TSTP Solution File: SET038-3 by Leo-III---1.7.12

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : SET038-3 : TPTP v8.2.0. Released v1.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n018.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 : Tue May 21 03:01:04 EDT 2024

% Result   : Unsatisfiable 6.04s 2.63s
% Output   : Refutation 6.52s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    2
%            Number of leaves      :  242
% Syntax   : Number of formulae    :  409 (  39 unt;  76 typ;   0 def)
%            Number of atoms       :  833 ( 118 equ;   0 cnn)
%            Maximal formula atoms :    8 (   2 avg)
%            Number of connectives : 3212 ( 468   ~; 500   |;   0   &;2244   @)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   8 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :  122 ( 122   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   78 (  76 usr;  11 con; 0-5 aty)
%            Number of variables   :  742 (   0   ^ 742   !;   0   ?; 742   :)

% Comments : 
%------------------------------------------------------------------------------
thf(member_type,type,
    member: $i > $i > $o ).

thf(apply_type,type,
    apply: $i > $i > $i ).

thf(a_function_type,type,
    a_function: $i ).

thf(a_type,type,
    a: $i ).

thf(a_range_type,type,
    a_range: $i ).

thf(proper_subset_type,type,
    proper_subset: $i > $i > $o ).

thf(homomorphism_type,type,
    homomorphism: $i > $i > $i > $i > $i > $o ).

thf(maps_type,type,
    maps: $i > $i > $i > $o ).

thf(compose_type,type,
    compose: $i > $i > $i ).

thf(one_to_one_function_type,type,
    one_to_one_function: $i > $o ).

thf(function_type,type,
    function: $i > $o ).

thf(little_set_type,type,
    little_set: $i > $o ).

thf(ordered_pair_type,type,
    ordered_pair: $i > $i > $i ).

thf(powerset_type,type,
    powerset: $i > $i ).

thf(closed_type,type,
    closed: $i > $i > $o ).

thf(relation_type,type,
    relation: $i > $o ).

thf(single_valued_set_type,type,
    single_valued_set: $i > $o ).

thf(subset_type,type,
    subset: $i > $i > $o ).

thf(sigma_type,type,
    sigma: $i > $i ).

thf(f20_type,type,
    f20: $i > $i ).

thf(f21_type,type,
    f21: $i > $i ).

thf(non_ordered_pair_type,type,
    non_ordered_pair: $i > $i > $i ).

thf(f25_type,type,
    f25: $i ).

thf(domain_of_type,type,
    domain_of: $i > $i ).

thf(cross_product_type,type,
    cross_product: $i > $i > $i ).

thf(ordered_pair_predicate_type,type,
    ordered_pair_predicate: $i > $o ).

thf(f3_type,type,
    f3: $i > $i ).

thf(range_of_type,type,
    range_of: $i > $i ).

thf(singleton_set_type,type,
    singleton_set: $i > $i ).

thf(converse_type,type,
    converse: $i > $i ).

thf(f18_type,type,
    f18: $i > $i ).

thf(first_type,type,
    first: $i > $i ).

thf(successor_type,type,
    successor: $i > $i ).

thf(union_type,type,
    union: $i > $i > $i ).

thf(f2_type,type,
    f2: $i > $i ).

thf(empty_set_type,type,
    empty_set: $i ).

thf(disjoint_type,type,
    disjoint: $i > $i > $o ).

thf(f24_type,type,
    f24: $i > $i ).

thf(f19_type,type,
    f19: $i > $i ).

thf(complement_type,type,
    complement: $i > $i ).

thf(intersection_type,type,
    intersection: $i > $i > $i ).

thf(infinity_type,type,
    infinity: $i ).

thf(f1_type,type,
    f1: $i > $i > $i ).

thf(second_type,type,
    second: $i > $i ).

thf(image_type,type,
    image: $i > $i > $i ).

thf(restrict_type,type,
    restrict: $i > $i > $i ).

thf(universal_set_type,type,
    universal_set: $i ).

thf(f22_type,type,
    f22: $i > $i > $i > $i ).

thf(flip_range_of_type,type,
    flip_range_of: $i > $i ).

thf(f13_type,type,
    f13: $i > $i > $i ).

thf(f30_type,type,
    f30: $i > $i > $i > $i ).

thf(f16_type,type,
    f16: $i > $i > $i ).

thf(rotate_right_type,type,
    rotate_right: $i > $i ).

thf(f9_type,type,
    f9: $i > $i > $i ).

thf(f10_type,type,
    f10: $i > $i > $i ).

thf(f11_type,type,
    f11: $i > $i > $i ).

thf(f23_type,type,
    f23: $i > $i > $i ).

thf(f5_type,type,
    f5: $i > $i > $i ).

thf(f31_type,type,
    f31: $i > $i > $i > $i ).

thf(f27_type,type,
    f27: $i > $i > $i ).

thf(f26_type,type,
    f26: $i > $i ).

thf(f12_type,type,
    f12: $i > $i > $i ).

thf(f14_type,type,
    f14: $i > $i > $i ).

thf(estin_type,type,
    estin: $i ).

thf(f4_type,type,
    f4: $i > $i > $i ).

thf(identity_relation_type,type,
    identity_relation: $i ).

thf(a_domain_type,type,
    a_domain: $i ).

thf(f17_type,type,
    f17: $i > $i > $i ).

thf(apply_to_two_arguments_type,type,
    apply_to_two_arguments: $i > $i > $i > $i ).

thf(f32_type,type,
    f32: $i > $i > $i > $i > $i > $i ).

thf(f33_type,type,
    f33: $i > $i > $i > $i > $i > $i ).

thf(f29_type,type,
    f29: $i > $i > $i > $i ).

thf(f28_type,type,
    f28: $i > $i > $i > $i ).

thf(f8_type,type,
    f8: $i > $i > $i ).

thf(f6_type,type,
    f6: $i > $i > $i ).

thf(f7_type,type,
    f7: $i > $i > $i ).

thf(87,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( first @ B ) )
      | ( B
        = ( ordered_pair @ ( f4 @ A @ B ) @ ( f5 @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',first3) ).

thf(365,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( first @ B ) )
      | ( B
        = ( ordered_pair @ ( f4 @ A @ B ) @ ( f5 @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(133,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( apply @ B @ C ) )
      | ( ( first @ ( f28 @ A @ B @ C ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply3) ).

thf(468,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( apply @ B @ C ) )
      | ( ( first @ ( f28 @ A @ B @ C ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(138,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( second @ B ) )
      | ( little_set @ ( f7 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',second2) ).

thf(481,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( second @ B ) )
      | ( little_set @ ( f7 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(82,axiom,
    ! [A: $i] :
      ( ( member @ A @ estin )
      | ~ ( little_set @ A )
      | ~ ( ordered_pair_predicate @ A )
      | ~ ( member @ ( first @ A ) @ ( second @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation3) ).

thf(352,plain,
    ! [A: $i] :
      ( ( member @ A @ estin )
      | ~ ( little_set @ A )
      | ~ ( ordered_pair_predicate @ A )
      | ~ ( member @ ( first @ A ) @ ( second @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(22,axiom,
    ! [B: $i,A: $i] : ( little_set @ ( ordered_pair @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pairs_are_small1) ).

thf(217,plain,
    ! [B: $i,A: $i] : ( little_set @ ( ordered_pair @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(16,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( maps @ A @ B @ C )
      | ( ( domain_of @ A )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps2) ).

thf(202,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( maps @ A @ B @ C )
      | ( ( domain_of @ A )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(121,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( rotate_right @ B ) )
      | ( little_set @ ( f11 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate_right3) ).

thf(440,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( rotate_right @ B ) )
      | ( little_set @ ( f11 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(150,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( sigma @ B ) )
      | ( member @ A @ ( f16 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sigma2) ).

thf(511,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( sigma @ B ) )
      | ( member @ A @ ( f16 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(97,axiom,
    maps @ a_function @ a_domain @ a_range,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a_mapping) ).

thf(385,plain,
    maps @ a_function @ a_domain @ a_range,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(166,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( apply @ B @ C ) )
      | ( ordered_pair_predicate @ ( f28 @ A @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply1) ).

thf(550,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( apply @ B @ C ) )
      | ( ordered_pair_predicate @ ( f28 @ A @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(145,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( complement @ B ) )
      | ~ ( member @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement1) ).

thf(500,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( complement @ B ) )
      | ~ ( member @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(77,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( powerset @ B ) )
      | ( subset @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',powerset1) ).

thf(340,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( powerset @ B ) )
      | ( subset @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(106,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( homomorphism @ A @ B @ C @ D @ E )
      | ~ ( closed @ B @ C )
      | ~ ( closed @ D @ E )
      | ~ ( maps @ A @ B @ D )
      | ( ( apply @ A @ ( apply_to_two_arguments @ C @ ( f32 @ A @ B @ C @ D @ E ) @ ( f33 @ A @ B @ C @ D @ E ) ) )
       != ( apply_to_two_arguments @ E @ ( apply @ A @ ( f32 @ A @ B @ C @ D @ E ) ) @ ( apply @ A @ ( f33 @ A @ B @ C @ D @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism7) ).

thf(404,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( homomorphism @ A @ B @ C @ D @ E )
      | ~ ( closed @ B @ C )
      | ~ ( closed @ D @ E )
      | ~ ( maps @ A @ B @ D )
      | ( ( apply @ A @ ( apply_to_two_arguments @ C @ ( f32 @ A @ B @ C @ D @ E ) @ ( f33 @ A @ B @ C @ D @ E ) ) )
       != ( apply_to_two_arguments @ E @ ( apply @ A @ ( f32 @ A @ B @ C @ D @ E ) ) @ ( apply @ A @ ( f33 @ A @ B @ C @ D @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(56,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( subset @ A @ B )
      | ~ ( subset @ B @ A )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',two_sets_equal) ).

thf(288,plain,
    ! [B: $i,A: $i] :
      ( ~ ( subset @ A @ B )
      | ~ ( subset @ B @ A )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(119,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( flip_range_of @ B ) )
      | ( little_set @ ( f14 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip_range3) ).

thf(436,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( flip_range_of @ B ) )
      | ( little_set @ ( f14 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(41,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( homomorphism @ A @ B @ C @ D @ E )
      | ( closed @ D @ E ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism2) ).

thf(257,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( homomorphism @ A @ B @ C @ D @ E )
      | ( closed @ D @ E ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(99,axiom,
    ! [A: $i] :
      ( ( member @ A @ universal_set )
      | ~ ( little_set @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',universal_set) ).

thf(389,plain,
    ! [A: $i] :
      ( ( member @ A @ universal_set )
      | ~ ( little_set @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(8,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( closed @ A @ B )
      | ( little_set @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closed1) ).

thf(184,plain,
    ! [B: $i,A: $i] :
      ( ~ ( closed @ A @ B )
      | ( little_set @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(90,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( relation @ A )
      | ~ ( member @ B @ A )
      | ( ordered_pair_predicate @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',relation1) ).

thf(372,plain,
    ! [B: $i,A: $i] :
      ( ~ ( relation @ A )
      | ~ ( member @ B @ A )
      | ( ordered_pair_predicate @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(147,axiom,
    ! [B: $i,A: $i] :
      ( ( subset @ A @ B )
      | ~ ( member @ ( f17 @ A @ B ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset3) ).

thf(504,plain,
    ! [B: $i,A: $i] :
      ( ( subset @ A @ B )
      | ~ ( member @ ( f17 @ A @ B ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(79,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ B )
      | ( little_set @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a2) ).

thf(344,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ B )
      | ( little_set @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(84,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( rotate_right @ B ) )
      | ~ ( little_set @ A )
      | ~ ( little_set @ C )
      | ~ ( little_set @ D )
      | ~ ( little_set @ E )
      | ( A
       != ( ordered_pair @ C @ ( ordered_pair @ D @ E ) ) )
      | ~ ( member @ ( ordered_pair @ D @ ( ordered_pair @ E @ C ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate_right6) ).

thf(357,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( rotate_right @ B ) )
      | ~ ( little_set @ A )
      | ~ ( little_set @ C )
      | ~ ( little_set @ D )
      | ~ ( little_set @ E )
      | ( A
       != ( ordered_pair @ C @ ( ordered_pair @ D @ E ) ) )
      | ~ ( member @ ( ordered_pair @ D @ ( ordered_pair @ E @ C ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(129,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( compose @ B @ C ) )
      | ( A
        = ( ordered_pair @ ( f29 @ A @ B @ C ) @ ( f30 @ A @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose4) ).

thf(460,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( compose @ B @ C ) )
      | ( A
        = ( ordered_pair @ ( f29 @ A @ B @ C ) @ ( f30 @ A @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(54,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( little_set @ A )
      | ~ ( little_set @ B )
      | ( ( second @ ( ordered_pair @ A @ B ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_of_second) ).

thf(284,plain,
    ! [B: $i,A: $i] :
      ( ~ ( little_set @ A )
      | ~ ( little_set @ B )
      | ( ( second @ ( ordered_pair @ A @ B ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(111,axiom,
    ! [A: $i] :
      ( ~ ( member @ A @ estin )
      | ( ordered_pair_predicate @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation1) ).

thf(416,plain,
    ! [A: $i] :
      ( ~ ( member @ A @ estin )
      | ( ordered_pair_predicate @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(68,axiom,
    ! [A: $i] :
      ( ( single_valued_set @ A )
      | ( member @ ( ordered_pair @ ( f19 @ A ) @ ( f20 @ A ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_set5) ).

thf(316,plain,
    ! [A: $i] :
      ( ( single_valued_set @ A )
      | ( member @ ( ordered_pair @ ( f19 @ A ) @ ( f20 @ A ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(43,axiom,
    ! [B: $i,A: $i] :
      ( ( union @ A @ B )
      = ( complement @ ( intersection @ ( complement @ A ) @ ( complement @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union) ).

thf(261,plain,
    ! [B: $i,A: $i] :
      ( ( union @ A @ B )
      = ( complement @ ( intersection @ ( complement @ A ) @ ( complement @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(136,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( cross_product @ B @ C ) )
      | ~ ( little_set @ A )
      | ~ ( ordered_pair_predicate @ A )
      | ~ ( member @ ( first @ A ) @ B )
      | ~ ( member @ ( second @ A ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cross_product4) ).

thf(475,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( cross_product @ B @ C ) )
      | ~ ( little_set @ A )
      | ~ ( ordered_pair_predicate @ A )
      | ~ ( member @ ( first @ A ) @ B )
      | ~ ( member @ ( second @ A ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(6,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( little_set @ A )
      | ~ ( little_set @ B )
      | ~ ( little_set @ C )
      | ~ ( little_set @ D )
      | ( ( ordered_pair @ A @ B )
       != ( ordered_pair @ C @ D ) )
      | ( B = D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',second_components_are_equal) ).

thf(179,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( little_set @ A )
      | ~ ( little_set @ B )
      | ~ ( little_set @ C )
      | ~ ( little_set @ D )
      | ( ( ordered_pair @ A @ B )
       != ( ordered_pair @ C @ D ) )
      | ( B = D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(70,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( rotate_right @ B ) )
      | ( A
        = ( ordered_pair @ ( f9 @ A @ B ) @ ( ordered_pair @ ( f10 @ A @ B ) @ ( f11 @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate_right4) ).

thf(321,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( rotate_right @ B ) )
      | ( A
        = ( ordered_pair @ ( f9 @ A @ B ) @ ( ordered_pair @ ( f10 @ A @ B ) @ ( f11 @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(39,axiom,
    ! [A: $i] :
      ( ~ ( ordered_pair_predicate @ A )
      | ( little_set @ ( first @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',first_component_is_small) ).

thf(253,plain,
    ! [A: $i] :
      ( ~ ( ordered_pair_predicate @ A )
      | ( little_set @ ( first @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(103,axiom,
    ! [B: $i,A: $i] :
      ( ( member @ A @ ( complement @ B ) )
      | ~ ( little_set @ A )
      | ( member @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement2) ).

thf(397,plain,
    ! [B: $i,A: $i] :
      ( ( member @ A @ ( complement @ B ) )
      | ~ ( little_set @ A )
      | ( member @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(53,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ordered_pair_predicate @ A )
      | ~ ( little_set @ B )
      | ~ ( little_set @ C )
      | ( A
       != ( ordered_pair @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_predicate4) ).

thf(280,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ordered_pair_predicate @ A )
      | ~ ( little_set @ B )
      | ~ ( little_set @ C )
      | ( A
       != ( ordered_pair @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(25,axiom,
    ! [A: $i] :
      ( ( relation @ A )
      | ~ ( ordered_pair_predicate @ ( f18 @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',relation3) ).

thf(221,plain,
    ! [A: $i] :
      ( ( relation @ A )
      | ~ ( ordered_pair_predicate @ ( f18 @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(58,axiom,
    ! [A: $i] :
      ( ~ ( ordered_pair_predicate @ A )
      | ( A
        = ( ordered_pair @ ( f2 @ A ) @ ( f3 @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_predicate3) ).

thf(293,plain,
    ! [A: $i] :
      ( ~ ( ordered_pair_predicate @ A )
      | ( A
        = ( ordered_pair @ ( f2 @ A ) @ ( f3 @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(116,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( image @ B @ C ) )
      | ( member @ ( f22 @ A @ B @ C ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image_and_substitution2) ).

thf(428,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( image @ B @ C ) )
      | ( member @ ( f22 @ A @ B @ C ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(45,axiom,
    ! [B: $i,A: $i] : ( little_set @ ( non_ordered_pair @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',non_ordered_pair4) ).

thf(265,plain,
    ! [B: $i,A: $i] : ( little_set @ ( non_ordered_pair @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(66,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( sigma @ B ) )
      | ( member @ ( f16 @ A @ B ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sigma1) ).

thf(312,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( sigma @ B ) )
      | ( member @ ( f16 @ A @ B ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(37,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( maps @ A @ B @ C )
      | ( subset @ ( range_of @ A ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps3) ).

thf(248,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( maps @ A @ B @ C )
      | ( subset @ ( range_of @ A ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(95,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( disjoint @ A @ B )
      | ~ ( member @ C @ A )
      | ~ ( member @ C @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',disjoint1) ).

thf(380,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( disjoint @ A @ B )
      | ~ ( member @ C @ A )
      | ~ ( member @ C @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(48,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( function @ A )
      | ~ ( function @ B )
      | ( function @ ( compose @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',composition_is_a_function) ).

thf(270,plain,
    ! [B: $i,A: $i] :
      ( ~ ( function @ A )
      | ~ ( function @ B )
      | ( function @ ( compose @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(15,axiom,
    function @ f25,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice1) ).

thf(201,plain,
    function @ f25,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(24,axiom,
    ! [B: $i,A: $i] : ( subset @ ( range_of @ ( compose @ A @ B ) ) @ ( range_of @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of_composition) ).

thf(220,plain,
    ! [B: $i,A: $i] : ( subset @ ( range_of @ ( compose @ A @ B ) ) @ ( range_of @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(30,axiom,
    ! [B: $i,A: $i] :
      ( ( ordered_pair @ A @ B )
      = ( non_ordered_pair @ ( singleton_set @ A ) @ ( non_ordered_pair @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair) ).

thf(233,plain,
    ! [B: $i,A: $i] :
      ( ( ordered_pair @ A @ B )
      = ( non_ordered_pair @ ( singleton_set @ A ) @ ( non_ordered_pair @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(125,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( rotate_right @ B ) )
      | ( member @ ( ordered_pair @ ( f10 @ A @ B ) @ ( ordered_pair @ ( f11 @ A @ B ) @ ( f9 @ A @ B ) ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate_right5) ).

thf(452,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( rotate_right @ B ) )
      | ( member @ ( ordered_pair @ ( f10 @ A @ B ) @ ( ordered_pair @ ( f11 @ A @ B ) @ ( f9 @ A @ B ) ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(74,axiom,
    ! [B: $i,A: $i] :
      ( ( disjoint @ A @ B )
      | ( member @ ( f23 @ A @ B ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',disjoint3) ).

thf(334,plain,
    ! [B: $i,A: $i] :
      ( ( disjoint @ A @ B )
      | ( member @ ( f23 @ A @ B ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(62,axiom,
    ! [A: $i] :
      ( ( single_valued_set @ A )
      | ( member @ ( ordered_pair @ ( f19 @ A ) @ ( f21 @ A ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_set6) ).

thf(302,plain,
    ! [A: $i] :
      ( ( single_valued_set @ A )
      | ( member @ ( ordered_pair @ ( f19 @ A ) @ ( f21 @ A ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(81,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( flip_range_of @ B ) )
      | ( A
        = ( ordered_pair @ ( f12 @ A @ B ) @ ( ordered_pair @ ( f13 @ A @ B ) @ ( f14 @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip_range4) ).

thf(349,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( flip_range_of @ B ) )
      | ( A
        = ( ordered_pair @ ( f12 @ A @ B ) @ ( ordered_pair @ ( f13 @ A @ B ) @ ( f14 @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(114,axiom,
    ! [B: $i,A: $i] :
      ( ( member @ A @ ( powerset @ B ) )
      | ~ ( little_set @ A )
      | ~ ( subset @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',powerset2) ).

thf(424,plain,
    ! [B: $i,A: $i] :
      ( ( member @ A @ ( powerset @ B ) )
      | ~ ( little_set @ A )
      | ~ ( subset @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(107,axiom,
    ! [A: $i] :
      ( ~ ( little_set @ A )
      | ( member @ A @ ( singleton_set @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_of_singleton_sets) ).

thf(407,plain,
    ! [A: $i] :
      ( ~ ( little_set @ A )
      | ( member @ A @ ( singleton_set @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(51,axiom,
    ! [B: $i,A: $i] :
      ( ( little_set @ ( f1 @ A @ B ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',extensionality1) ).

thf(275,plain,
    ! [B: $i,A: $i] :
      ( ( little_set @ ( f1 @ A @ B ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(85,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( intersection @ B @ C ) )
      | ( member @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection2) ).

thf(361,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( intersection @ B @ C ) )
      | ( member @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(155,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( compose @ B @ C ) )
      | ( little_set @ ( f29 @ A @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose1) ).

thf(522,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( compose @ B @ C ) )
      | ( little_set @ ( f29 @ A @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(143,axiom,
    ! [B: $i,A: $i] :
      ( ( member @ ( f1 @ A @ B ) @ A )
      | ( member @ ( f1 @ A @ B ) @ B )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',extensionality2) ).

thf(493,plain,
    ! [B: $i,A: $i] :
      ( ( member @ ( f1 @ A @ B ) @ A )
      | ( member @ ( f1 @ A @ B ) @ B )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(127,axiom,
    ! [B: $i,A: $i] :
      ( ( member @ A @ ( converse @ B ) )
      | ~ ( little_set @ A )
      | ~ ( ordered_pair_predicate @ A )
      | ~ ( member @ ( ordered_pair @ ( second @ A ) @ ( first @ A ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',converse3) ).

thf(456,plain,
    ! [B: $i,A: $i] :
      ( ( member @ A @ ( converse @ B ) )
      | ~ ( little_set @ A )
      | ~ ( ordered_pair_predicate @ A )
      | ~ ( member @ ( ordered_pair @ ( second @ A ) @ ( first @ A ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(71,axiom,
    ! [B: $i,A: $i] :
      ( ( disjoint @ A @ B )
      | ( member @ ( f23 @ A @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',disjoint2) ).

thf(324,plain,
    ! [B: $i,A: $i] :
      ( ( disjoint @ A @ B )
      | ( member @ ( f23 @ A @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(105,axiom,
    member @ a @ a_domain,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_of_domain) ).

thf(403,plain,
    member @ a @ a_domain,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(10,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( maps @ A @ B @ C )
      | ( function @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps1) ).

thf(188,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( maps @ A @ B @ C )
      | ( function @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(2,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( proper_subset @ A @ B )
      | ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',proper_subset2) ).

thf(169,plain,
    ! [B: $i,A: $i] :
      ( ~ ( proper_subset @ A @ B )
      | ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(113,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( flip_range_of @ B ) )
      | ~ ( little_set @ A )
      | ~ ( little_set @ C )
      | ~ ( little_set @ D )
      | ~ ( little_set @ E )
      | ( A
       != ( ordered_pair @ C @ ( ordered_pair @ D @ E ) ) )
      | ~ ( member @ ( ordered_pair @ C @ ( ordered_pair @ E @ D ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip_range6) ).

thf(420,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( flip_range_of @ B ) )
      | ~ ( little_set @ A )
      | ~ ( little_set @ C )
      | ~ ( little_set @ D )
      | ~ ( little_set @ E )
      | ( A
       != ( ordered_pair @ C @ ( ordered_pair @ D @ E ) ) )
      | ~ ( member @ ( ordered_pair @ C @ ( ordered_pair @ E @ D ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(134,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( apply_to_two_arguments @ A @ B @ C )
      = ( apply @ A @ ( ordered_pair @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply_to_two_arguments) ).

thf(471,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( apply_to_two_arguments @ A @ B @ C )
      = ( apply @ A @ ( ordered_pair @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(141,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( domain_of @ B ) )
      | ( A
        = ( first @ ( f8 @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain3) ).

thf(488,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( domain_of @ B ) )
      | ( A
        = ( first @ ( f8 @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(27,axiom,
    ! [A: $i] :
      ( ( one_to_one_function @ A )
      | ~ ( function @ A )
      | ~ ( function @ ( converse @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one_function3) ).

thf(225,plain,
    ! [A: $i] :
      ( ( one_to_one_function @ A )
      | ~ ( function @ A )
      | ~ ( function @ ( converse @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(12,axiom,
    ! [A: $i] :
      ( ~ ( little_set @ A )
      | ( little_set @ ( sigma @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sigma4) ).

thf(193,plain,
    ! [A: $i] :
      ( ~ ( little_set @ A )
      | ( little_set @ ( sigma @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(157,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( homomorphism @ A @ B @ C @ D @ E )
      | ~ ( closed @ B @ C )
      | ~ ( closed @ D @ E )
      | ~ ( maps @ A @ B @ D )
      | ( member @ ( f32 @ A @ B @ C @ D @ E ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism5) ).

thf(526,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( homomorphism @ A @ B @ C @ D @ E )
      | ~ ( closed @ B @ C )
      | ~ ( closed @ D @ E )
      | ~ ( maps @ A @ B @ D )
      | ( member @ ( f32 @ A @ B @ C @ D @ E ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(165,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( second @ B ) )
      | ( B
        = ( ordered_pair @ ( f6 @ A @ B ) @ ( f7 @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',second3) ).

thf(547,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( second @ B ) )
      | ( B
        = ( ordered_pair @ ( f6 @ A @ B ) @ ( f7 @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(101,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( rotate_right @ B ) )
      | ( little_set @ ( f10 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate_right2) ).

thf(393,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( rotate_right @ B ) )
      | ( little_set @ ( f10 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(152,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( subset @ A @ B )
      | ~ ( member @ C @ A )
      | ( member @ C @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset1) ).

thf(515,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( subset @ A @ B )
      | ~ ( member @ C @ A )
      | ( member @ C @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(3,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( homomorphism @ A @ B @ C @ D @ E )
      | ( maps @ A @ B @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism3) ).

thf(173,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( homomorphism @ A @ B @ C @ D @ E )
      | ( maps @ A @ B @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(32,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( little_set @ A )
      | ~ ( little_set @ B )
      | ( ( ordered_pair @ A @ C )
       != ( ordered_pair @ B @ D ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',first_components_are_equal) ).

thf(237,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( little_set @ A )
      | ~ ( little_set @ B )
      | ( ( ordered_pair @ A @ C )
       != ( ordered_pair @ B @ D ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(35,axiom,
    ! [A: $i] :
      ( ~ ( ordered_pair_predicate @ A )
      | ( little_set @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pairs_are_small2) ).

thf(244,plain,
    ! [A: $i] :
      ( ~ ( ordered_pair_predicate @ A )
      | ( little_set @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(46,axiom,
    ! [A: $i] :
      ( ~ ( function @ A )
      | ( relation @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function1) ).

thf(266,plain,
    ! [A: $i] :
      ( ~ ( function @ A )
      | ( relation @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(17,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( closed @ A @ B )
      | ( maps @ B @ ( cross_product @ A @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closed3) ).

thf(205,plain,
    ! [B: $i,A: $i] :
      ( ~ ( closed @ A @ B )
      | ( maps @ B @ ( cross_product @ A @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(149,axiom,
    ! [A: $i] :
      ( ( A = empty_set )
      | ( member @ ( f24 @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity1) ).

thf(508,plain,
    ! [A: $i] :
      ( ( A = empty_set )
      | ( member @ ( f24 @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(163,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( second @ B ) )
      | ( member @ A @ ( f7 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',second4) ).

thf(541,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( second @ B ) )
      | ( member @ A @ ( f7 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(159,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( flip_range_of @ B ) )
      | ( little_set @ ( f12 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip_range1) ).

thf(530,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( flip_range_of @ B ) )
      | ( little_set @ ( f12 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(60,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( image @ B @ C ) )
      | ( member @ ( first @ ( f22 @ A @ B @ C ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image_and_substitution3) ).

thf(298,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( image @ B @ C ) )
      | ( member @ ( first @ ( f22 @ A @ B @ C ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(109,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( image @ B @ C ) )
      | ( ordered_pair_predicate @ ( f22 @ A @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image_and_substitution1) ).

thf(412,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( image @ B @ C ) )
      | ( ordered_pair_predicate @ ( f22 @ A @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(59,axiom,
    ! [B: $i,A: $i] :
      ( ( restrict @ A @ B )
      = ( intersection @ A @ ( cross_product @ B @ universal_set ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',restrict) ).

thf(296,plain,
    ! [B: $i,A: $i] :
      ( ( restrict @ A @ B )
      = ( intersection @ A @ ( cross_product @ B @ universal_set ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(92,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( cross_product @ B @ C ) )
      | ( member @ ( second @ A ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cross_product3) ).

thf(375,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( cross_product @ B @ C ) )
      | ( member @ ( second @ A ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(151,axiom,
    ! [A: $i] :
      ( ~ ( member @ A @ estin )
      | ( member @ ( first @ A ) @ ( second @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation2) ).

thf(513,plain,
    ! [A: $i] :
      ( ~ ( member @ A @ estin )
      | ( member @ ( first @ A ) @ ( second @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(117,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( compose @ B @ C ) )
      | ( member @ ( ordered_pair @ ( f29 @ A @ B @ C ) @ ( f31 @ A @ B @ C ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose5) ).

thf(430,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( compose @ B @ C ) )
      | ( member @ ( ordered_pair @ ( f29 @ A @ B @ C ) @ ( f31 @ A @ B @ C ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(57,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( little_set @ A )
      | ~ ( function @ B )
      | ( little_set @ ( image @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image_and_substitution6) ).

thf(291,plain,
    ! [B: $i,A: $i] :
      ( ~ ( little_set @ A )
      | ~ ( function @ B )
      | ( little_set @ ( image @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(42,axiom,
    ! [A: $i] :
      ( ( single_valued_set @ A )
      | ( little_set @ ( f19 @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_set2) ).

thf(259,plain,
    ! [A: $i] :
      ( ( single_valued_set @ A )
      | ( little_set @ ( f19 @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(100,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( sigma @ B ) )
      | ~ ( member @ C @ B )
      | ~ ( member @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sigma3) ).

thf(391,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( sigma @ B ) )
      | ~ ( member @ C @ B )
      | ~ ( member @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(96,axiom,
    ! [A: $i] :
      ( ( member @ A @ identity_relation )
      | ~ ( little_set @ A )
      | ~ ( ordered_pair_predicate @ A )
      | ( ( first @ A )
       != ( second @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation3) ).

thf(382,plain,
    ! [A: $i] :
      ( ( member @ A @ identity_relation )
      | ~ ( little_set @ A )
      | ~ ( ordered_pair_predicate @ A )
      | ( ( first @ A )
       != ( second @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(132,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( domain_of @ B ) )
      | ( ordered_pair_predicate @ ( f8 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain1) ).

thf(466,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( domain_of @ B ) )
      | ( ordered_pair_predicate @ ( f8 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(93,axiom,
    ! [B: $i,A: $i] : ( subset @ ( image @ ( singleton_set @ A ) @ B ) @ ( apply @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image_and_apply2) ).

thf(377,plain,
    ! [B: $i,A: $i] : ( subset @ ( image @ ( singleton_set @ A ) @ B ) @ ( apply @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(29,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( subset @ ( range_of @ A ) @ ( domain_of @ B ) )
      | ( ( domain_of @ A )
        = ( domain_of @ ( compose @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain_of_composition) ).

thf(230,plain,
    ! [B: $i,A: $i] :
      ( ~ ( subset @ ( range_of @ A ) @ ( domain_of @ B ) )
      | ( ( domain_of @ A )
        = ( domain_of @ ( compose @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(69,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( non_ordered_pair @ B @ C ) )
      | ( A = B )
      | ( A = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',non_ordered_pair1) ).

thf(318,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( non_ordered_pair @ B @ C ) )
      | ( A = B )
      | ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(137,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( range_of @ B ) )
      | ~ ( little_set @ A )
      | ~ ( ordered_pair_predicate @ C )
      | ~ ( member @ C @ B )
      | ( A
       != ( second @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of4) ).

thf(477,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( range_of @ B ) )
      | ~ ( little_set @ A )
      | ~ ( ordered_pair_predicate @ C )
      | ~ ( member @ C @ B )
      | ( A
       != ( second @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(161,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( non_ordered_pair @ B @ C ) )
      | ~ ( little_set @ A )
      | ( A != C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',non_ordered_pair3) ).

thf(535,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( non_ordered_pair @ B @ C ) )
      | ~ ( little_set @ A )
      | ( A != C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(20,axiom,
    ! [A: $i] :
      ( ~ ( function @ A )
      | ( single_valued_set @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function2) ).

thf(213,plain,
    ! [A: $i] :
      ( ~ ( function @ A )
      | ( single_valued_set @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(139,axiom,
    ! [A: $i] :
      ( ~ ( member @ A @ identity_relation )
      | ( ( first @ A )
        = ( second @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation2) ).

thf(483,plain,
    ! [A: $i] :
      ( ~ ( member @ A @ identity_relation )
      | ( ( first @ A )
        = ( second @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(154,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( homomorphism @ A @ B @ C @ D @ E )
      | ~ ( closed @ B @ C )
      | ~ ( closed @ D @ E )
      | ~ ( maps @ A @ B @ D )
      | ( member @ ( f33 @ A @ B @ C @ D @ E ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism6) ).

thf(520,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( homomorphism @ A @ B @ C @ D @ E )
      | ~ ( closed @ B @ C )
      | ~ ( closed @ D @ E )
      | ~ ( maps @ A @ B @ D )
      | ( member @ ( f33 @ A @ B @ C @ D @ E ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(33,axiom,
    ! [A: $i] :
      ( ~ ( ordered_pair_predicate @ A )
      | ( little_set @ ( f2 @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_predicate1) ).

thf(240,plain,
    ! [A: $i] :
      ( ~ ( ordered_pair_predicate @ A )
      | ( little_set @ ( f2 @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(14,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( little_set @ A )
      | ~ ( little_set @ B )
      | ( ( non_ordered_pair @ C @ A )
       != ( non_ordered_pair @ C @ B ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',left_cancellation) ).

thf(198,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( little_set @ A )
      | ~ ( little_set @ B )
      | ( ( non_ordered_pair @ C @ A )
       != ( non_ordered_pair @ C @ B ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(73,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( apply @ B @ C ) )
      | ~ ( ordered_pair_predicate @ D )
      | ~ ( member @ D @ B )
      | ( ( first @ D )
       != C )
      | ~ ( member @ A @ ( second @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply5) ).

thf(330,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( apply @ B @ C ) )
      | ~ ( ordered_pair_predicate @ D )
      | ~ ( member @ D @ B )
      | ( ( first @ D )
       != C )
      | ~ ( member @ A @ ( second @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(104,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( compose @ B @ C ) )
      | ~ ( little_set @ A )
      | ~ ( little_set @ D )
      | ~ ( little_set @ E )
      | ~ ( little_set @ F )
      | ( A
       != ( ordered_pair @ D @ E ) )
      | ~ ( member @ ( ordered_pair @ D @ F ) @ B )
      | ~ ( member @ ( ordered_pair @ F @ E ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose7) ).

thf(399,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( compose @ B @ C ) )
      | ~ ( little_set @ A )
      | ~ ( little_set @ D )
      | ~ ( little_set @ E )
      | ~ ( little_set @ F )
      | ( A
       != ( ordered_pair @ D @ E ) )
      | ~ ( member @ ( ordered_pair @ D @ F ) @ B )
      | ~ ( member @ ( ordered_pair @ F @ E ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(52,axiom,
    ! [A: $i] :
      ( ~ ( ordered_pair_predicate @ A )
      | ( little_set @ ( second @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',second_component_is_small) ).

thf(278,plain,
    ! [A: $i] :
      ( ~ ( ordered_pair_predicate @ A )
      | ( little_set @ ( second @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(26,axiom,
    ! [B: $i,A: $i] :
      ( ( closed @ A @ B )
      | ~ ( little_set @ A )
      | ~ ( little_set @ B )
      | ~ ( maps @ B @ ( cross_product @ A @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closed4) ).

thf(223,plain,
    ! [B: $i,A: $i] :
      ( ( closed @ A @ B )
      | ~ ( little_set @ A )
      | ~ ( little_set @ B )
      | ~ ( maps @ B @ ( cross_product @ A @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(64,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( single_valued_set @ A )
      | ~ ( little_set @ B )
      | ~ ( little_set @ C )
      | ~ ( little_set @ D )
      | ~ ( member @ ( ordered_pair @ B @ C ) @ A )
      | ~ ( member @ ( ordered_pair @ B @ D ) @ A )
      | ( C = D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_set1) ).

thf(306,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( single_valued_set @ A )
      | ~ ( little_set @ B )
      | ~ ( little_set @ C )
      | ~ ( little_set @ D )
      | ~ ( member @ ( ordered_pair @ B @ C ) @ A )
      | ~ ( member @ ( ordered_pair @ B @ D ) @ A )
      | ( C = D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(120,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( apply @ B @ C ) )
      | ( member @ ( f28 @ A @ B @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply2) ).

thf(438,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( apply @ B @ C ) )
      | ( member @ ( f28 @ A @ B @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(1,negated_conjecture,
    ~ ( member @ ( apply @ a_function @ a ) @ a_range ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_mapping_in_range) ).

thf(167,plain,
    ~ ( member @ ( apply @ a_function @ a ) @ a_range ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).

thf(135,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( second @ B ) )
      | ( little_set @ ( f6 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',second1) ).

thf(473,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( second @ B ) )
      | ( little_set @ ( f6 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(115,axiom,
    ! [A: $i] :
      ~ ( member @ A @ empty_set ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',empty_set) ).

thf(426,plain,
    ! [A: $i] :
      ~ ( member @ A @ empty_set ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(91,axiom,
    member @ empty_set @ infinity,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',infinity2) ).

thf(374,plain,
    member @ empty_set @ infinity,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(40,axiom,
    ! [A: $i] :
      ( ( single_valued_set @ A )
      | ( little_set @ ( f20 @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_set3) ).

thf(255,plain,
    ! [A: $i] :
      ( ( single_valued_set @ A )
      | ( little_set @ ( f20 @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(19,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( maps @ A @ B @ C )
      | ~ ( function @ A )
      | ( ( domain_of @ A )
       != B )
      | ~ ( subset @ ( range_of @ A ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps4) ).

thf(209,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( maps @ A @ B @ C )
      | ~ ( function @ A )
      | ( ( domain_of @ A )
       != B )
      | ~ ( subset @ ( range_of @ A ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(78,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( range_of @ B ) )
      | ( ordered_pair_predicate @ ( f27 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of1) ).

thf(342,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( range_of @ B ) )
      | ( ordered_pair_predicate @ ( f27 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(128,axiom,
    ! [A: $i] :
      ( ( relation @ A )
      | ( member @ ( f18 @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',relation2) ).

thf(458,plain,
    ! [A: $i] :
      ( ( relation @ A )
      | ( member @ ( f18 @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(110,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( converse @ B ) )
      | ( ordered_pair_predicate @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',converse1) ).

thf(414,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( converse @ B ) )
      | ( ordered_pair_predicate @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(108,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( homomorphism @ A @ B @ C @ D @ E )
      | ~ ( member @ F @ B )
      | ~ ( member @ G @ B )
      | ( ( apply @ A @ ( apply_to_two_arguments @ C @ F @ G ) )
        = ( apply_to_two_arguments @ E @ ( apply @ A @ F ) @ ( apply @ A @ G ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism4) ).

thf(409,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( homomorphism @ A @ B @ C @ D @ E )
      | ~ ( member @ F @ B )
      | ~ ( member @ G @ B )
      | ( ( apply @ A @ ( apply_to_two_arguments @ C @ F @ G ) )
        = ( apply_to_two_arguments @ E @ ( apply @ A @ F ) @ ( apply @ A @ G ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(44,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( subset @ A @ B )
      | ~ ( subset @ B @ C )
      | ( subset @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',containment_is_transitive) ).

thf(263,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( subset @ A @ B )
      | ~ ( subset @ B @ C )
      | ( subset @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(89,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( range_of @ B ) )
      | ( member @ ( f27 @ A @ B ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of2) ).

thf(370,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( range_of @ B ) )
      | ( member @ ( f27 @ A @ B ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(146,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( converse @ B ) )
      | ( member @ ( ordered_pair @ ( second @ A ) @ ( first @ A ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',converse2) ).

thf(502,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( converse @ B ) )
      | ( member @ ( ordered_pair @ ( second @ A ) @ ( first @ A ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(130,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( apply @ B @ C ) )
      | ( member @ A @ ( second @ ( f28 @ A @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply4) ).

thf(463,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( apply @ B @ C ) )
      | ( member @ A @ ( second @ ( f28 @ A @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(76,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( compose @ B @ C ) )
      | ( little_set @ ( f31 @ A @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose3) ).

thf(338,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( compose @ B @ C ) )
      | ( little_set @ ( f31 @ A @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(5,axiom,
    ! [A: $i] :
      ( ~ ( one_to_one_function @ A )
      | ( function @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one_function1) ).

thf(177,plain,
    ! [A: $i] :
      ( ~ ( one_to_one_function @ A )
      | ( function @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(7,axiom,
    ! [A: $i] :
      ( ~ ( little_set @ A )
      | ( little_set @ ( powerset @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',powerset3) ).

thf(182,plain,
    ! [A: $i] :
      ( ~ ( little_set @ A )
      | ( little_set @ ( powerset @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(75,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( first @ B ) )
      | ( little_set @ ( f5 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',first2) ).

thf(336,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( first @ B ) )
      | ( little_set @ ( f5 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(67,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( intersection @ B @ C ) )
      | ( member @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection1) ).

thf(314,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( intersection @ B @ C ) )
      | ( member @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(123,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( function @ A )
      | ~ ( member @ B @ ( domain_of @ A ) )
      | ( ( apply @ A @ B )
       != C )
      | ( member @ ( ordered_pair @ B @ C ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply_for_functions2) ).

thf(446,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( function @ A )
      | ~ ( member @ B @ ( domain_of @ A ) )
      | ( ( apply @ A @ B )
       != C )
      | ( member @ ( ordered_pair @ B @ C ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(4,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( maps @ A @ B @ C )
      | ~ ( maps @ D @ C @ E )
      | ( maps @ ( compose @ D @ A ) @ B @ E ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps_for_composition) ).

thf(175,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( maps @ A @ B @ C )
      | ~ ( maps @ D @ C @ E )
      | ( maps @ ( compose @ D @ A ) @ B @ E ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(98,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( image @ B @ C ) )
      | ( ( second @ ( f22 @ A @ B @ C ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image_and_substitution4) ).

thf(386,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( image @ B @ C ) )
      | ( ( second @ ( f22 @ A @ B @ C ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(124,axiom,
    ! [A: $i] :
      ( ~ ( member @ A @ identity_relation )
      | ( ordered_pair_predicate @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation1) ).

thf(450,plain,
    ! [A: $i] :
      ( ~ ( member @ A @ identity_relation )
      | ( ordered_pair_predicate @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(86,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( first @ B ) )
      | ( member @ A @ ( f4 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',first4) ).

thf(363,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( first @ B ) )
      | ( member @ A @ ( f4 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(162,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( flip_range_of @ B ) )
      | ( member @ ( ordered_pair @ ( f12 @ A @ B ) @ ( ordered_pair @ ( f14 @ A @ B ) @ ( f13 @ A @ B ) ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip_range5) ).

thf(539,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( flip_range_of @ B ) )
      | ( member @ ( ordered_pair @ ( f12 @ A @ B ) @ ( ordered_pair @ ( f14 @ A @ B ) @ ( f13 @ A @ B ) ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(156,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( compose @ B @ C ) )
      | ( member @ ( ordered_pair @ ( f31 @ A @ B @ C ) @ ( f30 @ A @ B @ C ) ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose6) ).

thf(524,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( compose @ B @ C ) )
      | ( member @ ( ordered_pair @ ( f31 @ A @ B @ C ) @ ( f30 @ A @ B @ C ) ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(23,axiom,
    ! [A: $i] :
      ( ~ ( one_to_one_function @ A )
      | ( function @ ( converse @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one_function2) ).

thf(218,plain,
    ! [A: $i] :
      ( ~ ( one_to_one_function @ A )
      | ( function @ ( converse @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(88,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( intersection @ B @ C ) )
      | ~ ( member @ A @ B )
      | ~ ( member @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection3) ).

thf(368,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( intersection @ B @ C ) )
      | ~ ( member @ A @ B )
      | ~ ( member @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(38,axiom,
    ! [A: $i] :
      ( ( A = empty_set )
      | ( disjoint @ ( f24 @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity2) ).

thf(250,plain,
    ! [A: $i] :
      ( ( A = empty_set )
      | ( disjoint @ ( f24 @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(131,axiom,
    ! [B: $i,A: $i] : ( subset @ ( apply @ A @ B ) @ ( sigma @ ( image @ ( singleton_set @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image_and_apply1) ).

thf(465,plain,
    ! [B: $i,A: $i] : ( subset @ ( apply @ A @ B ) @ ( sigma @ ( image @ ( singleton_set @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(153,axiom,
    ! [A: $i] :
      ( ~ ( little_set @ A )
      | ( A = empty_set )
      | ( member @ ( ordered_pair @ A @ ( f26 @ A ) ) @ f25 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice3) ).

thf(517,plain,
    ! [A: $i] :
      ( ~ ( little_set @ A )
      | ( A = empty_set )
      | ( member @ ( ordered_pair @ A @ ( f26 @ A ) ) @ f25 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(63,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( compose @ B @ C ) )
      | ( little_set @ ( f30 @ A @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose2) ).

thf(304,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( compose @ B @ C ) )
      | ( little_set @ ( f30 @ A @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(9,axiom,
    ! [A: $i] :
      ( ( function @ A )
      | ~ ( relation @ A )
      | ~ ( single_valued_set @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function3) ).

thf(186,plain,
    ! [A: $i] :
      ( ( function @ A )
      | ~ ( relation @ A )
      | ~ ( single_valued_set @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(112,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( cross_product @ B @ C ) )
      | ( ordered_pair_predicate @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cross_product1) ).

thf(418,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( cross_product @ B @ C ) )
      | ( ordered_pair_predicate @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(94,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( rotate_right @ B ) )
      | ( little_set @ ( f9 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate_right1) ).

thf(378,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( rotate_right @ B ) )
      | ( little_set @ ( f9 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(18,axiom,
    ! [A: $i] :
      ( ~ ( ordered_pair_predicate @ A )
      | ( little_set @ ( f3 @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_predicate2) ).

thf(207,plain,
    ! [A: $i] :
      ( ~ ( ordered_pair_predicate @ A )
      | ( little_set @ ( f3 @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(49,axiom,
    little_set @ infinity,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',infinity1) ).

thf(272,plain,
    little_set @ infinity,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(80,axiom,
    ! [A: $i] :
      ( ~ ( little_set @ A )
      | ( A = empty_set )
      | ( member @ ( f26 @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice2) ).

thf(346,plain,
    ! [A: $i] :
      ( ~ ( little_set @ A )
      | ( A = empty_set )
      | ( member @ ( f26 @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(55,axiom,
    ! [B: $i,A: $i] : ( relation @ ( compose @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',composition_is_a_relation) ).

thf(287,plain,
    ! [B: $i,A: $i] : ( relation @ ( compose @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(31,axiom,
    ! [A: $i] :
      ( ( successor @ A )
      = ( union @ A @ ( singleton_set @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor) ).

thf(235,plain,
    ! [A: $i] :
      ( ( successor @ A )
      = ( union @ A @ ( singleton_set @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(36,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( homomorphism @ A @ B @ C @ D @ E )
      | ( closed @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism1) ).

thf(246,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( homomorphism @ A @ B @ C @ D @ E )
      | ( closed @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(61,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( flip_range_of @ B ) )
      | ( little_set @ ( f13 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip_range2) ).

thf(300,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( flip_range_of @ B ) )
      | ( little_set @ ( f13 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(118,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( image @ B @ C ) )
      | ~ ( little_set @ A )
      | ~ ( ordered_pair_predicate @ D )
      | ~ ( member @ D @ C )
      | ~ ( member @ ( first @ D ) @ B )
      | ( ( second @ D )
       != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image_and_substitution5) ).

thf(432,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( image @ B @ C ) )
      | ~ ( little_set @ A )
      | ~ ( ordered_pair_predicate @ D )
      | ~ ( member @ D @ C )
      | ~ ( member @ ( first @ D ) @ B )
      | ( ( second @ D )
       != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(11,axiom,
    ! [B: $i,A: $i] :
      ( ( proper_subset @ A @ B )
      | ~ ( subset @ A @ B )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',proper_subset3) ).

thf(190,plain,
    ! [B: $i,A: $i] :
      ( ( proper_subset @ A @ B )
      | ~ ( subset @ A @ B )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(50,axiom,
    ! [A: $i] :
      ( ( single_valued_set @ A )
      | ( little_set @ ( f21 @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_set4) ).

thf(273,plain,
    ! [A: $i] :
      ( ( single_valued_set @ A )
      | ( little_set @ ( f21 @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(72,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( second @ B ) )
      | ~ ( little_set @ C )
      | ~ ( little_set @ D )
      | ( B
       != ( ordered_pair @ C @ D ) )
      | ~ ( member @ A @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',second5) ).

thf(326,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( second @ B ) )
      | ~ ( little_set @ C )
      | ~ ( little_set @ D )
      | ( B
       != ( ordered_pair @ C @ D ) )
      | ~ ( member @ A @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(34,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( proper_subset @ A @ B )
      | ( subset @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',proper_subset1) ).

thf(242,plain,
    ! [B: $i,A: $i] :
      ( ~ ( proper_subset @ A @ B )
      | ( subset @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(13,axiom,
    ! [A: $i] :
      ( ( single_valued_set @ A )
      | ( ( f20 @ A )
       != ( f21 @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_set7) ).

thf(195,plain,
    ! [A: $i] :
      ( ( single_valued_set @ A )
      | ( ( f20 @ A )
       != ( f21 @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(160,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( range_of @ B ) )
      | ( A
        = ( second @ ( f27 @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of3) ).

thf(532,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( range_of @ B ) )
      | ( A
        = ( second @ ( f27 @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(148,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( domain_of @ B ) )
      | ( member @ ( f8 @ A @ B ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain2) ).

thf(506,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( domain_of @ B ) )
      | ( member @ ( f8 @ A @ B ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(47,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( closed @ A @ B )
      | ( little_set @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closed2) ).

thf(268,plain,
    ! [B: $i,A: $i] :
      ( ~ ( closed @ A @ B )
      | ( little_set @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(102,axiom,
    ! [B: $i,A: $i] :
      ( ( subset @ A @ B )
      | ( member @ ( f17 @ A @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset2) ).

thf(395,plain,
    ! [B: $i,A: $i] :
      ( ( subset @ A @ B )
      | ( member @ ( f17 @ A @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(158,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( first @ B ) )
      | ( little_set @ ( f4 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',first1) ).

thf(528,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ A @ ( first @ B ) )
      | ( little_set @ ( f4 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(122,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( non_ordered_pair @ B @ C ) )
      | ~ ( little_set @ A )
      | ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',non_ordered_pair2) ).

thf(442,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( non_ordered_pair @ B @ C ) )
      | ~ ( little_set @ A )
      | ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(28,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( little_set @ A )
      | ~ ( little_set @ B )
      | ( ( first @ ( ordered_pair @ A @ B ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_of_first) ).

thf(227,plain,
    ! [B: $i,A: $i] :
      ( ~ ( little_set @ A )
      | ~ ( little_set @ B )
      | ( ( first @ ( ordered_pair @ A @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(140,axiom,
    ! [A: $i] :
      ( ~ ( member @ A @ infinity )
      | ( member @ ( successor @ A ) @ infinity ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',infinity3) ).

thf(486,plain,
    ! [A: $i] :
      ( ~ ( member @ A @ infinity )
      | ( member @ ( successor @ A ) @ infinity ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(83,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( little_set @ A )
      | ~ ( little_set @ B )
      | ~ ( function @ C )
      | ~ ( member @ ( ordered_pair @ A @ B ) @ C )
      | ( ( apply @ C @ A )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply_for_functions1) ).

thf(354,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( little_set @ A )
      | ~ ( little_set @ B )
      | ~ ( function @ C )
      | ~ ( member @ ( ordered_pair @ A @ B ) @ C )
      | ( ( apply @ C @ A )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(65,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( member @ ( f1 @ A @ B ) @ A )
      | ~ ( member @ ( f1 @ A @ B ) @ B )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',extensionality3) ).

thf(309,plain,
    ! [B: $i,A: $i] :
      ( ~ ( member @ ( f1 @ A @ B ) @ A )
      | ~ ( member @ ( f1 @ A @ B ) @ B )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(164,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( domain_of @ B ) )
      | ~ ( little_set @ A )
      | ~ ( ordered_pair_predicate @ C )
      | ~ ( member @ C @ B )
      | ( A
       != ( first @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain4) ).

thf(543,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( domain_of @ B ) )
      | ~ ( little_set @ A )
      | ~ ( ordered_pair_predicate @ C )
      | ~ ( member @ C @ B )
      | ( A
       != ( first @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(21,axiom,
    ! [A: $i] :
      ( ( singleton_set @ A )
      = ( non_ordered_pair @ A @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_set) ).

thf(215,plain,
    ! [A: $i] :
      ( ( singleton_set @ A )
      = ( non_ordered_pair @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(142,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( function @ A )
      | ( little_set @ ( apply @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function_values_are_small) ).

thf(491,plain,
    ! [B: $i,A: $i] :
      ( ~ ( function @ A )
      | ( little_set @ ( apply @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(144,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( first @ B ) )
      | ~ ( little_set @ C )
      | ~ ( little_set @ D )
      | ( B
       != ( ordered_pair @ C @ D ) )
      | ~ ( member @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',first5) ).

thf(496,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( member @ A @ ( first @ B ) )
      | ~ ( little_set @ C )
      | ~ ( little_set @ D )
      | ( B
       != ( ordered_pair @ C @ D ) )
      | ~ ( member @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(126,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( cross_product @ B @ C ) )
      | ( member @ ( first @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cross_product2) ).

thf(454,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( member @ A @ ( cross_product @ B @ C ) )
      | ( member @ ( first @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(576,plain,
    $false,
    inference(cvc4,[status(thm)],[365,468,481,352,217,202,440,511,385,550,500,340,404,288,436,257,389,184,372,504,344,357,460,284,416,316,261,475,179,321,253,397,280,221,293,428,265,312,248,380,270,201,220,233,452,334,302,349,424,407,275,361,522,493,456,324,403,188,169,420,471,488,225,193,526,547,393,515,173,237,244,266,205,508,541,530,298,412,296,375,513,430,291,259,391,382,466,377,230,318,477,535,213,483,520,240,198,330,399,278,223,306,438,167,473,426,374,255,209,342,458,414,409,263,370,502,463,338,177,182,336,314,446,175,386,450,363,539,524,218,368,250,465,517,304,186,418,378,207,272,346,287,235,246,300,432,190,273,326,242,195,532,506,268,395,528,442,227,486,354,309,543,215,491,496,454]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12  % Problem  : SET038-3 : TPTP v8.2.0. Released v1.0.0.
% 0.13/0.15  % Command  : run_Leo-III %s %d
% 0.16/0.36  % Computer : n018.cluster.edu
% 0.16/0.36  % Model    : x86_64 x86_64
% 0.16/0.36  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.36  % Memory   : 8042.1875MB
% 0.16/0.36  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.16/0.36  % CPULimit : 300
% 0.16/0.36  % WCLimit  : 300
% 0.16/0.36  % DateTime : Mon May 20 11:18:54 EDT 2024
% 0.16/0.36  % CPUTime  : 
% 0.97/0.89  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 1.42/1.11  % [INFO] 	 Parsing done (209ms). 
% 1.42/1.12  % [INFO] 	 Running in sequential loop mode. 
% 1.97/1.37  % [INFO] 	 eprover registered as external prover. 
% 1.97/1.37  % [INFO] 	 cvc4 registered as external prover. 
% 1.97/1.37  % [INFO] 	 Scanning for conjecture ... 
% 2.22/1.46  % [INFO] 	 Found a conjecture (or negated_conjecture) and 165 axioms. Running axiom selection ... 
% 2.34/1.53  % [INFO] 	 Axiom selection finished. Selected 165 axioms (removed 0 axioms). 
% 2.85/1.63  % [INFO] 	 Problem is propositional (TPTP CNF). 
% 2.85/1.64  % [INFO] 	 Type checking passed. 
% 2.85/1.64  % [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 ... 
% 6.04/2.62  % External prover 'cvc4' found a proof!
% 6.04/2.62  % [INFO] 	 Killing All external provers ... 
% 6.04/2.63  % Time passed: 2105ms (effective reasoning time: 1503ms)
% 6.04/2.63  % Axioms used in derivation (165): cross_product4, complement1, single_valued_set4, image_and_substitution2, second2, function_values_are_small, maps4, first2, first1, function3, apply5, homomorphism4, flip_range5, closed3, infinity3, range_of3, apply1, regularity1, sigma2, proper_subset3, domain4, singleton_set, compose4, sigma1, relation1, ordered_pair_predicate2, single_valued_set7, one_to_one_function3, element_relation2, powerset1, union, rotate_right1, identity_relation2, apply_to_two_arguments, compose2, homomorphism3, image_and_substitution5, second5, rotate_right3, successor, non_ordered_pair2, empty_set, homomorphism7, member_of_domain, choice3, ordered_pair, image_and_substitution3, containment_is_transitive, maps_for_composition, closed4, homomorphism1, flip_range4, rotate_right6, disjoint2, range_of_composition, infinity2, property_of_first, subset3, compose3, apply4, image_and_substitution6, choice1, second3, single_valued_set3, range_of4, left_cancellation, converse3, domain3, regularity2, first3, cross_product3, rotate_right2, element_relation3, homomorphism6, extensionality2, intersection1, flip_range1, property_of_second, subset2, compose7, first_component_is_small, relation2, non_ordered_pair1, ordered_pair_predicate1, sigma4, proper_subset1, one_to_one_function2, maps1, property_of_singleton_sets, domain_of_composition, first4, compose6, function1, maps3, domain2, intersection3, second4, rotate_right5, identity_relation3, closed1, converse2, disjoint1, disjoint3, composition_is_a_function, second1, element_relation1, infinity1, image_and_substitution4, cross_product2, one_to_one_function1, extensionality1, non_ordered_pair4, image_and_apply1, sigma3, ordered_pair_predicate4, second_components_are_equal, single_valued_set2, powerset3, two_sets_equal, homomorphism5, apply3, flip_range2, maps2, ordered_pairs_are_small1, relation3, apply_for_functions2, image_and_substitution1, rotate_right4, second_component_is_small, complement2, flip_range6, cross_product1, function2, compose5, apply2, range_of2, universal_set, composition_is_a_relation, domain1, closed2, identity_relation1, subset1, proper_subset2, a_mapping, converse1, single_valued_set5, first_components_are_equal, intersection2, image_and_apply2, powerset2, first5, non_ordered_pair3, single_valued_set1, ordered_pair_predicate3, choice2, restrict, homomorphism2, a2, ordered_pairs_are_small2, single_valued_set6, extensionality3, apply_for_functions1, flip_range3, compose1, range_of1
% 6.04/2.63  % No. of inferences in proof: 333
% 6.04/2.63  % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : 2105 ms resp. 1503 ms w/o parsing
% 6.52/2.70  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 6.52/2.71  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------