%------------------------------------------------------------------------------
% File : FindProof---0.1
% Problem : SET042-3 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_findproof /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n004.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 02:41:41 PM UTC 2026
% Result : Unsatisfiable 118.13s 17.21s
% Output : Proof 118.13s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 18
% Syntax : Number of formulae : 115 ( 83 unt; 0 def)
% Number of atoms : 179 ( 80 equ)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 150 ( 86 ~; 64 |; 0 &)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 2 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 7 ( 5 usr; 1 prp; 0-2 aty)
% Number of functors : 16 ( 16 usr; 7 con; 0-4 aty)
% Number of variables : 137 ( 15 sgn 50 !; 0 ?)
% Comments :
%------------------------------------------------------------------------------
cnf(f40,axiom,
( ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z)
| ~ little_set(Z)
| member(Z,cross_product(X,Y)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cross_product4) ).
fof(f40_nnf,plain,
! [Z,X,Y] :
( ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z)
| ~ little_set(Z)
| member(Z,cross_product(X,Y)) ),
inference(nnf_transformation,[status(thm)],[f40]) ).
fof(f40_sk,plain,
! [Z,X,Y] :
( ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z)
| ~ little_set(Z)
| member(Z,cross_product(X,Y)) ),
inference(skolemisation,[status(esa)],[f40_nnf]) ).
cnf(c40,plain,
( ~ member(second(X0),X2)
| ~ member(first(X0),X1)
| ~ ordered_pair_predicate(X0)
| ~ little_set(X0)
| member(X0,cross_product(X1,X2)) ),
inference(cnf_transformation,[status(esa)],[f40_sk]) ).
cnf(t246,plain,
ifeq(little_set(X1),true,ifeq(ordered_pair_predicate(X1),true,ifeq(member(first(X1),X2),true,ifeq(member(second(X1),X3),true,member(X1,cross_product(X2,X3)),true),true),true),true) = true,
inference(equality_encoding,[status(esa)],[c40]) ).
cnf(t418,plain,
ifeq(little_set(X1),true,ifeq(ordered_pair_predicate(X1),true,ifeq(member(first(X1),X2),true,ifeq(member(second(X1),X3),true,member(X1,cross_product(X2,X3)),true),true),true),true) = true,
inference(orient,[status(thm)],[t246]) ).
cnf(f169,negated_conjecture,
~ member(ordered_pair(a,b),cross_product(set_a,set_b)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_ordered_pair_is_in_cross_product) ).
fof(f169_nnf,plain,
~ member(ordered_pair(a,b),cross_product(set_a,set_b)),
inference(nnf_transformation,[status(thm)],[f169]) ).
fof(f169_sk,plain,
~ member(ordered_pair(a,b),cross_product(set_a,set_b)),
inference(skolemisation,[status(esa)],[f169_nnf]) ).
cnf(c169,plain,
~ member(ordered_pair(a,b),cross_product(set_a,set_b)),
inference(cnf_transformation,[status(esa)],[f169_sk]) ).
cnf(t46,plain,
member(ordered_pair(a,b),cross_product(set_a,set_b)) = false,
inference(equality_encoding,[status(esa)],[c169]) ).
cnf(t773,plain,
member(ordered_pair(a,b),cross_product(set_a,set_b)) = false,
inference(orient,[status(thm)],[t46]) ).
cnf(t796,plain,
true = ifeq(little_set(ordered_pair(a,b)),true,ifeq(ordered_pair_predicate(ordered_pair(a,b)),true,ifeq(member(first(ordered_pair(a,b)),set_a),true,ifeq(member(second(ordered_pair(a,b)),set_b),true,false,true),true),true),true),
inference(cp,[status(thm)],[t418,t773]) ).
cnf(f150,axiom,
little_set(ordered_pair(X,Y)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pairs_are_small1) ).
fof(f150_nnf,plain,
! [X,Y] : little_set(ordered_pair(X,Y)),
inference(nnf_transformation,[status(thm)],[f150]) ).
fof(f150_sk,plain,
! [X,Y] : little_set(ordered_pair(X,Y)),
inference(skolemisation,[status(esa)],[f150_nnf]) ).
cnf(c150,plain,
little_set(ordered_pair(X0,X1)),
inference(cnf_transformation,[status(esa)],[f150_sk]) ).
cnf(t26,plain,
little_set(ordered_pair(X1,X2)) = true,
inference(equality_encoding,[status(esa)],[c150]) ).
cnf(t1251,plain,
little_set(ordered_pair(X1,X2)) = true,
inference(orient,[status(thm)],[t26]) ).
cnf(t88565,plain,
true = ifeq(true,true,ifeq(ordered_pair_predicate(ordered_pair(a,b)),true,ifeq(member(first(ordered_pair(a,b)),set_a),true,ifeq(member(second(ordered_pair(a,b)),set_b),true,false,true),true),true),true),
inference(step,[status(thm)],[t796,t1251]) ).
cnf(t31,plain,
ifeq(X1,X1,X2,X3) = X2,
introduced(definition) ).
cnf(t256,plain,
ifeq(X1,X1,X2,X3) = X2,
inference(orient,[status(thm)],[t31]) ).
cnf(t88566,plain,
true = ifeq(ordered_pair_predicate(ordered_pair(a,b)),true,ifeq(member(first(ordered_pair(a,b)),set_a),true,ifeq(member(second(ordered_pair(a,b)),set_b),true,false,true),true),true),
inference(step,[status(thm)],[t88565,t256]) ).
cnf(f0,axiom,
( little_set(X)
| ~ member(X,Y) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a2) ).
fof(f0_nnf,plain,
! [X,Y] :
( little_set(X)
| ~ member(X,Y) ),
inference(nnf_transformation,[status(thm)],[f0]) ).
fof(f0_sk,plain,
! [X,Y] :
( little_set(X)
| ~ member(X,Y) ),
inference(skolemisation,[status(esa)],[f0_nnf]) ).
cnf(c0,plain,
( little_set(X0)
| ~ member(X0,X1) ),
inference(cnf_transformation,[status(esa)],[f0_sk]) ).
cnf(hi0,axiom,
ifeq(member(X0,X1),true,little_set(X0),true) = true,
inference(equality_encoding,[status(esa)],[c0]) ).
cnf(f168,hypothesis,
member(b,set_b),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_of_set_b) ).
fof(f168_nnf,plain,
member(b,set_b),
inference(nnf_transformation,[status(thm)],[f168]) ).
cnf(c168,plain,
member(b,set_b),
inference(cnf_transformation,[status(esa)],[f168_nnf]) ).
cnf(hi163,axiom,
member(b,set_b) = true,
inference(equality_encoding,[status(esa)],[c168]) ).
cnf(h106,plain,
little_set(b) = true,
inference(hyper_resolution,[status(thm)],[hi0,hi163]) ).
cnf(f167,hypothesis,
member(a,set_a),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_of_set_a) ).
fof(f167_nnf,plain,
member(a,set_a),
inference(nnf_transformation,[status(thm)],[f167]) ).
cnf(c167,plain,
member(a,set_a),
inference(cnf_transformation,[status(esa)],[f167_nnf]) ).
cnf(hi162,axiom,
member(a,set_a) = true,
inference(equality_encoding,[status(esa)],[c167]) ).
cnf(h78,plain,
little_set(a) = true,
inference(hyper_resolution,[status(thm)],[hi0,hi162]) ).
cnf(f13,axiom,
( X != ordered_pair(Y,Z)
| ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(X) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair_predicate4) ).
fof(f13_nnf,plain,
! [X,Y,Z] :
( X != ordered_pair(Y,Z)
| ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(X) ),
inference(nnf_transformation,[status(thm)],[f13]) ).
fof(f13_sk,plain,
! [X,Y,Z] :
( X != ordered_pair(Y,Z)
| ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(X) ),
inference(skolemisation,[status(esa)],[f13_nnf]) ).
cnf(c13,plain,
( X0 != ordered_pair(X1,X2)
| ~ little_set(X2)
| ~ little_set(X1)
| ordered_pair_predicate(X0) ),
inference(cnf_transformation,[status(esa)],[f13_sk]) ).
cnf(hi13,axiom,
ifeq(little_set(X0),true,ifeq(little_set(X1),true,ifeq(X2,ordered_pair(X0,X1),ordered_pair_predicate(X2),true),true),true) = true,
inference(equality_encoding,[status(esa)],[c13]) ).
cnf(hi177,axiom,
or(X0,false) = X0,
introduced(definition) ).
cnf(f9,axiom,
ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair) ).
fof(f9_nnf,plain,
! [X,Y] : ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)),
inference(nnf_transformation,[status(thm)],[f9]) ).
fof(f9_sk,plain,
! [X,Y] : ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)),
inference(skolemisation,[status(esa)],[f9_nnf]) ).
cnf(c9,plain,
ordered_pair(X0,X1) = non_ordered_pair(singleton_set(X0),non_ordered_pair(X0,X1)),
inference(cnf_transformation,[status(esa)],[f9_sk]) ).
cnf(hi11,axiom,
ordered_pair(X0,X1) = non_ordered_pair(singleton_set(X0),non_ordered_pair(X0,X1)),
inference(equality_encoding,[status(esa)],[c9]) ).
cnf(f8,axiom,
singleton_set(X) = non_ordered_pair(X,X),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_set) ).
fof(f8_nnf,plain,
! [X] : singleton_set(X) = non_ordered_pair(X,X),
inference(nnf_transformation,[status(thm)],[f8]) ).
fof(f8_sk,plain,
! [X] : singleton_set(X) = non_ordered_pair(X,X),
inference(skolemisation,[status(esa)],[f8_nnf]) ).
cnf(c8,plain,
singleton_set(X0) = non_ordered_pair(X0,X0),
inference(cnf_transformation,[status(esa)],[f8_sk]) ).
cnf(hi12,axiom,
singleton_set(X0) = non_ordered_pair(X0,X0),
inference(equality_encoding,[status(esa)],[c8]) ).
cnf(t121,plain,
ordered_pair_predicate(or(non_ordered_pair(non_ordered_pair(a,a),non_ordered_pair(a,b)),false)) = true,
inference(hyper_resolution,[status(thm)],[hi13,h78,h106,hi177,hi11,hi12]) ).
cnf(t18,plain,
or(X1,false) = X1,
introduced(definition) ).
cnf(t275,plain,
or(X1,false) = X1,
inference(orient,[status(thm)],[t18]) ).
cnf(t86314,plain,
ordered_pair_predicate(non_ordered_pair(non_ordered_pair(a,a),non_ordered_pair(a,b))) = true,
inference(step,[status(thm)],[t121,t275]) ).
cnf(t27,plain,
non_ordered_pair(X1,X1) = singleton_set(X1),
inference(equality_encoding,[status(esa)],[c8]) ).
cnf(t808,plain,
non_ordered_pair(X1,X1) = singleton_set(X1),
inference(orient,[status(thm)],[t27]) ).
cnf(t86315,plain,
ordered_pair_predicate(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b))) = true,
inference(step,[status(thm)],[t86314,t808]) ).
cnf(t74,plain,
non_ordered_pair(singleton_set(X1),non_ordered_pair(X1,X2)) = ordered_pair(X1,X2),
inference(equality_encoding,[status(esa)],[c9]) ).
cnf(t819,plain,
non_ordered_pair(singleton_set(X1),non_ordered_pair(X1,X2)) = ordered_pair(X1,X2),
inference(orient,[status(thm)],[t74]) ).
cnf(t86316,plain,
ordered_pair_predicate(ordered_pair(a,b)) = true,
inference(step,[status(thm)],[t86315,t819]) ).
cnf(t2241,plain,
ordered_pair_predicate(ordered_pair(a,b)) = true,
inference(orient,[status(thm)],[t86316]) ).
cnf(t88567,plain,
true = ifeq(true,true,ifeq(member(first(ordered_pair(a,b)),set_a),true,ifeq(member(second(ordered_pair(a,b)),set_b),true,false,true),true),true),
inference(step,[status(thm)],[t88566,t2241]) ).
cnf(t88568,plain,
true = ifeq(member(first(ordered_pair(a,b)),set_a),true,ifeq(member(second(ordered_pair(a,b)),set_b),true,false,true),true),
inference(step,[status(thm)],[t88567,t256]) ).
cnf(f145,axiom,
( first(ordered_pair(X,Y)) = X
| ~ little_set(Y)
| ~ little_set(X) ),
file('/export/starexec/sandbox/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(hi140,axiom,
ifeq(little_set(X0),true,ifeq(little_set(X1),true,first(ordered_pair(X0,X1)),X0),X0) = X0,
inference(equality_encoding,[status(esa)],[c145]) ).
cnf(t53,plain,
first(non_ordered_pair(non_ordered_pair(a,a),non_ordered_pair(a,b))) = a,
inference(hyper_resolution,[status(thm)],[hi140,h78,h106,hi11,hi12]) ).
cnf(t86281,plain,
first(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b))) = a,
inference(step,[status(thm)],[t53,t808]) ).
cnf(t86282,plain,
first(ordered_pair(a,b)) = a,
inference(step,[status(thm)],[t86281,t819]) ).
cnf(t1677,plain,
first(ordered_pair(a,b)) = a,
inference(orient,[status(thm)],[t86282]) ).
cnf(t88569,plain,
true = ifeq(member(a,set_a),true,ifeq(member(second(ordered_pair(a,b)),set_b),true,false,true),true),
inference(step,[status(thm)],[t88568,t1677]) ).
cnf(t15,plain,
member(a,set_a) = true,
inference(equality_encoding,[status(esa)],[c167]) ).
cnf(t577,plain,
member(a,set_a) = true,
inference(orient,[status(thm)],[t15]) ).
cnf(t88570,plain,
true = ifeq(true,true,ifeq(member(second(ordered_pair(a,b)),set_b),true,false,true),true),
inference(step,[status(thm)],[t88569,t577]) ).
cnf(t88571,plain,
true = ifeq(member(second(ordered_pair(a,b)),set_b),true,false,true),
inference(step,[status(thm)],[t88570,t256]) ).
cnf(f146,axiom,
( second(ordered_pair(X,Y)) = Y
| ~ little_set(Y)
| ~ little_set(X) ),
file('/export/starexec/sandbox/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(hi141,axiom,
ifeq(little_set(X0),true,ifeq(little_set(X1),true,second(ordered_pair(X0,X1)),X1),X1) = X1,
inference(equality_encoding,[status(esa)],[c146]) ).
cnf(t85,plain,
second(non_ordered_pair(non_ordered_pair(a,a),non_ordered_pair(a,b))) = b,
inference(hyper_resolution,[status(thm)],[hi141,h78,h106,hi11,hi12]) ).
cnf(t86279,plain,
second(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b))) = b,
inference(step,[status(thm)],[t85,t808]) ).
cnf(t86280,plain,
second(ordered_pair(a,b)) = b,
inference(step,[status(thm)],[t86279,t819]) ).
cnf(t1659,plain,
second(ordered_pair(a,b)) = b,
inference(orient,[status(thm)],[t86280]) ).
cnf(t88572,plain,
true = ifeq(member(b,set_b),true,false,true),
inference(step,[status(thm)],[t88571,t1659]) ).
cnf(t16,plain,
member(b,set_b) = true,
inference(equality_encoding,[status(esa)],[c168]) ).
cnf(t553,plain,
member(b,set_b) = true,
inference(orient,[status(thm)],[t16]) ).
cnf(t88573,plain,
true = ifeq(true,true,false,true),
inference(step,[status(thm)],[t88572,t553]) ).
cnf(t88574,plain,
true = false,
inference(step,[status(thm)],[t88573,t256]) ).
cnf(t86044,plain,
false = true,
inference(orient,[status(thm)],[t88574]) ).
cnf(f30,axiom,
( ~ member(Z,X)
| ~ member(Z,complement(X)) ),
file('/export/starexec/sandbox/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/sandbox/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/sandbox/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/sandbox/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,c169]) ).
cnf(g0_0,plain,
true != true,
inference(rw,[status(thm)],[goal_0,t86044]) ).
cnf(contradiction_0,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[g0_0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET042-3 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.04 % Command : run_findproof /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.09/0.36 % Computer : n004.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 09:24:50 UTC 2026
% 0.13/0.37 % CPUTime :
% 0.13/0.37 Running run_findproof /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 118.13/17.21 % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p
% 118.13/17.21 % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
%------------------------------------------------------------------------------