%------------------------------------------------------------------------------
% File : Vampire---5.0.1
% Problem : NUM409+1 : TPTP v9.3.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% Computer : n020.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 : Tue Sep 29 12:15:04 PM UTC 2026
% Result : Theorem 2.07s 1.31s
% Output : Refutation 0.18s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 11
% Syntax : Number of formulae : 62 ( 18 unt; 1 def)
% Number of atoms : 171 ( 10 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 186 ( 77 ~; 65 |; 32 &)
% ( 7 <=>; 5 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 14 ( 12 usr; 2 prp; 0-2 aty)
% Number of functors : 4 ( 4 usr; 2 con; 0-1 aty)
% Number of variables : 43 ( 0 sgn 42 !; 1 ?)
% Comments :
%------------------------------------------------------------------------------
fof(f11,axiom,
! [X0] :
( ( relation(X0)
& function(X0) )
=> ( transfinite_sequence(X0)
<=> ordinal(relation_dom(X0)) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d7_ordinal1) ).
fof(f12,axiom,
! [X0,X1] :
( ( relation(X1)
& function(X1)
& transfinite_sequence(X1) )
=> ( transfinite_sequence_of(X1,X0)
<=> subset(relation_rng(X1),X0) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d8_ordinal1) ).
fof(f16,axiom,
( empty(empty_set)
& relation(empty_set)
& relation_empty_yielding(empty_set) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc12_relat_1) ).
fof(f19,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/sandbox2/benchmark/theBenchmark.p',fc2_ordinal1) ).
fof(f24,axiom,
! [X0] :
( empty(X0)
=> ( empty(relation_dom(X0))
& relation(relation_dom(X0)) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc7_relat_1) ).
fof(f25,axiom,
! [X0] :
( empty(X0)
=> ( empty(relation_rng(X0))
& relation(relation_rng(X0)) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc8_relat_1) ).
fof(f43,axiom,
! [X0] : subset(empty_set,X0),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_xboole_1) ).
fof(f45,conjecture,
! [X0] : transfinite_sequence_of(empty_set,X0),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t45_ordinal1) ).
fof(f46,negated_conjecture,
~ ! [X0] : transfinite_sequence_of(empty_set,X0),
inference(negated_conjecture,[status(cth)],[f45]) ).
fof(f50,axiom,
! [X0] :
( empty(X0)
=> X0 = empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_boole) ).
fof(f52,axiom,
! [X0,X1] :
~ ( empty(X0)
& X0 != X1
& empty(X1) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t8_boole) ).
fof(f57,plain,
( empty(empty_set)
& relation(empty_set) ),
inference(pure_predicate_removal,[],[f16]) ).
fof(f59,plain,
( relation(empty_set)
& function(empty_set)
& one_to_one(empty_set)
& empty(empty_set)
& epsilon_transitive(empty_set)
& epsilon_connected(empty_set)
& ordinal(empty_set) ),
inference(pure_predicate_removal,[],[f19]) ).
fof(f63,plain,
( relation(empty_set)
& function(empty_set)
& empty(empty_set)
& epsilon_transitive(empty_set)
& epsilon_connected(empty_set)
& ordinal(empty_set) ),
inference(pure_predicate_removal,[],[f59]) ).
fof(f74,plain,
! [X0] :
( ( transfinite_sequence(X0)
<=> ordinal(relation_dom(X0)) )
| ~ relation(X0)
| ~ function(X0) ),
inference(ennf_transformation,[],[f11]) ).
fof(f75,plain,
! [X0] :
( ( transfinite_sequence(X0)
<=> ordinal(relation_dom(X0)) )
| ~ relation(X0)
| ~ function(X0) ),
inference(flattening,[],[f74]) ).
fof(f76,plain,
! [X0,X1] :
( ( transfinite_sequence_of(X1,X0)
<=> subset(relation_rng(X1),X0) )
| ~ relation(X1)
| ~ function(X1)
| ~ transfinite_sequence(X1) ),
inference(ennf_transformation,[],[f12]) ).
fof(f77,plain,
! [X0,X1] :
( ( transfinite_sequence_of(X1,X0)
<=> subset(relation_rng(X1),X0) )
| ~ relation(X1)
| ~ function(X1)
| ~ transfinite_sequence(X1) ),
inference(flattening,[],[f76]) ).
fof(f83,plain,
! [X0] :
( ( empty(relation_dom(X0))
& relation(relation_dom(X0)) )
| ~ empty(X0) ),
inference(ennf_transformation,[],[f24]) ).
fof(f84,plain,
! [X0] :
( ( empty(relation_rng(X0))
& relation(relation_rng(X0)) )
| ~ empty(X0) ),
inference(ennf_transformation,[],[f25]) ).
fof(f88,plain,
? [X0] : ~ transfinite_sequence_of(empty_set,X0),
inference(ennf_transformation,[],[f46]) ).
fof(f93,plain,
! [X0] :
( X0 = empty_set
| ~ empty(X0) ),
inference(ennf_transformation,[],[f50]) ).
fof(f95,plain,
! [X0,X1] :
( ~ empty(X0)
| X0 = X1
| ~ empty(X1) ),
inference(ennf_transformation,[],[f52]) ).
fof(f99,plain,
! [X0] :
( ( ( transfinite_sequence(X0)
| ~ ordinal(relation_dom(X0)) )
& ( ordinal(relation_dom(X0))
| ~ transfinite_sequence(X0) ) )
| ~ relation(X0)
| ~ function(X0) ),
inference(nnf_transformation,[],[f75]) ).
fof(f100,plain,
! [X0,X1] :
( ( ( transfinite_sequence_of(X1,X0)
| ~ subset(relation_rng(X1),X0) )
& ( subset(relation_rng(X1),X0)
| ~ transfinite_sequence_of(X1,X0) ) )
| ~ relation(X1)
| ~ function(X1)
| ~ transfinite_sequence(X1) ),
inference(nnf_transformation,[],[f77]) ).
fof(f118,plain,
~ transfinite_sequence_of(empty_set,sK19),
inference(skolemize,[status(esa),new_symbols(skolem,[sK19]),skolemize(X0,sK19)],[f88]) ).
fof(f138,plain,
! [X0] :
( transfinite_sequence(X0)
| ~ ordinal(relation_dom(X0))
| ~ relation(X0)
| ~ function(X0) ),
inference(cnf_transformation,[],[f99]) ).
fof(f140,plain,
! [X0,X1] :
( ~ transfinite_sequence(X1)
| ~ subset(relation_rng(X1),X0)
| ~ relation(X1)
| ~ function(X1)
| transfinite_sequence_of(X1,X0) ),
inference(cnf_transformation,[],[f100]) ).
fof(f146,plain,
relation(empty_set),
inference(cnf_transformation,[],[f57]) ).
fof(f147,plain,
empty(empty_set),
inference(cnf_transformation,[],[f57]) ).
fof(f150,plain,
ordinal(empty_set),
inference(cnf_transformation,[],[f63]) ).
fof(f154,plain,
function(empty_set),
inference(cnf_transformation,[],[f63]) ).
fof(f161,plain,
! [X0] :
( empty(relation_dom(X0))
| ~ empty(X0) ),
inference(cnf_transformation,[],[f83]) ).
fof(f163,plain,
! [X0] :
( empty(relation_rng(X0))
| ~ empty(X0) ),
inference(cnf_transformation,[],[f84]) ).
fof(f201,plain,
! [X0] : subset(empty_set,X0),
inference(cnf_transformation,[],[f43]) ).
fof(f204,plain,
~ transfinite_sequence_of(empty_set,sK19),
inference(cnf_transformation,[],[f118]) ).
fof(f209,plain,
! [X0] :
( ~ empty(X0)
| empty_set = X0 ),
inference(cnf_transformation,[],[f93]) ).
fof(f211,plain,
! [X0,X1] :
( ~ empty(X0)
| X0 = X1
| ~ empty(X1) ),
inference(cnf_transformation,[],[f95]) ).
fof(f253,plain,
! [X0] :
( empty_set = relation_rng(X0)
| ~ empty(X0) ),
inference(resolution,[],[f163,f209]) ).
fof(f273,plain,
! [X0,X1] :
( ~ empty(X1)
| relation_dom(X0) = X1
| ~ empty(X0) ),
inference(resolution,[],[f211,f161]) ).
fof(f291,plain,
! [X0,X1] :
( ~ ordinal(relation_dom(X0))
| ~ relation(X0)
| ~ function(X0)
| ~ subset(relation_rng(X0),X1)
| ~ relation(X0)
| ~ function(X0)
| transfinite_sequence_of(X0,X1) ),
inference(resolution,[],[f138,f140]) ).
fof(f292,plain,
! [X0,X1] :
( transfinite_sequence_of(X0,X1)
| ~ relation(X0)
| ~ function(X0)
| ~ subset(relation_rng(X0),X1)
| ~ ordinal(relation_dom(X0)) ),
inference(duplicate_literal_removal,[],[f291]) ).
fof(f301,plain,
! [X0] :
( ~ empty(X0)
| relation_dom(X0) = X0 ),
inference(factoring,[],[f273]) ).
fof(f309,plain,
empty_set = relation_dom(empty_set),
inference(resolution,[],[f301,f147]) ).
fof(f323,definition,
( spl20_5
<=> subset(relation_rng(empty_set),sK19) ),
introduced(definition,[new_symbols(definition,[spl20_5])],[avatar_definition]) ).
fof(f324,plain,
( subset(relation_rng(empty_set),sK19)
| ~ spl20_5 ),
inference(avatar_component_clause,[],[f323]) ).
fof(f325,plain,
( ~ subset(relation_rng(empty_set),sK19)
| spl20_5 ),
inference(avatar_component_clause,[],[f323]) ).
fof(f329,plain,
( ~ subset(empty_set,sK19)
| ~ empty(empty_set)
| spl20_5 ),
inference(superposition,[],[f325,f253]) ).
fof(f330,plain,
( ~ empty(empty_set)
| spl20_5 ),
inference(forward_subsumption_resolution,[],[f329,f201]) ).
fof(f331,plain,
( $false
| spl20_5 ),
inference(forward_subsumption_resolution,[],[f330,f147]) ).
fof(f332,plain,
spl20_5,
inference(avatar_contradiction_clause,[],[f331]) ).
fof(f410,plain,
( ~ relation(empty_set)
| ~ function(empty_set)
| ~ subset(relation_rng(empty_set),sK19)
| ~ ordinal(relation_dom(empty_set)) ),
inference(resolution,[],[f292,f204]) ).
fof(f413,plain,
( ~ function(empty_set)
| ~ subset(relation_rng(empty_set),sK19)
| ~ ordinal(relation_dom(empty_set)) ),
inference(forward_subsumption_resolution,[],[f410,f146]) ).
fof(f415,plain,
( ~ subset(relation_rng(empty_set),sK19)
| ~ ordinal(relation_dom(empty_set)) ),
inference(forward_subsumption_resolution,[],[f413,f154]) ).
fof(f417,plain,
( ~ ordinal(relation_dom(empty_set))
| ~ spl20_5 ),
inference(forward_subsumption_resolution,[],[f415,f324]) ).
fof(f419,plain,
( ~ ordinal(empty_set)
| ~ spl20_5 ),
inference(forward_demodulation,[],[f417,f309]) ).
fof(f421,plain,
( $false
| ~ spl20_5 ),
inference(forward_subsumption_resolution,[],[f419,f150]) ).
fof(f422,plain,
~ spl20_5,
inference(avatar_contradiction_clause,[],[f421]) ).
cnf(s5,plain,
spl20_5,
inference(sat_conversion,[],[f332]) ).
cnf(s7,plain,
~ spl20_5,
inference(sat_conversion,[],[f422]) ).
cnf(s9,plain,
$false,
inference(rat,[],[s5,s7]) ).
fof(f425,plain,
$false,
inference(avatar_sat_refutation,[],[s9]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : NUM409+1 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.06 % Command : run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% 0.13/0.39 % Computer : n020.cluster.edu
% 0.13/0.39 % Model : x86_64 x86_64
% 0.13/0.39 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.39 % Memory : 8046.5625MB
% 0.13/0.39 % OS : Linux 6.8.0-71-generic
% 0.13/0.39 % CPULimit : 300
% 0.13/0.39 % WCLimit : 300
% 0.13/0.39 % DateTime : Sun Sep 27 19:49:20 UTC 2026
% 0.13/0.39 % CPUTime :
% 0.13/0.39 Running run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% 0.13/0.42 Running first-order theorem proving
% 0.13/0.42 Running: /export/starexec/sandbox2/solver/bin/vampire --input_syntax tptp --output_axiom_names on --mode casc -m 16384 --cores 7 -t 300 /export/starexec/sandbox2/benchmark/theBenchmark.p
% 2.07/1.31 % (3704356)Detected formulas, will run a generic FOF schedule.
% 2.07/1.31 % (3704362)lrs+11_1_ncem=casc2026/models/loop8.pt:sil=128000:npcc=on:lma=off:spb=units:urr=ec_only:bce=on:s2agt=64:updr=off:random_seed=3373169995:i=134677:sd=20:aac=none:nm=16:ss=included:sgt=10_2999 on theBenchmark for (2999ds/134677Mi)
% 2.07/1.31 % (3704365)dis-1010_2:3_sil=16000:sp=reverse_frequency:random_seed=3231108161:i=119:av=off:ss=axioms_2999 on theBenchmark for (2999ds/119Mi)
% 2.07/1.31 % (3704367)dis-21_1_sil=8000:lcm=predicate:random_seed=1655232745:st=5:avsq=on:i=129:avsqr=1,16:sd=3:aac=none:ep=RS:fsr=off:ss=included_2999 on theBenchmark for (2999ds/129Mi)
% 2.07/1.31 % (3704361)lrs+10_1_ncem=casc2026/models/loop8.pt:sil=128000:tgt=full:npcc=on:drc=off:sp=weighted_frequency:spb=goal:fd=preordered:foolp=on:random_seed=3107196860:i=141193_2999 on theBenchmark for (2999ds/141193Mi)
% 2.07/1.31 % (3704366)dis-1011_1_sil=16000:fde=unused:s2agt=70:random_seed=3881748881:s2a=on:i=139:gtg=position_2999 on theBenchmark for (2999ds/139Mi)
% 2.07/1.31 % (3704364)lrs+1010_1_to=lpo:sil=32000:sos=on:spb=goal_then_units:bce=on:random_seed=904477775:i=109:sd=1:ins=1:gsp=on:ss=axioms_2999 on theBenchmark for (2999ds/109Mi)
% 2.07/1.31 % (3704363)lrs+1010_1_anc=all:sfv=off:to=kbo:ncem=casc2026/models/loop7.pt:sil=128000:npcc=on:prc=on:sos=all:bsr=unit_only:sac=on:random_seed=389030515:i=141695:sd=1:nm=32:gsp=on:ss=included_2999 on theBenchmark for (2999ds/141695Mi)
% 2.07/1.31 % (3704364)Refutation not found, incomplete strategy
% 2.07/1.31 % (3704364)------------------------------
% 2.07/1.31 % (3704364)Version: Vampire 5.0.1 (Release build, commit ea8961452 on 2026-07-16 15:14:34 +0200)
% 2.07/1.31 % (3704364)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 2.07/1.31 % (3704364)CaDiCaL version: 2.1.3
% 2.07/1.31 % (3704364)Termination reason: Refutation not found, incomplete strategy
% 2.07/1.31 % (3704364)Time elapsed: 0.001 s
% 2.07/1.31 % (3704364)Peak memory usage: 87 MB
% 2.07/1.31 % (3704365)Refutation not found, incomplete strategy
% 2.07/1.31 % (3704365)------------------------------
% 2.07/1.31 % (3704365)Version: Vampire 5.0.1 (Release build, commit ea8961452 on 2026-07-16 15:14:34 +0200)
% 2.07/1.31 % (3704365)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 2.07/1.31 % (3704365)CaDiCaL version: 2.1.3
% 2.07/1.31 % (3704365)Termination reason: Refutation not found, incomplete strategy
% 2.07/1.31 % (3704365)Time elapsed: 0.003 s
% 2.07/1.31 % (3704365)Peak memory usage: 88 MB
% 2.07/1.31 % (3704365)Instructions burned: 2 (million)
% 2.07/1.31 % (3704366)First to succeed.
% 2.07/1.31 % (3704366)Solution written to "/export/starexec/sandbox2/tmp/vampire-proof-3704356"
% 2.07/1.31 % (3704367)Instruction limit reached!
% 2.07/1.31 % (3704367)------------------------------
% 2.07/1.31 % (3704367)Version: Vampire 5.0.1 (Release build, commit ea8961452 on 2026-07-16 15:14:34 +0200)
% 2.07/1.31 % (3704367)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 2.07/1.31 % (3704367)CaDiCaL version: 2.1.3
% 2.07/1.31 % (3704367)Termination reason: Instruction limit
% 2.07/1.31 % (3704367)Termination phase: Saturation
% 2.07/1.31 % (3704367)Time elapsed: 0.069 s
% 2.07/1.31 % (3704367)Peak memory usage: 88 MB
% 2.07/1.31 % (3704367)Instructions burned: 130 (million)
% 2.07/1.31 % (3704375)lrs+10_1_sil=8000:sp=occurrence:random_seed=667621048:i=285:sd=3:ss=axioms:sgt=8_2997 on theBenchmark for (2997ds/285Mi)
% 2.07/1.31 % (3704375)Also succeeded, but the first one will report.
% 2.07/1.31 % (3704365)------------------------------
% 2.07/1.31 % (3704365)------------------------------
% 2.07/1.31 % (3704364)------------------------------
% 2.07/1.31 % (3704364)------------------------------
% 2.07/1.31 % (3704366)Refutation found. Thanks to Tanya!
% 2.07/1.31 % SZS status Theorem for theBenchmark
% 2.07/1.31 % SZS output start Proof for theBenchmark
% See solution above
% 0.18/1.50 % (3704366)------------------------------
% 0.18/1.50 % (3704366)Version: Vampire 5.0.1 (Release build, commit ea8961452 on 2026-07-16 15:14:34 +0200)
% 0.18/1.50 % (3704366)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.18/1.50 % (3704366)CaDiCaL version: 2.1.3
% 0.18/1.50 % (3704366)Termination reason: Refutation
% 0.18/1.50 % (3704366)Time elapsed: 0.009 s
% 0.18/1.50 % (3704366)Peak memory usage: 90 MB
% 0.18/1.50 % (3704366)Instructions burned: 11 (million)
% 0.18/1.50 % (3704366)------------------------------
% 0.18/1.50 % (3704366)------------------------------
% 0.18/1.50 % (3704356)Success in time 0.44 s
% 0.18/1.50 % Vampire exiting
%------------------------------------------------------------------------------