%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SEU188+1 : TPTP v9.3.1. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n002.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 08:45:21 AM UTC 2026
% Result : Theorem 0.18s 0.51s
% Output : CNFRefutation 0.18s
% Verified :
% SZS Type : Refutation
% Derivation depth : 18
% Number of leaves : 13
% Syntax : Number of formulae : 65 ( 24 unt; 0 def)
% Number of atoms : 170 ( 53 equ)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 185 ( 80 ~; 74 |; 19 &)
% ( 2 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 3 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 6 ( 4 usr; 1 prp; 0-2 aty)
% Number of functors : 11 ( 11 usr; 2 con; 0-3 aty)
% Number of variables : 95 ( 3 sgn 44 !; 7 ?)
% Comments :
%------------------------------------------------------------------------------
fof(commutativity_k2_tarski,axiom,
! [A,B] : unordered_pair(A,B) = unordered_pair(B,A),
file('theBenchmark.p',commutativity_k2_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(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A),
file('theBenchmark.p',existence_m1_subset_1) ).
fof(fc5_relat_1,axiom,
! [A] :
( ( relation(A)
& ~ empty(A) )
=> ~ empty(relation_dom(A)) ),
file('theBenchmark.p',fc5_relat_1) ).
fof(fc6_relat_1,axiom,
! [A] :
( ( relation(A)
& ~ empty(A) )
=> ~ empty(relation_rng(A)) ),
file('theBenchmark.p',fc6_relat_1) ).
fof(fc7_relat_1,axiom,
! [A] :
( empty(A)
=> ( relation(relation_dom(A))
& empty(relation_dom(A)) ) ),
file('theBenchmark.p',fc7_relat_1) ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( in(A,B)
| empty(B) ) ),
file('theBenchmark.p',t2_subset) ).
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(t8_boole,axiom,
! [A,B] :
~ ( empty(B)
& A != B
& empty(A) ),
file('theBenchmark.p',t8_boole) ).
fof(t64_relat_1,conjecture,
! [A] :
( relation(A)
=> ( ( relation_rng(A) = empty_set
| relation_dom(A) = empty_set )
=> A = empty_set ) ),
file('theBenchmark.p',t64_relat_1) ).
fof(t64_relat_1_neg,negated_conjecture,
~ ! [A] :
( relation(A)
=> ( ( relation_rng(A) = empty_set
| relation_dom(A) = empty_set )
=> A = empty_set ) ),
inference(assume_negation,[status(cth)],[t64_relat_1]) ).
fof(nnf_4,plain,
! [A] :
( ! [B] :
( ( ? [C] :
( ( ! [D] : ~ in(ordered_pair(C,D),A)
| ~ in(C,B) )
& ( ? [X3] : in(ordered_pair(C,X3),A)
| in(C,B) ) )
| relation_dom(A) = B )
& ( ! [X0] :
( ( ! [X2] : ~ in(ordered_pair(X0,X2),A)
| in(X0,B) )
& ( ? [X1] : in(ordered_pair(X0,X1),A)
| ~ in(X0,B) ) )
| relation_dom(A) != B ) )
| ~ relation(A) ),
inference(fof_nnf,[status(thm)],[d4_relat_1]) ).
fof(sk_d4_relat_1_sk,plain,
! [A] :
( ! [B] :
( ( ( ( ! [D] : ~ in(ordered_pair(sK3(A,B),D),A)
| ~ in(sK3(A,B),B) )
& ( in(ordered_pair(sK3(A,B),sK4(A,B)),A)
| in(sK3(A,B),B) ) )
| relation_dom(A) = B )
& ( ! [X0] :
( ( ! [X2] : ~ in(ordered_pair(X0,X2),A)
| in(X0,B) )
& ( in(ordered_pair(X0,sK0(A,B,X0)),A)
| ~ in(X0,B) ) )
| relation_dom(A) != B ) )
| ~ relation(A) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK3,sK4]),skolemize(X1,sK0(A,B,X0)),skolemize(C,sK3(A,B)),skolemize(X3,sK4(A,B))],[nnf_4]) ).
fof(sk_d4_relat_1,plain,
( ! [VAR_0,VAR_1,VAR_2] :
( ~ in(ordered_pair(sK3(VAR_0,VAR_1),VAR_2),VAR_0)
| ~ in(sK3(VAR_0,VAR_1),VAR_1)
| relation_dom(VAR_0) = VAR_1
| ~ relation(VAR_0) )
& ! [VAR_0,VAR_1] :
( in(ordered_pair(sK3(VAR_0,VAR_1),sK4(VAR_0,VAR_1)),VAR_0)
| in(sK3(VAR_0,VAR_1),VAR_1)
| relation_dom(VAR_0) = VAR_1
| ~ relation(VAR_0) )
& ! [VAR_0,VAR_1,VAR_2,VAR_3] :
( ~ in(ordered_pair(VAR_2,VAR_3),VAR_0)
| in(VAR_2,VAR_1)
| relation_dom(VAR_0) != VAR_1
| ~ relation(VAR_0) )
& ! [VAR_0,VAR_1,VAR_2] :
( in(ordered_pair(VAR_2,sK0(VAR_0,VAR_1,VAR_2)),VAR_0)
| ~ in(VAR_2,VAR_1)
| relation_dom(VAR_0) != VAR_1
| ~ relation(VAR_0) ) ),
inference(cnf_transformation,[status(thm)],[sk_d4_relat_1_sk]) ).
cnf(c_34,plain,
( in(ordered_pair(C,sK0(A,B,C)),A)
| ~ in(C,B)
| relation_dom(A) != B
| ~ relation(A) ),
inference(split_conjunct,[status(thm)],[sk_d4_relat_1]) ).
cnf(c_44,plain,
( ~ empty(relation_dom(A))
| ~ relation(A)
| empty(A) ),
inference(clausify,[status(thm)],[fc5_relat_1]) ).
cnf(c_45,plain,
( ~ empty(relation_rng(A))
| ~ relation(A)
| empty(A) ),
inference(clausify,[status(thm)],[fc6_relat_1]) ).
cnf(c_46,plain,
( relation(relation_dom(A))
| ~ empty(A) ),
inference(clausify,[status(thm)],[fc7_relat_1]) ).
fof(nnf_33,plain,
? [A] :
( empty_set != A
& ( relation_rng(A) = empty_set
| relation_dom(A) = empty_set )
& relation(A) ),
inference(fof_nnf,[status(thm)],[t64_relat_1_neg]) ).
fof(sk_t64_relat_1_neg,plain,
( empty_set != sK1
& ( relation_rng(sK1) = empty_set
| relation_dom(sK1) = empty_set )
& relation(sK1) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1]),skolemize(A,sK1)],[nnf_33]) ).
cnf(c_51,negated_conjecture,
relation(sK1),
inference(split_conjunct,[status(thm)],[sk_t64_relat_1_neg]) ).
cnf(c_53,plain,
( in(A,B)
| empty(B)
| ~ element(A,B) ),
inference(clausify,[status(thm)],[t2_subset]) ).
fof(nnf_14,plain,
! [A] :
? [B] : element(B,A),
inference(fof_nnf,[status(thm)],[existence_m1_subset_1]) ).
fof(sk_existence_m1_subset_1,plain,
! [A] : element(sK2(A),A),
inference(skolemize,[status(esa),new_symbols(skolem,[sK2]),skolemize(B,sK2(A))],[nnf_14]) ).
cnf(c_54,plain,
element(sK2(A),A),
inference(split_conjunct,[status(thm)],[sk_existence_m1_subset_1]) ).
cnf(c_55,plain,
( in(sK2(A),A)
| empty(A) ),
inference(resolve,[status(thm)],[c_54,c_53]) ).
cnf(c_57,plain,
unordered_pair(A,B) = unordered_pair(B,A),
inference(clausify,[status(thm)],[commutativity_k2_tarski]) ).
cnf(c_58,plain,
ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)),
inference(clausify,[status(thm)],[d5_tarski]) ).
fof(fc1_xboole_0,axiom,
empty(empty_set),
file('theBenchmark.p',fc1_xboole_0) ).
cnf(c_59,plain,
empty(empty_set),
inference(clausify,[status(thm)],[fc1_xboole_0]) ).
cnf(c_64,plain,
( empty(relation_dom(A))
| ~ empty(A) ),
inference(clausify,[status(thm)],[fc7_relat_1]) ).
cnf(c_70,plain,
( empty_set = A
| ~ empty(A) ),
inference(clausify,[status(thm)],[t6_boole]) ).
cnf(c_71,plain,
( ~ empty(B)
| ~ in(A,B) ),
inference(clausify,[status(thm)],[t7_boole]) ).
cnf(c_72,plain,
( ~ empty(B)
| B = A
| ~ empty(A) ),
inference(clausify,[status(thm)],[t8_boole]) ).
cnf(c_73,negated_conjecture,
( relation_rng(sK1) = empty_set
| relation_dom(sK1) = empty_set ),
inference(split_conjunct,[status(thm)],[sk_t64_relat_1_neg]) ).
cnf(c_74,negated_conjecture,
empty_set != sK1,
inference(split_conjunct,[status(thm)],[sk_t64_relat_1_neg]) ).
cnf(c_75,negated_conjecture,
sK1 != empty_set,
inference(copy,[status(thm)],[c_74]) ).
cnf(c_114,plain,
( in(ordered_pair(C,sK0(relation_dom(A),B,C)),relation_dom(A))
| ~ in(C,B)
| relation_dom(relation_dom(A)) != B
| ~ empty(A) ),
inference(resolve,[status(thm)],[c_46,c_34]) ).
cnf(c_372,plain,
( in(unordered_pair(unordered_pair(C,sK0(relation_dom(A),B,C)),singleton(C)),relation_dom(A))
| ~ in(C,B)
| relation_dom(relation_dom(A)) != B
| ~ empty(A) ),
inference(paramod,[status(thm)],[c_58,c_114]) ).
cnf(c_115,plain,
( in(unordered_pair(singleton(C),unordered_pair(C,sK0(relation_dom(A),B,C))),relation_dom(A))
| ~ in(C,B)
| relation_dom(relation_dom(A)) != B
| ~ empty(A) ),
inference(paramod,[status(thm)],[c_57,c_372]) ).
cnf(c_179,negated_conjecture,
( ~ empty(relation_rng(sK1))
| empty(sK1) ),
inference(resolve,[status(thm)],[c_51,c_45]) ).
cnf(c_180,negated_conjecture,
( ~ empty(relation_dom(sK1))
| empty(sK1) ),
inference(resolve,[status(thm)],[c_51,c_44]) ).
cnf(c_199,plain,
empty(relation_dom(empty_set)),
inference(resolve,[status(thm)],[c_64,c_59]) ).
cnf(c_207,plain,
~ in(A,empty_set),
inference(resolve,[status(thm)],[c_71,c_59]) ).
cnf(c_373,plain,
( ~ empty(A)
| A = empty_set ),
inference(resolve,[status(thm)],[c_72,c_59]) ).
cnf(c_212,negated_conjecture,
~ empty(sK1),
inference(resolve,[status(thm)],[c_373,c_75]) ).
cnf(c_213,negated_conjecture,
~ empty(relation_dom(sK1)),
inference(resolve,[status(thm)],[c_212,c_180]) ).
cnf(c_214,negated_conjecture,
~ empty(relation_rng(sK1)),
inference(resolve,[status(thm)],[c_212,c_179]) ).
cnf(c_238,plain,
( in(unordered_pair(singleton(B),unordered_pair(B,sK0(relation_dom(empty_set),A,B))),relation_dom(empty_set))
| ~ in(B,A)
| relation_dom(relation_dom(empty_set)) != A ),
inference(resolve,[status(thm)],[c_115,c_59]) ).
cnf(c_374,plain,
empty_set = relation_dom(empty_set),
inference(resolve,[status(thm)],[c_199,c_70]) ).
cnf(c_340,plain,
relation_dom(empty_set) = empty_set,
inference(copy,[status(thm)],[c_374]) ).
cnf(c_375,plain,
( in(unordered_pair(singleton(B),unordered_pair(B,sK0(relation_dom(empty_set),A,B))),relation_dom(empty_set))
| ~ in(B,A)
| relation_dom(empty_set) != A ),
inference(paramod,[status(thm)],[c_340,c_238]) ).
cnf(c_376,plain,
( in(unordered_pair(singleton(B),unordered_pair(B,sK0(relation_dom(empty_set),A,B))),relation_dom(empty_set))
| ~ in(B,A)
| empty_set != A ),
inference(paramod,[status(thm)],[c_340,c_375]) ).
cnf(c_377,plain,
( in(unordered_pair(singleton(B),unordered_pair(B,sK0(empty_set,A,B))),relation_dom(empty_set))
| ~ in(B,A)
| empty_set != A ),
inference(paramod,[status(thm)],[c_340,c_376]) ).
cnf(c_378,plain,
( in(unordered_pair(singleton(B),unordered_pair(B,sK0(empty_set,A,B))),empty_set)
| ~ in(B,A)
| empty_set != A ),
inference(paramod,[status(thm)],[c_340,c_377]) ).
cnf(c_341,plain,
( ~ in(B,A)
| empty_set != A ),
inference(resolve,[status(thm)],[c_207,c_378]) ).
cnf(c_368,plain,
( empty(A)
| empty_set != A ),
inference(resolve,[status(thm)],[c_341,c_55]) ).
cnf(c_379,negated_conjecture,
( empty_set = relation_rng(sK1)
| relation_dom(sK1) = empty_set ),
inference(copy,[status(thm)],[c_73]) ).
cnf(c_380,negated_conjecture,
( relation_dom(sK1) = empty_set
| empty(relation_rng(sK1)) ),
inference(resolve,[status(thm)],[c_368,c_379]) ).
cnf(c_370,negated_conjecture,
relation_dom(sK1) = empty_set,
inference(resolve,[status(thm)],[c_214,c_380]) ).
cnf(c_381,negated_conjecture,
empty_set != relation_dom(sK1),
inference(resolve,[status(thm)],[c_368,c_213]) ).
cnf(c_382,negated_conjecture,
empty_set != empty_set,
inference(paramod,[status(thm)],[c_370,c_381]) ).
cnf(c_371,negated_conjecture,
$false,
inference(copy,[status(thm)],[c_382]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : SEU188+1 : TPTP v9.3.1. Released v3.3.0.
% 0.00/0.06 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.18/0.43 % Computer : n002.cluster.edu
% 0.18/0.43 % Model : x86_64 x86_64
% 0.18/0.43 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.43 % Memory : 8046.5625MB
% 0.18/0.43 % 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 08:49:35 UTC 2026
% 0.18/0.44 % CPUTime :
% 0.18/0.44 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.18/0.44 % Process 1301367 was started by sandbox2 on n002,
% 0.18/0.44 % Sat Sep 26 08:49:35 2026
% 0.18/0.44 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.18/0.46
% 0.18/0.46 % From the command line: assign(max_seconds, 300).
% 0.18/0.51
% 0.18/0.51 % SZS status Theorem for theBenchmark
% 0.18/0.51
% 0.18/0.51 % Proof 1 at 0.03 (+ 0.01) seconds.
% 0.18/0.51 % Length of proof is 45.
% 0.18/0.51 % Level of proof is 8.
% 0.18/0.51 % Maximum clause weight is 58.000.
% 0.18/0.51 % Given clauses 71.
% 0.18/0.51
% 0.18/0.51 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------