%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : NUM409+1 : TPTP v9.3.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n011.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 08:16:55 AM UTC 2026
% Result : Theorem 0.02s 0.40s
% Output : CNFRefutation 0.02s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 10
% Syntax : Number of formulae : 46 ( 19 unt; 0 def)
% Number of atoms : 100 ( 13 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 90 ( 36 ~; 32 |; 13 &)
% ( 3 <=>; 6 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 13 ( 11 usr; 1 prp; 0-2 aty)
% Number of functors : 4 ( 4 usr; 2 con; 0-1 aty)
% Number of variables : 26 ( 2 sgn 10 !; 1 ?)
% Comments :
%------------------------------------------------------------------------------
fof(cc1_funct_1,axiom,
! [A] :
( empty(A)
=> function(A) ),
file('theBenchmark.p',cc1_funct_1) ).
fof(d7_ordinal1,axiom,
! [A] :
( ( function(A)
& relation(A) )
=> ( transfinite_sequence(A)
<=> ordinal(relation_dom(A)) ) ),
file('theBenchmark.p',d7_ordinal1) ).
fof(d8_ordinal1,axiom,
! [A,B] :
( ( transfinite_sequence(B)
& function(B)
& relation(B) )
=> ( transfinite_sequence_of(B,A)
<=> subset(relation_rng(B),A) ) ),
file('theBenchmark.p',d8_ordinal1) ).
fof(fc12_relat_1,axiom,
( relation_empty_yielding(empty_set)
& relation(empty_set)
& empty(empty_set) ),
file('theBenchmark.p',fc12_relat_1) ).
fof(fc2_ordinal1,axiom,
( ordinal(empty_set)
& epsilon_connected(empty_set)
& epsilon_transitive(empty_set)
& empty(empty_set)
& one_to_one(empty_set)
& function(empty_set)
& relation_empty_yielding(empty_set)
& relation(empty_set) ),
file('theBenchmark.p',fc2_ordinal1) ).
fof(fc7_relat_1,axiom,
! [A] :
( empty(A)
=> ( relation(relation_dom(A))
& empty(relation_dom(A)) ) ),
file('theBenchmark.p',fc7_relat_1) ).
fof(t2_xboole_1,axiom,
! [A] : subset(empty_set,A),
file('theBenchmark.p',t2_xboole_1) ).
fof(t65_relat_1,axiom,
! [A] :
( relation(A)
=> ( relation_dom(A) = empty_set
<=> relation_rng(A) = empty_set ) ),
file('theBenchmark.p',t65_relat_1) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ),
file('theBenchmark.p',t6_boole) ).
fof(t45_ordinal1,conjecture,
! [A] : transfinite_sequence_of(empty_set,A),
file('theBenchmark.p',t45_ordinal1) ).
fof(t45_ordinal1_neg,negated_conjecture,
~ ! [A] : transfinite_sequence_of(empty_set,A),
inference(assume_negation,[status(cth)],[t45_ordinal1]) ).
cnf(c_52,plain,
( function(A)
| ~ empty(A) ),
inference(clausify,[status(thm)],[cc1_funct_1]) ).
cnf(c_54,plain,
( ~ ordinal(relation_dom(A))
| transfinite_sequence(A)
| ~ function(A)
| ~ relation(A) ),
inference(clausify,[status(thm)],[d7_ordinal1]) ).
cnf(c_56,plain,
( ~ subset(relation_rng(A),B)
| transfinite_sequence_of(A,B)
| ~ transfinite_sequence(A)
| ~ function(A)
| ~ relation(A) ),
inference(clausify,[status(thm)],[d8_ordinal1]) ).
cnf(c_58,plain,
function(empty_set),
inference(clausify,[status(thm)],[fc2_ordinal1]) ).
cnf(c_94,plain,
empty(empty_set),
inference(clausify,[status(thm)],[fc12_relat_1]) ).
cnf(c_95,plain,
relation(empty_set),
inference(clausify,[status(thm)],[fc12_relat_1]) ).
cnf(c_99,plain,
ordinal(empty_set),
inference(clausify,[status(thm)],[fc2_ordinal1]) ).
cnf(c_102,plain,
( empty(relation_dom(A))
| ~ empty(A) ),
inference(clausify,[status(thm)],[fc7_relat_1]) ).
cnf(c_103,plain,
( relation(relation_dom(A))
| ~ empty(A) ),
inference(clausify,[status(thm)],[fc7_relat_1]) ).
cnf(c_133,plain,
subset(empty_set,A),
inference(clausify,[status(thm)],[t2_xboole_1]) ).
cnf(c_138,plain,
( relation_rng(A) = empty_set
| relation_dom(A) != empty_set
| ~ relation(A) ),
inference(clausify,[status(thm)],[t65_relat_1]) ).
cnf(c_140,plain,
( empty_set = A
| ~ empty(A) ),
inference(clausify,[status(thm)],[t6_boole]) ).
fof(nnf_50,plain,
? [A] : ~ transfinite_sequence_of(empty_set,A),
inference(fof_nnf,[status(thm)],[t45_ordinal1_neg]) ).
fof(sk_t45_ordinal1_neg,plain,
~ transfinite_sequence_of(empty_set,sK0),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(A,sK0)],[nnf_50]) ).
cnf(c_143,negated_conjecture,
~ transfinite_sequence_of(empty_set,sK0),
inference(split_conjunct,[status(thm)],[sk_t45_ordinal1_neg]) ).
cnf(c_145,plain,
( ~ empty(A)
| ~ ordinal(relation_dom(A))
| transfinite_sequence(A)
| ~ relation(A) ),
inference(resolve,[status(thm)],[c_54,c_52]) ).
cnf(c_151,plain,
( ~ subset(relation_rng(empty_set),A)
| transfinite_sequence_of(empty_set,A)
| ~ transfinite_sequence(empty_set)
| ~ relation(empty_set) ),
inference(resolve,[status(thm)],[c_58,c_56]) ).
cnf(c_152,plain,
( ~ subset(relation_rng(empty_set),A)
| transfinite_sequence_of(empty_set,A)
| ~ transfinite_sequence(empty_set) ),
inference(resolve,[status(thm)],[c_95,c_151]) ).
cnf(c_207,plain,
empty(relation_dom(empty_set)),
inference(resolve,[status(thm)],[c_102,c_94]) ).
cnf(c_208,plain,
relation(relation_dom(empty_set)),
inference(resolve,[status(thm)],[c_103,c_94]) ).
cnf(c_269,plain,
( relation_rng(empty_set) = empty_set
| relation_dom(empty_set) != empty_set ),
inference(resolve,[status(thm)],[c_138,c_95]) ).
cnf(c_374,plain,
( ~ ordinal(relation_dom(relation_dom(empty_set)))
| transfinite_sequence(relation_dom(empty_set))
| ~ relation(relation_dom(empty_set)) ),
inference(resolve,[status(thm)],[c_207,c_145]) ).
cnf(c_361,plain,
( ~ ordinal(relation_dom(relation_dom(empty_set)))
| transfinite_sequence(relation_dom(empty_set)) ),
inference(resolve,[status(thm)],[c_208,c_374]) ).
cnf(c_375,plain,
empty_set = relation_dom(empty_set),
inference(resolve,[status(thm)],[c_207,c_140]) ).
cnf(c_365,plain,
relation_dom(empty_set) = empty_set,
inference(copy,[status(thm)],[c_375]) ).
cnf(c_376,plain,
( ~ ordinal(relation_dom(relation_dom(empty_set)))
| transfinite_sequence(empty_set) ),
inference(paramod,[status(thm)],[c_365,c_361]) ).
cnf(c_377,plain,
( ~ ordinal(relation_dom(empty_set))
| transfinite_sequence(empty_set) ),
inference(paramod,[status(thm)],[c_365,c_376]) ).
cnf(c_378,plain,
( ~ ordinal(empty_set)
| transfinite_sequence(empty_set) ),
inference(paramod,[status(thm)],[c_365,c_377]) ).
cnf(c_366,plain,
transfinite_sequence(empty_set),
inference(resolve,[status(thm)],[c_99,c_378]) ).
cnf(c_379,plain,
( relation_rng(empty_set) = empty_set
| empty_set != empty_set ),
inference(paramod,[status(thm)],[c_365,c_269]) ).
cnf(c_370,plain,
relation_rng(empty_set) = empty_set,
inference(copy,[status(thm)],[c_379]) ).
cnf(c_380,plain,
( ~ subset(empty_set,A)
| transfinite_sequence_of(empty_set,A)
| ~ transfinite_sequence(empty_set) ),
inference(paramod,[status(thm)],[c_370,c_152]) ).
cnf(c_381,plain,
( ~ subset(empty_set,A)
| transfinite_sequence_of(empty_set,A) ),
inference(resolve,[status(thm)],[c_366,c_380]) ).
cnf(c_372,plain,
transfinite_sequence_of(empty_set,A),
inference(resolve,[status(thm)],[c_133,c_381]) ).
cnf(c_373,plain,
$false,
inference(resolve,[status(thm)],[c_372,c_143]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : NUM409+1 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.36 % Computer : n011.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
% 0.02/0.36 % CPULimit : 300
% 0.02/0.36 % WCLimit : 300
% 0.02/0.36 % DateTime : Sat Sep 26 02:39:13 UTC 2026
% 0.02/0.36 % CPUTime :
% 0.02/0.36 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.02/0.36 % Process 3606127 was started by sandbox2 on n011,
% 0.02/0.36 % Sat Sep 26 02:39:13 2026
% 0.02/0.36 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.02/0.38
% 0.02/0.38 % From the command line: assign(max_seconds, 300).
% 0.02/0.40
% 0.02/0.40 % SZS status Theorem for theBenchmark
% 0.02/0.40
% 0.02/0.40 % Proof 1 at 0.01 (+ 0.01) seconds.
% 0.02/0.40 % Length of proof is 35.
% 0.02/0.40 % Level of proof is 6.
% 0.02/0.40 % Maximum clause weight is 27.000.
% 0.02/0.40 % Given clauses 100.
% 0.02/0.40
% 0.02/0.40 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------