%------------------------------------------------------------------------------
% File : Drodi---4.1.1
% Problem : NUM409+1 : TPTP v9.3.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : drodi -timeout(300) /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n026.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 : Thu Sep 24 01:40:27 PM UTC 2026
% Result : Theorem 0.14s 0.47s
% Output : CNFRefutation 0.14s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 15
% Syntax : Number of formulae : 63 ( 17 unt; 6 def)
% Number of atoms : 157 ( 7 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 156 ( 62 ~; 60 |; 19 &)
% ( 10 <=>; 5 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 3 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 19 ( 17 usr; 7 prp; 0-2 aty)
% Number of functors : 4 ( 4 usr; 2 con; 0-1 aty)
% Number of variables : 32 ( 31 !; 1 ?)
% Comments :
%------------------------------------------------------------------------------
fof(f11,axiom,
! [A] :
( ( function(A)
& relation(A) )
=> ( transfinite_sequence(A)
<=> ordinal(relation_dom(A)) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p') ).
fof(f12,axiom,
! [A,B] :
( ( transfinite_sequence(B)
& function(B)
& relation(B) )
=> ( transfinite_sequence_of(B,A)
<=> subset(relation_rng(B),A) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p') ).
fof(f16,axiom,
( relation_empty_yielding(empty_set)
& relation(empty_set)
& empty(empty_set) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p') ).
fof(f19,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('/export/starexec/sandbox/benchmark/theBenchmark.p') ).
fof(f24,axiom,
! [A] :
( empty(A)
=> ( relation(relation_dom(A))
& empty(relation_dom(A)) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p') ).
fof(f25,axiom,
! [A] :
( empty(A)
=> ( relation(relation_rng(A))
& empty(relation_rng(A)) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p') ).
fof(f43,axiom,
! [A] : subset(empty_set,A),
file('/export/starexec/sandbox/benchmark/theBenchmark.p') ).
fof(f45,conjecture,
! [A] : transfinite_sequence_of(empty_set,A),
file('/export/starexec/sandbox/benchmark/theBenchmark.p') ).
fof(f46,negated_conjecture,
~ ! [A] : transfinite_sequence_of(empty_set,A),
inference(negated_conjecture,[status(cth)],[f45]) ).
fof(f50,axiom,
! [A] :
( empty(A)
=> A = empty_set ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p') ).
fof(f82,plain,
! [A] :
( ( transfinite_sequence(A)
<=> ordinal(relation_dom(A)) )
| ~ function(A)
| ~ relation(A) ),
inference(pre_NNF_transformation,[status(thm)],[f11]) ).
fof(f83,plain,
! [A] :
( ( ( ~ ordinal(relation_dom(A))
| transfinite_sequence(A) )
& ( ordinal(relation_dom(A))
| ~ transfinite_sequence(A) ) )
| ~ function(A)
| ~ relation(A) ),
inference(NNF_transformation,[status(thm)],[f82]) ).
fof(f85,plain,
! [X0] :
( ~ ordinal(relation_dom(X0))
| transfinite_sequence(X0)
| ~ function(X0)
| ~ relation(X0) ),
inference(cnf_transformation,[status(thm)],[f83]) ).
fof(f86,plain,
! [A,B] :
( ( transfinite_sequence_of(B,A)
<=> subset(relation_rng(B),A) )
| ~ transfinite_sequence(B)
| ~ function(B)
| ~ relation(B) ),
inference(pre_NNF_transformation,[status(thm)],[f12]) ).
fof(f87,plain,
! [A,B] :
( ( ( ~ subset(relation_rng(B),A)
| transfinite_sequence_of(B,A) )
& ( subset(relation_rng(B),A)
| ~ transfinite_sequence_of(B,A) ) )
| ~ transfinite_sequence(B)
| ~ function(B)
| ~ relation(B) ),
inference(NNF_transformation,[status(thm)],[f86]) ).
fof(f88,plain,
! [B] :
( ( ! [A] :
( ~ subset(relation_rng(B),A)
| transfinite_sequence_of(B,A) )
& ! [A] :
( subset(relation_rng(B),A)
| ~ transfinite_sequence_of(B,A) ) )
| ~ transfinite_sequence(B)
| ~ function(B)
| ~ relation(B) ),
inference(miniscoping,[status(thm)],[f87]) ).
fof(f90,plain,
! [X0,X1] :
( ~ subset(relation_rng(X0),X1)
| transfinite_sequence_of(X0,X1)
| ~ transfinite_sequence(X0)
| ~ function(X0)
| ~ relation(X0) ),
inference(cnf_transformation,[status(thm)],[f88]) ).
fof(f100,plain,
empty(empty_set),
inference(cnf_transformation,[status(thm)],[f16]) ).
fof(f107,plain,
function(empty_set),
inference(cnf_transformation,[status(thm)],[f19]) ).
fof(f112,plain,
ordinal(empty_set),
inference(cnf_transformation,[status(thm)],[f19]) ).
fof(f121,plain,
! [A] :
( ( relation(relation_dom(A))
& empty(relation_dom(A)) )
| ~ empty(A) ),
inference(pre_NNF_transformation,[status(thm)],[f24]) ).
fof(f122,plain,
! [X0] :
( empty(relation_dom(X0))
| ~ empty(X0) ),
inference(cnf_transformation,[status(thm)],[f121]) ).
fof(f123,plain,
! [X0] :
( relation(relation_dom(X0))
| ~ empty(X0) ),
inference(cnf_transformation,[status(thm)],[f121]) ).
fof(f124,plain,
! [A] :
( ( relation(relation_rng(A))
& empty(relation_rng(A)) )
| ~ empty(A) ),
inference(pre_NNF_transformation,[status(thm)],[f25]) ).
fof(f125,plain,
! [X0] :
( empty(relation_rng(X0))
| ~ empty(X0) ),
inference(cnf_transformation,[status(thm)],[f124]) ).
fof(f186,plain,
! [X0] : subset(empty_set,X0),
inference(cnf_transformation,[status(thm)],[f43]) ).
fof(f191,plain,
? [A] : ~ transfinite_sequence_of(empty_set,A),
inference(pre_NNF_transformation,[status(thm)],[f46]) ).
fof(f192,plain,
~ transfinite_sequence_of(empty_set,sK19_skl),
inference(skolemize,[status(esa),new_symbols(skolem,[sK19_skl]),skolemize(A,sK19_skl)],[f191]) ).
fof(f193,plain,
~ transfinite_sequence_of(empty_set,sK19_skl),
inference(cnf_transformation,[status(thm)],[f192]) ).
fof(f204,plain,
! [A] :
( A = empty_set
| ~ empty(A) ),
inference(pre_NNF_transformation,[status(thm)],[f50]) ).
fof(f205,plain,
! [X0] :
( X0 = empty_set
| ~ empty(X0) ),
inference(cnf_transformation,[status(thm)],[f204]) ).
fof(f237,plain,
! [X0] :
( relation_dom(X0) = empty_set
| ~ empty(X0) ),
inference(resolution,[status(thm)],[f122,f205]) ).
fof(f238,plain,
! [X0] :
( relation_rng(X0) = empty_set
| ~ empty(X0) ),
inference(resolution,[status(thm)],[f125,f205]) ).
fof(f254,plain,
relation_dom(empty_set) = empty_set,
inference(resolution,[status(thm)],[f237,f100]) ).
fof(f256,plain,
( relation(empty_set)
| ~ empty(empty_set) ),
inference(paramodulation,[status(thm)],[f254,f123]) ).
fof(f259,definition,
( sQ0_spl
<=> relation(empty_set) ),
introduced(definition,[new_symbols(definition,[sQ0_spl])],[split_symbol_definition]) ).
fof(f262,definition,
( sQ1_spl
<=> function(empty_set) ),
introduced(definition,[new_symbols(definition,[sQ1_spl])],[split_symbol_definition]) ).
fof(f264,plain,
( sQ1_spl
| ~ function(empty_set) ),
inference(component_clause,[status(thm)],[f262]) ).
fof(f265,definition,
( sQ2_spl
<=> transfinite_sequence(empty_set) ),
introduced(definition,[new_symbols(definition,[sQ2_spl])],[split_symbol_definition]) ).
fof(f267,plain,
( sQ2_spl
| ~ transfinite_sequence(empty_set) ),
inference(component_clause,[status(thm)],[f265]) ).
fof(f272,definition,
( sQ4_spl
<=> empty(empty_set) ),
introduced(definition,[new_symbols(definition,[sQ4_spl])],[split_symbol_definition]) ).
fof(f274,plain,
( sQ4_spl
| ~ empty(empty_set) ),
inference(component_clause,[status(thm)],[f272]) ).
fof(f275,plain,
( sQ0_spl
| ~ sQ4_spl ),
inference(split_clause,[status(thm)],[f256,f272,f259]) ).
fof(f281,plain,
( sQ4_spl
| $false ),
inference(forward_subsumption_resolution,[status(thm)],[f274,f100]) ).
fof(f282,plain,
sQ4_spl,
inference(contradiction_clause,[status(thm)],[f281]) ).
fof(f286,plain,
relation_rng(empty_set) = empty_set,
inference(resolution,[status(thm)],[f238,f100]) ).
fof(f382,plain,
( ~ subset(relation_rng(empty_set),sK19_skl)
| ~ transfinite_sequence(empty_set)
| ~ function(empty_set)
| ~ relation(empty_set) ),
inference(resolution,[status(thm)],[f90,f193]) ).
fof(f383,definition,
( sQ8_spl
<=> subset(relation_rng(empty_set),sK19_skl) ),
introduced(definition,[new_symbols(definition,[sQ8_spl])],[split_symbol_definition]) ).
fof(f385,plain,
( sQ8_spl
| ~ subset(relation_rng(empty_set),sK19_skl) ),
inference(component_clause,[status(thm)],[f383]) ).
fof(f386,plain,
( ~ sQ8_spl
| ~ sQ2_spl
| ~ sQ1_spl
| ~ sQ0_spl ),
inference(split_clause,[status(thm)],[f382,f259,f262,f265,f383]) ).
fof(f387,plain,
( sQ2_spl
| ~ ordinal(relation_dom(empty_set))
| ~ function(empty_set)
| ~ relation(empty_set) ),
inference(resolution,[status(thm)],[f267,f85]) ).
fof(f389,definition,
( sQ9_spl
<=> ordinal(relation_dom(empty_set)) ),
introduced(definition,[new_symbols(definition,[sQ9_spl])],[split_symbol_definition]) ).
fof(f391,plain,
( sQ9_spl
| ~ ordinal(relation_dom(empty_set)) ),
inference(component_clause,[status(thm)],[f389]) ).
fof(f392,plain,
( sQ2_spl
| ~ sQ9_spl
| ~ sQ1_spl
| ~ sQ0_spl ),
inference(split_clause,[status(thm)],[f387,f259,f262,f389,f265]) ).
fof(f393,plain,
( sQ1_spl
| $false ),
inference(forward_subsumption_resolution,[status(thm)],[f264,f107]) ).
fof(f394,plain,
sQ1_spl,
inference(contradiction_clause,[status(thm)],[f393]) ).
fof(f395,plain,
( sQ9_spl
| ~ ordinal(empty_set) ),
inference(forward_demodulation,[status(thm)],[f254,f391]) ).
fof(f396,plain,
( sQ9_spl
| $false ),
inference(forward_subsumption_resolution,[status(thm)],[f395,f112]) ).
fof(f397,plain,
sQ9_spl,
inference(contradiction_clause,[status(thm)],[f396]) ).
fof(f398,plain,
( sQ8_spl
| ~ subset(empty_set,sK19_skl) ),
inference(forward_demodulation,[status(thm)],[f286,f385]) ).
fof(f399,plain,
( sQ8_spl
| $false ),
inference(forward_subsumption_resolution,[status(thm)],[f398,f186]) ).
fof(f400,plain,
sQ8_spl,
inference(contradiction_clause,[status(thm)],[f399]) ).
fof(f401,plain,
$false,
inference(sat_refutation,[status(thm)],[f275,f282,f386,f392,f394,f397,f400]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.04 % Problem : NUM409+1 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.05 % Command : drodi -timeout(300) /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.14/0.41 % Computer : n026.cluster.edu
% 0.14/0.41 % Model : x86_64 x86_64
% 0.14/0.41 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.41 % Memory : 8046.5625MB
% 0.14/0.41 % OS : Linux 6.8.0-71-generic
% 0.14/0.42 % CPULimit : 300
% 0.14/0.42 % WCLimit : 300
% 0.14/0.42 % DateTime : Mon Sep 21 02:29:51 UTC 2026
% 0.14/0.42 % CPUTime :
% 0.14/0.44 % Drodi V4.1.1
% 0.14/0.47 % Refutation found
% 0.14/0.47 % SZS status Theorem for theBenchmark: Theorem is valid
% 0.14/0.47 % SZS output start CNFRefutation for theBenchmark
% See solution above
% 0.14/0.52 % Elapsed time: 0.084270 seconds
% 0.14/0.52 % CPU time: 0.147887 seconds
% 0.14/0.52 % Total memory used: 29.978 MB
% 0.14/0.52 % Net memory used: 29.936 MB
%------------------------------------------------------------------------------