%------------------------------------------------------------------------------
% File : FindProof---0.1
% Problem : SET023-3 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_findproof /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n013.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 02:41:32 PM UTC 2026
% Result : Unsatisfiable 18.76s 2.82s
% Output : Proof 18.76s
% Verified :
% SZS Type : Refutation
% Derivation depth : 32
% Number of leaves : 13
% Syntax : Number of formulae : 102 ( 66 unt; 0 def)
% Number of atoms : 150 ( 70 equ)
% Maximal formula atoms : 3 ( 1 avg)
% Number of connectives : 118 ( 70 ~; 48 |; 0 &)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 2 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 7 ( 5 usr; 1 prp; 0-2 aty)
% Number of functors : 11 ( 11 usr; 4 con; 0-4 aty)
% Number of variables : 86 ( 4 sgn 32 !; 0 ?)
% Comments :
%------------------------------------------------------------------------------
cnf(f11,axiom,
( little_set(f3(X))
| ~ ordered_pair_predicate(X) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_predicate2) ).
fof(f11_nnf,plain,
! [X] :
( little_set(f3(X))
| ~ ordered_pair_predicate(X) ),
inference(nnf_transformation,[status(thm)],[f11]) ).
fof(f11_sk,plain,
! [X] :
( little_set(f3(X))
| ~ ordered_pair_predicate(X) ),
inference(skolemisation,[status(esa)],[f11_nnf]) ).
cnf(c11,plain,
( little_set(f3(X0))
| ~ ordered_pair_predicate(X0) ),
inference(cnf_transformation,[status(esa)],[f11_sk]) ).
cnf(t42,plain,
ifeq(ordered_pair_predicate(X1),true,little_set(f3(X1)),true) = true,
inference(equality_encoding,[status(esa)],[c11]) ).
cnf(t366,plain,
ifeq(ordered_pair_predicate(X1),true,little_set(f3(X1)),true) = true,
inference(orient,[status(thm)],[t42]) ).
cnf(f146,axiom,
( second(ordered_pair(X,Y)) = Y
| ~ little_set(Y)
| ~ little_set(X) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_of_second) ).
fof(f146_nnf,plain,
! [X,Y] :
( second(ordered_pair(X,Y)) = Y
| ~ little_set(Y)
| ~ little_set(X) ),
inference(nnf_transformation,[status(thm)],[f146]) ).
fof(f146_sk,plain,
! [X,Y] :
( second(ordered_pair(X,Y)) = Y
| ~ little_set(Y)
| ~ little_set(X) ),
inference(skolemisation,[status(esa)],[f146_nnf]) ).
cnf(c146,plain,
( second(ordered_pair(X0,X1)) = X1
| ~ little_set(X1)
| ~ little_set(X0) ),
inference(cnf_transformation,[status(esa)],[f146_sk]) ).
cnf(t112,plain,
ifeq(little_set(X1),true,ifeq(little_set(X2),true,second(ordered_pair(X1,X2)),X2),X2) = X2,
inference(equality_encoding,[status(esa)],[c146]) ).
cnf(t176,plain,
ifeq(little_set(X1),true,ifeq(little_set(X2),true,second(ordered_pair(X1,X2)),X2),X2) = X2,
inference(orient,[status(thm)],[t112]) ).
cnf(f12,axiom,
( X = ordered_pair(f2(X),f3(X))
| ~ ordered_pair_predicate(X) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_predicate3) ).
fof(f12_nnf,plain,
! [X] :
( X = ordered_pair(f2(X),f3(X))
| ~ ordered_pair_predicate(X) ),
inference(nnf_transformation,[status(thm)],[f12]) ).
fof(f12_sk,plain,
! [X] :
( X = ordered_pair(f2(X),f3(X))
| ~ ordered_pair_predicate(X) ),
inference(skolemisation,[status(esa)],[f12_nnf]) ).
cnf(c12,plain,
( X0 = ordered_pair(f2(X0),f3(X0))
| ~ ordered_pair_predicate(X0) ),
inference(cnf_transformation,[status(esa)],[f12_sk]) ).
cnf(t119,plain,
ifeq(ordered_pair_predicate(X1),true,X1,ordered_pair(f2(X1),f3(X1))) = ordered_pair(f2(X1),f3(X1)),
inference(equality_encoding,[status(esa)],[c12]) ).
cnf(t361,plain,
ifeq(ordered_pair_predicate(X1),true,X1,ordered_pair(f2(X1),f3(X1))) = ordered_pair(f2(X1),f3(X1)),
inference(orient,[status(thm)],[t119]) ).
cnf(f145,axiom,
( first(ordered_pair(X,Y)) = X
| ~ little_set(Y)
| ~ little_set(X) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_of_first) ).
fof(f145_nnf,plain,
! [X,Y] :
( first(ordered_pair(X,Y)) = X
| ~ little_set(Y)
| ~ little_set(X) ),
inference(nnf_transformation,[status(thm)],[f145]) ).
fof(f145_sk,plain,
! [X,Y] :
( first(ordered_pair(X,Y)) = X
| ~ little_set(Y)
| ~ little_set(X) ),
inference(skolemisation,[status(esa)],[f145_nnf]) ).
cnf(c145,plain,
( first(ordered_pair(X0,X1)) = X0
| ~ little_set(X1)
| ~ little_set(X0) ),
inference(cnf_transformation,[status(esa)],[f145_sk]) ).
cnf(t111,plain,
ifeq(little_set(X1),true,ifeq(little_set(X2),true,first(ordered_pair(X1,X2)),X1),X1) = X1,
inference(equality_encoding,[status(esa)],[c145]) ).
cnf(t177,plain,
ifeq(little_set(X1),true,ifeq(little_set(X2),true,first(ordered_pair(X1,X2)),X1),X1) = X1,
inference(orient,[status(thm)],[t111]) ).
cnf(f148,hypothesis,
ordered_pair_predicate(a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',an_ordered_pair_predicate) ).
fof(f148_nnf,plain,
ordered_pair_predicate(a),
inference(nnf_transformation,[status(thm)],[f148]) ).
cnf(c148,plain,
ordered_pair_predicate(a),
inference(cnf_transformation,[status(esa)],[f148_nnf]) ).
cnf(t4,plain,
ordered_pair_predicate(a) = true,
inference(equality_encoding,[status(esa)],[c148]) ).
cnf(t722,plain,
ordered_pair_predicate(a) = true,
inference(orient,[status(thm)],[t4]) ).
cnf(t729,plain,
ordered_pair(f2(a),f3(a)) = ifeq(true,true,a,ordered_pair(f2(a),f3(a))),
inference(cp,[status(thm)],[t361,t722]) ).
cnf(t15,plain,
ifeq(X1,X1,X2,X3) = X2,
introduced(definition) ).
cnf(t165,plain,
ifeq(X1,X1,X2,X3) = X2,
inference(orient,[status(thm)],[t15]) ).
cnf(t6234,plain,
ordered_pair(f2(a),f3(a)) = a,
inference(step,[status(thm)],[t729,t165]) ).
cnf(t3028,plain,
ordered_pair(f2(a),f3(a)) = a,
inference(orient,[status(thm)],[t6234]) ).
cnf(t3032,plain,
f2(a) = ifeq(little_set(f2(a)),true,ifeq(little_set(f3(a)),true,first(a),f2(a)),f2(a)),
inference(cp,[status(thm)],[t177,t3028]) ).
cnf(f10,axiom,
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_predicate1) ).
fof(f10_nnf,plain,
! [X] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ),
inference(nnf_transformation,[status(thm)],[f10]) ).
fof(f10_sk,plain,
! [X] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ),
inference(skolemisation,[status(esa)],[f10_nnf]) ).
cnf(c10,plain,
( little_set(f2(X0))
| ~ ordered_pair_predicate(X0) ),
inference(cnf_transformation,[status(esa)],[f10_sk]) ).
cnf(t41,plain,
ifeq(ordered_pair_predicate(X1),true,little_set(f2(X1)),true) = true,
inference(equality_encoding,[status(esa)],[c10]) ).
cnf(t365,plain,
ifeq(ordered_pair_predicate(X1),true,little_set(f2(X1)),true) = true,
inference(orient,[status(thm)],[t41]) ).
cnf(t727,plain,
true = ifeq(true,true,little_set(f2(a)),true),
inference(cp,[status(thm)],[t365,t722]) ).
cnf(t6188,plain,
true = little_set(f2(a)),
inference(step,[status(thm)],[t727,t165]) ).
cnf(t1080,plain,
little_set(f2(a)) = true,
inference(orient,[status(thm)],[t6188]) ).
cnf(t6235,plain,
f2(a) = ifeq(true,true,ifeq(little_set(f3(a)),true,first(a),f2(a)),f2(a)),
inference(step,[status(thm)],[t3032,t1080]) ).
cnf(t6236,plain,
f2(a) = ifeq(little_set(f3(a)),true,first(a),f2(a)),
inference(step,[status(thm)],[t6235,t165]) ).
cnf(t726,plain,
true = ifeq(true,true,little_set(f3(a)),true),
inference(cp,[status(thm)],[t366,t722]) ).
cnf(t6187,plain,
true = little_set(f3(a)),
inference(step,[status(thm)],[t726,t165]) ).
cnf(t1024,plain,
little_set(f3(a)) = true,
inference(orient,[status(thm)],[t6187]) ).
cnf(t6237,plain,
f2(a) = ifeq(true,true,first(a),f2(a)),
inference(step,[status(thm)],[t6236,t1024]) ).
cnf(t6238,plain,
f2(a) = first(a),
inference(step,[status(thm)],[t6237,t165]) ).
cnf(t3109,plain,
f2(a) = first(a),
inference(orient,[status(thm)],[t6238]) ).
cnf(t3110,plain,
ordered_pair(f2(a),f3(a)) = ifeq(ordered_pair_predicate(a),true,a,ordered_pair(first(a),f3(a))),
inference(cp,[status(thm)],[t361,t3109]) ).
cnf(t6312,plain,
ordered_pair(first(a),f3(a)) = ifeq(ordered_pair_predicate(a),true,a,ordered_pair(first(a),f3(a))),
inference(step,[status(thm)],[t3110,t3109]) ).
cnf(t6313,plain,
ordered_pair(first(a),f3(a)) = ifeq(true,true,a,ordered_pair(first(a),f3(a))),
inference(step,[status(thm)],[t6312,t722]) ).
cnf(t6314,plain,
ordered_pair(first(a),f3(a)) = a,
inference(step,[status(thm)],[t6313,t165]) ).
cnf(t6117,plain,
ordered_pair(first(a),f3(a)) = a,
inference(orient,[status(thm)],[t6314]) ).
cnf(t6120,plain,
f3(a) = ifeq(little_set(first(a)),true,ifeq(little_set(f3(a)),true,second(a),f3(a)),f3(a)),
inference(cp,[status(thm)],[t176,t6117]) ).
cnf(f147,axiom,
( little_set(first(X))
| ~ ordered_pair_predicate(X) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',first_component_is_small) ).
fof(f147_nnf,plain,
! [X] :
( little_set(first(X))
| ~ ordered_pair_predicate(X) ),
inference(nnf_transformation,[status(thm)],[f147]) ).
fof(f147_sk,plain,
! [X] :
( little_set(first(X))
| ~ ordered_pair_predicate(X) ),
inference(skolemisation,[status(esa)],[f147_nnf]) ).
cnf(c147,plain,
( little_set(first(X0))
| ~ ordered_pair_predicate(X0) ),
inference(cnf_transformation,[status(esa)],[f147_sk]) ).
cnf(t43,plain,
ifeq(ordered_pair_predicate(X1),true,little_set(first(X1)),true) = true,
inference(equality_encoding,[status(esa)],[c147]) ).
cnf(t364,plain,
ifeq(ordered_pair_predicate(X1),true,little_set(first(X1)),true) = true,
inference(orient,[status(thm)],[t43]) ).
cnf(t728,plain,
true = ifeq(true,true,little_set(first(a)),true),
inference(cp,[status(thm)],[t364,t722]) ).
cnf(t6189,plain,
true = little_set(first(a)),
inference(step,[status(thm)],[t728,t165]) ).
cnf(t1136,plain,
little_set(first(a)) = true,
inference(orient,[status(thm)],[t6189]) ).
cnf(t6315,plain,
f3(a) = ifeq(true,true,ifeq(little_set(f3(a)),true,second(a),f3(a)),f3(a)),
inference(step,[status(thm)],[t6120,t1136]) ).
cnf(t6316,plain,
f3(a) = ifeq(little_set(f3(a)),true,second(a),f3(a)),
inference(step,[status(thm)],[t6315,t165]) ).
cnf(t6317,plain,
f3(a) = ifeq(true,true,second(a),f3(a)),
inference(step,[status(thm)],[t6316,t1024]) ).
cnf(t6318,plain,
f3(a) = second(a),
inference(step,[status(thm)],[t6317,t165]) ).
cnf(t6136,plain,
f3(a) = second(a),
inference(orient,[status(thm)],[t6318]) ).
cnf(t6137,plain,
true = ifeq(ordered_pair_predicate(a),true,little_set(second(a)),true),
inference(cp,[status(thm)],[t366,t6136]) ).
cnf(t6331,plain,
true = ifeq(true,true,little_set(second(a)),true),
inference(step,[status(thm)],[t6137,t722]) ).
cnf(t6332,plain,
true = little_set(second(a)),
inference(step,[status(thm)],[t6331,t165]) ).
cnf(f149,negated_conjecture,
~ little_set(second(a)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_second_component_is_small) ).
fof(f149_nnf,plain,
~ little_set(second(a)),
inference(nnf_transformation,[status(thm)],[f149]) ).
fof(f149_sk,plain,
~ little_set(second(a)),
inference(skolemisation,[status(esa)],[f149_nnf]) ).
cnf(c149,plain,
~ little_set(second(a)),
inference(cnf_transformation,[status(esa)],[f149_sk]) ).
cnf(t6,plain,
little_set(second(a)) = false,
inference(equality_encoding,[status(esa)],[c149]) ).
cnf(t577,plain,
little_set(second(a)) = false,
inference(orient,[status(thm)],[t6]) ).
cnf(t6333,plain,
true = false,
inference(step,[status(thm)],[t6332,t577]) ).
cnf(t6150,plain,
false = true,
inference(orient,[status(thm)],[t6333]) ).
cnf(f30,axiom,
( ~ member(Z,X)
| ~ member(Z,complement(X)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement1) ).
fof(f30_nnf,plain,
! [Z,X] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ),
inference(nnf_transformation,[status(thm)],[f30]) ).
fof(f30_sk,plain,
! [Z,X] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ),
inference(skolemisation,[status(esa)],[f30_nnf]) ).
cnf(c30,plain,
( ~ member(X0,X1)
| ~ member(X0,complement(X1)) ),
inference(cnf_transformation,[status(esa)],[f30_sk]) ).
cnf(f57,axiom,
~ member(Z,empty_set),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',empty_set) ).
fof(f57_nnf,plain,
! [Z] : ~ member(Z,empty_set),
inference(nnf_transformation,[status(thm)],[f57]) ).
fof(f57_sk,plain,
! [Z] : ~ member(Z,empty_set),
inference(skolemisation,[status(esa)],[f57_nnf]) ).
cnf(c57,plain,
~ member(X0,empty_set),
inference(cnf_transformation,[status(esa)],[f57_sk]) ).
cnf(f70,axiom,
( X != Y
| ~ proper_subset(X,Y) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',proper_subset2) ).
fof(f70_nnf,plain,
! [X,Y] :
( X != Y
| ~ proper_subset(X,Y) ),
inference(nnf_transformation,[status(thm)],[f70]) ).
fof(f70_sk,plain,
! [X,Y] :
( X != Y
| ~ proper_subset(X,Y) ),
inference(skolemisation,[status(esa)],[f70_nnf]) ).
cnf(c70,plain,
( X0 != X1
| ~ proper_subset(X0,X1) ),
inference(cnf_transformation,[status(esa)],[f70_sk]) ).
cnf(f94,axiom,
( ~ member(U,Y)
| ~ member(U,X)
| ~ disjoint(X,Y) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',disjoint1) ).
fof(f94_nnf,plain,
! [X,Y,U] :
( ~ member(U,Y)
| ~ member(U,X)
| ~ disjoint(X,Y) ),
inference(nnf_transformation,[status(thm)],[f94]) ).
fof(f94_sk,plain,
! [X,Y,U] :
( ~ member(U,Y)
| ~ member(U,X)
| ~ disjoint(X,Y) ),
inference(skolemisation,[status(esa)],[f94_nnf]) ).
cnf(c94,plain,
( ~ member(X2,X1)
| ~ member(X2,X0)
| ~ disjoint(X0,X1) ),
inference(cnf_transformation,[status(esa)],[f94_sk]) ).
cnf(goal_0,negated_conjecture,
true != false,
inference(equality_encoding,[status(esa)],[c30,c57,c70,c94,c149]) ).
cnf(g0_0,plain,
true != true,
inference(rw,[status(thm)],[goal_0,t6150]) ).
cnf(contradiction_0,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[g0_0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : SET023-3 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.04 % Command : run_findproof /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.09/0.36 % Computer : n013.cluster.edu
% 0.09/0.36 % Model : x86_64 x86_64
% 0.09/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.36 % Memory : 8046.5625MB
% 0.09/0.36 % OS : Linux 6.8.0-71-generic
% 0.09/0.36 % CPULimit : 300
% 0.09/0.36 % WCLimit : 300
% 0.09/0.36 % DateTime : Thu Sep 24 09:14:06 UTC 2026
% 0.09/0.36 % CPUTime :
% 0.09/0.36 Running run_findproof /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 18.76/2.82 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p
% 18.76/2.82 % SZS output start Proof for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
%------------------------------------------------------------------------------