C @ B )
           => ( A
             != ( product_Pair @ TA @ TA @ C @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(77,axiom,
    ! [TA: $tType] :
      ( ( id_on @ TA @ ( bot_bot @ ( set @ TA ) ) )
      = ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_159_Id__on__empty) ).

thf(479,plain,
    ! [TA: $tType] :
      ( ( id_on @ TA @ ( bot_bot @ ( set @ TA ) ) )
      = ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(26,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( preorder @ TA )
     => ( preorder @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Opreorder) ).

thf(333,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( preorder @ TA )
     => ( preorder @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(174,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ( ( bNF_Ca1785829860lChain @ TA @ TB )
        = ( ^ [A: set @ ( product_prod @ TA @ TA ),B: TA > TB] :
            ! [C: TA,D: TA] :
              ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
             => ( ord_less_eq @ TB @ ( B @ C ) @ ( B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_158_relChain__def) ).

thf(741,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ( ( bNF_Ca1785829860lChain @ TA @ TB )
        = ( ^ [A: set @ ( product_prod @ TA @ TA ),B: TA > TB] :
            ! [C: TA,D: TA] :
              ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
             => ( ord_less_eq @ TB @ ( B @ C ) @ ( B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(7,axiom,
    ! [TA: $tType,TB: $tType,A: product_prod @ TA @ TB] :
      ( A
      = ( product_Pair @ TA @ TB @ ( product_fst @ TA @ TB @ A ) @ ( product_snd @ TA @ TB @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_surjective__pairing) ).

thf(269,plain,
    ! [TA: $tType,TB: $tType,A: product_prod @ TA @ TB] :
      ( A
      = ( product_Pair @ TA @ TB @ ( product_fst @ TA @ TB @ A ) @ ( product_snd @ TA @ TB @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(122,axiom,
    ! [TA: $tType,A: TA] : ( order_1409979114der_on @ TA @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) @ ( insert @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ A ) @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_linear__order__on__singleton) ).

thf(597,plain,
    ! [TA: $tType,A: TA] : ( order_1409979114der_on @ TA @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) @ ( insert @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ A ) @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(63,axiom,
    ! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ A @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_subset__refl) ).

thf(440,plain,
    ! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(143,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer199794634erence @ TA @ A @ B )
     => ( trans @ TA @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_155_preference_Otransitivity) ).

thf(655,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer199794634erence @ TA @ A @ B )
     => ( trans @ TA @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(164,axiom,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( member @ TA @ A @ ( uminus_uminus @ ( set @ TA ) @ B ) )
      = ( ~ ( member @ TA @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_249_Compl__iff) ).

thf(715,plain,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( member @ TA @ A @ ( uminus_uminus @ ( set @ TA ) @ B ) )
      = ( ~ ( member @ TA @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(235,axiom,
    ! [TA: $tType] :
      ( ( total_on @ TA )
      = ( ^ [A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
          ! [C: TA] :
            ( ( member @ TA @ C @ A )
           => ! [D: TA] :
                ( ( member @ TA @ D @ A )
               => ( ( C != D )
                 => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
                    | ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_total__on__def) ).

thf(913,plain,
    ! [TA: $tType] :
      ( ( total_on @ TA )
      = ( ^ [A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
          ! [C: TA] :
            ( ( member @ TA @ C @ A )
           => ! [D: TA] :
                ( ( member @ TA @ D @ A )
               => ( ( C != D )
                 => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
                    | ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(98,axiom,
    ! [TA: $tType,A: TA > $o] :
      ( ( ( collect @ TA @ A )
        = ( bot_bot @ ( set @ TA ) ) )
      = ( A
        = ( bot_bot @ ( TA > $o ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_Collect__empty__eq__bot) ).

thf(538,plain,
    ! [TA: $tType,A: TA > $o] :
      ( ( ( collect @ TA @ A )
        = ( bot_bot @ ( set @ TA ) ) )
      = ( A
        = ( bot_bot @ ( TA > $o ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(228,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( refl_on @ TA @ A @ B )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
       => ( member @ TA @ D @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_refl__onD2) ).

thf(898,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( refl_on @ TA @ A @ B )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
       => ( member @ TA @ D @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(53,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: TC > TA > TB,B: TC,C: TA] :
      ( ( produc2004651681e_prod @ TC @ TA @ TB @ A @ ( product_Pair @ TC @ TA @ B @ C ) )
      = ( A @ B @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_149_internal__case__prod__conv) ).

thf(417,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: TC > TA > TB,B: TC,C: TA] :
      ( ( produc2004651681e_prod @ TC @ TA @ TB @ A @ ( product_Pair @ TC @ TA @ B @ C ) )
      = ( A @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(205,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA] :
      ( ( refl_on @ TA @ A @ B )
     => ( ( member @ TA @ C @ A )
       => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ C ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_refl__onD) ).

thf(829,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA] :
      ( ( refl_on @ TA @ A @ B )
     => ( ( member @ TA @ C @ A )
       => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(49,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) )
      = ( A
        = ( bot_bot @ ( set @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_subset__empty) ).

thf(404,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) )
      = ( A
        = ( bot_bot @ ( set @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(168,axiom,
    ! [TA: $tType,A: TA] :
      ~ ( member @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_emptyE) ).

thf(724,plain,
    ! [TA: $tType,A: TA] :
      ~ ( member @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(166,axiom,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( member @ TA @ A @ B )
     => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ A ) @ ( id_on @ TA @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_160_Id__onI) ).

thf(719,plain,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( member @ TA @ A @ B )
     => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ A ) @ ( id_on @ TA @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(87,axiom,
    ! [TA: $tType] :
      ( ( bot_bot @ ( set @ TA ) )
      = ( collect @ TA @ ( bot_bot @ ( TA > $o ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_bot__set__def) ).

thf(504,plain,
    ! [TA: $tType] :
      ( ( bot_bot @ ( set @ TA ) )
      = ( collect @ TA @ ( bot_bot @ ( TA > $o ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(177,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( collect @ TA
        @ ^ [B: TA] : ( member @ TA @ B @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_Collect__mem__eq) ).

thf(751,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( collect @ TA
        @ ^ [B: TA] : ( member @ TA @ B @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(105,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB,C: TA] :
          ( ( ord_less_eq @ ( TA > TB ) @ A @ B )
         => ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_le__funD) ).

thf(559,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB,C: TA] :
          ( ( ord_less_eq @ ( TA > TB ) @ A @ B )
         => ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(55,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ TC ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ TC )] :
      ( ! [C: TA,D: TB,E: TC] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ TC ) @ C @ ( product_Pair @ TB @ TC @ D @ E ) ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_prod__induct3) ).

thf(421,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ TC ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ TC )] :
      ( ! [C: TA,D: TB,E: TC] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ TC ) @ C @ ( product_Pair @ TB @ TC @ D @ E ) ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(126,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( range @ TB @ TA @ ( bot_bot @ ( set @ ( product_prod @ TB @ TA ) ) ) )
      = ( bot_bot @ ( set @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_Range__empty) ).

thf(606,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( range @ TB @ TA @ ( bot_bot @ ( set @ ( product_prod @ TB @ TA ) ) ) )
      = ( bot_bot @ ( set @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(184,axiom,
    ! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: TA,C: TA] :
      ( ( trans @ TA @ A )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ C ) @ A )
       => ( ord_less_eq @ ( set @ TA ) @ ( order_under @ TA @ A @ B ) @ ( order_under @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_166_under__incr) ).

thf(770,plain,
    ! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: TA,C: TA] :
      ( ( trans @ TA @ A )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ C ) @ A )
       => ( ord_less_eq @ ( set @ TA ) @ ( order_under @ TA @ A @ B ) @ ( order_under @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(14,axiom,
    ! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA,B: TB,C: TA] :
      ( ( A
        = ( product_Pair @ TB @ TA @ B @ C ) )
     => ( ( product_snd @ TB @ TA @ A )
        = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_225_sndI) ).

thf(289,plain,
    ! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA,B: TB,C: TA] :
      ( ( A
        = ( product_Pair @ TB @ TA @ B @ C ) )
     => ( ( product_snd @ TB @ TA @ A )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(67,axiom,
    ! [TA: $tType,A: TA > $o] :
      ( ( ( bot_bot @ ( set @ TA ) )
        = ( collect @ TA @ A ) )
      = ( ! [B: TA] :
            ~ ( A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_empty__Collect__eq) ).

thf(448,plain,
    ! [TA: $tType,A: TA > $o] :
      ( ( ( bot_bot @ ( set @ TA ) )
        = ( collect @ TA @ A ) )
      = ( ! [B: TA] :
            ~ ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(202,axiom,
    ! [TA: $tType,A: TA,B: set @ TA,C: TA,D: set @ TA] :
      ( ~ ( member @ TA @ A @ B )
     => ( ~ ( member @ TA @ C @ D )
       => ( ( ( insert @ TA @ A @ B )
            = ( insert @ TA @ C @ D ) )
          = ( ( ( A = C )
             => ( B = D ) )
            & ( ( A != C )
             => ? [E: set @ TA] :
                  ( ( B
                    = ( insert @ TA @ C @ E ) )
                  & ~ ( member @ TA @ C @ E )
                  & ( D
                    = ( insert @ TA @ A @ E ) )
                  & ~ ( member @ TA @ A @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_insert__eq__iff) ).

thf(817,plain,
    ! [TA: $tType,A: TA,B: set @ TA,C: TA,D: set @ TA] :
      ( ~ ( member @ TA @ A @ B )
     => ( ~ ( member @ TA @ C @ D )
       => ( ( ( insert @ TA @ A @ B )
            = ( insert @ TA @ C @ D ) )
          = ( ( ( A = C )
             => ( B = D ) )
            & ( ( A != C )
             => ? [E: set @ TA] :
                  ( ( B
                    = ( insert @ TA @ C @ E ) )
                  & ~ ( member @ TA @ C @ E )
                  & ( D
                    = ( insert @ TA @ A @ E ) )
                  & ~ ( member @ TA @ A @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(230,axiom,
    prefer1997167224erence @ a @ carrier @ relation,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1_rational__preference__axioms) ).

thf(902,plain,
    prefer1997167224erence @ a @ carrier @ relation,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(179,axiom,
    ! [TA: $tType,A: set @ TA,B: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ ( uminus_uminus @ ( set @ TA ) @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) )
      = ( ~ ( member @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_subset__Compl__singleton) ).

thf(756,plain,
    ! [TA: $tType,A: set @ TA,B: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ ( uminus_uminus @ ( set @ TA ) @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) )
      = ( ~ ( member @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(133,axiom,
    linorder @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Olinorder) ).

thf(623,plain,
    linorder @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(79,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) )] :
      ( ! [C: TA,D: TB,E: TC,F: TD] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ TD ) @ D @ ( product_Pair @ TC @ TD @ E @ F ) ) ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_prod__induct4) ).

thf(485,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) )] :
      ( ! [C: TA,D: TB,E: TC,F: TD] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ TD ) @ D @ ( product_Pair @ TC @ TD @ E @ F ) ) ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(254,axiom,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( member @ TA @ A @ B )
     => ? [C: set @ TA] :
          ( ( B
            = ( insert @ TA @ A @ C ) )
          & ~ ( member @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_mk__disjoint__insert) ).

thf(966,plain,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( member @ TA @ A @ B )
     => ? [C: set @ TA] :
          ( ( B
            = ( insert @ TA @ A @ C ) )
          & ~ ( member @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(232,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( member @ TA @ C @ A )
       => ( ( member @ TA @ D @ A )
         => ( ( ord_less_eq @ ( set @ TA ) @ ( prefer1532642881r_than @ TA @ D @ A @ B ) @ ( prefer1532642881r_than @ TA @ C @ A @ B ) )
           => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_rational__preference_Ono__better__thansubset__rel) ).

thf(905,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( member @ TA @ C @ A )
       => ( ( member @ TA @ D @ A )
         => ( ( ord_less_eq @ ( set @ TA ) @ ( prefer1532642881r_than @ TA @ D @ A @ B ) @ ( prefer1532642881r_than @ TA @ C @ A @ B ) )
           => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(32,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
          | ( ord_less_eq @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_linear) ).

thf(348,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
          | ( ord_less_eq @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(112,axiom,
    ! [TA: $tType] :
      ( ( (=) @ ( set @ TA ) )
      = ( ^ [A: set @ TA,B: set @ TA] :
            ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
            & ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_set__eq__subset) ).

thf(574,plain,
    ! [TA: $tType] :
      ( ( (=) @ ( set @ TA ) )
      = ( ^ [A: set @ TA,B: set @ TA] :
            ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
            & ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(46,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
       => ( ord_less_eq @ ( set @ TA ) @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_subset__trans) ).

thf(397,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
       => ( ord_less_eq @ ( set @ TA ) @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(138,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( total_on @ TA @ A @ B )
     => ( prefer1801827867axioms @ TA @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_rational__preference__axioms_Ointro) ).

thf(634,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( total_on @ TA @ A @ B )
     => ( prefer1801827867axioms @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(222,axiom,
    ! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
      ( ( member @ TA @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ C @ B )
       => ( ord_less_eq @ ( set @ TA ) @ ( insert @ TA @ A @ C ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_insert__subsetI) ).

thf(877,plain,
    ! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
      ( ( member @ TA @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ C @ B )
       => ( ord_less_eq @ ( set @ TA ) @ ( insert @ TA @ A @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(15,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( bot @ TB )
     => ( ( bot_bot @ ( TA > TB ) )
        = ( ^ [A: TA] : ( bot_bot @ TB ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_bot__fun__def) ).

thf(293,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( bot @ TB )
     => ( ( bot_bot @ ( TA > TB ) )
        = ( ^ [A: TA] : ( bot_bot @ TB ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(186,axiom,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA,D: set @ ( product_prod @ TA @ TA )] :
      ( ( member @ TA @ A @ ( prefer310429814s_good @ TA @ B @ C @ D ) )
     => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ D ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_at__lst__asgd__ge) ).

thf(774,plain,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA,D: set @ ( product_prod @ TA @ TA )] :
      ( ( member @ TA @ A @ ( prefer310429814s_good @ TA @ B @ C @ D ) )
     => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(51,axiom,
    ! [TA: $tType] :
      ( ( prefer1997167224erence @ TA )
      = ( ^ [A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
            ( ( prefer199794634erence @ TA @ A @ B )
            & ( prefer1801827867axioms @ TA @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_rational__preference__def) ).

thf(411,plain,
    ! [TA: $tType] :
      ( ( prefer1997167224erence @ TA )
      = ( ^ [A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
            ( ( prefer199794634erence @ TA @ A @ B )
            & ( prefer1801827867axioms @ TA @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(88,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( domain @ TA @ TA @ ( id_on @ TA @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_253_Domain__Id__on) ).

thf(507,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( domain @ TA @ TA @ ( id_on @ TA @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(94,axiom,
    ! [TA: $tType] : ( ord @ ( set @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oord_4) ).

thf(527,plain,
    ! [TA: $tType] : ( ord @ ( set @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(6,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( A = B )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_eq__refl) ).

thf(265,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( A = B )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(212,axiom,
    ! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: set @ TA] :
      ( ! [C: TA,D: TA] :
          ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
         => ( member @ TA @ C @ B ) )
     => ( ! [C: TA,D: TA] :
            ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
           => ( member @ TA @ D @ B ) )
       => ( ( order_preorder_on @ TA @ B @ A )
         => ( prefer199794634erence @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_preference_Ointro) ).

thf(849,plain,
    ! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: set @ TA] :
      ( ! [C: TA,D: TA] :
          ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
         => ( member @ TA @ C @ B ) )
     => ( ! [C: TA,D: TA] :
            ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
           => ( member @ TA @ D @ B ) )
       => ( ( order_preorder_on @ TA @ B @ A )
         => ( prefer199794634erence @ TA @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(40,axiom,
    ! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
      ( ( ( product_Pair @ TB @ TA @ A @ B )
        = ( product_Pair @ TB @ TA @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_prod_Oinject) ).

thf(380,plain,
    ! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
      ( ( ( product_Pair @ TB @ TA @ A @ B )
        = ( product_Pair @ TB @ TA @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(95,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( is_singleton @ TA @ A )
     => ~ ! [B: TA] :
            ( A
           != ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_222_is__singletonE) ).

thf(529,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( is_singleton @ TA @ A )
     => ~ ! [B: TA] :
            ( A
           != ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(145,axiom,
    ! [TA: $tType,A: TA > $o,B: TA > $o] :
      ( ! [C: TA] :
          ( ( A @ C )
          = ( B @ C ) )
     => ( ( collect @ TA @ A )
        = ( collect @ TA @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_Collect__cong) ).

thf(661,plain,
    ! [TA: $tType,A: TA > $o,B: TA > $o] :
      ( ! [C: TA] :
          ( ( A @ C )
          = ( B @ C ) )
     => ( ( collect @ TA @ A )
        = ( collect @ TA @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(101,axiom,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) )
        = ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_singleton__inject) ).

thf(546,plain,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) )
        = ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(239,axiom,
    total_on @ a @ carrier @ relation,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_total) ).

thf(925,plain,
    total_on @ a @ carrier @ relation,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(200,axiom,
    ! [TA: $tType,A: TA] :
      ~ ( member @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_empty__iff) ).

thf(810,plain,
    ! [TA: $tType,A: TA] :
      ~ ( member @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(234,axiom,
    ! [TA: $tType,TB: $tType,A: TA,B: set @ ( product_prod @ TB @ TA ),C: TA > $o] :
      ( ( member @ TA @ A @ ( range @ TB @ TA @ B ) )
     => ( ! [D: TB,E: TA] :
            ( ( member @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ D @ E ) @ B )
           => ( C @ E ) )
       => ( C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_Range_Oinducts) ).

thf(910,plain,
    ! [TA: $tType,TB: $tType,A: TA,B: set @ ( product_prod @ TB @ TA ),C: TA > $o] :
      ( ( member @ TA @ A @ ( range @ TB @ TA @ B ) )
     => ( ! [D: TB,E: TA] :
            ( ( member @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ D @ E ) @ B )
           => ( C @ E ) )
       => ( C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(18,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( preorder @ TB )
        & ( preorder @ TA ) )
     => ( preorder @ ( product_prod @ TB @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Orderings_Opreorder_12) ).

thf(302,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( preorder @ TB )
        & ( preorder @ TA ) )
     => ( preorder @ ( product_prod @ TB @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(203,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( A
       != ( bot_bot @ ( set @ TA ) ) )
     => ( ! [B: TA,C: TA] :
            ( ( member @ TA @ B @ A )
           => ( ( member @ TA @ C @ A )
             => ( B = C ) ) )
       => ( is_singleton @ TA @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_221_is__singletonI_H) ).

thf(820,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( A
       != ( bot_bot @ ( set @ TA ) ) )
     => ( ! [B: TA,C: TA] :
            ( ( member @ TA @ B @ A )
           => ( ( member @ TA @ C @ A )
             => ( B = C ) ) )
       => ( is_singleton @ TA @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(106,axiom,
    ! [TA: $tType,A: TA] : ( is_singleton @ TA @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_is__singletonI) ).

thf(561,plain,
    ! [TA: $tType,A: TA] : ( is_singleton @ TA @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(58,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ ( prefer1532642881r_than @ TA @ C @ A @ B ) @ ( prefer1532642881r_than @ TA @ D @ A @ B ) )
        | ( ord_less_eq @ ( set @ TA ) @ ( prefer1532642881r_than @ TA @ D @ A @ B ) @ ( prefer1532642881r_than @ TA @ C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_rational__preference_Onbt__nest) ).

thf(429,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ ( prefer1532642881r_than @ TA @ C @ A @ B ) @ ( prefer1532642881r_than @ TA @ D @ A @ B ) )
        | ( ord_less_eq @ ( set @ TA ) @ ( prefer1532642881r_than @ TA @ D @ A @ B ) @ ( prefer1532642881r_than @ TA @ C @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(9,axiom,
    ! [TA: $tType] : ( order_bot @ ( set @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oorder__bot_1) ).

thf(275,plain,
    ! [TA: $tType] : ( order_bot @ ( set @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(170,axiom,
    ! [A: a,B: a,C: a] :
      ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
        & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) )
     => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ C ) @ relation )
          & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ B ) @ relation ) )
       => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ C ) @ relation )
          & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ A ) @ relation ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_strict__trans) ).

thf(729,plain,
    ! [A: a,B: a,C: a] :
      ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
        & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) )
     => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ C ) @ relation )
          & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ B ) @ relation ) )
       => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ C ) @ relation )
          & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ A ) @ relation ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(4,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( bot @ TB )
     => ( ( bot_bot @ ( TA > TB ) )
        = ( ^ [A: TA] : ( bot_bot @ TB ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_bot__apply) ).

thf(260,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( bot @ TB )
     => ( ( bot_bot @ ( TA > TB ) )
        = ( ^ [A: TA] : ( bot_bot @ TB ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(217,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA,E: TA] :
      ( ( prefer199794634erence @ TA @ A @ B )
     => ( ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
          & ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) )
       => ( ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ E ) @ B )
            & ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ D ) @ B ) )
         => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ E ) @ B )
            & ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_preference_Oindiff__trans) ).

thf(862,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA,E: TA] :
      ( ( prefer199794634erence @ TA @ A @ B )
     => ( ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
          & ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) )
       => ( ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ E ) @ B )
            & ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ D ) @ B ) )
         => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ E ) @ B )
            & ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(50,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( A = B )
     => ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_equalityD1) ).

thf(407,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( A = B )
     => ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(157,axiom,
    ! [A: a,B: a] :
      ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
     => ( member @ a @ A @ carrier ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_not__outside) ).

thf(693,plain,
    ! [A: a,B: a] :
      ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
     => ( member @ a @ A @ carrier ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(37,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
           => ~ ( ord_less_eq @ TA @ B @ C ) )
         => ( ( ( ord_less_eq @ TA @ B @ A )
             => ~ ( ord_less_eq @ TA @ A @ C ) )
           => ( ( ( ord_less_eq @ TA @ A @ C )
               => ~ ( ord_less_eq @ TA @ C @ B ) )
             => ( ( ( ord_less_eq @ TA @ C @ B )
                 => ~ ( ord_less_eq @ TA @ B @ A ) )
               => ( ( ( ord_less_eq @ TA @ B @ C )
                   => ~ ( ord_less_eq @ TA @ C @ A ) )
                 => ~ ( ( ord_less_eq @ TA @ C @ A )
                     => ~ ( ord_less_eq @ TA @ A @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_le__cases3) ).

thf(361,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
           => ~ ( ord_less_eq @ TA @ B @ C ) )
         => ( ( ( ord_less_eq @ TA @ B @ A )
             => ~ ( ord_less_eq @ TA @ A @ C ) )
           => ( ( ( ord_less_eq @ TA @ A @ C )
               => ~ ( ord_less_eq @ TA @ C @ B ) )
             => ( ( ( ord_less_eq @ TA @ C @ B )
                 => ~ ( ord_less_eq @ TA @ B @ A ) )
               => ( ( ( ord_less_eq @ TA @ B @ C )
                   => ~ ( ord_less_eq @ TA @ C @ A ) )
                 => ~ ( ( ord_less_eq @ TA @ C @ A )
                     => ~ ( ord_less_eq @ TA @ A @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(223,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ! [C: TA,D: TA] :
          ( ( member @ TA @ C @ A )
         => ( ( member @ TA @ D @ A )
           => ( ( C != D )
             => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
                | ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) ) ) ) )
     => ( total_on @ TA @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_total__onI) ).

thf(879,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ! [C: TA,D: TA] :
          ( ( member @ TA @ C @ A )
         => ( ( member @ TA @ D @ A )
           => ( ( C != D )
             => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
                | ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) ) ) ) )
     => ( total_on @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(61,axiom,
    ! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ ( product_prod @ TA @ TB )] :
      ( ( domain @ TA @ TB @ ( insert @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ C ) )
      = ( insert @ TA @ A @ ( domain @ TA @ TB @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_Domain__insert) ).

thf(435,plain,
    ! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ ( product_prod @ TA @ TB )] :
      ( ( domain @ TA @ TB @ ( insert @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ C ) )
      = ( insert @ TA @ A @ ( domain @ TA @ TB @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(225,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( refl_on @ TA @ A @ B )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
       => ( ( member @ TA @ C @ A )
          & ( member @ TA @ D @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_refl__on__domain) ).

thf(889,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( refl_on @ TA @ A @ B )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
       => ( ( member @ TA @ C @ A )
          & ( member @ TA @ D @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(71,axiom,
    ! [TA: $tType] :
      ( ( is_singleton @ TA )
      = ( ^ [A: set @ TA] :
          ? [B: TA] :
            ( A
            = ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_223_is__singleton__def) ).

thf(461,plain,
    ! [TA: $tType] :
      ( ( is_singleton @ TA )
      = ( ^ [A: set @ TA] :
          ? [B: TA] :
            ( A
            = ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(83,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( bot @ TA )
        & ( bot @ TB ) )
     => ( ( bot_bot @ ( product_prod @ TB @ TA ) )
        = ( product_Pair @ TB @ TA @ ( bot_bot @ TB ) @ ( bot_bot @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_bot__prod__def) ).

thf(493,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( bot @ TA )
        & ( bot @ TB ) )
     => ( ( bot_bot @ ( product_prod @ TB @ TA ) )
        = ( product_Pair @ TB @ TA @ ( bot_bot @ TB ) @ ( bot_bot @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(209,axiom,
    ! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
      ( ! [C: TA,D: TB] :
          ( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ A )
         => ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) )
     => ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_subrelI) ).

thf(842,plain,
    ! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
      ( ! [C: TA,D: TB] :
          ( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ A )
         => ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) )
     => ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(25,axiom,
    ! [TA: $tType,TB: $tType,A: TB > $o,B: TB,C: TA > $o,D: TA] :
      ( ( A @ B )
     => ( ( C @ D )
       => ( ( A @ ( product_fst @ TB @ TA @ ( product_Pair @ TB @ TA @ B @ D ) ) )
          & ( C @ ( product_snd @ TB @ TA @ ( product_Pair @ TB @ TA @ B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_216_conjI__realizer) ).

thf(324,plain,
    ! [TA: $tType,TB: $tType,A: TB > $o,B: TB,C: TA > $o,D: TA] :
      ( ( A @ B )
     => ( ( C @ D )
       => ( ( A @ ( product_fst @ TB @ TA @ ( product_Pair @ TB @ TA @ B @ D ) ) )
          & ( C @ ( product_snd @ TB @ TA @ ( product_Pair @ TB @ TA @ B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(236,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ! [D: TA] :
              ( ( member @ TA @ D @ A )
             => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
                | ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_rational__preference_Ocompl) ).

thf(916,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ! [D: TA] :
              ( ( member @ TA @ D @ A )
             => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
                | ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(73,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: product_prod @ TC @ ( product_prod @ TB @ TA )] :
      ~ ! [B: TC,C: TB,D: TA] :
          ( A
         != ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ B @ ( product_Pair @ TB @ TA @ C @ D ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_prod__cases3) ).

thf(467,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: product_prod @ TC @ ( product_prod @ TB @ TA )] :
      ~ ! [B: TC,C: TB,D: TA] :
          ( A
         != ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ B @ ( product_Pair @ TB @ TA @ C @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(211,axiom,
    ! [TA: $tType,A: TA] : ( member @ TA @ A @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_singletonI) ).

thf(847,plain,
    ! [TA: $tType,A: TA] : ( member @ TA @ A @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(108,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( prefer199794634erence @ TA @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_rational__preference_Oaxioms_I1_J) ).

thf(566,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( prefer199794634erence @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(116,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_subset__insertI2) ).

thf(583,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(92,axiom,
    ! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
      ( ( ( product_Pair @ TB @ TA @ A @ B )
        = ( product_Pair @ TB @ TA @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_old_Oprod_Oinject) ).

thf(522,plain,
    ! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
      ( ( ( product_Pair @ TB @ TA @ A @ B )
        = ( product_Pair @ TB @ TA @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(159,axiom,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( member @ TA @ A @ ( insert @ TA @ B @ C ) )
      = ( ( A = B )
        | ( member @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_insert__iff) ).

thf(697,plain,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( member @ TA @ A @ ( insert @ TA @ B @ C ) )
      = ( ( A = B )
        | ( member @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(221,axiom,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( member @ TA @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_singleton__iff) ).

thf(874,plain,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( member @ TA @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(130,axiom,
    ! [TA: $tType] :
      ( ( order_bot @ TA )
     => ! [A: TA] :
          ( ( ord_less_eq @ TA @ A @ ( bot_bot @ TA ) )
          = ( A
            = ( bot_bot @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_bot_Oextremum__unique) ).

thf(615,plain,
    ! [TA: $tType] :
      ( ( order_bot @ TA )
     => ! [A: TA] :
          ( ( ord_less_eq @ TA @ A @ ( bot_bot @ TA ) )
          = ( A
            = ( bot_bot @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(43,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ( ( ord_less_eq @ ( product_prod @ TA @ TB ) )
        = ( ^ [A: product_prod @ TA @ TB,B: product_prod @ TA @ TB] :
              ( ( ord_less_eq @ TA @ ( product_fst @ TA @ TB @ A ) @ ( product_fst @ TA @ TB @ B ) )
              & ( ord_less_eq @ TB @ ( product_snd @ TA @ TB @ A ) @ ( product_snd @ TA @ TB @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181_less__eq__prod__def) ).

thf(388,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ( ( ord_less_eq @ ( product_prod @ TA @ TB ) )
        = ( ^ [A: product_prod @ TA @ TB,B: product_prod @ TA @ TB] :
              ( ( ord_less_eq @ TA @ ( product_fst @ TA @ TB @ A ) @ ( product_fst @ TA @ TB @ B ) )
              & ( ord_less_eq @ TB @ ( product_snd @ TA @ TB @ A ) @ ( product_snd @ TA @ TB @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(35,axiom,
    ! [TA: $tType] : ( bot @ ( set @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Obot_5) ).

thf(356,plain,
    ! [TA: $tType] : ( bot @ ( set @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(85,axiom,
    ! [TA: $tType,A: set @ ( product_prod @ TA @ TA )] : ( total_on @ TA @ ( bot_bot @ ( set @ TA ) ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_total__on__empty) ).

thf(499,plain,
    ! [TA: $tType,A: set @ ( product_prod @ TA @ TA )] : ( total_on @ TA @ ( bot_bot @ ( set @ TA ) ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(189,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( member @ TA @ C @ A )
       => ( ( member @ TA @ D @ A )
         => ( ( C != D )
           => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
              | ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_rational__preference_OcompleteD) ).

thf(783,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( member @ TA @ C @ A )
       => ( ( member @ TA @ D @ A )
         => ( ( C != D )
           => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
              | ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(54,axiom,
    order_bot @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oorder__bot_6) ).

thf(420,plain,
    order_bot @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(1,conjecture,
    ( ( member @ a @ z @ carrier )
    & ( member @ a @ y @ carrier ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).

thf(2,negated_conjecture,
    ~ ( ( member @ a @ z @ carrier )
      & ( member @ a @ y @ carrier ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(256,plain,
    ~ ( ( member @ a @ z @ carrier )
      & ( member @ a @ y @ carrier ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(29,axiom,
    ! [TA: $tType,TB: $tType,A: product_prod @ TA @ TB,B: product_prod @ TA @ TB] :
      ( ( ( product_fst @ TA @ TB @ A )
        = ( product_fst @ TA @ TB @ B ) )
     => ( ( ( product_snd @ TA @ TB @ A )
          = ( product_snd @ TA @ TB @ B ) )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_prod__eqI) ).

thf(339,plain,
    ! [TA: $tType,TB: $tType,A: product_prod @ TA @ TB,B: product_prod @ TA @ TB] :
      ( ( ( product_fst @ TA @ TB @ A )
        = ( product_fst @ TA @ TB @ B ) )
     => ( ( ( product_snd @ TA @ TB @ A )
          = ( product_snd @ TA @ TB @ B ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(243,axiom,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( member @ TA @ A @ ( insert @ TA @ B @ C ) )
     => ( ( A != B )
       => ( member @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_insertE) ).

thf(933,plain,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( member @ TA @ A @ ( insert @ TA @ B @ C ) )
     => ( ( A != B )
       => ( member @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(148,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) )] :
      ( ! [C: TA,D: TB,E: TC,F: TD,G: TE] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ TE ) @ E @ ( product_Pair @ TD @ TE @ F @ G ) ) ) ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_prod__induct5) ).

thf(669,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) )] :
      ( ! [C: TA,D: TB,E: TC,F: TD,G: TE] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ TE ) @ E @ ( product_Pair @ TD @ TE @ F @ G ) ) ) ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(76,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) )] :
      ~ ! [B: TD,C: TC,D: TB,E: TA] :
          ( A
         != ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ B @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ C @ ( product_Pair @ TB @ TA @ D @ E ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_prod__cases4) ).

thf(476,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) )] :
      ~ ! [B: TD,C: TC,D: TB,E: TA] :
          ( A
         != ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ B @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ C @ ( product_Pair @ TB @ TA @ D @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(81,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer199794634erence @ TA @ A @ B )
     => ( ( prefer1801827867axioms @ TA @ A @ B )
       => ( prefer1997167224erence @ TA @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_rational__preference_Ointro) ).

thf(489,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer199794634erence @ TA @ A @ B )
     => ( ( prefer1801827867axioms @ TA @ A @ B )
       => ( prefer1997167224erence @ TA @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(119,axiom,
    ! [TA: $tType,A: set @ TA] : ( refl_on @ TA @ A @ ( id_on @ TA @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_165_refl__on__Id__on) ).

thf(590,plain,
    ! [TA: $tType,A: set @ TA] : ( refl_on @ TA @ A @ ( id_on @ TA @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(152,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TB )
        & ( ord @ TA ) )
     => ( ord @ ( product_prod @ TB @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Orderings_Oord_14) ).

thf(679,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TB )
        & ( ord @ TA ) )
     => ( ord @ ( product_prod @ TB @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(255,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ! [B: TA] :
          ~ ( member @ TA @ B @ A )
     => ( ord_less_eq @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_subset__emptyI) ).

thf(970,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ! [B: TA] :
          ~ ( member @ TA @ B @ A )
     => ( ord_less_eq @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(141,axiom,
    ord @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oord_9) ).

thf(647,plain,
    ord @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(173,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ! [B: TA] :
          ~ ( member @ TA @ B @ A )
     => ( A
        = ( bot_bot @ ( set @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_equals0I) ).

thf(738,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ! [B: TA] :
          ~ ( member @ TA @ B @ A )
     => ( A
        = ( bot_bot @ ( set @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(167,axiom,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( prefer951318096ood_as @ a @ B @ carrier @ relation ) )
      = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
        & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_as__good__as__sameIff) ).

thf(721,plain,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( prefer951318096ood_as @ a @ B @ carrier @ relation ) )
      = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
        & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(192,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
       => ( ord_less_eq @ ( set @ TA ) @ ( prefer1532642881r_than @ TA @ D @ A @ B ) @ ( prefer1532642881r_than @ TA @ C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_rational__preference_Ono__better__subset__pref) ).

thf(792,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
       => ( ord_less_eq @ ( set @ TA ) @ ( prefer1532642881r_than @ TA @ D @ A @ B ) @ ( prefer1532642881r_than @ TA @ C @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(198,axiom,
    ! [TA: $tType] :
      ( ( ord_less_eq @ ( set @ TA ) )
      = ( ^ [A: set @ TA,B: set @ TA] :
          ! [C: TA] :
            ( ( member @ TA @ C @ A )
           => ( member @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_subset__iff) ).

thf(805,plain,
    ! [TA: $tType] :
      ( ( ord_less_eq @ ( set @ TA ) )
      = ( ^ [A: set @ TA,B: set @ TA] :
          ! [C: TA] :
            ( ( member @ TA @ C @ A )
           => ( member @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(21,axiom,
    ! [TA: $tType,TB: $tType,A: TA,B: TB] :
      ( ( product_fst @ TA @ TB @ ( product_Pair @ TA @ TB @ A @ B ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_fst__conv) ).

thf(311,plain,
    ! [TA: $tType,TB: $tType,A: TA,B: TB] :
      ( ( product_fst @ TA @ TB @ ( product_Pair @ TA @ TB @ A @ B ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(99,axiom,
    ! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
      ( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_prod__cases) ).

thf(541,plain,
    ! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
      ( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(144,axiom,
    ! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: set @ ( product_prod @ TA @ TA )] :
      ( ( A
        = ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) )
     => ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_146_less__by__empty) ).

thf(657,plain,
    ! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: set @ ( product_prod @ TA @ TA )] :
      ( ( A
        = ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) )
     => ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(249,axiom,
    ! [TA: $tType,A: TA,B: set @ TA,C: TA] :
      ( ( member @ TA @ A @ B )
     => ( member @ TA @ A @ ( insert @ TA @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_insertI2) ).

thf(953,plain,
    ! [TA: $tType,A: TA,B: set @ TA,C: TA] :
      ( ( member @ TA @ A @ B )
     => ( member @ TA @ A @ ( insert @ TA @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(193,axiom,
    ! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: TA,C: TA,D: TA] :
      ( ( trans @ TA @ A )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ C ) @ A )
       => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
         => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_transD) ).

thf(794,plain,
    ! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: TA,C: TA,D: TA] :
      ( ( trans @ TA @ A )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ C ) @ A )
       => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
         => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(165,axiom,
    refl_on @ a @ carrier @ relation,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_reflexivity) ).

thf(718,plain,
    refl_on @ a @ carrier @ relation,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(240,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( member @ TA @ C @ B ) )
     => ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_85_subsetI) ).

thf(926,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( member @ TA @ C @ B ) )
     => ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(131,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( domain @ TA @ TB @ ( bot_bot @ ( set @ ( product_prod @ TA @ TB ) ) ) )
      = ( bot_bot @ ( set @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_254_Domain__empty) ).

thf(618,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( domain @ TA @ TB @ ( bot_bot @ ( set @ ( product_prod @ TA @ TB ) ) ) )
      = ( bot_bot @ ( set @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(214,axiom,
    ! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
      ( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
      = ( ? [C: TA] : ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_239_Range__iff) ).

thf(856,plain,
    ! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
      ( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
      = ( ? [C: TA] : ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(111,axiom,
    preorder @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Opreorder_7) ).

thf(573,plain,
    preorder @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(16,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( (=) @ ( product_prod @ TA @ TB ) )
      = ( ^ [A: product_prod @ TA @ TB,B: product_prod @ TA @ TB] :
            ( ( ( product_fst @ TA @ TB @ A )
              = ( product_fst @ TA @ TB @ B ) )
            & ( ( product_snd @ TA @ TB @ A )
              = ( product_snd @ TA @ TB @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_prod__eq__iff) ).

thf(296,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( (=) @ ( product_prod @ TA @ TB ) )
      = ( ^ [A: product_prod @ TA @ TB,B: product_prod @ TA @ TB] :
            ( ( ( product_fst @ TA @ TB @ A )
              = ( product_fst @ TA @ TB @ B ) )
            & ( ( product_snd @ TA @ TB @ A )
              = ( product_snd @ TA @ TB @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(65,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ ( uminus_uminus @ ( set @ TA ) @ A ) @ ( uminus_uminus @ ( set @ TA ) @ B ) )
      = ( ord_less_eq @ ( set @ TA ) @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_252_Compl__subset__Compl__iff) ).

thf(444,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ ( uminus_uminus @ ( set @ TA ) @ A ) @ ( uminus_uminus @ ( set @ TA ) @ B ) )
      = ( ord_less_eq @ ( set @ TA ) @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(124,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) )] :
      ~ ! [B: TF,C: TE,D: TD,E: TC,F: TB,G: TA] :
          ( A
         != ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ B @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ C @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ E @ ( product_Pair @ TB @ TA @ F @ G ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_prod__cases6) ).

thf(601,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) )] :
      ~ ! [B: TF,C: TE,D: TD,E: TC,F: TB,G: TA] :
          ( A
         != ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ B @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ C @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ E @ ( product_Pair @ TB @ TA @ F @ G ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(38,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( bot @ TA )
        & ( bot @ TB ) )
     => ( ( product_fst @ TA @ TB @ ( bot_bot @ ( product_prod @ TA @ TB ) ) )
        = ( bot_bot @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_fst__bot) ).

thf(375,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( bot @ TA )
        & ( bot @ TB ) )
     => ( ( product_fst @ TA @ TB @ ( bot_bot @ ( product_prod @ TA @ TB ) ) )
        = ( bot_bot @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(171,axiom,
    ! [TA: $tType] :
      ( ( prefer199794634erence @ TA )
      = ( ^ [A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
            ( ! [C: TA,D: TA] :
                ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
               => ( member @ TA @ C @ A ) )
            & ! [C: TA,D: TA] :
                ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
               => ( member @ TA @ D @ A ) )
            & ( order_preorder_on @ TA @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_preference__def) ).

thf(733,plain,
    ! [TA: $tType] :
      ( ( prefer199794634erence @ TA )
      = ( ^ [A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
            ( ! [C: TA,D: TA] :
                ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
               => ( member @ TA @ C @ A ) )
            & ! [C: TA,D: TA] :
                ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
               => ( member @ TA @ D @ A ) )
            & ( order_preorder_on @ TA @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(56,axiom,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( insert @ TA @ A @ B )
     != ( bot_bot @ ( set @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_insert__not__empty) ).

thf(423,plain,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( insert @ TA @ A @ B )
     != ( bot_bot @ ( set @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(48,axiom,
    ! [TA: $tType] :
      ( ( order_bot @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ ( bot_bot @ TA ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_bot_Oextremum) ).

thf(402,plain,
    ! [TA: $tType] :
      ( ( order_bot @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ ( bot_bot @ TA ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(185,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA,E: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( ( member @ TA @ C @ A )
          & ( member @ TA @ D @ A )
          & ( member @ TA @ E @ A ) )
       => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
         => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ E ) @ B )
           => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ E ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_rational__preference_Oweak__is__transitive) ).

thf(772,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA,E: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( ( member @ TA @ C @ A )
          & ( member @ TA @ D @ A )
          & ( member @ TA @ E @ A ) )
       => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
         => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ E ) @ B )
           => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ E ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(13,axiom,
    ! [TA: $tType,TB: $tType,A: product_prod @ TA @ TB] :
      ( A
      = ( product_Pair @ TA @ TB @ ( product_fst @ TA @ TB @ A ) @ ( product_snd @ TA @ TB @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_prod_Oexhaust__sel) ).

thf(286,plain,
    ! [TA: $tType,TB: $tType,A: product_prod @ TA @ TB] :
      ( A
      = ( product_Pair @ TA @ TB @ ( product_fst @ TA @ TB @ A ) @ ( product_snd @ TA @ TB @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(113,axiom,
    ! [TA: $tType] : ( order_1409979114der_on @ TA @ ( bot_bot @ ( set @ TA ) ) @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_lnear__order__on__empty) ).

thf(577,plain,
    ! [TA: $tType] : ( order_1409979114der_on @ TA @ ( bot_bot @ ( set @ TA ) ) @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(114,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: product_prod @ TA @ TB,B: product_prod @ TA @ TB] :
          ( ( ord_less_eq @ ( product_prod @ TA @ TB ) @ A @ B )
         => ( ord_less_eq @ TA @ ( product_fst @ TA @ TB @ A ) @ ( product_fst @ TA @ TB @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_214_fst__mono) ).

thf(579,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: product_prod @ TA @ TB,B: product_prod @ TA @ TB] :
          ( ( ord_less_eq @ ( product_prod @ TA @ TB ) @ A @ B )
         => ( ord_less_eq @ TA @ ( product_fst @ TA @ TB @ A ) @ ( product_fst @ TA @ TB @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(11,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( bot @ TA )
     => ( bot @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Obot) ).

thf(281,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( bot @ TA )
     => ( bot @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(44,axiom,
    ! [TA: $tType,A: set @ TA,B: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
      = ( ( A
          = ( bot_bot @ ( set @ TA ) ) )
        | ( A
          = ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_212_subset__singleton__iff) ).

thf(391,plain,
    ! [TA: $tType,A: set @ TA,B: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
      = ( ( A
          = ( bot_bot @ ( set @ TA ) ) )
        | ( A
          = ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(244,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( ( member @ TA @ C @ A )
          & ( member @ TA @ D @ A ) )
       => ( ( ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) )
          = ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
            & ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_rational__preference_Ostrict__not__refl__weak) ).

thf(936,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( ( member @ TA @ C @ A )
          & ( member @ TA @ D @ A ) )
       => ( ( ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) )
          = ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
            & ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(128,axiom,
    ! [TA: $tType,A: TA] : ( total_on @ TA @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) @ ( insert @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ A ) @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_total__on__singleton) ).

thf(611,plain,
    ! [TA: $tType,A: TA] : ( total_on @ TA @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) @ ( insert @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ A ) @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(27,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( order_bot @ TA )
     => ( order_bot @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oorder__bot) ).

thf(335,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( order_bot @ TA )
     => ( order_bot @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(161,axiom,
    ! [TA: $tType,A: set @ ( product_prod @ TA @ TA )] :
      ( ! [B: TA,C: TA,D: TA] :
          ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ C ) @ A )
         => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
           => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ D ) @ A ) ) )
     => ( trans @ TA @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_transI) ).

thf(704,plain,
    ! [TA: $tType,A: set @ ( product_prod @ TA @ TA )] :
      ( ! [B: TA,C: TA,D: TA] :
          ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ C ) @ A )
         => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
           => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ D ) @ A ) ) )
     => ( trans @ TA @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(247,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( A
         != ( bot_bot @ ( set @ TA ) ) )
       => ( ( member @ TA @ C @ A )
         => ( ( prefer1532642881r_than @ TA @ C @ A @ B )
           != ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_rational__preference_Ono__better__than__nonepty) ).

thf(947,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( A
         != ( bot_bot @ ( set @ TA ) ) )
       => ( ( member @ TA @ C @ A )
         => ( ( prefer1532642881r_than @ TA @ C @ A @ B )
           != ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(33,axiom,
    ! [TA: $tType,TB: $tType,A: TB,B: TA,C: TA] :
      ( ( ( product_snd @ TB @ TA @ ( product_Pair @ TB @ TA @ A @ B ) )
        = C )
     => ( B = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_snd__eqD) ).

thf(350,plain,
    ! [TA: $tType,TB: $tType,A: TB,B: TA,C: TA] :
      ( ( ( product_snd @ TB @ TA @ ( product_Pair @ TB @ TA @ A @ B ) )
        = C )
     => ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(229,axiom,
    ! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
      ( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
     => ~ ! [C: TA] :
            ~ ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_Range_Ocases) ).

thf(900,plain,
    ! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
      ( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
     => ~ ! [C: TA] :
            ~ ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(206,axiom,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( A = B )
     => ( ( member @ TA @ A @ C )
       => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( id_on @ TA @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_Id__on__eqI) ).

thf(831,plain,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( A = B )
     => ( ( member @ TA @ A @ C )
       => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( id_on @ TA @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(153,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( A = B )
     => ( ord_less_eq @ ( set @ TA ) @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_equalityD2) ).

thf(681,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( A = B )
     => ( ord_less_eq @ ( set @ TA ) @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(172,axiom,
    ! [A: a,B: a,C: a] :
      ( ( ( member @ a @ A @ carrier )
        & ( member @ a @ B @ carrier )
        & ( member @ a @ C @ carrier ) )
     => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
       => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ C ) @ relation )
         => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ C ) @ relation ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_weak__is__transitive) ).

thf(736,plain,
    ! [A: a,B: a,C: a] :
      ( ( ( member @ a @ A @ carrier )
        & ( member @ a @ B @ carrier )
        & ( member @ a @ C @ carrier ) )
     => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
       => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ C ) @ relation )
         => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ C ) @ relation ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(68,axiom,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( B = C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_ord__le__eq__trans) ).

thf(451,plain,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( B = C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(23,axiom,
    ! [TA: $tType,TB: $tType,A: TA > TB > $o,B: TA,C: TB,D: product_prod @ TA @ TB] :
      ( ( A @ B @ C )
     => ( ( D
          = ( product_Pair @ TA @ TB @ B @ C ) )
       => ( A @ ( product_fst @ TA @ TB @ D ) @ ( product_snd @ TA @ TB @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_215_BNF__Greatest__Fixpoint_Osubst__Pair) ).

thf(318,plain,
    ! [TA: $tType,TB: $tType,A: TA > TB > $o,B: TA,C: TB,D: product_prod @ TA @ TB] :
      ( ( A @ B @ C )
     => ( ( D
          = ( product_Pair @ TA @ TB @ B @ C ) )
       => ( A @ ( product_fst @ TA @ TB @ D ) @ ( product_snd @ TA @ TB @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(123,axiom,
    ! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_empty__subsetI) ).

thf(599,plain,
    ! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(182,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA,E: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
          & ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) )
       => ( ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ E ) @ B )
            & ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ D ) @ B ) )
         => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ E ) @ B )
            & ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_rational__preference_Ostrict__trans) ).

thf(764,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA,E: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
          & ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) )
       => ( ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ E ) @ B )
            & ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ D ) @ B ) )
         => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ E ) @ B )
            & ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(125,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ord_less_eq @ ( set @ TA ) @ ( insert @ TA @ C @ A ) @ ( insert @ TA @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_insert__mono) ).

thf(604,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ord_less_eq @ ( set @ TA ) @ ( insert @ TA @ C @ A ) @ ( insert @ TA @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(137,axiom,
    ! [TA: $tType] :
      ( ( prefer1801827867axioms @ TA )
      = ( total_on @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_rational__preference__axioms__def) ).

thf(631,plain,
    ! [TA: $tType] :
      ( ( prefer1801827867axioms @ TA )
      = ( total_on @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(97,axiom,
    ! [TA: $tType,A: TA,B: TA,C: TA,D: TA] :
      ( ( ( insert @ TA @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
        = ( insert @ TA @ C @ ( insert @ TA @ D @ ( bot_bot @ ( set @ TA ) ) ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_doubleton__eq__iff) ).

thf(535,plain,
    ! [TA: $tType,A: TA,B: TA,C: TA,D: TA] :
      ( ( ( insert @ TA @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
        = ( insert @ TA @ C @ ( insert @ TA @ D @ ( bot_bot @ ( set @ TA ) ) ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(127,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( prefer1801827867axioms @ TA @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_rational__preference_Oaxioms_I2_J) ).

thf(609,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( prefer1801827867axioms @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(132,axiom,
    ! [TA: $tType] : ( refl_on @ TA @ ( bot_bot @ ( set @ TA ) ) @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_refl__on__empty) ).

thf(621,plain,
    ! [TA: $tType] : ( refl_on @ TA @ ( bot_bot @ ( set @ TA ) ) @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(227,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( ? [B: TA] : ( member @ TA @ B @ A ) )
      = ( A
       != ( bot_bot @ ( set @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_ex__in__conv) ).

thf(895,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( ? [B: TA] : ( member @ TA @ B @ A ) )
      = ( A
       != ( bot_bot @ ( set @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(31,axiom,
    ! [TA: $tType,TB: $tType,A: product_prod @ TA @ TB,B: product_prod @ TA @ TB] :
      ( ( ( ( product_fst @ TA @ TB @ A )
          = ( product_fst @ TA @ TB @ B ) )
        & ( ( product_snd @ TA @ TB @ A )
          = ( product_snd @ TA @ TB @ B ) ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_prod_Oexpand) ).

thf(345,plain,
    ! [TA: $tType,TB: $tType,A: product_prod @ TA @ TB,B: product_prod @ TA @ TB] :
      ( ( ( ( product_fst @ TA @ TB @ A )
          = ( product_fst @ TA @ TB @ B ) )
        & ( ( product_snd @ TA @ TB @ A )
          = ( product_snd @ TA @ TB @ B ) ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(103,axiom,
    ! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
      ( ( ( product_Pair @ TB @ TA @ A @ B )
        = ( product_Pair @ TB @ TA @ C @ D ) )
     => ~ ( ( A = C )
         => ( B != D ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_Pair__inject) ).

thf(552,plain,
    ! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
      ( ( ( product_Pair @ TB @ TA @ A @ B )
        = ( product_Pair @ TB @ TA @ C @ D ) )
     => ~ ( ( A = C )
         => ( B != D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(195,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( refl_on @ TA @ A @ B )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
       => ( member @ TA @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_refl__onD1) ).

thf(799,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( refl_on @ TA @ A @ B )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
       => ( member @ TA @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(169,axiom,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( member @ TA @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_singletonD) ).

thf(726,plain,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( member @ TA @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(160,axiom,
    ! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ ( product_prod @ TA @ TB ),D: TB] :
      ( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ C )
     => ( ( D = B )
       => ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ D ) @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_ssubst__Pair__rhs) ).

thf(700,plain,
    ! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ ( product_prod @ TA @ TB ),D: TB] :
      ( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ C )
     => ( ( D = B )
       => ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ D ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(91,axiom,
    ! [TA: $tType] : ( order_preorder_on @ TA @ ( bot_bot @ ( set @ TA ) ) @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_preorder__on__empty) ).

thf(520,plain,
    ! [TA: $tType] : ( order_preorder_on @ TA @ ( bot_bot @ ( set @ TA ) ) @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(154,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: product_prod @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) )] :
      ~ ! [B: TG,C: TF,D: TE,E: TD,F: TC,G: TB,H: TA] :
          ( A
         != ( product_Pair @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) ) @ B @ ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ C @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ D @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ E @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ F @ ( product_Pair @ TB @ TA @ G @ H ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_prod__cases7) ).

thf(685,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: product_prod @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) )] :
      ~ ! [B: TG,C: TF,D: TE,E: TD,F: TC,G: TB,H: TA] :
          ( A
         != ( product_Pair @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) ) @ B @ ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ C @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ D @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ E @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ F @ ( product_Pair @ TB @ TA @ G @ H ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(47,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) )] :
      ~ ! [B: TE,C: TD,D: TC,E: TB,F: TA] :
          ( A
         != ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ B @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ C @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ D @ ( product_Pair @ TB @ TA @ E @ F ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_prod__cases5) ).

thf(399,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) )] :
      ~ ! [B: TE,C: TD,D: TC,E: TB,F: TA] :
          ( A
         != ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ B @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ C @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ D @ ( product_Pair @ TB @ TA @ E @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(251,axiom,
    ! [A: a,B: a] :
      ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
     => ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ B @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ A @ carrier @ relation ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_no__better__subset__pref) ).

thf(959,plain,
    ! [A: a,B: a] :
      ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
     => ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ B @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ A @ carrier @ relation ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(204,axiom,
    ! [A: a] :
      ( ( member @ a @ A @ carrier )
     => ! [B: a] :
          ( ( member @ a @ B @ carrier )
         => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
            | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_compl) ).

thf(827,plain,
    ! [A: a] :
      ( ( member @ a @ A @ carrier )
     => ! [B: a] :
          ( ( member @ a @ B @ carrier )
         => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
            | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(139,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( A = B )
     => ~ ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
         => ~ ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_105_equalityE) ).

thf(636,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( A = B )
     => ~ ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
         => ~ ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(219,axiom,
    ! [TA: $tType] :
      ( ( bot_bot @ ( TA > $o ) )
      = ( ^ [A: TA] : ( member @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_bot__empty__eq) ).

thf(868,plain,
    ! [TA: $tType] :
      ( ( bot_bot @ ( TA > $o ) )
      = ( ^ [A: TA] : ( member @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(188,axiom,
    ! [A: a,B: a] :
      ( ( ( member @ a @ A @ carrier )
        & ( member @ a @ B @ carrier ) )
     => ( ( ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) )
        = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
          & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_strict__not__refl__weak) ).

thf(780,plain,
    ! [A: a,B: a] :
      ( ( ( member @ a @ A @ carrier )
        & ( member @ a @ B @ carrier ) )
     => ( ( ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) )
        = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
          & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(69,axiom,
    ! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
    ? [B: TB,C: TA] :
      ( A
      = ( product_Pair @ TB @ TA @ B @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_surj__pair) ).

thf(455,plain,
    ! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
    ? [B: TB,C: TA] :
      ( A
      = ( product_Pair @ TB @ TA @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(107,axiom,
    ! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB )] :
      ( ( ( range @ TA @ TB @ A )
        = ( bot_bot @ ( set @ TB ) ) )
      = ( A
        = ( bot_bot @ ( set @ ( product_prod @ TA @ TB ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_Range__empty__iff) ).

thf(563,plain,
    ! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB )] :
      ( ( ( range @ TA @ TB @ A )
        = ( bot_bot @ ( set @ TB ) ) )
      = ( A
        = ( bot_bot @ ( set @ ( product_prod @ TA @ TB ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(17,axiom,
    ! [TA: $tType,TB: $tType,A: TB,B: product_prod @ TB @ TA] :
      ( ( A
        = ( product_fst @ TB @ TA @ B ) )
      = ( ? [C: TA] :
            ( B
            = ( product_Pair @ TB @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_229_eq__fst__iff) ).

thf(299,plain,
    ! [TA: $tType,TB: $tType,A: TB,B: product_prod @ TB @ TA] :
      ( ( A
        = ( product_fst @ TB @ TA @ B ) )
      = ( ? [C: TA] :
            ( B
            = ( product_Pair @ TB @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(62,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB] :
          ( ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) )
         => ( ord_less_eq @ ( TA > TB ) @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_le__funI) ).

thf(438,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB] :
          ( ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) )
         => ( ord_less_eq @ ( TA > TB ) @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(59,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TA )
     => ( ord @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oord) ).

thf(431,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TA )
     => ( ord @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(238,axiom,
    ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ z @ y ) @ relation )
    & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ y @ z ) @ relation ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0_assms) ).

thf(922,plain,
    ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ z @ y ) @ relation )
    & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ y @ z ) @ relation ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(45,axiom,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( insert @ TA @ A @ ( insert @ TA @ A @ B ) )
      = ( insert @ TA @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_insert__absorb2) ).

thf(394,plain,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( insert @ TA @ A @ ( insert @ TA @ A @ B ) )
      = ( insert @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(20,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA > TA > $o,B: TA,C: TA] :
          ( ! [D: TA,E: TA] :
              ( ( ord_less_eq @ TA @ D @ E )
             => ( A @ D @ E ) )
         => ( ! [D: TA,E: TA] :
                ( ( A @ E @ D )
               => ( A @ D @ E ) )
           => ( A @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_linorder__wlog) ).

thf(306,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA > TA > $o,B: TA,C: TA] :
          ( ! [D: TA,E: TA] :
              ( ( ord_less_eq @ TA @ D @ E )
             => ( A @ D @ E ) )
         => ( ! [D: TA,E: TA] :
                ( ( A @ E @ D )
               => ( A @ D @ E ) )
           => ( A @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(75,axiom,
    ! [TA: $tType,A: TA > $o,B: TA > $o] :
      ( ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) )
      = ( ! [C: TA] :
            ( ( A @ C )
           => ( B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_Collect__mono__iff) ).

thf(473,plain,
    ! [TA: $tType,A: TA > $o,B: TA > $o] :
      ( ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) )
      = ( ! [C: TA] :
            ( ( A @ C )
           => ( B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(57,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: TC > TA > TB,B: TC,C: TA] :
      ( ( product_rec_prod @ TC @ TA @ TB @ A @ ( product_Pair @ TC @ TA @ B @ C ) )
      = ( A @ B @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_old_Oprod_Orec) ).

thf(426,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: TC > TA > TB,B: TC,C: TA] :
      ( ( product_rec_prod @ TC @ TA @ TB @ A @ ( product_Pair @ TC @ TA @ B @ C ) )
      = ( A @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(246,axiom,
    ! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ ( insert @ TA @ A @ B ) @ C )
      = ( ( member @ TA @ A @ C )
        & ( ord_less_eq @ ( set @ TA ) @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_insert__subset) ).

thf(944,plain,
    ! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ ( insert @ TA @ A @ B ) @ C )
      = ( ( member @ TA @ A @ C )
        & ( ord_less_eq @ ( set @ TA ) @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(201,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA,E: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( ( member @ TA @ C @ A )
          & ( member @ TA @ D @ A )
          & ( member @ TA @ E @ A ) )
       => ( ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
            & ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) )
         => ( ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ E ) @ B )
              & ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ C ) @ B ) )
            | ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ D ) @ B )
              & ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ E ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_rational__preference_Ostrict__is__neg__transitive) ).

thf(812,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA,E: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( ( member @ TA @ C @ A )
          & ( member @ TA @ D @ A )
          & ( member @ TA @ E @ A ) )
       => ( ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
            & ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) )
         => ( ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ E ) @ B )
              & ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ C ) @ B ) )
            | ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ D ) @ B )
              & ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ E ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(245,axiom,
    ! [TA: $tType,A: TA,B: set @ TA,C: TA] :
      ( ( ~ ( member @ TA @ A @ B )
       => ( A = C ) )
     => ( member @ TA @ A @ ( insert @ TA @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_insertCI) ).

thf(939,plain,
    ! [TA: $tType,A: TA,B: set @ TA,C: TA] :
      ( ( ~ ( member @ TA @ A @ B )
       => ( A = C ) )
     => ( member @ TA @ A @ ( insert @ TA @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(30,axiom,
    ! [TA: $tType,TB: $tType,A: product_prod @ TA @ TB] :
      ( ( product_Pair @ TA @ TB @ ( product_fst @ TA @ TB @ A ) @ ( product_snd @ TA @ TB @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_prod_Ocollapse) ).

thf(342,plain,
    ! [TA: $tType,TB: $tType,A: product_prod @ TA @ TB] :
      ( ( product_Pair @ TA @ TB @ ( product_fst @ TA @ TB @ A ) @ ( product_snd @ TA @ TB @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(178,axiom,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ~ ( member @ TA @ A @ B )
     => ( member @ TA @ A @ ( uminus_uminus @ ( set @ TA ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_ComplI) ).

thf(754,plain,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ~ ( member @ TA @ A @ B )
     => ( member @ TA @ A @ ( uminus_uminus @ ( set @ TA ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(70,axiom,
    ! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
      ~ ! [B: TB,C: TA] :
          ( A
         != ( product_Pair @ TB @ TA @ B @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_old_Oprod_Oexhaust) ).

thf(458,plain,
    ! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
      ~ ! [B: TB,C: TA] :
          ( A
         != ( product_Pair @ TB @ TA @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(142,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( A
            = ( B @ C ) )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less_eq @ TA @ E @ F )
                 => ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_ord__eq__le__subst) ).

thf(648,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( A
            = ( B @ C ) )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less_eq @ TA @ E @ F )
                 => ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(80,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: product_prod @ TA @ TB,B: product_prod @ TA @ TB] :
          ( ( ord_less_eq @ ( product_prod @ TA @ TB ) @ A @ B )
         => ( ord_less_eq @ TB @ ( product_snd @ TA @ TB @ A ) @ ( product_snd @ TA @ TB @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_213_snd__mono) ).

thf(487,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: product_prod @ TA @ TB,B: product_prod @ TA @ TB] :
          ( ( ord_less_eq @ ( product_prod @ TA @ TB ) @ A @ B )
         => ( ord_less_eq @ TB @ ( product_snd @ TA @ TB @ A ) @ ( product_snd @ TA @ TB @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(187,axiom,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( member @ TA @ A @ B )
     => ~ ! [C: set @ TA] :
            ( ( B
              = ( insert @ TA @ A @ C ) )
           => ( member @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_Set_Oset__insert) ).

thf(776,plain,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( member @ TA @ A @ B )
     => ~ ! [C: set @ TA] :
            ( ( B
              = ( insert @ TA @ A @ C ) )
           => ( member @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(93,axiom,
    ! [TA: $tType,A: set @ TA] : ( trans @ TA @ ( id_on @ TA @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_164_trans__Id__on) ).

thf(525,plain,
    ! [TA: $tType,A: set @ TA] : ( trans @ TA @ ( id_on @ TA @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(52,axiom,
    ! [TA: $tType] :
      ( ( is_empty @ TA )
      = ( ^ [A: set @ TA] :
            ( A
            = ( bot_bot @ ( set @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_148_Set_Ois__empty__def) ).

thf(414,plain,
    ! [TA: $tType] :
      ( ( is_empty @ TA )
      = ( ^ [A: set @ TA] :
            ( A
            = ( bot_bot @ ( set @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(5,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( bot @ TB )
        & ( bot @ TA ) )
     => ( bot @ ( product_prod @ TB @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Orderings_Obot_15) ).

thf(263,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( bot @ TB )
        & ( bot @ TA ) )
     => ( bot @ ( product_prod @ TB @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(163,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( ! [B: TA] :
            ~ ( member @ TA @ B @ A ) )
      = ( A
        = ( bot_bot @ ( set @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_all__not__in__conv) ).

thf(712,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( ! [B: TA] :
            ~ ( member @ TA @ B @ A ) )
      = ( A
        = ( bot_bot @ ( set @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(104,axiom,
    ! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
      ( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B )
     => ( ord_less_eq @ ( set @ TB ) @ ( range @ TA @ TB @ A ) @ ( range @ TA @ TB @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_Range__mono) ).

thf(557,plain,
    ! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
      ( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B )
     => ( ord_less_eq @ ( set @ TB ) @ ( range @ TA @ TB @ A ) @ ( range @ TA @ TB @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(150,axiom,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) )
        = ( insert @ TA @ B @ C ) )
      = ( ( B = A )
        & ( ord_less_eq @ ( set @ TA ) @ C @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_singleton__insert__inj__eq) ).

thf(673,plain,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) )
        = ( insert @ TA @ B @ C ) )
      = ( ( B = A )
        & ( ord_less_eq @ ( set @ TA ) @ C @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(117,axiom,
    ! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ ( product_prod @ TA @ TB )] :
      ( ( range @ TA @ TB @ ( insert @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ C ) )
      = ( insert @ TB @ B @ ( range @ TA @ TB @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_231_Range__insert) ).

thf(585,plain,
    ! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ ( product_prod @ TA @ TB )] :
      ( ( range @ TA @ TB @ ( insert @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ C ) )
      = ( insert @ TB @ B @ ( range @ TA @ TB @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(208,axiom,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA,D: set @ ( product_prod @ TA @ TA )] :
      ( ( member @ TA @ A @ ( prefer310429814s_good @ TA @ B @ C @ D ) )
     => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ D ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_at__least__as__goodD) ).

thf(840,plain,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA,D: set @ ( product_prod @ TA @ TA )] :
      ( ( member @ TA @ A @ ( prefer310429814s_good @ TA @ B @ C @ D ) )
     => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(74,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ ( uminus_uminus @ ( set @ TA ) @ A ) )
      = ( A
        = ( bot_bot @ ( set @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_255_subset__Compl__self__eq) ).

thf(470,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ ( uminus_uminus @ ( set @ TA ) @ A ) )
      = ( A
        = ( bot_bot @ ( set @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(199,axiom,
    ! [TA: $tType,A: TA,B: set @ TA] : ( member @ TA @ A @ ( insert @ TA @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_insertI1) ).

thf(808,plain,
    ! [TA: $tType,A: TA,B: set @ TA] : ( member @ TA @ A @ ( insert @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(140,axiom,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_ord__eq__le__trans) ).

thf(643,plain,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(233,axiom,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ carrier )
     => ( ( member @ a @ B @ carrier )
       => ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ B @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ A @ carrier @ relation ) )
         => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_no__better__thansubset__rel) ).

thf(907,plain,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ carrier )
     => ( ( member @ a @ B @ carrier )
       => ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ B @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ A @ carrier @ relation ) )
         => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(175,axiom,
    ! [A: a,B: a] :
      ( ( carrier
       != ( bot_bot @ ( set @ a ) ) )
     => ( ( member @ a @ A @ carrier )
       => ( ( member @ a @ B @ carrier )
         => ( ~ ( member @ a @ A @ ( prefer310429814s_good @ a @ B @ carrier @ relation ) )
           => ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_at__lst__asgd__not__ge) ).

thf(744,plain,
    ! [A: a,B: a] :
      ( ( carrier
       != ( bot_bot @ ( set @ a ) ) )
     => ( ( member @ a @ A @ carrier )
       => ( ( member @ a @ B @ carrier )
         => ( ~ ( member @ a @ A @ ( prefer310429814s_good @ a @ B @ carrier @ relation ) )
           => ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(129,axiom,
    ! [TA: $tType,A: set @ TA,B: TA] : ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_subset__insertI) ).

thf(613,plain,
    ! [TA: $tType,A: set @ TA,B: TA] : ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(42,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TB,C: TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ord_less_eq @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ A @ C ) @ ( product_Pair @ TB @ TA @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_Pair__mono) ).

thf(386,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TB,C: TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ord_less_eq @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ A @ C ) @ ( product_Pair @ TB @ TA @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(22,axiom,
    ! [TA: $tType,TB: $tType,A: product_prod @ TA @ TB,B: TA,C: TB] :
      ( ( A
        = ( product_Pair @ TA @ TB @ B @ C ) )
     => ( ( product_fst @ TA @ TB @ A )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_227_fstI) ).

thf(314,plain,
    ! [TA: $tType,TB: $tType,A: product_prod @ TA @ TB,B: TA,C: TB] :
      ( ( A
        = ( product_Pair @ TA @ TB @ B @ C ) )
     => ( ( product_fst @ TA @ TB @ A )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(110,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( total_on @ TA @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_rational__preference_Ototal) ).

thf(571,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( total_on @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(121,axiom,
    ! [TA: $tType,A: TA] : ( refl_on @ TA @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) @ ( insert @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ A ) @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_refl__on__singleton) ).

thf(595,plain,
    ! [TA: $tType,A: TA] : ( refl_on @ TA @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) @ ( insert @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ A ) @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(215,axiom,
    prefer199794634erence @ a @ carrier @ relation,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_preference__axioms) ).

thf(859,plain,
    prefer199794634erence @ a @ carrier @ relation,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(210,axiom,
    ! [A: a,B: a] :
      ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
     => ( member @ a @ A @ ( prefer310429814s_good @ a @ B @ carrier @ relation ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_pref__in__at__least__as) ).

thf(845,plain,
    ! [A: a,B: a] :
      ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
     => ( member @ a @ A @ ( prefer310429814s_good @ a @ B @ carrier @ relation ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(135,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer199794634erence @ TA @ A @ B )
     => ( order_preorder_on @ TA @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_preference_Otrans__refl) ).

thf(627,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer199794634erence @ TA @ A @ B )
     => ( order_preorder_on @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(242,axiom,
    ! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
      ( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
     => ~ ! [C: TA] :
            ~ ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_238_RangeE) ).

thf(931,plain,
    ! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
      ( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
     => ~ ! [C: TA] :
            ~ ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(78,axiom,
    ! [TA: $tType] :
      ( ( order_bot @ TA )
     => ! [A: TA] :
          ( ( ord_less_eq @ TA @ A @ ( bot_bot @ TA ) )
         => ( A
            = ( bot_bot @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_bot_Oextremum__uniqueI) ).

thf(482,plain,
    ! [TA: $tType] :
      ( ( order_bot @ TA )
     => ! [A: TA] :
          ( ( ord_less_eq @ TA @ A @ ( bot_bot @ TA ) )
         => ( A
            = ( bot_bot @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(149,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer199794634erence @ TA @ A @ B )
     => ( refl_on @ TA @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_preference_Oreflexivity) ).

thf(671,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
      ( ( prefer199794634erence @ TA @ A @ B )
     => ( refl_on @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(115,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ord_less_eq @ ( set @ TA ) @ ( uminus_uminus @ ( set @ TA ) @ B ) @ ( uminus_uminus @ ( set @ TA ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_Compl__anti__mono) ).

thf(581,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ord_less_eq @ ( set @ TA ) @ ( uminus_uminus @ ( set @ TA ) @ B ) @ ( uminus_uminus @ ( set @ TA ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(180,axiom,
    order_preorder_on @ a @ carrier @ relation,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_trans__refl) ).

thf(759,plain,
    order_preorder_on @ a @ carrier @ relation,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(155,axiom,
    ! [TA: $tType,A: TA,B: TA > $o] :
      ( ( member @ TA @ A @ ( collect @ TA @ B ) )
      = ( B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_mem__Collect__eq) ).

thf(688,plain,
    ! [TA: $tType,A: TA,B: TA > $o] :
      ( ( member @ TA @ A @ ( collect @ TA @ B ) )
      = ( B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(231,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( member @ TA @ C @ A )
       => ( member @ TA @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_in__mono) ).

thf(903,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( member @ TA @ C @ A )
       => ( member @ TA @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(118,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) )] :
      ( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ TF ) @ F @ ( product_Pair @ TE @ TF @ G @ H ) ) ) ) ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_prod__induct6) ).

thf(588,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) )] :
      ( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ TF ) @ F @ ( product_Pair @ TE @ TF @ G @ H ) ) ) ) ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(252,axiom,
    ! [A: a,B: a] :
      ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ A @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ B @ carrier @ relation ) )
      | ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ B @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ A @ carrier @ relation ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_67_nbt__nest) ).

thf(961,plain,
    ! [A: a,B: a] :
      ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ A @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ B @ carrier @ relation ) )
      | ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ B @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ A @ carrier @ relation ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(176,axiom,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( id_on @ TA @ C ) )
      = ( ( A = B )
        & ( member @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_161_Id__on__iff) ).

thf(748,plain,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( id_on @ TA @ C ) )
      = ( ( A = B )
        & ( member @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(19,axiom,
    ! [TA: $tType] : ( preorder @ ( set @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Opreorder_2) ).

thf(304,plain,
    ! [TA: $tType] : ( preorder @ ( set @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(3,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_order__refl) ).

thf(258,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(190,axiom,
    ! [A: a,B: a,C: a] :
      ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
        & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) )
     => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ C ) @ relation )
          & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ B ) @ relation ) )
       => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ C ) @ relation )
          & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ A ) @ relation ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_indiff__trans) ).

thf(786,plain,
    ! [A: a,B: a,C: a] :
      ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
        & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) )
     => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ C ) @ relation )
          & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ B ) @ relation ) )
       => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ C ) @ relation )
          & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ A ) @ relation ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(253,axiom,
    ! [TA: $tType] :
      ( ( trans @ TA )
      = ( ^ [A: set @ ( product_prod @ TA @ TA )] :
          ! [B: TA,C: TA,D: TA] :
            ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ C ) @ A )
           => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
             => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_trans__def) ).

thf(963,plain,
    ! [TA: $tType] :
      ( ( trans @ TA )
      = ( ^ [A: set @ ( product_prod @ TA @ TA )] :
          ! [B: TA,C: TA,D: TA] :
            ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ C ) @ A )
           => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
             => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(90,axiom,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( insert @ TA @ A @ ( insert @ TA @ B @ C ) )
      = ( insert @ TA @ B @ ( insert @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_insert__commute) ).

thf(517,plain,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( insert @ TA @ A @ ( insert @ TA @ B @ C ) )
      = ( insert @ TA @ B @ ( insert @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(109,axiom,
    ! [TA: $tType] :
      ( ( is_singleton @ TA )
      = ( ^ [A: set @ TA] :
            ( A
            = ( insert @ TA @ ( the_elem @ TA @ A ) @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_220_is__singleton__the__elem) ).

thf(568,plain,
    ! [TA: $tType] :
      ( ( is_singleton @ TA )
      = ( ^ [A: set @ TA] :
            ( A
            = ( insert @ TA @ ( the_elem @ TA @ A ) @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(39,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ~ ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_le__cases) ).

thf(378,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ~ ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(147,axiom,
    ! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
      ( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_old_Oprod_Oinducts) ).

thf(667,plain,
    ! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
      ( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(241,axiom,
    ! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: TA,C: TA,D: TA] :
      ( ( trans @ TA @ A )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ C ) @ A )
       => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
         => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_transE) ).

thf(929,plain,
    ! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: TA,C: TA,D: TA] :
      ( ( trans @ TA @ A )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ C ) @ A )
       => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
         => ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(8,axiom,
    ! [TA: $tType,TB: $tType,A: TB,B: product_prod @ TA @ TB] :
      ( ( A
        = ( product_snd @ TA @ TB @ B ) )
      = ( ? [C: TA] :
            ( B
            = ( product_Pair @ TA @ TB @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_eq__snd__iff) ).

thf(272,plain,
    ! [TA: $tType,TB: $tType,A: TB,B: product_prod @ TA @ TB] :
      ( ( A
        = ( product_snd @ TA @ TB @ B ) )
      = ( ? [C: TA] :
            ( B
            = ( product_Pair @ TA @ TB @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(41,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ( uminus_uminus @ ( set @ TA ) @ A )
        = ( uminus_uminus @ ( set @ TA ) @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_248_Compl__eq__Compl__iff) ).

thf(383,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ( uminus_uminus @ ( set @ TA ) @ A )
        = ( uminus_uminus @ ( set @ TA ) @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(102,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ( ( ord_less_eq @ ( TA > TB ) )
        = ( ^ [A: TA > TB,B: TA > TB] :
            ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_le__fun__def) ).

thf(549,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ( ( ord_less_eq @ ( TA > TB ) )
        = ( ^ [A: TA > TB,B: TA > TB] :
            ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(60,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) )] :
      ( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF,I: TG] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) @ F @ ( product_Pair @ TE @ ( product_prod @ TF @ TG ) @ G @ ( product_Pair @ TF @ TG @ H @ I ) ) ) ) ) ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_prod__induct7) ).

thf(433,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) )] :
      ( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF,I: TG] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) @ F @ ( product_Pair @ TE @ ( product_prod @ TF @ TG ) @ G @ ( product_Pair @ TF @ TG @ H @ I ) ) ) ) ) ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(89,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ B )
         => ( ( ( C @ B )
              = D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less_eq @ TB @ E @ F )
                 => ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_ord__le__eq__subst) ).

thf(510,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ B )
         => ( ( ( C @ B )
              = D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less_eq @ TB @ E @ F )
                 => ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(66,axiom,
    bot @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Obot_10) ).

thf(447,plain,
    bot @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(213,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( member @ TA @ C @ A )
       => ( member @ TA @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_subsetD) ).

thf(854,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( member @ TA @ C @ A )
       => ( member @ TA @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(191,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer199794634erence @ TA @ A @ B )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
       => ( member @ TA @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_preference_Onot__outside) ).

thf(790,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer199794634erence @ TA @ A @ B )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
       => ( member @ TA @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(218,axiom,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( member @ TA @ A @ B )
     => ( ( insert @ TA @ A @ B )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_insert__absorb) ).

thf(865,plain,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( member @ TA @ A @ B )
     => ( ( insert @ TA @ A @ B )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(248,axiom,
    ! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
      ( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
      = ( ? [C: TA,D: TB] :
            ( ( A = D )
            & ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_Range_Osimps) ).

thf(950,plain,
    ! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
      ( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
      = ( ? [C: TA,D: TB] :
            ( ( A = D )
            & ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(207,axiom,
    ! [A: a,B: a,C: a] :
      ( ( ( member @ a @ A @ carrier )
        & ( member @ a @ B @ carrier )
        & ( member @ a @ C @ carrier ) )
     => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
          & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) )
       => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ C ) @ relation )
            & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ A ) @ relation ) )
          | ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ B ) @ relation )
            & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ C ) @ relation ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_strict__is__neg__transitive) ).

thf(835,plain,
    ! [A: a,B: a,C: a] :
      ( ( ( member @ a @ A @ carrier )
        & ( member @ a @ B @ carrier )
        & ( member @ a @ C @ carrier ) )
     => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
          & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) )
       => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ C ) @ relation )
            & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ A ) @ relation ) )
          | ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ B ) @ relation )
            & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ C ) @ relation ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(36,axiom,
    ! [TA: $tType,TB: $tType,A: TB,B: TA] :
      ( ( product_snd @ TB @ TA @ ( product_Pair @ TB @ TA @ A @ B ) )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_snd__conv) ).

thf(358,plain,
    ! [TA: $tType,TB: $tType,A: TB,B: TA] :
      ( ( product_snd @ TB @ TA @ ( product_Pair @ TB @ TA @ A @ B ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(183,axiom,
    ! [TA: $tType] :
      ( ( ord_less_eq @ ( set @ TA ) )
      = ( ^ [A: set @ TA,B: set @ TA] :
          ! [C: TA] :
            ( ( member @ TA @ C @ A )
           => ( member @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_subset__eq) ).

thf(767,plain,
    ! [TA: $tType] :
      ( ( ord_less_eq @ ( set @ TA ) )
      = ( ^ [A: set @ TA,B: set @ TA] :
          ! [C: TA] :
            ( ( member @ TA @ C @ A )
           => ( member @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(146,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( range @ TA @ TA @ ( id_on @ TA @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_Range__Id__on) ).

thf(664,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( range @ TA @ TA @ ( id_on @ TA @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(96,axiom,
    ! [TA: $tType,A: set @ TA,B: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
     => ( ( A
          = ( bot_bot @ ( set @ TA ) ) )
        | ( A
          = ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_subset__singletonD) ).

thf(532,plain,
    ! [TA: $tType,A: set @ TA,B: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
     => ( ( A
          = ( bot_bot @ ( set @ TA ) ) )
        | ( A
          = ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(158,axiom,
    trans @ a @ relation,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_147_transitivity) ).

thf(696,plain,
    trans @ a @ relation,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(250,axiom,
    ! [A: a] :
      ( ( carrier
       != ( bot_bot @ ( set @ a ) ) )
     => ( ( member @ a @ A @ carrier )
       => ( ( prefer1532642881r_than @ a @ A @ carrier @ relation )
         != ( bot_bot @ ( set @ a ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_no__better__than__nonepty) ).

thf(955,plain,
    ! [A: a] :
      ( ( carrier
       != ( bot_bot @ ( set @ a ) ) )
     => ( ( member @ a @ A @ carrier )
       => ( ( prefer1532642881r_than @ a @ A @ carrier @ relation )
         != ( bot_bot @ ( set @ a ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(224,axiom,
    ! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
      ( ~ ( member @ TA @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ ( insert @ TA @ A @ C ) )
        = ( ord_less_eq @ ( set @ TA ) @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_subset__insert) ).

thf(886,plain,
    ! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
      ( ~ ( member @ TA @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ ( insert @ TA @ A @ C ) )
        = ( ord_less_eq @ ( set @ TA ) @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(237,axiom,
    ! [TA: $tType,A: set @ TA,B: TA] :
      ( ( A
        = ( bot_bot @ ( set @ TA ) ) )
     => ~ ( member @ TA @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_equals0D) ).

thf(918,plain,
    ! [TA: $tType,A: set @ TA,B: TA] :
      ( ( A
        = ( bot_bot @ ( set @ TA ) ) )
     => ~ ( member @ TA @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(196,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
       => ( member @ TA @ D @ ( prefer1532642881r_than @ TA @ D @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_rational__preference_Oworse__in__no__better) ).

thf(801,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
       => ( member @ TA @ D @ ( prefer1532642881r_than @ TA @ D @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(194,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( A
         != ( bot_bot @ ( set @ TA ) ) )
       => ( ( member @ TA @ C @ A )
         => ( ( member @ TA @ D @ A )
           => ( ~ ( member @ TA @ C @ ( prefer310429814s_good @ TA @ D @ A @ B ) )
             => ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_rational__preference_Oat__lst__asgd__not__ge) ).

thf(796,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( A
         != ( bot_bot @ ( set @ TA ) ) )
       => ( ( member @ TA @ C @ A )
         => ( ( member @ TA @ D @ A )
           => ( ~ ( member @ TA @ C @ ( prefer310429814s_good @ TA @ D @ A @ B ) )
             => ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(12,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( bot @ TA )
        & ( bot @ TB ) )
     => ( ( product_snd @ TB @ TA @ ( bot_bot @ ( product_prod @ TB @ TA ) ) )
        = ( bot_bot @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_snd__bot) ).

thf(283,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( bot @ TA )
        & ( bot @ TB ) )
     => ( ( product_snd @ TB @ TA @ ( bot_bot @ ( product_prod @ TB @ TA ) ) )
        = ( bot_bot @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(134,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TA,C: TB,D: TA] :
          ( ( ord_less_eq @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ A @ B ) @ ( product_Pair @ TB @ TA @ C @ D ) )
          = ( ( ord_less_eq @ TB @ A @ C )
            & ( ord_less_eq @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_Pair__le) ).

thf(624,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TA,C: TB,D: TA] :
          ( ( ord_less_eq @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ A @ B ) @ ( product_Pair @ TB @ TA @ C @ D ) )
          = ( ( ord_less_eq @ TB @ A @ C )
            & ( ord_less_eq @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(72,axiom,
    ! [TA: $tType,A: TA > $o,B: TA > $o] :
      ( ! [C: TA] :
          ( ( A @ C )
         => ( B @ C ) )
     => ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_Collect__mono) ).

thf(464,plain,
    ! [TA: $tType,A: TA > $o,B: TA > $o] :
      ( ! [C: TA] :
          ( ( A @ C )
         => ( B @ C ) )
     => ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(216,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
       => ( member @ TA @ C @ ( prefer310429814s_good @ TA @ D @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_rational__preference_Opref__in__at__least__as) ).

thf(860,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
       => ( member @ TA @ C @ ( prefer310429814s_good @ TA @ D @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(64,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB,C: TA] :
          ( ( ord_less_eq @ ( TA > TB ) @ A @ B )
         => ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_le__funE) ).

thf(442,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB,C: TA] :
          ( ( ord_less_eq @ ( TA > TB ) @ A @ B )
         => ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(220,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( member @ TA @ C @ ( prefer951318096ood_as @ TA @ D @ A @ B ) )
        = ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
          & ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_rational__preference_Oas__good__as__sameIff) ).

thf(871,plain,
    ! [TA: $tType,A: set @ TA,B: set @ ( product_prod @ TA @ TA ),C: TA,D: TA] :
      ( ( prefer1997167224erence @ TA @ A @ B )
     => ( ( member @ TA @ C @ ( prefer951318096ood_as @ TA @ D @ A @ B ) )
        = ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ B )
          & ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(86,axiom,
    ! [TA: $tType] :
      ( ( order_preorder_on @ TA )
      = ( ^ [A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
            ( ( refl_on @ TA @ A @ B )
            & ( trans @ TA @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_156_preorder__on__def) ).

thf(501,plain,
    ! [TA: $tType] :
      ( ( order_preorder_on @ TA )
      = ( ^ [A: set @ TA,B: set @ ( product_prod @ TA @ TA )] :
            ( ( refl_on @ TA @ A @ B )
            & ( trans @ TA @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(10,axiom,
    ! [TA: $tType,TB: $tType,A: TA,B: TB,C: TA] :
      ( ( ( product_fst @ TA @ TB @ ( product_Pair @ TA @ TB @ A @ B ) )
        = C )
     => ( A = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_fst__eqD) ).

thf(277,plain,
    ! [TA: $tType,TB: $tType,A: TA,B: TB,C: TA] :
      ( ( ( product_fst @ TA @ TB @ ( product_Pair @ TA @ TB @ A @ B ) )
        = C )
     => ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(156,axiom,
    ! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ ( product_prod @ TA @ TB )] :
      ( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ C )
     => ( member @ TB @ B @ ( range @ TA @ TB @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_Range_Ointros) ).

thf(691,plain,
    ! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ ( product_prod @ TA @ TB )] :
      ( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ C )
     => ( member @ TB @ B @ ( range @ TA @ TB @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(24,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_order__trans) ).

thf(322,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(197,axiom,
    ! [A: a,B: a] :
      ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
     => ( member @ a @ B @ ( prefer1532642881r_than @ a @ B @ carrier @ relation ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_worse__in__no__better) ).

thf(803,plain,
    ! [A: a,B: a] :
      ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
     => ( member @ a @ B @ ( prefer1532642881r_than @ a @ B @ carrier @ relation ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(28,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order_bot @ TB )
        & ( order_bot @ TA ) )
     => ( order_bot @ ( product_prod @ TB @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Orderings_Oorder__bot_11) ).

thf(337,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order_bot @ TB )
        & ( order_bot @ TA ) )
     => ( order_bot @ ( product_prod @ TB @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(34,axiom,
    ! [TA: $tType,TB: $tType,A: TA > TB > $o,B: TA,C: TB] :
      ( ( A @ B @ C )
     => ( A @ ( product_snd @ TB @ TA @ ( product_Pair @ TB @ TA @ C @ B ) ) @ ( product_fst @ TB @ TA @ ( product_Pair @ TB @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_exI__realizer) ).

thf(354,plain,
    ! [TA: $tType,TB: $tType,A: TA > TB > $o,B: TA,C: TB] :
      ( ( A @ B @ C )
     => ( A @ ( product_snd @ TB @ TA @ ( product_Pair @ TB @ TA @ C @ B ) ) @ ( product_fst @ TB @ TA @ ( product_Pair @ TB @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(100,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_86_subset__antisym) ).

thf(543,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(151,axiom,
    ! [TA: $tType,A: TA,B: set @ TA,C: TA] :
      ( ( ( insert @ TA @ A @ B )
        = ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) )
      = ( ( A = C )
        & ( ord_less_eq @ ( set @ TA ) @ B @ ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_singleton__insert__inj__eq_H) ).

thf(676,plain,
    ! [TA: $tType,A: TA,B: set @ TA,C: TA] :
      ( ( ( insert @ TA @ A @ B )
        = ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) )
      = ( ( A = C )
        & ( ord_less_eq @ ( set @ TA ) @ B @ ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(82,axiom,
    ! [TA: $tType,A: TA] : ( trans @ TA @ ( insert @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ A ) @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_trans__singleton) ).

thf(491,plain,
    ! [TA: $tType,A: TA] : ( trans @ TA @ ( insert @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ A ) @ ( bot_bot @ ( set @ ( product_prod @ TA @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(120,axiom,
    ! [TA: $tType,A: TA > $o] :
      ( ( ( collect @ TA @ A )
        = ( bot_bot @ ( set @ TA ) ) )
      = ( ! [B: TA] :
            ~ ( A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_Collect__empty__eq) ).

thf(592,plain,
    ! [TA: $tType,A: TA > $o] :
      ( ( ( collect @ TA @ A )
        = ( bot_bot @ ( set @ TA ) ) )
      = ( ! [B: TA] :
            ~ ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(84,axiom,
    ! [TA: $tType,A: TA] :
      ( ( the_elem @ TA @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_the__elem__eq) ).

thf(496,plain,
    ! [TA: $tType,A: TA] :
      ( ( the_elem @ TA @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(162,axiom,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ carrier )
     => ( ( member @ a @ B @ carrier )
       => ( ( A != B )
         => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
            | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_completeD) ).

thf(708,plain,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ carrier )
     => ( ( member @ a @ B @ carrier )
       => ( ( A != B )
         => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ relation )
            | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ B @ A ) @ relation ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(972,plain,
    $false,
    inference(e,[status(thm)],[892,629,760,479,333,741,269,597,440,655,715,913,538,898,417,829,404,724,719,504,751,559,421,606,770,289,448,817,902,756,623,485,966,905,348,574,397,634,877,293,774,411,507,527,265,849,380,529,661,546,925,810,910,302,820,561,429,275,729,260,862,407,693,361,879,435,889,461,493,842,324,916,467,847,566,583,522,697,874,615,388,356,499,783,420,256,339,933,669,476,489,590,679,970,647,738,721,792,805,311,541,657,953,794,718,926,618,856,573,296,444,601,375,733,423,402,772,286,577,579,281,391,936,611,335,704,947,350,900,831,681,736,451,318,599,764,604,631,535,609,621,895,345,552,799,726,700,520,685,399,959,827,636,868,780,455,563,299,438,431,922,394,306,473,426,944,812,939,342,754,458,648,487,776,525,414,263,712,557,673,585,840,470,808,643,907,744,613,386,314,571,595,859,845,627,931,482,671,581,759,688,903,588,961,748,304,258,786,963,517,568,378,667,929,272,383,549,433,510,447,854,790,865,950,835,358,767,664,532,696,955,886,918,801,796,283,624,464,860,442,871,501,277,691,322,803,337,354,543,676,491,592,496,708]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.09/0.10  % Problem  : ITP153^2 : TPTP v8.2.0. Released v7.5.0.
% 0.09/0.13  % Command  : run_Leo-III %s %d
% 0.12/0.33  % Computer : n008.cluster.edu
% 0.12/0.33  % Model    : x86_64 x86_64
% 0.12/0.33  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33  % Memory   : 8042.1875MB
% 0.12/0.33  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33  % CPULimit : 300
% 0.12/0.33  % WCLimit  : 300
% 0.12/0.33  % DateTime : Sat May 18 15:18:09 EDT 2024
% 0.12/0.33  % CPUTime  : 
% 1.05/1.04  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 2.42/1.45  % [INFO] 	 Parsing done (403ms). 
% 2.42/1.46  % [INFO] 	 Running in sequential loop mode. 
% 3.57/1.88  % [INFO] 	 eprover registered as external prover. 
% 3.57/1.88  % [INFO] 	 cvc4 registered as external prover. 
% 3.57/1.89  % [INFO] 	 Scanning for conjecture ... 
% 3.99/2.06  % [INFO] 	 Found a conjecture (or negated_conjecture) and 277 axioms. Running axiom selection ... 
% 4.53/2.23  % [INFO] 	 Axiom selection finished. Selected 253 axioms (removed 24 axioms). 
% 5.60/2.57  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 5.82/2.61  % [INFO] 	 Type checking passed. 
% 5.82/2.61  % [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 ... 
% 175.48/152.86  % External prover 'e' found a proof!
% 175.48/152.86  % [INFO] 	 Killing All external provers ... 
% 175.48/152.86  % Time passed: 152369ms (effective reasoning time: 151397ms)
% 175.48/152.86  % 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)>
% 175.48/152.87  % Axioms used in derivation (253): fact_210_total__on__singleton, fact_226_lnear__order__on__empty, tcon_fun___Orderings_Obot, fact_110_bot__apply, fact_2_preference__axioms, fact_59_prod__cases, fact_197_insertI2, fact_244_Range__empty__iff, fact_121_le__fun__def, fact_206_fst__eqD, fact_44_prod__induct7, fact_21_preference_Oindiff__trans, fact_65_no__better__thansubset__rel, fact_35_total__on__def, tcon_HOL_Obool___Orderings_Opreorder_7, fact_218_the__elem__eq, fact_1_rational__preference__axioms, fact_29_worse__in__no__better, fact_53_prod__cases7, fact_152_transI, fact_149_internal__case__prod__conv, fact_182_singleton__inject, fact_222_is__singletonE, fact_37_refl__onD1, fact_104_subset__eq, fact_87_empty__iff, fact_113_bot_Oextremum__uniqueI, fact_234_Range__Id__on, fact_14_pref__in__at__least__as, tcon_HOL_Obool___Orderings_Oord_9, tcon_Set_Oset___Orderings_Oorder__bot_1, fact_64_no__better__subset__pref, fact_27_rational__preference_Ostrict__not__refl__weak, fact_229_eq__fst__iff, fact_74_rational__preference_Onbt__nest, fact_36_refl__onD, fact_250_ComplI, fact_187_subset__insertI2, fact_34_total__onI, fact_227_fstI, fact_84_Pair__le, fact_50_prod__induct5, fact_71_refl__on__empty, tcon_fun___Orderings_Oord, fact_97_set__eq__subset, fact_238_RangeE, fact_83_empty__subsetI, fact_161_Id__on__iff, fact_5_reflexivity, fact_100_subset__refl, fact_200_prod__eq__iff, tcon_HOL_Obool___Orderings_Olinorder, tcon_Product__Type_Oprod___Orderings_Oorder__bot_11, fact_224_linear__order__on__singleton, fact_13_strict__trans, fact_148_Set_Ois__empty__def, fact_94_equals0D, fact_135_ord__eq__le__trans, fact_185_singleton__iff, fact_235_Range__empty, fact_162_Id__on__eqI, tcon_HOL_Obool___Orderings_Oorder__bot_6, fact_128_linear, fact_228_eq__snd__iff, fact_252_Compl__subset__Compl__iff, fact_106_subsetD, fact_107_in__mono, fact_79_rational__preference_Ono__better__subset__pref, fact_240_Range_Ocases, fact_231_Range__insert, fact_201_prod_Oexpand, fact_221_is__singletonI_H, fact_173_insert__iff, fact_46_Collect__mem__eq, fact_57_prod__cases3, fact_204_prod_Oexhaust__sel, fact_255_subset__Compl__self__eq, fact_32_old_Oprod_Oinject, fact_220_is__singleton__the__elem, fact_144_bot__fun__def, fact_191_mk__disjoint__insert, fact_98_subset__trans, fact_88_all__not__in__conv, fact_80_rational__preference_Ono__better__thansubset__rel, fact_69_rational__preference_Ointro, fact_73_total__on__empty, fact_155_preference_Otransitivity, fact_190_insert__mono, fact_195_insert__ident, fact_146_less__by__empty, fact_202_prod__eqI, fact_41_at__lst__asgd__ge, fact_223_is__singleton__def, fact_251_Compl__anti__mono, fact_249_Compl__iff, fact_115_bot_Oextremum, fact_26_rational__preference_Oweak__is__transitive, fact_246_Domain__insert, fact_33_rational__preference_Oas__good__as__sameIff, fact_96_Collect__mono__iff, fact_208_snd__eqD, tcon_Product__Type_Oprod___Orderings_Oord_14, fact_63_at__lst__asgd__not__ge, fact_68_old_Oprod_Orec, fact_7_completeD, fact_18_preference_Otrans__refl, fact_196_Set_Oset__insert, fact_170_snd__bot, fact_76_rational__preference__axioms__def, fact_213_snd__mono, fact_217_exI__realizer, fact_179_prod_Ocollapse, fact_119_le__funE, fact_90_Collect__empty__eq, fact_243_Range_Oinducts, fact_85_subsetI, fact_11_strict__not__refl__weak, fact_254_Domain__empty, fact_164_trans__Id__on, fact_156_preorder__on__def, fact_55_prod__cases5, fact_177_singleton__insert__inj__eq, fact_61_rational__preference_Oworse__in__no__better, fact_58_Pair__inject, fact_45_mem__Collect__eq, fact_95_emptyE, fact_67_nbt__nest, fact_105_equalityE, fact_4_total, fact_86_subset__antisym, fact_101_subset__iff, fact_207_snd__conv, fact_3_trans__refl, fact_160_Id__onI, fact_103_equalityD1, fact_245_Range__mono, fact_151_transE, tcon_Product__Type_Oprod___Orderings_Obot_15, fact_75_rational__preference_Ono__better__than__nonepty, fact_180_insert__subsetI, fact_117_bot__empty__eq, fact_214_fst__mono, fact_6_compl, fact_150_transD, fact_109_bot__prod__def, fact_93_equals0I, fact_178_singleton__insert__inj__eq_H, fact_129_eq__refl, fact_203_surjective__pairing, fact_125_ord__le__eq__subst, fact_241_Range_Osimps, fact_111_order__refl, fact_8_not__outside, fact_199_insertE, fact_183_insert__not__empty, fact_102_equalityD2, fact_39_refl__on__domain, fact_49_prod__induct6, fact_20_preference_Oreflexivity, fact_108_Pair__mono, fact_157_Collect__empty__eq__bot, fact_81_rational__preference_Oat__lst__asgd__not__ge, fact_17_preference_Ointro, fact_47_Collect__cong, fact_82_subset__empty, fact_91_subrelI, fact_112_subset__emptyI, fact_19_preference_Onot__outside, fact_165_refl__on__Id__on, fact_163_Id__onE, fact_158_relChain__def, fact_136_ord__le__eq__trans, fact_216_conjI__realizer, tcon_fun___Orderings_Oorder__bot, fact_132_le__cases3, fact_194_insert__absorb, fact_54_prod__cases6, fact_51_prod__induct4, fact_31_prod_Oinject, fact_166_under__incr, fact_212_subset__singleton__iff, fact_154_trans__empty, fact_15_rational__preference_Oaxioms_I1_J, fact_198_insertI1, fact_205_fst__conv, fact_248_Compl__eq__Compl__iff, fact_169_fst__bot, fact_16_preference__def, fact_23_rational__preference_Ototal, fact_114_bot_Oextremum__unique, fact_184_doubleton__eq__iff, fact_188_subset__insertI, fact_24_rational__preference_OcompleteD, fact_25_rational__preference_Ostrict__trans, tcon_fun___Orderings_Opreorder, fact_239_Range__iff, fact_242_Range_Ointros, fact_70_rational__preference__def, fact_40_at__least__as__goodD, fact_118_le__funD, fact_171_trans__singleton, fact_120_le__funI, fact_247_subset__Compl__singleton, fact_193_insert__eq__iff, fact_189_subset__insert, tcon_Set_Oset___Orderings_Obot_5, fact_116_bot__set__def, fact_176_insert__subset, fact_175_singletonI, fact_10_weak__is__transitive, fact_43_old_Oprod_Oexhaust, fact_77_rational__preference__axioms_Ointro, fact_140_linorder__wlog, fact_138_order__trans, fact_60_surj__pair, fact_172_insert__absorb2, fact_78_rational__preference_Oaxioms_I2_J, fact_66_no__better__than__nonepty, tcon_HOL_Obool___Orderings_Obot_10, fact_153_trans__def, fact_145_ssubst__Pair__rhs, fact_225_sndI, tcon_Set_Oset___Orderings_Opreorder_2, fact_174_insertCI, fact_28_rational__preference_Ostrict__is__neg__transitive, fact_253_Domain__Id__on, fact_209_refl__on__singleton, fact_89_empty__Collect__eq, fact_124_ord__eq__le__subst, fact_159_Id__on__empty, fact_30_as__good__as__sameIff, fact_72_preorder__on__empty, fact_0_assms, fact_52_prod__induct3, fact_186_singletonD, fact_99_Collect__mono, fact_42_old_Oprod_Oinducts, fact_22_rational__preference_Ocompl, fact_38_refl__onD2, fact_9_strict__is__neg__transitive, tcon_Set_Oset___Orderings_Oord_4, tcon_Product__Type_Oprod___Orderings_Opreorder_12, fact_62_rational__preference_Opref__in__at__least__as, fact_92_ex__in__conv, fact_219_is__singletonI, fact_192_insert__commute, fact_215_BNF__Greatest__Fixpoint_Osubst__Pair, fact_147_transitivity, fact_130_le__cases, fact_181_less__eq__prod__def, fact_12_indiff__trans, fact_211_subset__singletonD, fact_56_prod__cases4
% 175.48/152.87  % No. of inferences in proof: 510
% 175.48/152.87  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 152369 ms resp. 151397 ms w/o parsing
% 176.05/153.00  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 176.05/153.00  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------