TSTP Solution File: SET752+4 by Leo-III---1.7.7
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%------------------------------------------------------------------------------
% File : Leo-III---1.7.7
% Problem : SET752+4 : TPTP v8.1.2. Bugfixed v2.2.1.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n001.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 : Fri May 19 11:54:13 EDT 2023
% Result : Theorem 74.65s 13.46s
% Output : Refutation 74.65s
% Verified :
% SZS Type : Refutation
% Derivation depth : 24
% Number of leaves : 68
% Syntax : Number of formulae : 191 ( 40 unt; 39 typ; 0 def)
% Number of atoms : 614 ( 89 equ; 0 cnn)
% Maximal formula atoms : 26 ( 4 avg)
% Number of connectives : 2190 ( 108 ~; 69 |; 204 &;1642 @)
% ( 30 <=>; 137 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 96 ( 96 >; 0 *; 0 +; 0 <<)
% Number of symbols : 41 ( 39 usr; 7 con; 0-6 aty)
% Number of variables : 555 ( 0 ^; 525 !; 30 ?; 555 :)
% Comments :
%------------------------------------------------------------------------------
thf(maps_type,type,
maps: $i > $i > $i > $o ).
thf(subset_type,type,
subset: $i > $i > $o ).
thf(equal_set_type,type,
equal_set: $i > $i > $o ).
thf(image2_type,type,
image2: $i > $i > $i ).
thf(union_type,type,
union: $i > $i > $i ).
thf(equal_maps_type,type,
equal_maps: $i > $i > $i > $i > $o ).
thf(member_type,type,
member: $i > $i > $o ).
thf(apply_type,type,
apply: $i > $i > $i > $o ).
thf(compose_function_type,type,
compose_function: $i > $i > $i > $i > $i > $i ).
thf(increasing_type,type,
increasing: $i > $i > $i > $i > $i > $o ).
thf(injective_type,type,
injective: $i > $i > $i > $o ).
thf(empty_set_type,type,
empty_set: $i ).
thf(intersection_type,type,
intersection: $i > $i > $i ).
thf(inverse_predicate_type,type,
inverse_predicate: $i > $i > $i > $i > $o ).
thf(unordered_pair_type,type,
unordered_pair: $i > $i > $i ).
thf(inverse_function_type,type,
inverse_function: $i > $i > $i > $i ).
thf(singleton_type,type,
singleton: $i > $i ).
thf(one_to_one_type,type,
one_to_one: $i > $i > $i > $o ).
thf(surjective_type,type,
surjective: $i > $i > $i > $o ).
thf(product_type,type,
product: $i > $i ).
thf(compose_predicate_type,type,
compose_predicate: $i > $i > $i > $i > $i > $i > $o ).
thf(decreasing_type,type,
decreasing: $i > $i > $i > $i > $i > $o ).
thf(inverse_image3_type,type,
inverse_image3: $i > $i > $i > $i ).
thf(difference_type,type,
difference: $i > $i > $i ).
thf(inverse_image2_type,type,
inverse_image2: $i > $i > $i ).
thf(identity_type,type,
identity: $i > $i > $o ).
thf(sum_type,type,
sum: $i > $i ).
thf(image3_type,type,
image3: $i > $i > $i > $i ).
thf(isomorphism_type,type,
isomorphism: $i > $i > $i > $i > $i > $o ).
thf(power_set_type,type,
power_set: $i > $i ).
thf(sk1_type,type,
sk1: $i ).
thf(sk2_type,type,
sk2: $i ).
thf(sk3_type,type,
sk3: $i ).
thf(sk4_type,type,
sk4: $i ).
thf(sk5_type,type,
sk5: $i ).
thf(sk6_type,type,
sk6: $i > $i > $i > $i > $i ).
thf(sk7_type,type,
sk7: $i > $i > $i > $i > $i ).
thf(sk8_type,type,
sk8: $i > $i > $i > $i > $i ).
thf(sk33_type,type,
sk33: $i > $i > $i ).
thf(16,axiom,
! [A: $i,B: $i,C: $i] :
( ( surjective @ A @ B @ C )
<=> ! [D: $i] :
( ( member @ D @ C )
=> ? [E: $i] :
( ( member @ E @ B )
& ( apply @ A @ E @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',surjective) ).
thf(174,plain,
! [A: $i,B: $i,C: $i] :
( ( ( surjective @ A @ B @ C )
=> ! [D: $i] :
( ( member @ D @ C )
=> ? [E: $i] :
( ( member @ E @ B )
& ( apply @ A @ E @ D ) ) ) )
& ( ! [D: $i] :
( ( member @ D @ C )
=> ? [E: $i] :
( ( member @ E @ B )
& ( apply @ A @ E @ D ) ) )
=> ( surjective @ A @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(10,axiom,
! [A: $i,B: $i,C: $i] :
( ( member @ A @ ( unordered_pair @ B @ C ) )
<=> ( ( A = B )
| ( A = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair) ).
thf(119,plain,
! [A: $i,B: $i,C: $i] :
( ( ( member @ A @ ( unordered_pair @ B @ C ) )
=> ( ( A = B )
| ( A = C ) ) )
& ( ( ( A = B )
| ( A = C ) )
=> ( member @ A @ ( unordered_pair @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(120,plain,
( ! [A: $i,B: $i,C: $i] :
( ( member @ A @ ( unordered_pair @ B @ C ) )
=> ( ( A = B )
| ( A = C ) ) )
& ! [A: $i,B: $i,C: $i] :
( ( ( A = B )
| ( A = C ) )
=> ( member @ A @ ( unordered_pair @ B @ C ) ) ) ),
inference(miniscope,[status(thm)],[119]) ).
thf(121,plain,
! [C: $i,B: $i,A: $i] :
( ( A != B )
| ( member @ A @ ( unordered_pair @ B @ C ) ) ),
inference(cnf,[status(esa)],[120]) ).
thf(124,plain,
! [C: $i,B: $i,A: $i] :
( ( A != B )
| ( member @ A @ ( unordered_pair @ B @ C ) ) ),
inference(lifteq,[status(thm)],[121]) ).
thf(125,plain,
! [B: $i,A: $i] : ( member @ A @ ( unordered_pair @ A @ B ) ),
inference(simp,[status(thm)],[124]) ).
thf(12,axiom,
! [A: $i,B: $i] :
( ( member @ A @ ( singleton @ B ) )
<=> ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton) ).
thf(132,plain,
! [A: $i,B: $i] :
( ( ( member @ A @ ( singleton @ B ) )
=> ( A = B ) )
& ( ( A = B )
=> ( member @ A @ ( singleton @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(8,axiom,
! [A: $i,B: $i,C: $i] :
( ( member @ A @ ( intersection @ B @ C ) )
<=> ( ( member @ A @ B )
& ( member @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection) ).
thf(97,plain,
! [A: $i,B: $i,C: $i] :
( ( ( member @ A @ ( intersection @ B @ C ) )
=> ( ( member @ A @ B )
& ( member @ A @ C ) ) )
& ( ( ( member @ A @ B )
& ( member @ A @ C ) )
=> ( member @ A @ ( intersection @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(4,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
( ( ( member @ F @ C )
& ( member @ G @ E ) )
=> ( ( apply @ ( compose_function @ A @ B @ C @ D @ E ) @ F @ G )
<=> ? [H: $i] :
( ( member @ H @ D )
& ( apply @ B @ F @ H )
& ( apply @ A @ H @ G ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose_function) ).
thf(53,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
( ( ( member @ F @ C )
& ( member @ G @ E ) )
=> ( ( ( apply @ ( compose_function @ A @ B @ C @ D @ E ) @ F @ G )
=> ? [H: $i] :
( ( member @ H @ D )
& ( apply @ B @ F @ H )
& ( apply @ A @ H @ G ) ) )
& ( ? [H: $i] :
( ( member @ H @ D )
& ( apply @ B @ F @ H )
& ( apply @ A @ H @ G ) )
=> ( apply @ ( compose_function @ A @ B @ C @ D @ E ) @ F @ G ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(1,conjecture,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( maps @ A @ B @ C )
& ( subset @ D @ B )
& ( subset @ E @ B ) )
=> ( equal_set @ ( image2 @ A @ ( union @ D @ E ) ) @ ( union @ ( image2 @ A @ D ) @ ( image2 @ A @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',thIIa02) ).
thf(2,negated_conjecture,
~ ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( maps @ A @ B @ C )
& ( subset @ D @ B )
& ( subset @ E @ B ) )
=> ( equal_set @ ( image2 @ A @ ( union @ D @ E ) ) @ ( union @ ( image2 @ A @ D ) @ ( image2 @ A @ E ) ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(31,plain,
~ ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( maps @ A @ B @ C )
& ( subset @ D @ B )
& ( subset @ E @ B ) )
=> ( equal_set @ ( image2 @ A @ ( union @ D @ E ) ) @ ( union @ ( image2 @ A @ D ) @ ( image2 @ A @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(35,plain,
maps @ sk1 @ sk2 @ sk3,
inference(cnf,[status(esa)],[31]) ).
thf(29,axiom,
! [A: $i,B: $i] :
( ( equal_set @ A @ B )
<=> ( ( subset @ A @ B )
& ( subset @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_set) ).
thf(310,plain,
! [A: $i,B: $i] :
( ( ( equal_set @ A @ B )
=> ( ( subset @ A @ B )
& ( subset @ B @ A ) ) )
& ( ( ( subset @ A @ B )
& ( subset @ B @ A ) )
=> ( equal_set @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(3,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( equal_maps @ A @ B @ C @ D )
<=> ! [E: $i,F: $i,G: $i] :
( ( ( member @ E @ C )
& ( member @ F @ D )
& ( member @ G @ D ) )
=> ( ( ( apply @ A @ E @ F )
& ( apply @ B @ E @ G ) )
=> ( F = G ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_maps) ).
thf(36,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( equal_maps @ A @ B @ C @ D )
=> ! [E: $i,F: $i,G: $i] :
( ( ( member @ E @ C )
& ( member @ F @ D )
& ( member @ G @ D ) )
=> ( ( ( apply @ A @ E @ F )
& ( apply @ B @ E @ G ) )
=> ( F = G ) ) ) )
& ( ! [E: $i,F: $i,G: $i] :
( ( ( member @ E @ C )
& ( member @ F @ D )
& ( member @ G @ D ) )
=> ( ( ( apply @ A @ E @ F )
& ( apply @ B @ E @ G ) )
=> ( F = G ) ) )
=> ( equal_maps @ A @ B @ C @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(37,plain,
( ! [A: $i,B: $i,C: $i,D: $i] :
( ( equal_maps @ A @ B @ C @ D )
=> ! [E: $i,F: $i,G: $i] :
( ( ( member @ E @ C )
& ( member @ F @ D )
& ( member @ G @ D ) )
=> ( ( ( apply @ A @ E @ F )
& ( apply @ B @ E @ G ) )
=> ( F = G ) ) ) )
& ! [A: $i,B: $i,C: $i,D: $i] :
( ! [E: $i,F: $i,G: $i] :
( ( ( member @ E @ C )
& ( member @ F @ D )
& ( member @ G @ D ) )
=> ( ( ( apply @ A @ E @ F )
& ( apply @ B @ E @ G ) )
=> ( F = G ) ) )
=> ( equal_maps @ A @ B @ C @ D ) ) ),
inference(miniscope,[status(thm)],[36]) ).
thf(41,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( member @ ( sk8 @ D @ C @ B @ A ) @ D )
| ( equal_maps @ A @ B @ C @ D ) ),
inference(cnf,[status(esa)],[37]) ).
thf(47,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( member @ ( sk8 @ D @ C @ B @ A ) @ D )
| ( equal_maps @ A @ B @ C @ D ) ),
inference(simp,[status(thm)],[41]) ).
thf(22,axiom,
! [A: $i,B: $i] :
( ( member @ A @ ( sum @ B ) )
<=> ? [C: $i] :
( ( member @ C @ B )
& ( member @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum) ).
thf(227,plain,
! [A: $i,B: $i] :
( ( ( member @ A @ ( sum @ B ) )
=> ? [C: $i] :
( ( member @ C @ B )
& ( member @ A @ C ) ) )
& ( ? [C: $i] :
( ( member @ C @ B )
& ( member @ A @ C ) )
=> ( member @ A @ ( sum @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(133,plain,
( ! [A: $i,B: $i] :
( ( member @ A @ ( singleton @ B ) )
=> ( A = B ) )
& ! [A: $i,B: $i] :
( ( A = B )
=> ( member @ A @ ( singleton @ B ) ) ) ),
inference(miniscope,[status(thm)],[132]) ).
thf(135,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ B ) )
| ( A = B ) ),
inference(cnf,[status(esa)],[133]) ).
thf(138,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( member @ A @ ( singleton @ B ) ) ),
inference(lifteq,[status(thm)],[135]) ).
thf(134,plain,
! [B: $i,A: $i] :
( ( A != B )
| ( member @ A @ ( singleton @ B ) ) ),
inference(cnf,[status(esa)],[133]) ).
thf(136,plain,
! [B: $i,A: $i] :
( ( A != B )
| ( member @ A @ ( singleton @ B ) ) ),
inference(lifteq,[status(thm)],[134]) ).
thf(137,plain,
! [A: $i] : ( member @ A @ ( singleton @ A ) ),
inference(simp,[status(thm)],[136]) ).
thf(7,axiom,
! [A: $i] :
~ ( member @ A @ empty_set ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',empty_set) ).
thf(94,plain,
! [A: $i] :
~ ( member @ A @ empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(95,plain,
~ ? [A: $i] : ( member @ A @ empty_set ),
inference(miniscope,[status(thm)],[94]) ).
thf(96,plain,
! [A: $i] :
~ ( member @ A @ empty_set ),
inference(cnf,[status(esa)],[95]) ).
thf(328,plain,
! [B: $i,A: $i] :
( ( member @ A @ ( singleton @ A ) )
!= ( member @ B @ empty_set ) ),
inference(paramod_ordered,[status(thm)],[137,96]) ).
thf(330,plain,
! [B: $i,A: $i] :
( ( A != B )
| ( ( singleton @ A )
!= empty_set ) ),
inference(simp,[status(thm)],[328]) ).
thf(333,plain,
! [A: $i] :
( ( singleton @ A )
!= empty_set ),
inference(simp,[status(thm)],[330]) ).
thf(122,plain,
! [C: $i,B: $i,A: $i] :
( ( A != C )
| ( member @ A @ ( unordered_pair @ B @ C ) ) ),
inference(cnf,[status(esa)],[120]) ).
thf(126,plain,
! [C: $i,B: $i,A: $i] :
( ( A != C )
| ( member @ A @ ( unordered_pair @ B @ C ) ) ),
inference(lifteq,[status(thm)],[122]) ).
thf(127,plain,
! [B: $i,A: $i] : ( member @ B @ ( unordered_pair @ A @ B ) ),
inference(simp,[status(thm)],[126]) ).
thf(355,plain,
! [C: $i,B: $i,A: $i] :
( ( member @ B @ ( unordered_pair @ A @ B ) )
!= ( member @ C @ empty_set ) ),
inference(paramod_ordered,[status(thm)],[127,96]) ).
thf(357,plain,
! [C: $i,B: $i,A: $i] :
( ( B != C )
| ( ( unordered_pair @ A @ B )
!= empty_set ) ),
inference(simp,[status(thm)],[355]) ).
thf(360,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ A @ B )
!= empty_set ),
inference(simp,[status(thm)],[357]) ).
thf(400,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ B ) )
| ( A != empty_set )
| ( B
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[138,360]) ).
thf(401,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ ( unordered_pair @ B @ C ) ) )
| ( A != empty_set ) ),
inference(pattern_uni,[status(thm)],[400:[bind(A,$thf( A )),bind(B,$thf( unordered_pair @ E @ F )),bind(C,$thf( E )),bind(D,$thf( F ))]]) ).
thf(550,plain,
! [B: $i,A: $i] :
~ ( member @ empty_set @ ( singleton @ ( unordered_pair @ A @ B ) ) ),
inference(simp,[status(thm)],[401]) ).
thf(430,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ B ) )
| ( B != empty_set )
| ( A
!= ( singleton @ C ) ) ),
inference(paramod_ordered,[status(thm)],[138,333]) ).
thf(431,plain,
! [B: $i,A: $i] :
( ~ ( member @ ( singleton @ B ) @ ( singleton @ A ) )
| ( A != empty_set ) ),
inference(pattern_uni,[status(thm)],[430:[bind(A,$thf( singleton @ D )),bind(B,$thf( B )),bind(C,$thf( D ))]]) ).
thf(568,plain,
! [A: $i] :
~ ( member @ ( singleton @ A ) @ ( singleton @ empty_set ) ),
inference(simp,[status(thm)],[431]) ).
thf(631,plain,
! [B: $i,A: $i] :
( ( member @ A @ ( singleton @ A ) )
!= ( member @ ( singleton @ B ) @ ( singleton @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[137,568]) ).
thf(640,plain,
! [B: $i,A: $i] :
( ( A
!= ( singleton @ B ) )
| ( ( singleton @ A )
!= ( singleton @ empty_set ) ) ),
inference(simp,[status(thm)],[631]) ).
thf(641,plain,
! [A: $i] :
( ( singleton @ ( singleton @ A ) )
!= ( singleton @ empty_set ) ),
inference(simp,[status(thm)],[640]) ).
thf(690,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ B ) )
| ( B
!= ( singleton @ empty_set ) )
| ( A
!= ( singleton @ ( singleton @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[138,641]) ).
thf(691,plain,
! [B: $i,A: $i] :
( ~ ( member @ ( singleton @ ( singleton @ B ) ) @ ( singleton @ A ) )
| ( A
!= ( singleton @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[690:[bind(A,$thf( singleton @ ( singleton @ E ) )),bind(B,$thf( B )),bind(C,$thf( E ))]]) ).
thf(698,plain,
! [A: $i] :
~ ( member @ ( singleton @ ( singleton @ A ) ) @ ( singleton @ ( singleton @ empty_set ) ) ),
inference(simp,[status(thm)],[691]) ).
thf(38,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( member @ ( sk6 @ D @ C @ B @ A ) @ C )
| ( equal_maps @ A @ B @ C @ D ) ),
inference(cnf,[status(esa)],[37]) ).
thf(46,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( member @ ( sk6 @ D @ C @ B @ A ) @ C )
| ( equal_maps @ A @ B @ C @ D ) ),
inference(simp,[status(thm)],[38]) ).
thf(15,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( compose_predicate @ A @ B @ C @ D @ E @ F )
<=> ! [G: $i,H: $i] :
( ( ( member @ G @ D )
& ( member @ H @ F ) )
=> ( ( apply @ A @ G @ H )
<=> ? [I: $i] :
( ( member @ I @ E )
& ( apply @ C @ G @ I )
& ( apply @ B @ I @ H ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose_predicate) ).
thf(152,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( compose_predicate @ A @ B @ C @ D @ E @ F )
=> ! [G: $i,H: $i] :
( ( ( member @ G @ D )
& ( member @ H @ F ) )
=> ( ( ( apply @ A @ G @ H )
=> ? [I: $i] :
( ( member @ I @ E )
& ( apply @ C @ G @ I )
& ( apply @ B @ I @ H ) ) )
& ( ? [I: $i] :
( ( member @ I @ E )
& ( apply @ C @ G @ I )
& ( apply @ B @ I @ H ) )
=> ( apply @ A @ G @ H ) ) ) ) )
& ( ! [G: $i,H: $i] :
( ( ( member @ G @ D )
& ( member @ H @ F ) )
=> ( ( ( apply @ A @ G @ H )
=> ? [I: $i] :
( ( member @ I @ E )
& ( apply @ C @ G @ I )
& ( apply @ B @ I @ H ) ) )
& ( ? [I: $i] :
( ( member @ I @ E )
& ( apply @ C @ G @ I )
& ( apply @ B @ I @ H ) )
=> ( apply @ A @ G @ H ) ) ) )
=> ( compose_predicate @ A @ B @ C @ D @ E @ F ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(30,axiom,
! [A: $i,B: $i] :
( ( member @ A @ ( power_set @ B ) )
<=> ( subset @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_set) ).
thf(316,plain,
! [A: $i,B: $i] :
( ( ( member @ A @ ( power_set @ B ) )
=> ( subset @ A @ B ) )
& ( ( subset @ A @ B )
=> ( member @ A @ ( power_set @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(23,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( member @ D @ ( image3 @ A @ B @ C ) )
<=> ( ( member @ D @ C )
& ? [E: $i] :
( ( member @ E @ B )
& ( apply @ A @ E @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',image3) ).
thf(233,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( member @ D @ ( image3 @ A @ B @ C ) )
=> ( ( member @ D @ C )
& ? [E: $i] :
( ( member @ E @ B )
& ( apply @ A @ E @ D ) ) ) )
& ( ( ( member @ D @ C )
& ? [E: $i] :
( ( member @ E @ B )
& ( apply @ A @ E @ D ) ) )
=> ( member @ D @ ( image3 @ A @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(18,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( member @ D @ ( inverse_image3 @ A @ B @ C ) )
<=> ( ( member @ D @ C )
& ? [E: $i] :
( ( member @ E @ B )
& ( apply @ A @ D @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inverse_image3) ).
thf(201,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( member @ D @ ( inverse_image3 @ A @ B @ C ) )
=> ( ( member @ D @ C )
& ? [E: $i] :
( ( member @ E @ B )
& ( apply @ A @ D @ E ) ) ) )
& ( ( ( member @ D @ C )
& ? [E: $i] :
( ( member @ E @ B )
& ( apply @ A @ D @ E ) ) )
=> ( member @ D @ ( inverse_image3 @ A @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(21,axiom,
! [A: $i,B: $i] :
( ( identity @ A @ B )
<=> ! [C: $i] :
( ( member @ C @ B )
=> ( apply @ A @ C @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',identity) ).
thf(220,plain,
! [A: $i,B: $i] :
( ( ( identity @ A @ B )
=> ! [C: $i] :
( ( member @ C @ B )
=> ( apply @ A @ C @ C ) ) )
& ( ! [C: $i] :
( ( member @ C @ B )
=> ( apply @ A @ C @ C ) )
=> ( identity @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(724,plain,
! [B: $i,A: $i] :
( ( member @ A @ ( singleton @ A ) )
!= ( member @ ( singleton @ ( singleton @ B ) ) @ ( singleton @ ( singleton @ empty_set ) ) ) ),
inference(paramod_ordered,[status(thm)],[137,698]) ).
thf(730,plain,
! [B: $i,A: $i] :
( ( A
!= ( singleton @ ( singleton @ B ) ) )
| ( ( singleton @ A )
!= ( singleton @ ( singleton @ empty_set ) ) ) ),
inference(simp,[status(thm)],[724]) ).
thf(746,plain,
! [A: $i] :
( ( singleton @ ( singleton @ ( singleton @ A ) ) )
!= ( singleton @ ( singleton @ empty_set ) ) ),
inference(simp,[status(thm)],[730]) ).
thf(33,plain,
subset @ sk5 @ sk2,
inference(cnf,[status(esa)],[31]) ).
thf(221,plain,
( ! [A: $i,B: $i] :
( ( identity @ A @ B )
=> ! [C: $i] :
( ( member @ C @ B )
=> ( apply @ A @ C @ C ) ) )
& ! [A: $i,B: $i] :
( ! [C: $i] :
( ( member @ C @ B )
=> ( apply @ A @ C @ C ) )
=> ( identity @ A @ B ) ) ),
inference(miniscope,[status(thm)],[220]) ).
thf(222,plain,
! [B: $i,A: $i] :
( ( member @ ( sk33 @ B @ A ) @ B )
| ( identity @ A @ B ) ),
inference(cnf,[status(esa)],[221]) ).
thf(225,plain,
! [B: $i,A: $i] :
( ( member @ ( sk33 @ B @ A ) @ B )
| ( identity @ A @ B ) ),
inference(simp,[status(thm)],[222]) ).
thf(6,axiom,
! [A: $i,B: $i,C: $i] :
( ( injective @ A @ B @ C )
<=> ! [D: $i,E: $i,F: $i] :
( ( ( member @ D @ B )
& ( member @ E @ B )
& ( member @ F @ C ) )
=> ( ( ( apply @ A @ D @ F )
& ( apply @ A @ E @ F ) )
=> ( D = E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',injective) ).
thf(77,plain,
! [A: $i,B: $i,C: $i] :
( ( ( injective @ A @ B @ C )
=> ! [D: $i,E: $i,F: $i] :
( ( ( member @ D @ B )
& ( member @ E @ B )
& ( member @ F @ C ) )
=> ( ( ( apply @ A @ D @ F )
& ( apply @ A @ E @ F ) )
=> ( D = E ) ) ) )
& ( ! [D: $i,E: $i,F: $i] :
( ( ( member @ D @ B )
& ( member @ E @ B )
& ( member @ F @ C ) )
=> ( ( ( apply @ A @ D @ F )
& ( apply @ A @ E @ F ) )
=> ( D = E ) ) )
=> ( injective @ A @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(19,axiom,
! [A: $i,B: $i,C: $i] :
( ( member @ A @ ( difference @ C @ B ) )
<=> ( ( member @ A @ C )
& ~ ( member @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',difference) ).
thf(208,plain,
! [A: $i,B: $i,C: $i] :
( ( ( member @ A @ ( difference @ C @ B ) )
=> ( ( member @ A @ C )
& ~ ( member @ A @ B ) ) )
& ( ( ( member @ A @ C )
& ~ ( member @ A @ B ) )
=> ( member @ A @ ( difference @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(209,plain,
( ! [A: $i,B: $i,C: $i] :
( ( member @ A @ ( difference @ C @ B ) )
=> ( ( member @ A @ C )
& ~ ( member @ A @ B ) ) )
& ! [A: $i,B: $i,C: $i] :
( ( ( member @ A @ C )
& ~ ( member @ A @ B ) )
=> ( member @ A @ ( difference @ C @ B ) ) ) ),
inference(miniscope,[status(thm)],[208]) ).
thf(212,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( difference @ C @ B ) )
| ~ ( member @ A @ B ) ),
inference(cnf,[status(esa)],[209]) ).
thf(11,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( member @ D @ B )
& ( member @ E @ C ) )
=> ( ( apply @ A @ D @ E )
<=> ( apply @ ( inverse_function @ A @ B @ C ) @ E @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inverse_function) ).
thf(129,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( member @ D @ B )
& ( member @ E @ C ) )
=> ( ( ( apply @ A @ D @ E )
=> ( apply @ ( inverse_function @ A @ B @ C ) @ E @ D ) )
& ( ( apply @ ( inverse_function @ A @ B @ C ) @ E @ D )
=> ( apply @ A @ D @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(432,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ B ) )
| ( A != empty_set )
| ( B
!= ( singleton @ C ) ) ),
inference(paramod_ordered,[status(thm)],[138,333]) ).
thf(433,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ ( singleton @ B ) ) )
| ( A != empty_set ) ),
inference(pattern_uni,[status(thm)],[432:[bind(A,$thf( A )),bind(B,$thf( singleton @ D )),bind(C,$thf( D ))]]) ).
thf(569,plain,
! [A: $i] :
~ ( member @ empty_set @ ( singleton @ ( singleton @ A ) ) ),
inference(simp,[status(thm)],[433]) ).
thf(326,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ B @ ( difference @ D @ C ) )
| ( ( member @ A @ ( singleton @ A ) )
!= ( member @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[137,212]) ).
thf(327,plain,
! [B: $i,A: $i] :
~ ( member @ B @ ( difference @ A @ ( singleton @ B ) ) ),
inference(pattern_uni,[status(thm)],[326:[bind(A,$thf( E )),bind(B,$thf( E )),bind(C,$thf( singleton @ E ))]]) ).
thf(331,plain,
! [B: $i,A: $i] :
~ ( member @ B @ ( difference @ A @ ( singleton @ B ) ) ),
inference(simp,[status(thm)],[327]) ).
thf(362,plain,
! [C: $i,B: $i,A: $i] :
( ( member @ A @ ( singleton @ A ) )
!= ( member @ C @ ( difference @ B @ ( singleton @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[137,331]) ).
thf(366,plain,
! [C: $i,B: $i,A: $i] :
( ( A != C )
| ( ( singleton @ A )
!= ( difference @ B @ ( singleton @ C ) ) ) ),
inference(simp,[status(thm)],[362]) ).
thf(371,plain,
! [B: $i,A: $i] :
( ( difference @ A @ ( singleton @ B ) )
!= ( singleton @ B ) ),
inference(simp,[status(thm)],[366]) ).
thf(25,axiom,
! [A: $i,B: $i,C: $i] :
( ( member @ C @ ( image2 @ A @ B ) )
<=> ? [D: $i] :
( ( member @ D @ B )
& ( apply @ A @ D @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',image2) ).
thf(266,plain,
! [A: $i,B: $i,C: $i] :
( ( ( member @ C @ ( image2 @ A @ B ) )
=> ? [D: $i] :
( ( member @ D @ B )
& ( apply @ A @ D @ C ) ) )
& ( ? [D: $i] :
( ( member @ D @ B )
& ( apply @ A @ D @ C ) )
=> ( member @ C @ ( image2 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(444,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ B ) )
| ( subset @ A @ sk2 )
| ( B != sk5 ) ),
inference(paramod_ordered,[status(thm)],[138,33]) ).
thf(445,plain,
! [A: $i] :
( ~ ( member @ A @ ( singleton @ sk5 ) )
| ( subset @ A @ sk2 ) ),
inference(pattern_uni,[status(thm)],[444:[bind(A,$thf( A )),bind(B,$thf( sk5 ))]]) ).
thf(817,plain,
! [C: $i,B: $i,A: $i] :
( ( identity @ A @ B )
| ( subset @ C @ sk2 )
| ( ( member @ ( sk33 @ B @ A ) @ B )
!= ( member @ C @ ( singleton @ sk5 ) ) ) ),
inference(paramod_ordered,[status(thm)],[225,445]) ).
thf(818,plain,
! [A: $i] :
( ( identity @ A @ ( singleton @ sk5 ) )
| ( subset @ ( sk33 @ ( singleton @ sk5 ) @ A ) @ sk2 ) ),
inference(pattern_uni,[status(thm)],[817:[bind(A,$thf( E )),bind(B,$thf( singleton @ sk5 )),bind(C,$thf( sk33 @ ( singleton @ sk5 ) @ E ))]]) ).
thf(824,plain,
! [A: $i] :
( ( identity @ A @ ( singleton @ sk5 ) )
| ( subset @ ( sk33 @ ( singleton @ sk5 ) @ A ) @ sk2 ) ),
inference(simp,[status(thm)],[818]) ).
thf(42,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( member @ ( sk7 @ D @ C @ B @ A ) @ D )
| ( equal_maps @ A @ B @ C @ D ) ),
inference(cnf,[status(esa)],[37]) ).
thf(45,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( member @ ( sk7 @ D @ C @ B @ A ) @ D )
| ( equal_maps @ A @ B @ C @ D ) ),
inference(simp,[status(thm)],[42]) ).
thf(27,axiom,
! [A: $i,B: $i,C: $i] :
( ( maps @ A @ B @ C )
<=> ( ! [D: $i] :
( ( member @ D @ B )
=> ? [E: $i] :
( ( member @ E @ C )
& ( apply @ A @ D @ E ) ) )
& ! [D: $i,E: $i,F: $i] :
( ( ( member @ D @ B )
& ( member @ E @ C )
& ( member @ F @ C ) )
=> ( ( ( apply @ A @ D @ E )
& ( apply @ A @ D @ F ) )
=> ( E = F ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',maps) ).
thf(279,plain,
! [A: $i,B: $i,C: $i] :
( ( ( maps @ A @ B @ C )
=> ( ! [D: $i] :
( ( member @ D @ B )
=> ? [E: $i] :
( ( member @ E @ C )
& ( apply @ A @ D @ E ) ) )
& ! [D: $i,E: $i,F: $i] :
( ( ( member @ D @ B )
& ( member @ E @ C )
& ( member @ F @ C ) )
=> ( ( ( apply @ A @ D @ E )
& ( apply @ A @ D @ F ) )
=> ( E = F ) ) ) ) )
& ( ( ! [D: $i] :
( ( member @ D @ B )
=> ? [E: $i] :
( ( member @ E @ C )
& ( apply @ A @ D @ E ) ) )
& ! [D: $i,E: $i,F: $i] :
( ( ( member @ D @ B )
& ( member @ E @ C )
& ( member @ F @ C ) )
=> ( ( ( apply @ A @ D @ E )
& ( apply @ A @ D @ F ) )
=> ( E = F ) ) ) )
=> ( maps @ A @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(32,plain,
~ ( equal_set @ ( image2 @ sk1 @ ( union @ sk4 @ sk5 ) ) @ ( union @ ( image2 @ sk1 @ sk4 ) @ ( image2 @ sk1 @ sk5 ) ) ),
inference(cnf,[status(esa)],[31]) ).
thf(34,plain,
subset @ sk4 @ sk2,
inference(cnf,[status(esa)],[31]) ).
thf(606,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( equal_maps @ A @ B @ C @ D )
| ( ( member @ ( sk6 @ D @ C @ B @ A ) @ C )
!= ( member @ E @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[46,96]) ).
thf(607,plain,
! [C: $i,B: $i,A: $i] : ( equal_maps @ C @ B @ empty_set @ A ),
inference(pattern_uni,[status(thm)],[606:[bind(A,$thf( I )),bind(B,$thf( H )),bind(C,$thf( empty_set )),bind(D,$thf( F )),bind(E,$thf( sk6 @ F @ empty_set @ H @ I ))]]) ).
thf(616,plain,
! [C: $i,B: $i,A: $i] : ( equal_maps @ C @ B @ empty_set @ A ),
inference(simp,[status(thm)],[607]) ).
thf(28,axiom,
! [A: $i,B: $i,C: $i] :
( ( member @ A @ ( union @ B @ C ) )
<=> ( ( member @ A @ B )
| ( member @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',union) ).
thf(303,plain,
! [A: $i,B: $i,C: $i] :
( ( ( member @ A @ ( union @ B @ C ) )
=> ( ( member @ A @ B )
| ( member @ A @ C ) ) )
& ( ( ( member @ A @ B )
| ( member @ A @ C ) )
=> ( member @ A @ ( union @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(9,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( inverse_predicate @ A @ B @ C @ D )
<=> ! [E: $i,F: $i] :
( ( ( member @ E @ C )
& ( member @ F @ D ) )
=> ( ( apply @ B @ E @ F )
<=> ( apply @ A @ F @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inverse_predicate) ).
thf(103,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( inverse_predicate @ A @ B @ C @ D )
=> ! [E: $i,F: $i] :
( ( ( member @ E @ C )
& ( member @ F @ D ) )
=> ( ( ( apply @ B @ E @ F )
=> ( apply @ A @ F @ E ) )
& ( ( apply @ A @ F @ E )
=> ( apply @ B @ E @ F ) ) ) ) )
& ( ! [E: $i,F: $i] :
( ( ( member @ E @ C )
& ( member @ F @ D ) )
=> ( ( ( apply @ B @ E @ F )
=> ( apply @ A @ F @ E ) )
& ( ( apply @ A @ F @ E )
=> ( apply @ B @ E @ F ) ) ) )
=> ( inverse_predicate @ A @ B @ C @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(694,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ B ) )
| ( A
!= ( singleton @ empty_set ) )
| ( B
!= ( singleton @ ( singleton @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[138,641]) ).
thf(695,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) )
| ( A
!= ( singleton @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[694:[bind(A,$thf( A )),bind(B,$thf( singleton @ ( singleton @ E ) )),bind(C,$thf( E ))]]) ).
thf(700,plain,
! [A: $i] :
~ ( member @ ( singleton @ empty_set ) @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) ),
inference(simp,[status(thm)],[695]) ).
thf(24,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( isomorphism @ A @ B @ C @ D @ E )
<=> ( ( maps @ A @ B @ D )
& ( one_to_one @ A @ B @ D )
& ! [F: $i,G: $i,H: $i,I: $i] :
( ( ( member @ F @ B )
& ( member @ G @ D )
& ( member @ H @ B )
& ( member @ I @ D )
& ( apply @ A @ F @ G )
& ( apply @ A @ H @ I ) )
=> ( ( apply @ C @ F @ H )
<=> ( apply @ E @ G @ I ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',isomorphism) ).
thf(240,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( isomorphism @ A @ B @ C @ D @ E )
=> ( ( maps @ A @ B @ D )
& ( one_to_one @ A @ B @ D )
& ! [F: $i,G: $i,H: $i,I: $i] :
( ( ( member @ F @ B )
& ( member @ G @ D )
& ( member @ H @ B )
& ( member @ I @ D )
& ( apply @ A @ F @ G )
& ( apply @ A @ H @ I ) )
=> ( ( ( apply @ C @ F @ H )
=> ( apply @ E @ G @ I ) )
& ( ( apply @ E @ G @ I )
=> ( apply @ C @ F @ H ) ) ) ) ) )
& ( ( ( maps @ A @ B @ D )
& ( one_to_one @ A @ B @ D )
& ! [F: $i,G: $i,H: $i,I: $i] :
( ( ( member @ F @ B )
& ( member @ G @ D )
& ( member @ H @ B )
& ( member @ I @ D )
& ( apply @ A @ F @ G )
& ( apply @ A @ H @ I ) )
=> ( ( ( apply @ C @ F @ H )
=> ( apply @ E @ G @ I ) )
& ( ( apply @ E @ G @ I )
=> ( apply @ C @ F @ H ) ) ) ) )
=> ( isomorphism @ A @ B @ C @ D @ E ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(902,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ B ) )
| ( B
!= ( singleton @ ( singleton @ empty_set ) ) )
| ( A
!= ( singleton @ ( singleton @ ( singleton @ C ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[138,746]) ).
thf(903,plain,
! [B: $i,A: $i] :
( ~ ( member @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) @ ( singleton @ A ) )
| ( A
!= ( singleton @ ( singleton @ empty_set ) ) ) ),
inference(pattern_uni,[status(thm)],[902:[bind(A,$thf( singleton @ ( singleton @ ( singleton @ F ) ) )),bind(B,$thf( B )),bind(C,$thf( F ))]]) ).
thf(915,plain,
! [A: $i] :
~ ( member @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) @ ( singleton @ ( singleton @ ( singleton @ empty_set ) ) ) ),
inference(simp,[status(thm)],[903]) ).
thf(398,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ B ) )
| ( B != empty_set )
| ( A
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[138,360]) ).
thf(399,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ ( unordered_pair @ B @ C ) @ ( singleton @ A ) )
| ( A != empty_set ) ),
inference(pattern_uni,[status(thm)],[398:[bind(A,$thf( unordered_pair @ E @ F )),bind(B,$thf( B )),bind(C,$thf( E )),bind(D,$thf( F ))]]) ).
thf(548,plain,
! [B: $i,A: $i] :
~ ( member @ ( unordered_pair @ A @ B ) @ ( singleton @ empty_set ) ),
inference(simp,[status(thm)],[399]) ).
thf(337,plain,
! [C: $i,B: $i,A: $i] :
( ( member @ A @ ( unordered_pair @ A @ B ) )
!= ( member @ C @ empty_set ) ),
inference(paramod_ordered,[status(thm)],[125,96]) ).
thf(339,plain,
! [C: $i,B: $i,A: $i] :
( ( A != C )
| ( ( unordered_pair @ A @ B )
!= empty_set ) ),
inference(simp,[status(thm)],[337]) ).
thf(342,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ B @ A )
!= empty_set ),
inference(simp,[status(thm)],[339]) ).
thf(14,axiom,
! [A: $i,B: $i] :
( ( member @ A @ ( product @ B ) )
<=> ! [C: $i] :
( ( member @ C @ B )
=> ( member @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',product) ).
thf(145,plain,
! [A: $i,B: $i] :
( ( ( member @ A @ ( product @ B ) )
=> ! [C: $i] :
( ( member @ C @ B )
=> ( member @ A @ C ) ) )
& ( ! [C: $i] :
( ( member @ C @ B )
=> ( member @ A @ C ) )
=> ( member @ A @ ( product @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(17,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( decreasing @ A @ B @ C @ D @ E )
<=> ! [F: $i,G: $i,H: $i,I: $i] :
( ( ( member @ F @ B )
& ( member @ G @ D )
& ( member @ H @ B )
& ( member @ I @ D )
& ( apply @ C @ F @ H )
& ( apply @ A @ F @ G )
& ( apply @ A @ H @ I ) )
=> ( apply @ E @ I @ G ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',decreasing_function) ).
thf(182,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( decreasing @ A @ B @ C @ D @ E )
=> ! [F: $i,G: $i,H: $i,I: $i] :
( ( ( member @ F @ B )
& ( member @ G @ D )
& ( member @ H @ B )
& ( member @ I @ D )
& ( apply @ C @ F @ H )
& ( apply @ A @ F @ G )
& ( apply @ A @ H @ I ) )
=> ( apply @ E @ I @ G ) ) )
& ( ! [F: $i,G: $i,H: $i,I: $i] :
( ( ( member @ F @ B )
& ( member @ G @ D )
& ( member @ H @ B )
& ( member @ I @ D )
& ( apply @ C @ F @ H )
& ( apply @ A @ F @ G )
& ( apply @ A @ H @ I ) )
=> ( apply @ E @ I @ G ) )
=> ( decreasing @ A @ B @ C @ D @ E ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(347,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( equal_maps @ A @ B @ C @ D )
| ( ( member @ ( sk7 @ D @ C @ B @ A ) @ D )
!= ( member @ E @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[45,96]) ).
thf(348,plain,
! [C: $i,B: $i,A: $i] : ( equal_maps @ C @ B @ A @ empty_set ),
inference(pattern_uni,[status(thm)],[347:[bind(A,$thf( I )),bind(B,$thf( H )),bind(C,$thf( G )),bind(D,$thf( empty_set )),bind(E,$thf( sk7 @ empty_set @ G @ H @ I ))]]) ).
thf(351,plain,
! [C: $i,B: $i,A: $i] : ( equal_maps @ C @ B @ A @ empty_set ),
inference(simp,[status(thm)],[348]) ).
thf(13,axiom,
! [A: $i,B: $i,C: $i] :
( ( one_to_one @ A @ B @ C )
<=> ( ( injective @ A @ B @ C )
& ( surjective @ A @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one) ).
thf(139,plain,
! [A: $i,B: $i,C: $i] :
( ( ( one_to_one @ A @ B @ C )
=> ( ( injective @ A @ B @ C )
& ( surjective @ A @ B @ C ) ) )
& ( ( ( injective @ A @ B @ C )
& ( surjective @ A @ B @ C ) )
=> ( one_to_one @ A @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(26,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
<=> ! [C: $i] :
( ( member @ C @ A )
=> ( member @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subset) ).
thf(272,plain,
! [A: $i,B: $i] :
( ( ( subset @ A @ B )
=> ! [C: $i] :
( ( member @ C @ A )
=> ( member @ C @ B ) ) )
& ( ! [C: $i] :
( ( member @ C @ A )
=> ( member @ C @ B ) )
=> ( subset @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(377,plain,
! [C: $i,B: $i,A: $i] :
( ( identity @ A @ B )
| ( ( member @ ( sk33 @ B @ A ) @ B )
!= ( member @ C @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[225,96]) ).
thf(378,plain,
! [A: $i] : ( identity @ A @ empty_set ),
inference(pattern_uni,[status(thm)],[377:[bind(A,$thf( E )),bind(B,$thf( empty_set )),bind(C,$thf( sk33 @ empty_set @ E ))]]) ).
thf(383,plain,
! [A: $i] : ( identity @ A @ empty_set ),
inference(simp,[status(thm)],[378]) ).
thf(20,axiom,
! [A: $i,B: $i,C: $i] :
( ( member @ C @ ( inverse_image2 @ A @ B ) )
<=> ? [D: $i] :
( ( member @ D @ B )
& ( apply @ A @ C @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inverse_image2) ).
thf(214,plain,
! [A: $i,B: $i,C: $i] :
( ( ( member @ C @ ( inverse_image2 @ A @ B ) )
=> ? [D: $i] :
( ( member @ D @ B )
& ( apply @ A @ C @ D ) ) )
& ( ? [D: $i] :
( ( member @ D @ B )
& ( apply @ A @ C @ D ) )
=> ( member @ C @ ( inverse_image2 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(5,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( increasing @ A @ B @ C @ D @ E )
<=> ! [F: $i,G: $i,H: $i,I: $i] :
( ( ( member @ F @ B )
& ( member @ G @ D )
& ( member @ H @ B )
& ( member @ I @ D )
& ( apply @ C @ F @ H )
& ( apply @ A @ F @ G )
& ( apply @ A @ H @ I ) )
=> ( apply @ E @ G @ I ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',increasing_function) ).
thf(58,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( increasing @ A @ B @ C @ D @ E )
=> ! [F: $i,G: $i,H: $i,I: $i] :
( ( ( member @ F @ B )
& ( member @ G @ D )
& ( member @ H @ B )
& ( member @ I @ D )
& ( apply @ C @ F @ H )
& ( apply @ A @ F @ G )
& ( apply @ A @ H @ I ) )
=> ( apply @ E @ G @ I ) ) )
& ( ! [F: $i,G: $i,H: $i,I: $i] :
( ( ( member @ F @ B )
& ( member @ G @ D )
& ( member @ H @ B )
& ( member @ I @ D )
& ( apply @ C @ F @ H )
& ( apply @ A @ F @ G )
& ( apply @ A @ H @ I ) )
=> ( apply @ E @ G @ I ) )
=> ( increasing @ A @ B @ C @ D @ E ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(446,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ B ) )
| ( subset @ sk5 @ A )
| ( B != sk2 ) ),
inference(paramod_ordered,[status(thm)],[138,33]) ).
thf(447,plain,
! [A: $i] :
( ~ ( member @ A @ ( singleton @ sk2 ) )
| ( subset @ sk5 @ A ) ),
inference(pattern_uni,[status(thm)],[446:[bind(A,$thf( A )),bind(B,$thf( sk2 ))]]) ).
thf(908,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ B ) )
| ( A
!= ( singleton @ ( singleton @ empty_set ) ) )
| ( B
!= ( singleton @ ( singleton @ ( singleton @ C ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[138,746]) ).
thf(909,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) ) )
| ( A
!= ( singleton @ ( singleton @ empty_set ) ) ) ),
inference(pattern_uni,[status(thm)],[908:[bind(A,$thf( A )),bind(B,$thf( singleton @ ( singleton @ ( singleton @ F ) ) )),bind(C,$thf( F ))]]) ).
thf(918,plain,
! [A: $i] :
~ ( member @ ( singleton @ ( singleton @ empty_set ) ) @ ( singleton @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) ) ),
inference(simp,[status(thm)],[909]) ).
thf(26674,plain,
$false,
inference(cvc4,[status(thm)],[174,125,132,97,53,35,310,47,227,138,333,550,698,46,152,316,233,201,220,746,137,33,225,77,212,96,129,569,371,266,360,824,45,279,32,34,641,445,208,616,303,103,700,240,915,548,342,145,182,331,127,31,351,139,568,272,383,214,119,58,447,36,918,94]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SET752+4 : TPTP v8.1.2. Bugfixed v2.2.1.
% 0.12/0.14 % Command : run_Leo-III %s %d
% 0.14/0.34 % Computer : n001.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Thu May 18 19:36:20 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.99/0.86 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.43/1.02 % [INFO] Parsing done (156ms).
% 1.43/1.02 % [INFO] Running in sequential loop mode.
% 1.93/1.22 % [INFO] eprover registered as external prover.
% 1.93/1.22 % [INFO] cvc4 registered as external prover.
% 1.93/1.23 % [INFO] Scanning for conjecture ...
% 1.93/1.29 % [INFO] Found a conjecture and 28 axioms. Running axiom selection ...
% 2.20/1.34 % [INFO] Axiom selection finished. Selected 28 axioms (removed 0 axioms).
% 2.49/1.40 % [INFO] Problem is first-order (TPTP FOF).
% 2.49/1.41 % [INFO] Type checking passed.
% 2.49/1.41 % [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 ...
% 74.65/13.45 % External prover 'cvc4' found a proof!
% 74.65/13.45 % [INFO] Killing All external provers ...
% 74.65/13.45 % Time passed: 12942ms (effective reasoning time: 12427ms)
% 74.65/13.45 % 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)>
% 74.65/13.46 % Axioms used in derivation (28): power_set, unordered_pair, inverse_image3, union, intersection, one_to_one, surjective, injective, increasing_function, image2, identity, difference, sum, product, empty_set, inverse_predicate, isomorphism, decreasing_function, image3, subset, equal_maps, inverse_function, equal_set, compose_function, inverse_image2, singleton, maps, compose_predicate
% 74.65/13.46 % No. of inferences in proof: 152
% 74.65/13.46 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 12942 ms resp. 12427 ms w/o parsing
% 74.65/13.51 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 74.65/13.51 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------