%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET813+4 : TPTP v9.3.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sun Sep 27 08:30:57 AM UTC 2026
% Result : Theorem 5.13s 7.93s
% Output : Refutation 5.13s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 20
% Syntax : Number of formulae : 42 ( 3 unt; 0 typ; 0 def)
% Number of atoms : 198 ( 12 equ; 0 cnn)
% Maximal formula atoms : 22 ( 4 avg)
% Number of connectives : 619 ( 10 ~; 9 |; 65 &; 453 @)
% ( 17 <=>; 65 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 8 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 23 ( 21 usr; 4 con; 0-3 aty)
% Number of variables : 128 ( 0 ^; 119 !; 9 ?; 128 :)
% Comments :
%------------------------------------------------------------------------------
thf(member_decl,type,
member: $i > $i > $o ).
thf(on_decl,type,
on: $i ).
thf(suc_decl,type,
suc: $i > $i ).
thf(equal_set_decl,type,
equal_set: $i > $i > $o ).
thf(subset_decl,type,
subset: $i > $i > $o ).
thf(least_decl,type,
least: $i > $i > $i > $o ).
thf(apply_decl,type,
apply: $i > $i > $i > $o ).
thf(initial_segment_decl,type,
initial_segment: $i > $i > $i > $i ).
thf(power_set_decl,type,
power_set: $i > $i ).
thf(set_decl,type,
set: $i > $o ).
thf(strict_well_order_decl,type,
strict_well_order: $i > $i > $o ).
thf(member_predicate_decl,type,
member_predicate: $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(product_decl,type,
product: $i > $i ).
thf(strict_order_decl,type,
strict_order: $i > $i > $o ).
thf(empty_set_decl,type,
empty_set: $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(difference_decl,type,
difference: $i > $i > $i ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(sum_decl,type,
sum: $i > $i ).
thf(8,axiom,
! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ member ) )
<=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton) ).
thf(69,plain,
! [A: $i,B: $i] :
( ( ( A = B )
=> ( B @ singleton @ ( A @ member ) ) )
& ( ( B @ singleton @ ( A @ member ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(20,axiom,
! [A: $i,B: $i] :
( ( B @ sum @ ( A @ member ) )
<=> ? [C: $i] :
( ( C @ ( A @ member ) )
& ( B @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum) ).
thf(202,plain,
! [A: $i,B: $i] :
( ( ? [C: $i] :
( ( C @ ( A @ member ) )
& ( B @ ( C @ member ) ) )
=> ( B @ sum @ ( A @ member ) ) )
& ( ( B @ sum @ ( A @ member ) )
=> ? [C: $i] :
( ( C @ ( A @ member ) )
& ( B @ ( C @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(16,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
<=> ( ( C @ ( A @ member ) )
& ( B @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection) ).
thf(174,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( C @ ( A @ member ) )
& ( B @ ( A @ member ) ) )
=> ( C @ ( B @ intersection ) @ ( A @ member ) ) )
& ( ( C @ ( B @ intersection ) @ ( A @ member ) )
=> ( ( C @ ( A @ member ) )
& ( B @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(5,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) )
<=> ( ( A @ ( D @ ( B @ apply ) ) )
& ( C @ ( D @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',initial_segment) ).
thf(46,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( A @ ( D @ ( B @ apply ) ) )
& ( C @ ( D @ member ) ) )
=> ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) ) )
& ( ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) )
=> ( ( A @ ( D @ ( B @ apply ) ) )
& ( C @ ( D @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(10,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ strict_well_order ) )
<=> ( ! [C: $i] :
( ( ? [D: $i] : ( C @ ( D @ member ) )
& ( B @ ( C @ subset ) ) )
=> ? [D: $i] : ( C @ ( A @ ( D @ least ) ) ) )
& ( B @ ( A @ strict_order ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',strict_well_order) ).
thf(85,plain,
! [A: $i,B: $i] :
( ( ( ! [C: $i] :
( ( ? [D: $i] : ( C @ ( D @ member ) )
& ( B @ ( C @ subset ) ) )
=> ? [D: $i] : ( C @ ( A @ ( D @ least ) ) ) )
& ( B @ ( A @ strict_order ) ) )
=> ( B @ ( A @ strict_well_order ) ) )
& ( ( B @ ( A @ strict_well_order ) )
=> ( ! [C: $i] :
( ( ? [D: $i] : ( C @ ( D @ member ) )
& ( B @ ( C @ subset ) ) )
=> ? [D: $i] : ( C @ ( A @ ( D @ least ) ) ) )
& ( B @ ( A @ strict_order ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(4,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ least ) ) )
<=> ( ! [D: $i] :
( ( B @ ( D @ member ) )
=> ( ( D @ ( C @ ( A @ apply ) ) )
| ( C = D ) ) )
& ( B @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',least) ).
thf(33,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ! [D: $i] :
( ( B @ ( D @ member ) )
=> ( ( D @ ( C @ ( A @ apply ) ) )
| ( C = D ) ) )
& ( B @ ( C @ member ) ) )
=> ( B @ ( A @ ( C @ least ) ) ) )
& ( ( B @ ( A @ ( C @ least ) ) )
=> ( ! [D: $i] :
( ( B @ ( D @ member ) )
=> ( ( D @ ( C @ ( A @ apply ) ) )
| ( C = D ) ) )
& ( B @ ( C @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(6,axiom,
! [A: $i,B: $i] :
( ( B @ power_set @ ( A @ member ) )
<=> ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_set) ).
thf(53,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ subset ) )
=> ( B @ power_set @ ( A @ member ) ) )
& ( ( B @ power_set @ ( A @ member ) )
=> ( B @ ( A @ subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(14,axiom,
! [A: $i,B: $i] :
( ( A @ suc @ ( B @ member ) )
<=> ( A @ singleton @ ( A @ union ) @ ( B @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor) ).
thf(161,plain,
! [A: $i,B: $i] :
( ( ( A @ singleton @ ( A @ union ) @ ( B @ member ) )
=> ( A @ suc @ ( B @ member ) ) )
& ( ( A @ suc @ ( B @ member ) )
=> ( A @ singleton @ ( A @ union ) @ ( B @ member ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(17,axiom,
! [A: $i] :
( ( A @ set )
=> ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( B @ set ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',set_member) ).
thf(181,plain,
! [A: $i] :
( ( A @ set )
=> ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( B @ set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(15,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( C @ difference ) @ ( A @ member ) )
<=> ( ~ ( B @ ( A @ member ) )
& ( C @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',difference) ).
thf(167,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ~ ( B @ ( A @ member ) )
& ( C @ ( A @ member ) ) )
=> ( B @ ( C @ difference ) @ ( A @ member ) ) )
& ( ( B @ ( C @ difference ) @ ( A @ member ) )
=> ( ~ ( B @ ( A @ member ) )
& ( C @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(21,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( member_predicate @ apply ) ) )
<=> ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rel_member) ).
thf(209,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ member ) )
=> ( B @ ( A @ ( member_predicate @ apply ) ) ) )
& ( ( B @ ( A @ ( member_predicate @ apply ) ) )
=> ( B @ ( A @ member ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(12,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
<=> ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset) ).
thf(150,plain,
! [A: $i,B: $i] :
( ( ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) )
=> ( B @ ( A @ subset ) ) )
& ( ( B @ ( A @ subset ) )
=> ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(18,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ union ) @ ( A @ member ) )
<=> ( ( C @ ( A @ member ) )
| ( B @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union) ).
thf(183,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( C @ ( A @ member ) )
| ( B @ ( A @ member ) ) )
=> ( C @ ( B @ union ) @ ( A @ member ) ) )
& ( ( C @ ( B @ union ) @ ( A @ member ) )
=> ( ( C @ ( A @ member ) )
| ( B @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(9,axiom,
! [A: $i,B: $i] :
( ( B @ product @ ( A @ member ) )
<=> ! [C: $i] :
( ( B @ ( C @ member ) )
=> ( C @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',product) ).
thf(77,plain,
! [A: $i,B: $i] :
( ( ! [C: $i] :
( ( B @ ( C @ member ) )
=> ( C @ ( A @ member ) ) )
=> ( B @ product @ ( A @ member ) ) )
& ( ( B @ product @ ( A @ member ) )
=> ! [C: $i] :
( ( B @ ( C @ member ) )
=> ( C @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(11,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ strict_order ) )
<=> ( ! [C: $i,D: $i,E: $i] :
( ( ( B @ ( E @ member ) )
& ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ( ( ( E @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) )
=> ( E @ ( C @ ( A @ apply ) ) ) ) )
& ! [C: $i,D: $i] :
( ( ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ~ ( ( C @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',strict_order) ).
thf(96,plain,
! [A: $i,B: $i] :
( ( ( ! [C: $i,D: $i,E: $i] :
( ( ( B @ ( E @ member ) )
& ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ( ( ( E @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) )
=> ( E @ ( C @ ( A @ apply ) ) ) ) )
& ! [C: $i,D: $i] :
( ( ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ~ ( ( C @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) ) ) )
=> ( B @ ( A @ strict_order ) ) )
& ( ( B @ ( A @ strict_order ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( B @ ( E @ member ) )
& ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ( ( ( E @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) )
=> ( E @ ( C @ ( A @ apply ) ) ) ) )
& ! [C: $i,D: $i] :
( ( ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ~ ( ( C @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(19,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
<=> ( ( A = C )
| ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair) ).
thf(191,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( A = C )
| ( A = B ) )
=> ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
& ( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
=> ( ( A = C )
| ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(1,conjecture,
! [A: $i] :
( ( on @ ( A @ member ) )
=> ( A @ suc @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',thV12) ).
thf(2,negated_conjecture,
~ ! [A: $i] :
( ( on @ ( A @ member ) )
=> ( A @ suc @ ( A @ member ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(22,plain,
~ ! [A: $i] :
( ( on @ ( A @ member ) )
=> ( A @ suc @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(7,axiom,
! [A: $i] :
( ( on @ ( A @ member ) )
<=> ( ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( A @ ( B @ subset ) ) )
& ( A @ ( member_predicate @ strict_well_order ) )
& ( A @ set ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordinal_number) ).
thf(59,plain,
! [A: $i] :
( ( ( ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( A @ ( B @ subset ) ) )
& ( A @ ( member_predicate @ strict_well_order ) )
& ( A @ set ) )
=> ( on @ ( A @ member ) ) )
& ( ( on @ ( A @ member ) )
=> ( ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( A @ ( B @ subset ) ) )
& ( A @ ( member_predicate @ strict_well_order ) )
& ( A @ set ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(3,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ equal_set ) )
<=> ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_set) ).
thf(26,plain,
! [A: $i,B: $i] :
( ( ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( B @ ( A @ equal_set ) ) )
& ( ( B @ ( A @ equal_set ) )
=> ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(13,axiom,
! [A: $i] :
~ ( empty_set @ ( A @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',empty_set) ).
thf(158,plain,
! [A: $i] :
~ ( empty_set @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(219,plain,
$false,
inference(e,[status(thm)],[69,202,174,46,85,33,53,161,181,167,209,150,183,77,96,191,22,59,26,158]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET813+4 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.08 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.13/5.39 % Computer : n003.cluster.edu
% 0.13/5.39 % Model : x86_64 x86_64
% 0.13/5.39 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/5.39 % Memory : 8046.5625MB
% 0.13/5.39 % OS : Linux 6.8.0-71-generic
% 0.13/5.39 % CPULimit : 300
% 0.13/5.39 % WCLimit : 300
% 0.13/5.39 % DateTime : Sat Sep 26 08:19:29 UTC 2026
% 0.13/5.39 % CPUTime :
% 0.13/5.39 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.17/5.97 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 1.25/6.19 % [INFO] Parsing done (213ms).
% 1.45/6.20 % [INFO] Running in sequential loop mode.
% 2.09/6.61 % [INFO] eprover registered as external prover.
% 2.09/6.62 % [INFO] Scanning for conjecture ...
% 2.26/6.71 % [INFO] Found a conjecture (or negated_conjecture) and 19 axioms. Running axiom selection ...
% 2.26/6.78 % [INFO] Axiom selection finished. Selected 19 axioms (removed 0 axioms).
% 2.45/6.84 % [INFO] Problem is first-order (TPTP FOF).
% 2.45/6.85 % [INFO] Type checking passed.
% 2.45/6.86 % [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 ...
% 5.13/7.93 % External prover 'e' found a proof!
% 5.13/7.93 % [INFO] Killing All external provers ...
% 5.13/7.93 % Time passed: 2391ms (effective reasoning time: 1717ms)
% 5.13/7.93 % 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)>
% 5.13/7.93 % Axioms used in derivation (19): power_set, unordered_pair, rel_member, union, strict_well_order, intersection, set_member, initial_segment, least, subset, equal_set, singleton, ordinal_number, difference, strict_order, sum, product, successor, empty_set
% 5.13/7.93 % No. of inferences in proof: 42
% 5.13/7.93 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 2391 ms resp. 1717 ms w/o parsing
% 5.13/8.01 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 5.13/8.01 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------