%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET002^3 : TPTP v9.3.1. Released v8.1.0.
% 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 : n009.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 : Wed Sep 30 08:22:50 AM UTC 2026
% Result : Theorem 5.86s 3.29s
% Output : Refutation 6.40s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 23
% Syntax : Number of formulae : 42 ( 15 unt; 0 typ; 6 def)
% Number of atoms : 180 ( 12 equ; 0 cnn)
% Maximal formula atoms : 9 ( 4 avg)
% Number of connectives : 372 ( 2 ~; 2 |; 7 &; 350 @)
% ( 0 <=>; 11 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 8 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 11 ( 11 >; 0 *; 0 +; 0 <<)
% Number of symbols : 13 ( 11 usr; 2 con; 0-3 aty)
% Number of variables : 99 ( 57 ^; 42 !; 0 ?; 99 :)
% Comments :
%------------------------------------------------------------------------------
thf(mworld_type,type,
mworld: $tType ).
thf(mactual_decl,type,
mactual: mworld ).
thf(mlocal_decl,type,
mlocal: ( mworld > $o ) > $o ).
thf(mand_decl,type,
mand: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mor_decl,type,
mor: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mimplies_decl,type,
mimplies: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mequiv_decl,type,
mequiv: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mforall_di_decl,type,
mforall_di: ( $i > mworld > $o ) > mworld > $o ).
thf(qmltpeq_decl,type,
qmltpeq: $i > $i > mworld > $o ).
thf(member_decl,type,
member: $i > $i > mworld > $o ).
thf(subset_decl,type,
subset: $i > $i > mworld > $o ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(mlocal_def,definition,
( mlocal
= ( ^ [A: mworld > $o] : ( A @ mactual ) ) ) ).
thf(mand_def,definition,
( mand
= ( ^ [A: mworld > $o,B: mworld > $o,C: mworld] :
( ( A @ C )
& ( B @ C ) ) ) ) ).
thf(mor_def,definition,
( mor
= ( ^ [A: mworld > $o,B: mworld > $o,C: mworld] :
( ( A @ C )
| ( B @ C ) ) ) ) ).
thf(mimplies_def,definition,
( mimplies
= ( ^ [A: mworld > $o,B: mworld > $o,C: mworld] :
( ( A @ C )
=> ( B @ C ) ) ) ) ).
thf(mequiv_def,definition,
( mequiv
= ( ^ [A: mworld > $o,B: mworld > $o,C: mworld] :
( ( A @ C )
= ( B @ C ) ) ) ) ).
thf(mforall_di_def,definition,
( mforall_di
= ( ^ [A: $i > mworld > $o,B: mworld] :
! [C: $i] : ( A @ C @ B ) ) ) ).
thf(16,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ A @ C ) @ ( union @ B @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_substitution_1) ).
thf(52,plain,
! [A: $i,B: $i,C: $i] :
( ( qmltpeq @ A @ B @ mactual )
=> ( qmltpeq @ ( union @ A @ C ) @ ( union @ B @ C ) @ mactual ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(8,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mequiv @ ( qmltpeq @ A @ B )
@ ( mforall_di
@ ^ [C: $i] : ( mequiv @ ( member @ C @ A ) @ ( member @ C @ B ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_member_defn) ).
thf(34,plain,
! [A: $i,B: $i] :
( ( qmltpeq @ A @ B @ mactual )
= ( ! [C: $i] :
( ( member @ C @ A @ mactual )
= ( member @ C @ B @ mactual ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(12,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( subset @ A @ B ) @ ( qmltpeq @ ( union @ A @ B ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset_union) ).
thf(44,plain,
! [A: $i,B: $i] :
( ( subset @ A @ B @ mactual )
=> ( qmltpeq @ ( union @ A @ B ) @ B @ mactual ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(14,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ C @ A ) ) @ ( member @ C @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_substitution_2) ).
thf(48,plain,
! [A: $i,B: $i,C: $i] :
( ( ( qmltpeq @ A @ B @ mactual )
& ( member @ C @ A @ mactual ) )
=> ( member @ C @ B @ mactual ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(1,conjecture,
( mlocal
@ ( mforall_di
@ ^ [A: $i] : ( qmltpeq @ ( union @ A @ A ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_idempotency_of_union) ).
thf(2,negated_conjecture,
~ ( mlocal
@ ( mforall_di
@ ^ [A: $i] : ( qmltpeq @ ( union @ A @ A ) @ A ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(19,plain,
~ ! [A: $i] : ( qmltpeq @ ( union @ A @ A ) @ A @ mactual ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(11,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] : ( subset @ A @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_of_subset) ).
thf(42,plain,
! [A: $i] : ( subset @ A @ A @ mactual ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(9,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mequiv @ ( qmltpeq @ A @ B ) @ ( mand @ ( subset @ A @ B ) @ ( subset @ B @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_defn) ).
thf(37,plain,
! [A: $i,B: $i] :
( ( qmltpeq @ A @ B @ mactual )
= ( ( subset @ A @ B @ mactual )
& ( subset @ B @ A @ mactual ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(5,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mequiv @ ( member @ C @ ( union @ A @ B ) ) @ ( mor @ ( member @ C @ A ) @ ( member @ C @ B ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_defn) ).
thf(25,plain,
! [A: $i,B: $i,C: $i] :
( ( member @ C @ ( union @ A @ B ) @ mactual )
= ( ( member @ C @ A @ mactual )
| ( member @ C @ B @ mactual ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(13,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ A @ C ) ) @ ( member @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_substitution_1) ).
thf(46,plain,
! [A: $i,B: $i,C: $i] :
( ( ( qmltpeq @ A @ B @ mactual )
& ( member @ A @ C @ mactual ) )
=> ( member @ B @ C @ mactual ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(18,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',symmetry) ).
thf(57,plain,
! [A: $i,B: $i] :
( ( qmltpeq @ A @ B @ mactual )
=> ( qmltpeq @ B @ A @ mactual ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(6,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mequiv @ ( subset @ A @ B )
@ ( mforall_di
@ ^ [C: $i] : ( mimplies @ ( member @ C @ A ) @ ( member @ C @ B ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset_defn) ).
thf(28,plain,
! [A: $i,B: $i] :
( ( subset @ A @ B @ mactual )
= ( ! [C: $i] :
( ( member @ C @ A @ mactual )
=> ( member @ C @ B @ mactual ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(3,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ B @ C ) ) @ ( qmltpeq @ A @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',transitivity) ).
thf(21,plain,
! [A: $i,B: $i,C: $i] :
( ( ( qmltpeq @ A @ B @ mactual )
& ( qmltpeq @ B @ C @ mactual ) )
=> ( qmltpeq @ A @ C @ mactual ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(15,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subset @ C @ A ) ) @ ( subset @ C @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset_substitution_2) ).
thf(50,plain,
! [A: $i,B: $i,C: $i] :
( ( ( qmltpeq @ A @ B @ mactual )
& ( subset @ C @ A @ mactual ) )
=> ( subset @ C @ B @ mactual ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(7,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ C @ A ) @ ( union @ C @ B ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_substitution_2) ).
thf(31,plain,
! [A: $i,B: $i,C: $i] :
( ( qmltpeq @ A @ B @ mactual )
=> ( qmltpeq @ ( union @ C @ A ) @ ( union @ C @ B ) @ mactual ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(10,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( qmltpeq @ ( union @ A @ B ) @ ( union @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_of_union) ).
thf(40,plain,
! [A: $i,B: $i] : ( qmltpeq @ ( union @ A @ B ) @ ( union @ B @ A ) @ mactual ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(17,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] : ( qmltpeq @ A @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity) ).
thf(55,plain,
! [A: $i] : ( qmltpeq @ A @ A @ mactual ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(4,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subset @ A @ C ) ) @ ( subset @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset_substitution_1) ).
thf(23,plain,
! [A: $i,B: $i,C: $i] :
( ( ( qmltpeq @ A @ B @ mactual )
& ( subset @ A @ C @ mactual ) )
=> ( subset @ B @ C @ mactual ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(59,plain,
$false,
inference(e,[status(thm)],[52,34,44,48,19,42,37,25,46,57,28,21,50,31,40,55,23]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET002^3 : TPTP v9.3.1. Released v8.1.0.
% 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
% 0.10/0.23 % Computer : n009.cluster.edu
% 0.10/0.23 % Model : x86_64 x86_64
% 0.10/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.23 % Memory : 8046.5625MB
% 0.10/0.23 % OS : Linux 6.8.0-71-generic
% 0.10/0.24 % CPULimit : 300
% 0.10/0.24 % WCLimit : 300
% 0.10/0.24 % DateTime : Tue Sep 29 13:35:45 UTC 2026
% 0.10/0.24 % CPUTime :
% 0.10/0.24 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 1.18/1.04 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 2.02/1.35 % [INFO] Parsing done (301ms).
% 2.02/1.37 % [INFO] Running in sequential loop mode.
% 3.07/1.91 % [INFO] eprover registered as external prover.
% 3.07/1.92 % [INFO] Scanning for conjecture ...
% 3.35/2.11 % [INFO] Found a conjecture (or negated_conjecture) and 17 axioms. Running axiom selection ...
% 3.61/2.21 % [INFO] Axiom selection finished. Selected 16 axioms (removed 1 axioms).
% 3.61/2.26 % [INFO] Problem is higher-order (TPTP THF).
% 3.61/2.27 % [INFO] Type checking passed.
% 3.61/2.28 % [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 ...
% 5.86/3.28 % External prover 'e' found a proof!
% 5.86/3.28 % [INFO] Killing All external provers ...
% 5.86/3.28 % Time passed: 2830ms (effective reasoning time: 1902ms)
% 5.86/3.28 % 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)>
% 5.86/3.29 % Axioms used in derivation (16): member_substitution_1, reflexivity_of_subset, equal_member_defn, subset_substitution_2, union_substitution_2, subset_union, equal_defn, member_substitution_2, symmetry, reflexivity, commutativity_of_union, subset_substitution_1, union_substitution_1, transitivity, union_defn, subset_defn
% 5.86/3.29 % No. of inferences in proof: 36
% 5.86/3.29 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 2830 ms resp. 1902 ms w/o parsing
% 6.40/3.42 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 6.40/3.42 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------