%------------------------------------------------------------------------------
% File : iProver---3.9.4
% Problem : SET960+1 : TPTP v9.3.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : run_iprover 300 /export/starexec/sandbox/benchmark/theBenchmark.p THM
% Computer : n014.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 : Fri Sep 25 02:49:12 PM UTC 2026
% Result : Theorem 3.21s 1.27s
% Output : CNFRefutation 3.21s
% Verified :
% SZS Type : Refutation
% Derivation depth : 19
% Number of leaves : 6
% Syntax : Number of formulae : 70 ( 20 unt; 1 def)
% Number of atoms : 240 ( 114 equ)
% Maximal formula atoms : 18 ( 3 avg)
% Number of connectives : 242 ( 100 ~; 104 |; 32 &)
% ( 5 <=>; 0 =>; 0 <=; 1 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 4 ( 2 usr; 2 prp; 0-2 aty)
% Number of functors : 13 ( 13 usr; 3 con; 0-3 aty)
% Number of variables : 149 ( 0 sgn 129 !; 20 ?; 39 :)
% Comments :
%------------------------------------------------------------------------------
fof(f2,axiom,
! [X0,X1] : unordered_pair(X0,X1) = unordered_pair(X1,X0),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_k2_tarski) ).
fof(f3,axiom,
! [X0] :
( X0 = empty_set
<=> ! [X1] : ~ in(X1,X0) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_xboole_0) ).
fof(f4,axiom,
! [X0,X1,X2] :
( X2 = cartesian_product2(X0,X1)
<=> ! [X3] :
( in(X3,X2)
<=> ? [X4,X5] :
( X3 = ordered_pair(X4,X5)
& in(X5,X1)
& in(X4,X0) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d2_zfmisc_1) ).
fof(f5,axiom,
! [X0,X1] : ordered_pair(X0,X1) = unordered_pair(unordered_pair(X0,X1),singleton(X0)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d5_tarski) ).
fof(f10,conjecture,
! [X0,X1] :
( cartesian_product2(X0,X1) = empty_set
<=> ( X1 = empty_set
| X0 = empty_set ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t113_zfmisc_1) ).
fof(f11,negated_conjecture,
~ ! [X0,X1] :
( cartesian_product2(X0,X1) = empty_set
<=> ( X1 = empty_set
| X0 = empty_set ) ),
inference(negated_conjecture,[status(cth)],[f10]) ).
fof(f13,plain,
? [X0,X1] :
( cartesian_product2(X0,X1) = empty_set
<~> ( X1 = empty_set
| X0 = empty_set ) ),
inference(ennf_transformation,[],[f11]) ).
fof(f14,plain,
! [X0] :
( ( empty_set != X0
| ! [X1] : ~ in(X1,X0) )
& ( ? [X1] : in(X1,X0)
| X0 = empty_set ) ),
inference(nnf_transformation,[],[f3]) ).
fof(f15,plain,
! [X0] :
( ( empty_set != X0
| ! [X2] : ~ in(X2,X0) )
& ( ? [X1] : in(X1,X0)
| X0 = empty_set ) ),
inference(rectify,[],[f14]) ).
fof(f16,plain,
! [X0] :
( ( empty_set != X0
| ! [X2] : ~ in(X2,X0) )
& ( in(sK0(X0),X0)
| X0 = empty_set ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(X1,sK0(X0))],[f15]) ).
fof(f17,plain,
! [X0,X1,X2] :
( ( cartesian_product2(X0,X1) != X2
| ! [X3] :
( ( ~ in(X3,X2)
| ? [X4,X5] :
( X3 = ordered_pair(X4,X5)
& in(X5,X1)
& in(X4,X0) ) )
& ( ! [X4,X5] :
( ordered_pair(X4,X5) != X3
| ~ in(X5,X1)
| ~ in(X4,X0) )
| in(X3,X2) ) ) )
& ( ? [X3] :
( ( in(X3,X2)
| ? [X4,X5] :
( X3 = ordered_pair(X4,X5)
& in(X5,X1)
& in(X4,X0) ) )
& ( ~ in(X3,X2)
| ! [X4,X5] :
( ordered_pair(X4,X5) != X3
| ~ in(X5,X1)
| ~ in(X4,X0) ) ) )
| X2 = cartesian_product2(X0,X1) ) ),
inference(nnf_transformation,[],[f4]) ).
fof(f18,plain,
! [X0,X1,X2] :
( ( cartesian_product2(X0,X1) != X2
| ! [X8] :
( ( ~ in(X8,X2)
| ? [X11,X12] :
( ordered_pair(X11,X12) = X8
& in(X12,X1)
& in(X11,X0) ) )
& ( ! [X9,X10] :
( ordered_pair(X9,X10) != X8
| ~ in(X10,X1)
| ~ in(X9,X0) )
| in(X8,X2) ) ) )
& ( ? [X3] :
( ( in(X3,X2)
| ? [X6,X7] :
( ordered_pair(X6,X7) = X3
& in(X7,X1)
& in(X6,X0) ) )
& ( ~ in(X3,X2)
| ! [X4,X5] :
( ordered_pair(X4,X5) != X3
| ~ in(X5,X1)
| ~ in(X4,X0) ) ) )
| X2 = cartesian_product2(X0,X1) ) ),
inference(rectify,[],[f17]) ).
fof(f19,plain,
! [X0,X1,X2] :
( ( cartesian_product2(X0,X1) != X2
| ! [X8] :
( ( ~ in(X8,X2)
| ( ordered_pair(sK4(X0,X1,X8),sK5(X0,X1,X8)) = X8
& in(sK5(X0,X1,X8),X1)
& in(sK4(X0,X1,X8),X0) ) )
& ( ! [X9,X10] :
( ordered_pair(X9,X10) != X8
| ~ in(X10,X1)
| ~ in(X9,X0) )
| in(X8,X2) ) ) )
& ( ( ( in(sK1(X0,X1,X2),X2)
| ( sK1(X0,X1,X2) = ordered_pair(sK2(X0,X1,X2),sK3(X0,X1,X2))
& in(sK3(X0,X1,X2),X1)
& in(sK2(X0,X1,X2),X0) ) )
& ( ~ in(sK1(X0,X1,X2),X2)
| ! [X4,X5] :
( ordered_pair(X4,X5) != sK1(X0,X1,X2)
| ~ in(X5,X1)
| ~ in(X4,X0) ) ) )
| X2 = cartesian_product2(X0,X1) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1,sK2,sK3,sK4,sK5]),skolemize(X3,sK1(X0,X1,X2)),skolemize(X6,sK2(X0,X1,X2)),skolemize(X7,sK3(X0,X1,X2)),skolemize(X11,sK4(X0,X1,X8)),skolemize(X12,sK5(X0,X1,X8))],[f18]) ).
fof(f22,plain,
? [X0,X1] :
( ( cartesian_product2(X0,X1) = empty_set
| X1 = empty_set
| X0 = empty_set )
& ( empty_set != cartesian_product2(X0,X1)
| ( empty_set != X1
& empty_set != X0 ) ) ),
inference(nnf_transformation,[],[f13]) ).
fof(f23,plain,
? [X0,X1] :
( ( cartesian_product2(X0,X1) = empty_set
| X1 = empty_set
| X0 = empty_set )
& ( empty_set != cartesian_product2(X0,X1)
| ( empty_set != X1
& empty_set != X0 ) ) ),
inference(flattening,[],[f22]) ).
fof(f24,plain,
( ( empty_set = cartesian_product2(sK8,sK9)
| empty_set = sK9
| empty_set = sK8 )
& ( empty_set != cartesian_product2(sK8,sK9)
| ( empty_set != sK9
& empty_set != sK8 ) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK8,sK9]),skolemize(X0,sK8),skolemize(X1,sK9)],[f23]) ).
fof(f26,plain,
! [X0,X1] : unordered_pair(X0,X1) = unordered_pair(X1,X0),
inference(cnf_transformation,[],[f2]) ).
fof(f27,plain,
! [X2,X0] :
( empty_set != X0
| ~ in(X2,X0) ),
inference(cnf_transformation,[],[f16]) ).
fof(f28,plain,
! [X0] :
( in(sK0(X0),X0)
| empty_set = X0 ),
inference(cnf_transformation,[],[f16]) ).
fof(f30,plain,
! [X2,X0,X1,X8] :
( cartesian_product2(X0,X1) != X2
| ~ in(X8,X2)
| in(sK5(X0,X1,X8),X1) ),
inference(cnf_transformation,[],[f19]) ).
fof(f31,plain,
! [X2,X0,X1,X8] :
( cartesian_product2(X0,X1) != X2
| ~ in(X8,X2)
| in(sK4(X0,X1,X8),X0) ),
inference(cnf_transformation,[],[f19]) ).
fof(f32,plain,
! [X2,X10,X0,X1,X8,X9] :
( cartesian_product2(X0,X1) != X2
| ordered_pair(X9,X10) != X8
| ~ in(X10,X1)
| ~ in(X9,X0)
| in(X8,X2) ),
inference(cnf_transformation,[],[f19]) ).
fof(f34,plain,
! [X2,X0,X1] :
( in(sK1(X0,X1,X2),X2)
| in(sK3(X0,X1,X2),X1)
| cartesian_product2(X0,X1) = X2 ),
inference(cnf_transformation,[],[f19]) ).
fof(f37,plain,
! [X0,X1] : ordered_pair(X0,X1) = unordered_pair(unordered_pair(X0,X1),singleton(X0)),
inference(cnf_transformation,[],[f5]) ).
fof(f42,plain,
( empty_set = cartesian_product2(sK8,sK9)
| empty_set = sK9
| empty_set = sK8 ),
inference(cnf_transformation,[],[f24]) ).
fof(f43,plain,
( empty_set != cartesian_product2(sK8,sK9)
| empty_set != sK9 ),
inference(cnf_transformation,[],[f24]) ).
fof(f44,plain,
( empty_set != cartesian_product2(sK8,sK9)
| empty_set != sK8 ),
inference(cnf_transformation,[],[f24]) ).
fof(f47,plain,
! [X2,X10,X0,X1,X8,X9] :
( cartesian_product2(X0,X1) != X2
| unordered_pair(unordered_pair(X9,X10),singleton(X9)) != X8
| ~ in(X10,X1)
| ~ in(X9,X0)
| in(X8,X2) ),
inference(definition_unfolding,[],[f32,f37]) ).
fof(f50,plain,
! [X2] : ~ in(X2,empty_set),
inference(equality_resolution,[],[f27]) ).
fof(f51,plain,
! [X2,X10,X0,X1,X9] :
( cartesian_product2(X0,X1) != X2
| ~ in(X10,X1)
| ~ in(X9,X0)
| in(unordered_pair(unordered_pair(X9,X10),singleton(X9)),X2) ),
inference(equality_resolution,[],[f47]) ).
fof(f52,plain,
! [X10,X0,X1,X9] :
( ~ in(X10,X1)
| ~ in(X9,X0)
| in(unordered_pair(unordered_pair(X9,X10),singleton(X9)),cartesian_product2(X0,X1)) ),
inference(equality_resolution,[],[f51]) ).
fof(f53,plain,
! [X0,X1,X8] :
( ~ in(X8,cartesian_product2(X0,X1))
| in(sK4(X0,X1,X8),X0) ),
inference(equality_resolution,[],[f31]) ).
fof(f54,plain,
! [X0,X1,X8] :
( ~ in(X8,cartesian_product2(X0,X1))
| in(sK5(X0,X1,X8),X1) ),
inference(equality_resolution,[],[f30]) ).
tcf(c_50,plain,
! [X0: $i,X1: $i] : unordered_pair(X0,X1) = unordered_pair(X1,X0),
inference(cnf_transformation,[],[f26]) ).
tcf(c_51,plain,
! [X0: $i] :
( in(sK0(X0),X0)
| ( X0 = empty_set ) ),
inference(cnf_transformation,[],[f28]) ).
tcf(c_52,plain,
! [X0: $i] : ~ in(X0,empty_set),
inference(cnf_transformation,[],[f50]) ).
tcf(c_55,plain,
! [X0: $i,X1: $i,X2: $i] :
( in(sK3(X0,X1,X2),X1)
| in(sK1(X0,X1,X2),X2)
| ( cartesian_product2(X0,X1) = X2 ) ),
inference(cnf_transformation,[],[f34]) ).
tcf(c_57,plain,
! [X0: $i,X1: $i,X2: $i,X3: $i] :
( in(unordered_pair(unordered_pair(X0,X2),singleton(X0)),cartesian_product2(X1,X3))
| ~ in(X2,X3)
| ~ in(X0,X1) ),
inference(cnf_transformation,[],[f52]) ).
tcf(c_58,plain,
! [X0: $i,X1: $i,X2: $i] :
( in(sK4(X1,X2,X0),X1)
| ~ in(X0,cartesian_product2(X1,X2)) ),
inference(cnf_transformation,[],[f53]) ).
tcf(c_59,plain,
! [X0: $i,X1: $i,X2: $i] :
( in(sK5(X1,X2,X0),X2)
| ~ in(X0,cartesian_product2(X1,X2)) ),
inference(cnf_transformation,[],[f54]) ).
tcf(c_65,negated_conjecture,
( ( empty_set != sK8 )
| ( cartesian_product2(sK8,sK9) != empty_set ) ),
inference(cnf_transformation,[],[f44]) ).
tcf(c_66,negated_conjecture,
( ( empty_set != sK9 )
| ( cartesian_product2(sK8,sK9) != empty_set ) ),
inference(cnf_transformation,[],[f43]) ).
tcf(c_67,negated_conjecture,
( ( empty_set = sK9 )
| ( empty_set = sK8 )
| ( cartesian_product2(sK8,sK9) = empty_set ) ),
inference(cnf_transformation,[],[f42]) ).
tcf(c_187,plain,
! [X0: $i,X1: $i,X2: $i,X3: $i] :
( in(unordered_pair(singleton(X0),unordered_pair(X0,X2)),cartesian_product2(X1,X3))
| ~ in(X2,X3)
| ~ in(X0,X1) ),
inference(demodulation,[status(thm)],[c_57,c_50]) ).
tcf(c_275,definition,
iPr_def_10 = cartesian_product2(sK8,sK9),
introduced(definition,[new_symbols(definition,[iPr_def_10])],[]) ).
tcf(c_276,negated_conjecture,
( ( iPr_def_10 = empty_set )
| ( empty_set = sK9 )
| ( empty_set = sK8 ) ),
inference(demodulation,[status(thm)],[c_67,c_275]) ).
tcf(c_277,negated_conjecture,
( ( iPr_def_10 != empty_set )
| ( empty_set != sK9 ) ),
inference(demodulation,[status(thm)],[c_66]) ).
tcf(c_278,negated_conjecture,
( ( iPr_def_10 != empty_set )
| ( empty_set != sK8 ) ),
inference(demodulation,[status(thm)],[c_65]) ).
tcf(c_546,plain,
( ( empty_set = iPr_def_10 )
| ( empty_set = sK8 )
| ( cartesian_product2(sK8,empty_set) = iPr_def_10 ) ),
inference(superposition,[status(thm)],[c_276,c_275]) ).
tcf(c_581,plain,
! [X0: $i,X1: $i] : ~ in(X0,cartesian_product2(empty_set,X1)),
inference(superposition,[status(thm)],[c_58,c_52]) ).
tcf(c_585,plain,
! [X0: $i] : cartesian_product2(empty_set,X0) = empty_set,
inference(superposition,[status(thm)],[c_51,c_581]) ).
tcf(c_596,plain,
! [X0: $i,X1: $i] : ~ in(X0,cartesian_product2(X1,empty_set)),
inference(superposition,[status(thm)],[c_59,c_52]) ).
tcf(c_599,plain,
! [X0: $i] : cartesian_product2(X0,empty_set) = empty_set,
inference(superposition,[status(thm)],[c_51,c_596]) ).
tcf(c_608,plain,
( ( empty_set = iPr_def_10 )
| ( empty_set = sK8 ) ),
inference(demodulation,[status(thm)],[c_546,c_599]) ).
tcf(c_616,plain,
( ( empty_set = iPr_def_10 )
| ( cartesian_product2(empty_set,sK9) = iPr_def_10 ) ),
inference(superposition,[status(thm)],[c_608,c_275]) ).
tcf(c_620,plain,
empty_set = iPr_def_10,
inference(demodulation,[status(thm)],[c_616,c_585]) ).
tcf(c_625,plain,
! [X0: $i] : ~ in(X0,iPr_def_10),
inference(demodulation,[status(thm)],[c_52,c_620]) ).
tcf(c_628,plain,
( ( iPr_def_10 != iPr_def_10 )
| ( sK8 != iPr_def_10 ) ),
inference(demodulation,[status(thm)],[c_278,c_620]) ).
tcf(c_629,plain,
( ( iPr_def_10 != iPr_def_10 )
| ( sK9 != iPr_def_10 ) ),
inference(demodulation,[status(thm)],[c_277,c_620]) ).
tcf(c_634,plain,
sK9 != iPr_def_10,
inference(equality_resolution_simp,[status(thm)],[c_629]) ).
tcf(c_635,plain,
sK8 != iPr_def_10,
inference(equality_resolution_simp,[status(thm)],[c_628]) ).
tcf(c_640,plain,
! [X0: $i] : cartesian_product2(X0,iPr_def_10) = iPr_def_10,
inference(light_normalisation,[status(thm)],[c_599,c_620]) ).
tcf(c_677,plain,
! [X0: $i,X1: $i] :
( in(unordered_pair(singleton(X0),unordered_pair(X0,X1)),iPr_def_10)
| ~ in(X1,sK9)
| ~ in(X0,sK8) ),
inference(superposition,[status(thm)],[c_275,c_187]) ).
tcf(c_732,plain,
! [X0: $i,X1: $i] :
( in(sK1(X0,iPr_def_10,X1),X1)
| ( cartesian_product2(X0,iPr_def_10) = X1 ) ),
inference(superposition,[status(thm)],[c_55,c_625]) ).
tcf(c_735,plain,
! [X0: $i,X1: $i] :
( in(sK1(X1,iPr_def_10,X0),X0)
| ( X0 = iPr_def_10 ) ),
inference(light_normalisation,[status(thm)],[c_732,c_640]) ).
tcf(c_1125,plain,
! [X0: $i,X1: $i] :
( ~ in(X1,sK9)
| ~ in(X0,sK8) ),
inference(forward_subsumption_resolution,[status(thm)],[c_677,c_625]) ).
tcf(c_1142,plain,
! [X0: $i] :
( ( sK8 = iPr_def_10 )
| ~ in(X0,sK9) ),
inference(superposition,[status(thm)],[c_735,c_1125]) ).
tcf(c_1144,plain,
! [X0: $i] : ~ in(X0,sK9),
inference(forward_subsumption_resolution,[status(thm)],[c_1142,c_635]) ).
tcf(c_1156,plain,
sK9 = iPr_def_10,
inference(superposition,[status(thm)],[c_735,c_1144]) ).
tcf(c_1158,plain,
$false,
inference(forward_subsumption_resolution,[status(thm)],[c_1156,c_634]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : SET960+1 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.04 % Command : run_iprover 300 /export/starexec/sandbox/benchmark/theBenchmark.p THM
% 0.08/0.35 % Computer : n014.cluster.edu
% 0.08/0.35 % Model : x86_64 x86_64
% 0.08/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.35 % Memory : 8046.5625MB
% 0.08/0.35 % OS : Linux 6.8.0-71-generic
% 0.08/0.35 % CPULimit : 300
% 0.08/0.35 % WCLimit : 300
% 0.08/0.35 % DateTime : Thu Sep 24 12:26:32 UTC 2026
% 0.08/0.36 % CPUTime :
% 0.08/0.36 Running run_iprover 300 /export/starexec/sandbox/benchmark/theBenchmark.p THM
% 0.14/0.39 Running first-order theorem proving
% 0.14/0.39 Running: /export/starexec/sandbox/solver/bin/iproveropt-multi-core.sh -d -n -l tptp -s fof_schedule -t 300 /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.14/0.40
% 0.14/0.40 % ======== iProver multi-core TPTP/SMT =========
% 0.14/0.40
% 0.14/0.40 % Detected problem language: tptp
% 0.14/0.41 % Proving...
% 3.21/1.27 % SZS status Started for theBenchmark.p
% 3.21/1.27 % SZS status Theorem for theBenchmark.p
% 3.21/1.27
% 3.21/1.27 %---------------- iProver v3.9.4 (pre CASC 2026/SMT-COMP 2026) ----------------%
% 3.21/1.27
% 3.21/1.27 % ------ iProver source info
% 3.21/1.27
% 3.21/1.27 % git: date: 2026-07-19 20:42:38 +0200
% 3.21/1.27 % git: sha1: 804e7d636a263075307957e923b7a22a4035de61
% 3.21/1.27 % git: non_committed_changes: false
% 3.21/1.27
% 3.21/1.27 % ------ Parsing...
% 3.21/1.27 % ------ Clausification by vclausify_rel & Parsing by iProver...%
% 3.21/1.27
% 3.21/1.27 % ------ Preprocessing... sup_sim: 5 sf_s rm: 1 0s sf_e pe_s pe_e %
% 3.21/1.27
% 3.21/1.27 % ------ Preprocessing... gs_s sp: 0 0s gs_e snvd_s sp: 0 0s snvd_e %
% 3.21/1.27
% 3.21/1.27 % ------ Preprocessing... sf_s rm: 1 0s sf_e sf_s rm: 0 0s sf_e
% 3.21/1.27 % ------ Proving...
% 3.21/1.27 % ------ Problem Properties
% 3.21/1.27
% 3.21/1.27 %
% 3.21/1.27 % clauses 20
% 3.21/1.27 % conjectures 3
% 3.21/1.27 % EPR 8
% 3.21/1.27 % Horn 15
% 3.21/1.27 % unary 7
% 3.21/1.27 % binary 7
% 3.21/1.27 % lits 41
% 3.21/1.27 % lits eq 17
% 3.21/1.27 % fd_pure 0
% 3.21/1.27 % fd_pseudo 0
% 3.21/1.27 % fd_cond 1
% 3.21/1.27 % fd_pseudo_cond 4
% 3.21/1.27 % AC symbols 0
% 3.21/1.27
% 3.21/1.27 % ------ Schedule dynamic 5 is on
% 3.21/1.27
% 3.21/1.27 % ------ Input Options "--resolution_flag false --inst_lit_sel_side none" Time Limit: 10.
% 3.21/1.27
% 3.21/1.27
% 3.21/1.27 % ------
% 3.21/1.27 % Current options:
% 3.21/1.27 % ------
% 3.21/1.27
% 3.21/1.27
% 3.21/1.27 %
% 3.21/1.27
% 3.21/1.27 % ------ Proving...
% 3.21/1.27 %
% 3.21/1.27
% 3.21/1.27 % SZS status Theorem for theBenchmark.p
% 3.21/1.27
% 3.21/1.27 % SZS output start CNFRefutation for theBenchmark.p
% See solution above
% 3.21/1.27
% 3.21/1.27
%------------------------------------------------------------------------------