%------------------------------------------------------------------------------
% File : CSE_E---1.7
% Problem : NUM409+1 : TPTP v9.2.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/lemma_parallel_prover %s --lemma-prover /export/starexec/sandbox/solver/bin/cse --final-prover /export/starexec/sandbox/solver/bin/eprover --proof-time %d --global-time-limit %d
% Computer : n024.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue May 5 04:57:15 PM UTC 2026
% Result : Theorem 1.11s 1.61s
% Output : CNFRefutation 1.11s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 17
% Syntax : Number of formulae : 78 ( 30 unt; 0 def)
% Number of atoms : 202 ( 21 equ)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 217 ( 93 ~; 85 |; 26 &)
% ( 5 <=>; 8 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 4 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 15 ( 13 usr; 1 prp; 0-2 aty)
% Number of functors : 13 ( 13 usr; 2 con; 0-3 aty)
% Number of variables : 116 ( 15 sgn 55 !; 3 ?)
% Comments :
%------------------------------------------------------------------------------
fof(t5_subset,axiom,
! [X1,X2,X3] :
~ ( in(X1,X2)
& element(X2,powerset(X3))
& empty(X3) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t5_subset) ).
fof(t3_subset,axiom,
! [X1,X2] :
( element(X1,powerset(X2))
<=> subset(X1,X2) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t3_subset) ).
fof(d8_ordinal1,axiom,
! [X1,X2] :
( ( relation(X2)
& function(X2)
& transfinite_sequence(X2) )
=> ( transfinite_sequence_of(X2,X1)
<=> subset(relation_rng(X2),X1) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d8_ordinal1) ).
fof(dt_m1_ordinal1,axiom,
! [X1,X2] :
( transfinite_sequence_of(X2,X1)
=> ( relation(X2)
& function(X2)
& transfinite_sequence(X2) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_m1_ordinal1) ).
fof(t2_subset,axiom,
! [X1,X2] :
( element(X1,X2)
=> ( empty(X2)
| in(X1,X2) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t2_subset) ).
fof(existence_m1_subset_1,axiom,
! [X1] :
? [X2] : element(X2,X1),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_m1_subset_1) ).
fof(d4_relat_1,axiom,
! [X1] :
( relation(X1)
=> ! [X2] :
( X2 = relation_dom(X1)
<=> ! [X3] :
( in(X3,X2)
<=> ? [X4] : in(ordered_pair(X3,X4),X1) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d4_relat_1) ).
fof(d5_tarski,axiom,
! [X1,X2] : ordered_pair(X1,X2) = unordered_pair(unordered_pair(X1,X2),singleton(X1)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d5_tarski) ).
fof(t2_xboole_1,axiom,
! [X1] : subset(empty_set,X1),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t2_xboole_1) ).
fof(commutativity_k2_tarski,axiom,
! [X1,X2] : unordered_pair(X1,X2) = unordered_pair(X2,X1),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_k2_tarski) ).
fof(fc6_relat_1,axiom,
! [X1] :
( ( ~ empty(X1)
& relation(X1) )
=> ~ empty(relation_rng(X1)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc6_relat_1) ).
fof(existence_m1_ordinal1,axiom,
! [X1] :
? [X2] : transfinite_sequence_of(X2,X1),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_m1_ordinal1) ).
fof(fc12_relat_1,axiom,
( empty(empty_set)
& relation(empty_set)
& relation_empty_yielding(empty_set) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc12_relat_1) ).
fof(t6_boole,axiom,
! [X1] :
( empty(X1)
=> X1 = empty_set ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t6_boole) ).
fof(d7_ordinal1,axiom,
! [X1] :
( ( relation(X1)
& function(X1) )
=> ( transfinite_sequence(X1)
<=> ordinal(relation_dom(X1)) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d7_ordinal1) ).
fof(t45_ordinal1,conjecture,
! [X1] : transfinite_sequence_of(empty_set,X1),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t45_ordinal1) ).
fof(fc2_ordinal1,axiom,
( relation(empty_set)
& relation_empty_yielding(empty_set)
& function(empty_set)
& one_to_one(empty_set)
& empty(empty_set)
& epsilon_transitive(empty_set)
& epsilon_connected(empty_set)
& ordinal(empty_set) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc2_ordinal1) ).
fof(c_0_17,plain,
! [X137,X138,X139] :
( ~ in(X137,X138)
| ~ element(X138,powerset(X139))
| ~ empty(X139) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[t5_subset])]) ).
fof(c_0_18,plain,
! [X131,X132] :
( ( ~ element(X131,powerset(X132))
| subset(X131,X132) )
& ( ~ subset(X131,X132)
| element(X131,powerset(X132)) ) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[t3_subset])]) ).
fof(c_0_19,plain,
! [X96,X97] :
( ( ~ transfinite_sequence_of(X97,X96)
| subset(relation_rng(X97),X96)
| ~ relation(X97)
| ~ function(X97)
| ~ transfinite_sequence(X97) )
& ( ~ subset(relation_rng(X97),X96)
| transfinite_sequence_of(X97,X96)
| ~ relation(X97)
| ~ function(X97)
| ~ transfinite_sequence(X97) ) ),
inference(distribute,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[d8_ordinal1])])]) ).
fof(c_0_20,plain,
! [X98,X99] :
( ( relation(X99)
| ~ transfinite_sequence_of(X99,X98) )
& ( function(X99)
| ~ transfinite_sequence_of(X99,X98) )
& ( transfinite_sequence(X99)
| ~ transfinite_sequence_of(X99,X98) ) ),
inference(distribute,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[dt_m1_ordinal1])])]) ).
cnf(c_0_21,plain,
( ~ in(X1,X2)
| ~ element(X2,powerset(X3))
| ~ empty(X3) ),
inference(split_conjunct,[status(thm)],[c_0_17]) ).
cnf(c_0_22,plain,
( element(X1,powerset(X2))
| ~ subset(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_18]) ).
cnf(c_0_23,plain,
( subset(relation_rng(X1),X2)
| ~ transfinite_sequence_of(X1,X2)
| ~ relation(X1)
| ~ function(X1)
| ~ transfinite_sequence(X1) ),
inference(split_conjunct,[status(thm)],[c_0_19]) ).
cnf(c_0_24,plain,
( function(X1)
| ~ transfinite_sequence_of(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_20]) ).
cnf(c_0_25,plain,
( relation(X1)
| ~ transfinite_sequence_of(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_20]) ).
cnf(c_0_26,plain,
( transfinite_sequence(X1)
| ~ transfinite_sequence_of(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_20]) ).
fof(c_0_27,plain,
! [X128,X129] :
( ~ element(X128,X129)
| empty(X129)
| in(X128,X129) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[t2_subset])]) ).
fof(c_0_28,plain,
! [X102] : element(esk5_1(X102),X102),
inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[existence_m1_subset_1])]) ).
fof(c_0_29,plain,
! [X83,X84,X85,X87,X88,X89,X91] :
( ( ~ in(X85,X84)
| in(ordered_pair(X85,esk1_3(X83,X84,X85)),X83)
| X84 != relation_dom(X83)
| ~ relation(X83) )
& ( ~ in(ordered_pair(X87,X88),X83)
| in(X87,X84)
| X84 != relation_dom(X83)
| ~ relation(X83) )
& ( ~ in(esk2_2(X83,X89),X89)
| ~ in(ordered_pair(esk2_2(X83,X89),X91),X83)
| X89 = relation_dom(X83)
| ~ relation(X83) )
& ( in(esk2_2(X83,X89),X89)
| in(ordered_pair(esk2_2(X83,X89),esk3_2(X83,X89)),X83)
| X89 = relation_dom(X83)
| ~ relation(X83) ) ),
inference(distribute,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(fof_nnf,[status(thm)],[d4_relat_1])])])])])]) ).
fof(c_0_30,plain,
! [X93,X94] : ordered_pair(X93,X94) = unordered_pair(unordered_pair(X93,X94),singleton(X93)),
inference(variable_rename,[status(thm)],[d5_tarski]) ).
cnf(c_0_31,plain,
( ~ subset(X1,X2)
| ~ empty(X2)
| ~ in(X3,X1) ),
inference(spm,[status(thm)],[c_0_21,c_0_22]) ).
cnf(c_0_32,plain,
( subset(relation_rng(X1),X2)
| ~ transfinite_sequence_of(X1,X2) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(csr,[status(thm)],[c_0_23,c_0_24]),c_0_25]),c_0_26]) ).
cnf(c_0_33,plain,
( empty(X2)
| in(X1,X2)
| ~ element(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_27]) ).
cnf(c_0_34,plain,
element(esk5_1(X1),X1),
inference(split_conjunct,[status(thm)],[c_0_28]) ).
fof(c_0_35,plain,
! [X130] : subset(empty_set,X130),
inference(variable_rename,[status(thm)],[t2_xboole_1]) ).
cnf(c_0_36,plain,
( in(ordered_pair(X1,esk1_3(X3,X2,X1)),X3)
| ~ in(X1,X2)
| X2 != relation_dom(X3)
| ~ relation(X3) ),
inference(split_conjunct,[status(thm)],[c_0_29]) ).
cnf(c_0_37,plain,
ordered_pair(X1,X2) = unordered_pair(unordered_pair(X1,X2),singleton(X1)),
inference(split_conjunct,[status(thm)],[c_0_30]) ).
fof(c_0_38,plain,
! [X81,X82] : unordered_pair(X81,X82) = unordered_pair(X82,X81),
inference(variable_rename,[status(thm)],[commutativity_k2_tarski]) ).
fof(c_0_39,plain,
! [X1] :
( ( ~ empty(X1)
& relation(X1) )
=> ~ empty(relation_rng(X1)) ),
inference(fof_simplification,[status(thm)],[fc6_relat_1]) ).
cnf(c_0_40,plain,
( ~ transfinite_sequence_of(X1,X2)
| ~ empty(X2)
| ~ in(X3,relation_rng(X1)) ),
inference(spm,[status(thm)],[c_0_31,c_0_32]) ).
cnf(c_0_41,plain,
( empty(X1)
| in(esk5_1(X1),X1) ),
inference(spm,[status(thm)],[c_0_33,c_0_34]) ).
fof(c_0_42,plain,
! [X100] : transfinite_sequence_of(esk4_1(X100),X100),
inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[existence_m1_ordinal1])]) ).
cnf(c_0_43,plain,
subset(empty_set,X1),
inference(split_conjunct,[status(thm)],[c_0_35]) ).
cnf(c_0_44,plain,
( in(unordered_pair(unordered_pair(X1,esk1_3(X3,X2,X1)),singleton(X1)),X3)
| X2 != relation_dom(X3)
| ~ relation(X3)
| ~ in(X1,X2) ),
inference(rw,[status(thm)],[c_0_36,c_0_37]) ).
cnf(c_0_45,plain,
unordered_pair(X1,X2) = unordered_pair(X2,X1),
inference(split_conjunct,[status(thm)],[c_0_38]) ).
fof(c_0_46,plain,
! [X108] :
( empty(X108)
| ~ relation(X108)
| ~ empty(relation_rng(X108)) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_39])]) ).
cnf(c_0_47,plain,
( empty(relation_rng(X1))
| ~ transfinite_sequence_of(X1,X2)
| ~ empty(X2) ),
inference(spm,[status(thm)],[c_0_40,c_0_41]) ).
cnf(c_0_48,plain,
transfinite_sequence_of(esk4_1(X1),X1),
inference(split_conjunct,[status(thm)],[c_0_42]) ).
cnf(c_0_49,plain,
( ~ empty(X1)
| ~ in(X2,empty_set) ),
inference(spm,[status(thm)],[c_0_31,c_0_43]) ).
cnf(c_0_50,plain,
( in(unordered_pair(singleton(X1),unordered_pair(X1,esk1_3(X2,relation_dom(X2),X1))),X2)
| ~ relation(X2)
| ~ in(X1,relation_dom(X2)) ),
inference(er,[status(thm)],[inference(rw,[status(thm)],[c_0_44,c_0_45])]) ).
cnf(c_0_51,plain,
relation(empty_set),
inference(split_conjunct,[status(thm)],[fc12_relat_1]) ).
fof(c_0_52,plain,
! [X141] :
( ~ empty(X141)
| X141 = empty_set ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[t6_boole])]) ).
cnf(c_0_53,plain,
( empty(X1)
| ~ relation(X1)
| ~ empty(relation_rng(X1)) ),
inference(split_conjunct,[status(thm)],[c_0_46]) ).
cnf(c_0_54,plain,
( empty(relation_rng(esk4_1(X1)))
| ~ empty(X1) ),
inference(spm,[status(thm)],[c_0_47,c_0_48]) ).
cnf(c_0_55,plain,
relation(esk4_1(X1)),
inference(spm,[status(thm)],[c_0_25,c_0_48]) ).
cnf(c_0_56,plain,
( ~ empty(X1)
| ~ in(X2,relation_dom(empty_set)) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_49,c_0_50]),c_0_51])]) ).
cnf(c_0_57,plain,
( X1 = empty_set
| ~ empty(X1) ),
inference(split_conjunct,[status(thm)],[c_0_52]) ).
cnf(c_0_58,plain,
( empty(esk4_1(X1))
| ~ empty(X1) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_53,c_0_54]),c_0_55])]) ).
cnf(c_0_59,plain,
( empty(relation_dom(empty_set))
| ~ empty(X1) ),
inference(spm,[status(thm)],[c_0_56,c_0_41]) ).
cnf(c_0_60,plain,
empty(empty_set),
inference(split_conjunct,[status(thm)],[fc12_relat_1]) ).
cnf(c_0_61,plain,
( relation_rng(esk4_1(X1)) = empty_set
| ~ empty(X1) ),
inference(spm,[status(thm)],[c_0_57,c_0_54]) ).
cnf(c_0_62,plain,
( esk4_1(X1) = empty_set
| ~ empty(X1) ),
inference(spm,[status(thm)],[c_0_57,c_0_58]) ).
fof(c_0_63,plain,
! [X95] :
( ( ~ transfinite_sequence(X95)
| ordinal(relation_dom(X95))
| ~ relation(X95)
| ~ function(X95) )
& ( ~ ordinal(relation_dom(X95))
| transfinite_sequence(X95)
| ~ relation(X95)
| ~ function(X95) ) ),
inference(distribute,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[d7_ordinal1])])]) ).
cnf(c_0_64,plain,
empty(relation_dom(empty_set)),
inference(spm,[status(thm)],[c_0_59,c_0_60]) ).
fof(c_0_65,negated_conjecture,
~ ! [X1] : transfinite_sequence_of(empty_set,X1),
inference(assume_negation,[status(cth)],[t45_ordinal1]) ).
cnf(c_0_66,plain,
( relation_rng(empty_set) = empty_set
| ~ empty(X1) ),
inference(spm,[status(thm)],[c_0_61,c_0_62]) ).
cnf(c_0_67,plain,
( transfinite_sequence(X1)
| ~ ordinal(relation_dom(X1))
| ~ relation(X1)
| ~ function(X1) ),
inference(split_conjunct,[status(thm)],[c_0_63]) ).
cnf(c_0_68,plain,
relation_dom(empty_set) = empty_set,
inference(spm,[status(thm)],[c_0_57,c_0_64]) ).
cnf(c_0_69,plain,
ordinal(empty_set),
inference(split_conjunct,[status(thm)],[fc2_ordinal1]) ).
cnf(c_0_70,plain,
function(empty_set),
inference(split_conjunct,[status(thm)],[fc2_ordinal1]) ).
fof(c_0_71,negated_conjecture,
~ transfinite_sequence_of(empty_set,esk20_0),
inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_65])])]) ).
cnf(c_0_72,plain,
( transfinite_sequence_of(X1,X2)
| ~ subset(relation_rng(X1),X2)
| ~ relation(X1)
| ~ function(X1)
| ~ transfinite_sequence(X1) ),
inference(split_conjunct,[status(thm)],[c_0_19]) ).
cnf(c_0_73,plain,
relation_rng(empty_set) = empty_set,
inference(spm,[status(thm)],[c_0_66,c_0_60]) ).
cnf(c_0_74,plain,
transfinite_sequence(empty_set),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_67,c_0_68]),c_0_51]),c_0_69]),c_0_70])]) ).
cnf(c_0_75,negated_conjecture,
~ transfinite_sequence_of(empty_set,esk20_0),
inference(split_conjunct,[status(thm)],[c_0_71]) ).
cnf(c_0_76,plain,
transfinite_sequence_of(empty_set,X1),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_72,c_0_73]),c_0_43]),c_0_74]),c_0_51]),c_0_70])]) ).
cnf(c_0_77,negated_conjecture,
$false,
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_75,c_0_76])]),
[proof] ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : NUM409+1 : TPTP v9.2.1. Released v3.2.0.
% 0.00/0.06 % Command : /export/starexec/sandbox/solver/bin/lemma_parallel_prover %s --lemma-prover /export/starexec/sandbox/solver/bin/cse --final-prover /export/starexec/sandbox/solver/bin/eprover --proof-time %d --global-time-limit %d
% 0.06/0.24 % Computer : n024.cluster.edu
% 0.06/0.24 % Model : x86_64 x86_64
% 0.06/0.24 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.06/0.24 % Memory : 8042.1875MB
% 0.06/0.24 % OS : Linux 3.10.0-693.el7.x86_64
% 0.06/0.24 % CPULimit : 300
% 0.06/0.24 % WCLimit : 300
% 0.06/0.24 % DateTime : Mon May 4 19:31:26 EDT 2026
% 0.06/0.24 % CPUTime :
% 0.06/0.25 % start to proof: theBenchmark
% 1.11/1.61 % Version : CSE_E---1.7
% 1.11/1.61 % Problem : theBenchmark.p
% 1.11/1.61 % SZS status Theorem for theBenchmark.p
% 1.11/1.61 % SZS output start CNFRefutation
% See solution above
% 1.11/1.61 % Total time : 1.361s
%------------------------------------------------------------------------------