%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SEU055+1 : TPTP v9.3.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n018.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 08:44:57 AM UTC 2026
% Result : Theorem 2.70s 0.86s
% Output : CNFRefutation 2.70s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 10
% Syntax : Number of formulae : 69 ( 28 unt; 0 def)
% Number of atoms : 184 ( 64 equ)
% Maximal formula atoms : 23 ( 2 avg)
% Number of connectives : 186 ( 71 ~; 73 |; 28 &)
% ( 5 <=>; 9 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 3 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 9 ( 7 usr; 1 prp; 0-2 aty)
% Number of functors : 11 ( 11 usr; 2 con; 0-2 aty)
% Number of variables : 83 ( 9 sgn 43 !; 5 ?)
% Comments :
%------------------------------------------------------------------------------
fof(d1_tarski,axiom,
! [A,B] :
( B = singleton(A)
<=> ! [C] :
( in(C,B)
<=> C = A ) ),
file('theBenchmark.p',d1_tarski) ).
fof(d8_funct_1,axiom,
! [A] :
( ( function(A)
& relation(A) )
=> ( one_to_one(A)
<=> ! [B,C] :
( ( apply(A,B) = apply(A,C)
& in(C,relation_dom(A))
& in(B,relation_dom(A)) )
=> B = C ) ) ),
file('theBenchmark.p',d8_funct_1) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( empty(B)
& element(B,powerset(A)) ),
file('theBenchmark.p',rc2_subset_1) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A),
file('theBenchmark.p',reflexivity_r1_tarski) ).
fof(t117_funct_1,axiom,
! [A,B] :
( ( function(B)
& relation(B) )
=> ( in(A,relation_dom(B))
=> relation_image(B,singleton(A)) = singleton(apply(B,A)) ) ),
file('theBenchmark.p',t117_funct_1) ).
fof(t20_zfmisc_1,axiom,
! [A,B] :
( set_difference(singleton(A),singleton(B)) = singleton(A)
<=> A != B ),
file('theBenchmark.p',t20_zfmisc_1) ).
fof(t37_xboole_1,axiom,
! [A,B] :
( set_difference(A,B) = empty_set
<=> subset(A,B) ),
file('theBenchmark.p',t37_xboole_1) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ),
file('theBenchmark.p',t6_boole) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( empty(B)
& in(A,B) ),
file('theBenchmark.p',t7_boole) ).
fof(t124_funct_1,conjecture,
! [A] :
( ( function(A)
& relation(A) )
=> ( ! [B,C] : relation_image(A,set_difference(B,C)) = set_difference(relation_image(A,B),relation_image(A,C))
=> one_to_one(A) ) ),
file('theBenchmark.p',t124_funct_1) ).
fof(t124_funct_1_neg,negated_conjecture,
~ ! [A] :
( ( function(A)
& relation(A) )
=> ( ! [B,C] : relation_image(A,set_difference(B,C)) = set_difference(relation_image(A,B),relation_image(A,C))
=> one_to_one(A) ) ),
inference(assume_negation,[status(cth)],[t124_funct_1]) ).
fof(nnf_6,plain,
! [A] :
( ( ( ? [B,C] :
( C != B
& apply(A,C) = apply(A,B)
& in(C,relation_dom(A))
& in(B,relation_dom(A)) )
| one_to_one(A) )
& ( ! [X0,X1] :
( X1 = X0
| apply(A,X1) != apply(A,X0)
| ~ in(X1,relation_dom(A))
| ~ in(X0,relation_dom(A)) )
| ~ one_to_one(A) ) )
| ~ function(A)
| ~ relation(A) ),
inference(fof_nnf,[status(thm)],[d8_funct_1]) ).
fof(sk_d8_funct_1_sk,plain,
! [A] :
( ( ( ( sK1(A) != sK0(A)
& apply(A,sK1(A)) = apply(A,sK0(A))
& in(sK1(A),relation_dom(A))
& in(sK0(A),relation_dom(A)) )
| one_to_one(A) )
& ( ! [X0,X1] :
( X1 = X0
| apply(A,X1) != apply(A,X0)
| ~ in(X1,relation_dom(A))
| ~ in(X0,relation_dom(A)) )
| ~ one_to_one(A) ) )
| ~ function(A)
| ~ relation(A) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK1]),skolemize(B,sK0(A)),skolemize(C,sK1(A))],[nnf_6]) ).
fof(sk_d8_funct_1,plain,
( ! [VAR_0] :
( sK1(VAR_0) != sK0(VAR_0)
| one_to_one(VAR_0)
| ~ function(VAR_0)
| ~ relation(VAR_0) )
& ! [VAR_0] :
( apply(VAR_0,sK1(VAR_0)) = apply(VAR_0,sK0(VAR_0))
| one_to_one(VAR_0)
| ~ function(VAR_0)
| ~ relation(VAR_0) )
& ! [VAR_0] :
( in(sK1(VAR_0),relation_dom(VAR_0))
| one_to_one(VAR_0)
| ~ function(VAR_0)
| ~ relation(VAR_0) )
& ! [VAR_0] :
( in(sK0(VAR_0),relation_dom(VAR_0))
| one_to_one(VAR_0)
| ~ function(VAR_0)
| ~ relation(VAR_0) )
& ! [VAR_0,VAR_1,VAR_2] :
( VAR_2 = VAR_1
| apply(VAR_0,VAR_2) != apply(VAR_0,VAR_1)
| ~ in(VAR_2,relation_dom(VAR_0))
| ~ in(VAR_1,relation_dom(VAR_0))
| ~ one_to_one(VAR_0)
| ~ function(VAR_0)
| ~ relation(VAR_0) ) ),
inference(cnf_transformation,[status(thm)],[sk_d8_funct_1_sk]) ).
cnf(c_43,plain,
( in(sK0(A),relation_dom(A))
| one_to_one(A)
| ~ function(A)
| ~ relation(A) ),
inference(split_conjunct,[status(thm)],[sk_d8_funct_1]) ).
cnf(c_44,plain,
( in(sK1(A),relation_dom(A))
| one_to_one(A)
| ~ function(A)
| ~ relation(A) ),
inference(split_conjunct,[status(thm)],[sk_d8_funct_1]) ).
cnf(c_45,plain,
( apply(A,sK1(A)) = apply(A,sK0(A))
| one_to_one(A)
| ~ function(A)
| ~ relation(A) ),
inference(split_conjunct,[status(thm)],[sk_d8_funct_1]) ).
cnf(c_46,plain,
( sK1(A) != sK0(A)
| one_to_one(A)
| ~ function(A)
| ~ relation(A) ),
inference(split_conjunct,[status(thm)],[sk_d8_funct_1]) ).
cnf(c_50,plain,
( relation_image(A,singleton(B)) = singleton(apply(A,B))
| ~ in(B,relation_dom(A))
| ~ function(A)
| ~ relation(A) ),
inference(clausify,[status(thm)],[t117_funct_1]) ).
fof(nnf_39,plain,
? [A] :
( ~ one_to_one(A)
& ! [B,C] : relation_image(A,set_difference(B,C)) = set_difference(relation_image(A,B),relation_image(A,C))
& function(A)
& relation(A) ),
inference(fof_nnf,[status(thm)],[t124_funct_1_neg]) ).
fof(sk_t124_funct_1_neg,plain,
( ~ one_to_one(sK2)
& ! [B,C] : relation_image(sK2,set_difference(B,C)) = set_difference(relation_image(sK2,B),relation_image(sK2,C))
& function(sK2)
& relation(sK2) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK2]),skolemize(A,sK2)],[nnf_39]) ).
cnf(c_51,negated_conjecture,
function(sK2),
inference(split_conjunct,[status(thm)],[sk_t124_funct_1_neg]) ).
cnf(c_52,plain,
( ~ subset(A,B)
| set_difference(A,B) = empty_set ),
inference(clausify,[status(thm)],[t37_xboole_1]) ).
cnf(c_54,plain,
subset(A,A),
inference(clausify,[status(thm)],[reflexivity_r1_tarski]) ).
cnf(c_60,plain,
set_difference(A,A) = empty_set,
inference(resolve,[status(thm)],[c_52,c_54]) ).
cnf(c_61,negated_conjecture,
( in(sK0(sK2),relation_dom(sK2))
| one_to_one(sK2)
| ~ relation(sK2) ),
inference(resolve,[status(thm)],[c_51,c_43]) ).
cnf(c_62,negated_conjecture,
( in(sK1(sK2),relation_dom(sK2))
| one_to_one(sK2)
| ~ relation(sK2) ),
inference(resolve,[status(thm)],[c_51,c_44]) ).
cnf(c_63,negated_conjecture,
( apply(sK2,sK1(sK2)) = apply(sK2,sK0(sK2))
| one_to_one(sK2)
| ~ relation(sK2) ),
inference(resolve,[status(thm)],[c_51,c_45]) ).
cnf(c_64,negated_conjecture,
( sK1(sK2) != sK0(sK2)
| one_to_one(sK2)
| ~ relation(sK2) ),
inference(resolve,[status(thm)],[c_51,c_46]) ).
cnf(c_65,negated_conjecture,
( relation_image(sK2,singleton(A)) = singleton(apply(sK2,A))
| ~ in(A,relation_dom(sK2))
| ~ relation(sK2) ),
inference(resolve,[status(thm)],[c_51,c_50]) ).
cnf(c_66,negated_conjecture,
( singleton(apply(sK2,A)) = relation_image(sK2,singleton(A))
| ~ in(A,relation_dom(sK2))
| ~ relation(sK2) ),
inference(copy,[status(thm)],[c_65]) ).
cnf(c_89,negated_conjecture,
~ one_to_one(sK2),
inference(split_conjunct,[status(thm)],[sk_t124_funct_1_neg]) ).
cnf(c_90,negated_conjecture,
relation_image(sK2,set_difference(A,B)) = set_difference(relation_image(sK2,A),relation_image(sK2,B)),
inference(split_conjunct,[status(thm)],[sk_t124_funct_1_neg]) ).
cnf(c_91,negated_conjecture,
relation(sK2),
inference(split_conjunct,[status(thm)],[sk_t124_funct_1_neg]) ).
cnf(c_93,plain,
( ~ empty(B)
| ~ in(A,B) ),
inference(clausify,[status(thm)],[t7_boole]) ).
cnf(c_94,plain,
( empty_set = A
| ~ empty(A) ),
inference(clausify,[status(thm)],[t6_boole]) ).
cnf(c_100,plain,
( B = A
| set_difference(singleton(A),singleton(B)) = singleton(A) ),
inference(clausify,[status(thm)],[t20_zfmisc_1]) ).
fof(nnf_21,plain,
! [A] :
? [B] :
( empty(B)
& element(B,powerset(A)) ),
inference(fof_nnf,[status(thm)],[rc2_subset_1]) ).
fof(sk_rc2_subset_1,plain,
! [A] :
( empty(sK3(A))
& element(sK3(A),powerset(A)) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK3]),skolemize(B,sK3(A))],[nnf_21]) ).
cnf(c_107,plain,
empty(sK3(A)),
inference(split_conjunct,[status(thm)],[sk_rc2_subset_1]) ).
cnf(c_130,plain,
( C != A
| in(C,B)
| singleton(A) != B ),
inference(clausify,[status(thm)],[d1_tarski]) ).
cnf(c_1417,negated_conjecture,
( sK1(sK2) != sK0(sK2)
| one_to_one(sK2) ),
inference(resolve,[status(thm)],[c_91,c_64]) ).
cnf(c_134,negated_conjecture,
sK1(sK2) != sK0(sK2),
inference(resolve,[status(thm)],[c_89,c_1417]) ).
cnf(c_1418,negated_conjecture,
( apply(sK2,sK1(sK2)) = apply(sK2,sK0(sK2))
| one_to_one(sK2) ),
inference(resolve,[status(thm)],[c_91,c_63]) ).
cnf(c_135,negated_conjecture,
apply(sK2,sK1(sK2)) = apply(sK2,sK0(sK2)),
inference(resolve,[status(thm)],[c_89,c_1418]) ).
cnf(c_1419,negated_conjecture,
( in(sK1(sK2),relation_dom(sK2))
| one_to_one(sK2) ),
inference(resolve,[status(thm)],[c_91,c_62]) ).
cnf(c_136,negated_conjecture,
in(sK1(sK2),relation_dom(sK2)),
inference(resolve,[status(thm)],[c_89,c_1419]) ).
cnf(c_1420,negated_conjecture,
( in(sK0(sK2),relation_dom(sK2))
| one_to_one(sK2) ),
inference(resolve,[status(thm)],[c_91,c_61]) ).
cnf(c_137,negated_conjecture,
in(sK0(sK2),relation_dom(sK2)),
inference(resolve,[status(thm)],[c_89,c_1420]) ).
cnf(c_138,negated_conjecture,
( singleton(apply(sK2,A)) = relation_image(sK2,singleton(A))
| ~ in(A,relation_dom(sK2)) ),
inference(resolve,[status(thm)],[c_91,c_66]) ).
cnf(c_1421,plain,
( relation_image(sK2,singleton(B)) = set_difference(relation_image(sK2,singleton(B)),relation_image(sK2,singleton(A)))
| A = B ),
inference(paramod,[status(thm)],[c_100,c_90]) ).
cnf(c_174,plain,
( set_difference(relation_image(sK2,singleton(B)),relation_image(sK2,singleton(A))) = relation_image(sK2,singleton(B))
| A = B ),
inference(copy,[status(thm)],[c_1421]) ).
cnf(c_1422,plain,
empty_set = sK3(A),
inference(resolve,[status(thm)],[c_107,c_94]) ).
cnf(c_216,plain,
sK3(A) = empty_set,
inference(copy,[status(thm)],[c_1422]) ).
cnf(c_1423,plain,
~ in(A,sK3(B)),
inference(resolve,[status(thm)],[c_107,c_93]) ).
cnf(c_217,plain,
~ in(A,empty_set),
inference(paramod,[status(thm)],[c_216,c_1423]) ).
cnf(c_456,plain,
singleton(apply(sK2,sK0(sK2))) = relation_image(sK2,singleton(sK0(sK2))),
inference(resolve,[status(thm)],[c_138,c_137]) ).
cnf(c_1424,negated_conjecture,
singleton(apply(sK2,sK1(sK2))) = relation_image(sK2,singleton(sK1(sK2))),
inference(resolve,[status(thm)],[c_138,c_136]) ).
cnf(c_1425,negated_conjecture,
singleton(apply(sK2,sK0(sK2))) = relation_image(sK2,singleton(sK1(sK2))),
inference(paramod,[status(thm)],[c_135,c_1424]) ).
cnf(c_1426,negated_conjecture,
relation_image(sK2,singleton(sK0(sK2))) = relation_image(sK2,singleton(sK1(sK2))),
inference(paramod,[status(thm)],[c_456,c_1425]) ).
cnf(c_457,negated_conjecture,
relation_image(sK2,singleton(sK1(sK2))) = relation_image(sK2,singleton(sK0(sK2))),
inference(copy,[status(thm)],[c_1426]) ).
cnf(c_1427,plain,
( B != A
| singleton(A) != empty_set ),
inference(resolve,[status(thm)],[c_130,c_217]) ).
cnf(c_1428,negated_conjecture,
singleton(apply(sK2,sK0(sK2))) != empty_set,
inference(resolve,[status(thm)],[c_1427,c_135]) ).
cnf(c_582,negated_conjecture,
relation_image(sK2,singleton(sK0(sK2))) != empty_set,
inference(paramod,[status(thm)],[c_456,c_1428]) ).
cnf(c_1429,negated_conjecture,
set_difference(relation_image(sK2,singleton(sK0(sK2))),relation_image(sK2,singleton(sK1(sK2)))) = relation_image(sK2,singleton(sK0(sK2))),
inference(resolve,[status(thm)],[c_174,c_134]) ).
cnf(c_1430,negated_conjecture,
set_difference(relation_image(sK2,singleton(sK0(sK2))),relation_image(sK2,singleton(sK0(sK2)))) = relation_image(sK2,singleton(sK0(sK2))),
inference(paramod,[status(thm)],[c_457,c_1429]) ).
cnf(c_1431,negated_conjecture,
empty_set = relation_image(sK2,singleton(sK0(sK2))),
inference(paramod,[status(thm)],[c_60,c_1430]) ).
cnf(c_1432,negated_conjecture,
relation_image(sK2,singleton(sK0(sK2))) = empty_set,
inference(copy,[status(thm)],[c_1431]) ).
cnf(c_1416,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_582,c_1432]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.04 % Problem : SEU055+1 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.06 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.18/0.44 % Computer : n018.cluster.edu
% 0.18/0.44 % Model : x86_64 x86_64
% 0.18/0.44 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.44 % Memory : 8046.5625MB
% 0.18/0.44 % OS : Linux 6.8.0-71-generic
% 0.18/0.44 % CPULimit : 300
% 0.18/0.44 % WCLimit : 300
% 0.18/0.44 % DateTime : Sat Sep 26 08:35:06 UTC 2026
% 0.18/0.45 % CPUTime :
% 0.18/0.45 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.18/0.45 % Process 16017 was started by sandbox on n018,
% 0.18/0.45 % Sat Sep 26 08:35:06 2026
% 0.18/0.45 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.18/0.47
% 0.18/0.47 % From the command line: assign(max_seconds, 300).
% 2.70/0.86
% 2.70/0.86 % SZS status Theorem for theBenchmark
% 2.70/0.86
% 2.70/0.86 % Proof 1 at 0.20 (+ 0.07) seconds.
% 2.70/0.86 % Length of proof is 45.
% 2.70/0.86 % Level of proof is 7.
% 2.70/0.86 % Maximum clause weight is 25.000.
% 2.70/0.86 % Given clauses 154.
% 2.70/0.86
% 2.70/0.86 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------