%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SEU209+1 : TPTP v9.3.1. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 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 : Sun Sep 27 08:45:25 AM UTC 2026
% Result : Theorem 6.49s 1.26s
% Output : CNFRefutation 6.49s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 6
% Syntax : Number of formulae : 37 ( 11 unt; 0 def)
% Number of atoms : 142 ( 27 equ)
% Maximal formula atoms : 26 ( 3 avg)
% Number of connectives : 175 ( 70 ~; 71 |; 24 &)
% ( 5 <=>; 5 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 5 ( 3 usr; 1 prp; 0-2 aty)
% Number of functors : 11 ( 11 usr; 2 con; 0-4 aty)
% Number of variables : 114 ( 7 sgn 66 !; 8 ?)
% Comments :
%------------------------------------------------------------------------------
fof(commutativity_k2_tarski,axiom,
! [A,B] : unordered_pair(A,B) = unordered_pair(B,A),
file('theBenchmark.p',commutativity_k2_tarski) ).
fof(d14_relat_1,axiom,
! [A] :
( relation(A)
=> ! [B,C] :
( C = relation_inverse_image(A,B)
<=> ! [D] :
( in(D,C)
<=> ? [E] :
( in(E,B)
& in(ordered_pair(D,E),A) ) ) ) ),
file('theBenchmark.p',d14_relat_1) ).
fof(d3_tarski,axiom,
! [A,B] :
( subset(A,B)
<=> ! [C] :
( in(C,A)
=> in(C,B) ) ),
file('theBenchmark.p',d3_tarski) ).
fof(d4_relat_1,axiom,
! [A] :
( relation(A)
=> ! [B] :
( B = relation_dom(A)
<=> ! [C] :
( in(C,B)
<=> ? [D] : in(ordered_pair(C,D),A) ) ) ),
file('theBenchmark.p',d4_relat_1) ).
fof(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)),
file('theBenchmark.p',d5_tarski) ).
fof(t167_relat_1,conjecture,
! [A,B] :
( relation(B)
=> subset(relation_inverse_image(B,A),relation_dom(B)) ),
file('theBenchmark.p',t167_relat_1) ).
fof(t167_relat_1_neg,negated_conjecture,
~ ! [A,B] :
( relation(B)
=> subset(relation_inverse_image(B,A),relation_dom(B)) ),
inference(assume_negation,[status(cth)],[t167_relat_1]) ).
fof(nnf_4,plain,
! [A] :
( ! [B,C] :
( ( ? [D] :
( ( ! [E] :
( ~ in(E,B)
| ~ in(ordered_pair(D,E),A) )
| ~ in(D,C) )
& ( ? [X3] :
( in(X3,B)
& in(ordered_pair(D,X3),A) )
| in(D,C) ) )
| relation_inverse_image(A,B) = C )
& ( ! [X0] :
( ( ! [X2] :
( ~ in(X2,B)
| ~ in(ordered_pair(X0,X2),A) )
| in(X0,C) )
& ( ? [X1] :
( in(X1,B)
& in(ordered_pair(X0,X1),A) )
| ~ in(X0,C) ) )
| relation_inverse_image(A,B) != C ) )
| ~ relation(A) ),
inference(fof_nnf,[status(thm)],[d14_relat_1]) ).
fof(sk_d14_relat_1_sk,plain,
! [A] :
( ! [B,C] :
( ( ( ( ! [E] :
( ~ in(E,B)
| ~ in(ordered_pair(sK4(A,B,C),E),A) )
| ~ in(sK4(A,B,C),C) )
& ( ( in(sK5(A,B,C),B)
& in(ordered_pair(sK4(A,B,C),sK5(A,B,C)),A) )
| in(sK4(A,B,C),C) ) )
| relation_inverse_image(A,B) = C )
& ( ! [X0] :
( ( ! [X2] :
( ~ in(X2,B)
| ~ in(ordered_pair(X0,X2),A) )
| in(X0,C) )
& ( ( in(sK0(A,B,C,X0),B)
& in(ordered_pair(X0,sK0(A,B,C,X0)),A) )
| ~ in(X0,C) ) )
| relation_inverse_image(A,B) != C ) )
| ~ relation(A) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK4,sK5]),skolemize(X1,sK0(A,B,C,X0)),skolemize(D,sK4(A,B,C)),skolemize(X3,sK5(A,B,C))],[nnf_4]) ).
fof(sk_d14_relat_1,plain,
( ! [VAR_0,VAR_1,VAR_2,VAR_3] :
( ~ in(VAR_3,VAR_1)
| ~ in(ordered_pair(sK4(VAR_0,VAR_1,VAR_2),VAR_3),VAR_0)
| ~ in(sK4(VAR_0,VAR_1,VAR_2),VAR_2)
| relation_inverse_image(VAR_0,VAR_1) = VAR_2
| ~ relation(VAR_0) )
& ! [VAR_0,VAR_1,VAR_2] :
( in(sK5(VAR_0,VAR_1,VAR_2),VAR_1)
| in(sK4(VAR_0,VAR_1,VAR_2),VAR_2)
| relation_inverse_image(VAR_0,VAR_1) = VAR_2
| ~ relation(VAR_0) )
& ! [VAR_0,VAR_1,VAR_2] :
( in(ordered_pair(sK4(VAR_0,VAR_1,VAR_2),sK5(VAR_0,VAR_1,VAR_2)),VAR_0)
| in(sK4(VAR_0,VAR_1,VAR_2),VAR_2)
| relation_inverse_image(VAR_0,VAR_1) = VAR_2
| ~ relation(VAR_0) )
& ! [VAR_0,VAR_1,VAR_2,VAR_3,VAR_4] :
( ~ in(VAR_4,VAR_1)
| ~ in(ordered_pair(VAR_3,VAR_4),VAR_0)
| in(VAR_3,VAR_2)
| relation_inverse_image(VAR_0,VAR_1) != VAR_2
| ~ relation(VAR_0) )
& ! [VAR_0,VAR_1,VAR_2,VAR_3] :
( in(sK0(VAR_0,VAR_1,VAR_2,VAR_3),VAR_1)
| ~ in(VAR_3,VAR_2)
| relation_inverse_image(VAR_0,VAR_1) != VAR_2
| ~ relation(VAR_0) )
& ! [VAR_0,VAR_1,VAR_2,VAR_3] :
( in(ordered_pair(VAR_3,sK0(VAR_0,VAR_1,VAR_2,VAR_3)),VAR_0)
| ~ in(VAR_3,VAR_2)
| relation_inverse_image(VAR_0,VAR_1) != VAR_2
| ~ relation(VAR_0) ) ),
inference(cnf_transformation,[status(thm)],[sk_d14_relat_1_sk]) ).
cnf(c_40,plain,
( in(ordered_pair(D,sK0(A,B,C,D)),A)
| ~ in(D,C)
| relation_inverse_image(A,B) != C
| ~ relation(A) ),
inference(split_conjunct,[status(thm)],[sk_d14_relat_1]) ).
cnf(c_48,plain,
( ~ in(ordered_pair(C,D),A)
| in(C,B)
| relation_dom(A) != B
| ~ relation(A) ),
inference(clausify,[status(thm)],[d4_relat_1]) ).
fof(nnf_39,plain,
? [A,B] :
( ~ subset(relation_inverse_image(B,A),relation_dom(B))
& relation(B) ),
inference(fof_nnf,[status(thm)],[t167_relat_1_neg]) ).
fof(sk_t167_relat_1_neg,plain,
( ~ subset(relation_inverse_image(sK1,sK2),relation_dom(sK1))
& relation(sK1) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1,sK2]),skolemize(A,sK2),skolemize(B,sK1)],[nnf_39]) ).
cnf(c_56,negated_conjecture,
relation(sK1),
inference(split_conjunct,[status(thm)],[sk_t167_relat_1_neg]) ).
cnf(c_57,negated_conjecture,
( in(ordered_pair(C,sK0(sK1,A,B,C)),sK1)
| ~ in(C,B)
| relation_inverse_image(sK1,A) != B ),
inference(resolve,[status(thm)],[c_56,c_40]) ).
cnf(c_64,negated_conjecture,
( ~ in(ordered_pair(B,C),sK1)
| in(B,A)
| relation_dom(sK1) != A ),
inference(resolve,[status(thm)],[c_56,c_48]) ).
cnf(c_118,negated_conjecture,
~ subset(relation_inverse_image(sK1,sK2),relation_dom(sK1)),
inference(split_conjunct,[status(thm)],[sk_t167_relat_1_neg]) ).
cnf(c_144,plain,
ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)),
inference(clausify,[status(thm)],[d5_tarski]) ).
fof(nnf_5,plain,
! [A,B] :
( ( ? [C] :
( ~ in(C,B)
& in(C,A) )
| subset(A,B) )
& ( ! [X3] :
( in(X3,B)
| ~ in(X3,A) )
| ~ subset(A,B) ) ),
inference(fof_nnf,[status(thm)],[d3_tarski]) ).
fof(sk_d3_tarski_sk,plain,
! [A,B] :
( ( ( ~ in(sK3(A,B),B)
& in(sK3(A,B),A) )
| subset(A,B) )
& ( ! [X3] :
( in(X3,B)
| ~ in(X3,A) )
| ~ subset(A,B) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK3]),skolemize(C,sK3(A,B))],[nnf_5]) ).
fof(sk_d3_tarski,plain,
( ! [VAR_0,VAR_1] :
( ~ in(sK3(VAR_0,VAR_1),VAR_1)
| subset(VAR_0,VAR_1) )
& ! [VAR_0,VAR_1] :
( in(sK3(VAR_0,VAR_1),VAR_0)
| subset(VAR_0,VAR_1) )
& ! [VAR_0,VAR_1,VAR_2] :
( in(VAR_2,VAR_1)
| ~ in(VAR_2,VAR_0)
| ~ subset(VAR_0,VAR_1) ) ),
inference(cnf_transformation,[status(thm)],[sk_d3_tarski_sk]) ).
cnf(c_145,plain,
( ~ in(sK3(A,B),B)
| subset(A,B) ),
inference(split_conjunct,[status(thm)],[sk_d3_tarski]) ).
cnf(c_146,plain,
( in(sK3(A,B),A)
| subset(A,B) ),
inference(split_conjunct,[status(thm)],[sk_d3_tarski]) ).
cnf(c_148,plain,
unordered_pair(A,B) = unordered_pair(B,A),
inference(clausify,[status(thm)],[commutativity_k2_tarski]) ).
cnf(c_4872,negated_conjecture,
( ~ in(unordered_pair(unordered_pair(B,C),singleton(B)),sK1)
| in(B,A)
| relation_dom(sK1) != A ),
inference(paramod,[status(thm)],[c_144,c_64]) ).
cnf(c_194,negated_conjecture,
( ~ in(unordered_pair(singleton(B),unordered_pair(B,C)),sK1)
| in(B,A)
| relation_dom(sK1) != A ),
inference(paramod,[status(thm)],[c_148,c_4872]) ).
cnf(c_4873,negated_conjecture,
( in(unordered_pair(unordered_pair(C,sK0(sK1,A,B,C)),singleton(C)),sK1)
| ~ in(C,B)
| relation_inverse_image(sK1,A) != B ),
inference(paramod,[status(thm)],[c_144,c_57]) ).
cnf(c_199,negated_conjecture,
( in(unordered_pair(singleton(C),unordered_pair(C,sK0(sK1,A,B,C))),sK1)
| ~ in(C,B)
| relation_inverse_image(sK1,A) != B ),
inference(paramod,[status(thm)],[c_148,c_4873]) ).
cnf(c_319,negated_conjecture,
~ in(sK3(relation_inverse_image(sK1,sK2),relation_dom(sK1)),relation_dom(sK1)),
inference(resolve,[status(thm)],[c_145,c_118]) ).
cnf(c_321,plain,
in(sK3(relation_inverse_image(sK1,sK2),relation_dom(sK1)),relation_inverse_image(sK1,sK2)),
inference(resolve,[status(thm)],[c_146,c_118]) ).
cnf(c_4874,plain,
( ~ in(unordered_pair(singleton(A),unordered_pair(A,B)),sK1)
| in(A,relation_dom(sK1)) ),
inference(xxres,[status(thm)],[c_194]) ).
cnf(c_4004,plain,
~ in(unordered_pair(singleton(sK3(relation_inverse_image(sK1,sK2),relation_dom(sK1))),unordered_pair(sK3(relation_inverse_image(sK1,sK2),relation_dom(sK1)),A)),sK1),
inference(resolve,[status(thm)],[c_4874,c_319]) ).
cnf(c_4875,negated_conjecture,
( in(unordered_pair(singleton(sK3(relation_inverse_image(sK1,sK2),relation_dom(sK1))),unordered_pair(sK3(relation_inverse_image(sK1,sK2),relation_dom(sK1)),sK0(sK1,A,relation_inverse_image(sK1,sK2),sK3(relation_inverse_image(sK1,sK2),relation_dom(sK1))))),sK1)
| relation_inverse_image(sK1,A) != relation_inverse_image(sK1,sK2) ),
inference(resolve,[status(thm)],[c_321,c_199]) ).
cnf(c_4876,negated_conjecture,
( in(unordered_pair(singleton(sK3(relation_inverse_image(sK1,sK2),relation_dom(sK1))),unordered_pair(sK3(relation_inverse_image(sK1,sK2),relation_dom(sK1)),sK0(sK1,A,relation_inverse_image(sK1,sK2),sK3(relation_inverse_image(sK1,sK2),relation_dom(sK1))))),sK1)
| relation_inverse_image(sK1,sK2) != relation_inverse_image(sK1,A) ),
inference(copy,[status(thm)],[c_4875]) ).
cnf(c_4072,negated_conjecture,
relation_inverse_image(sK1,sK2) != relation_inverse_image(sK1,A),
inference(resolve,[status(thm)],[c_4004,c_4876]) ).
cnf(c_4871,plain,
$false,
inference(xxres,[status(thm)],[c_4072]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SEU209+1 : TPTP v9.3.1. Released v3.3.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.08/0.35 % Computer : n009.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 : Sat Sep 26 08:50:15 UTC 2026
% 0.08/0.35 % CPUTime :
% 0.13/0.36 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.13/0.36 % Process 3899106 was started by sandbox on n009,
% 0.13/0.36 % Sat Sep 26 08:50:15 2026
% 0.13/0.36 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.13/0.36
% 0.13/0.36 % From the command line: assign(max_seconds, 300).
% 6.49/1.26
% 6.49/1.26 % given #500 (T,wt=22): 1380 f2(empty_set,A,powerset(empty_set)) = empty_set. [resolve(1198,b,857,a),flip(a),flip(b),unit_del(a,576)].
% 6.49/1.26
% 6.49/1.26 % SZS status Theorem for theBenchmark
% 6.49/1.26
% 6.49/1.26 % Proof 1 at 0.58 (+ 0.14) seconds.
% 6.49/1.26 % Length of proof is 23.
% 6.49/1.26 % Level of proof is 6.
% 6.49/1.26 % Maximum clause weight is 45.000.
% 6.49/1.26 % Given clauses 596.
% 6.49/1.26
% 6.49/1.26 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------