%------------------------------------------------------------------------------
% File : Drodi---4.1.1
% Problem : GRA007+2 : TPTP v9.3.1. Bugfixed v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : drodi -timeout(300) /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n016.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 12:30:39 PM UTC 2026
% Result : Theorem 0.14s 0.56s
% Output : CNFRefutation 0.14s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 28
% Syntax : Number of formulae : 155 ( 10 unt; 19 def)
% Number of atoms : 636 ( 138 equ)
% Maximal formula atoms : 11 ( 4 avg)
% Number of connectives : 798 ( 317 ~; 333 |; 107 &)
% ( 22 <=>; 13 =>; 1 <=; 5 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 31 ( 29 usr; 20 prp; 0-3 aty)
% Number of functors : 12 ( 12 usr; 5 con; 0-3 aty)
% Number of variables : 320 ( 293 !; 27 ?)
% Comments :
%------------------------------------------------------------------------------
fof(f3,axiom,
( complete
=> ! [V1,V2] :
( ( V1 != V2
& vertex(V2)
& vertex(V1) )
=> ? [E] :
( ( ( V2 = tail_of(E)
& V1 = head_of(E) )
<~> ( V1 = tail_of(E)
& V2 = head_of(E) ) )
& edge(E) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p') ).
fof(f6,axiom,
! [V1,V2,P,E] :
( ( on_path(E,P)
& path(V1,V2,P) )
=> ( in_path(tail_of(E),P)
& in_path(head_of(E),P)
& edge(E) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p') ).
fof(f7,axiom,
! [V1,V2,P,V] :
( ( in_path(V,P)
& path(V1,V2,P) )
=> ( ? [E] :
( ( V = tail_of(E)
| V = head_of(E) )
& on_path(E,P) )
& vertex(V) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p') ).
fof(f8,axiom,
! [E1,E2] :
( sequential(E1,E2)
<=> ( head_of(E1) = tail_of(E2)
& E1 != E2
& edge(E2)
& edge(E1) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p') ).
fof(f9,axiom,
! [P,V1,V2] :
( path(V1,V2,P)
=> ! [E1,E2] :
( precedes(E1,E2,P)
<= ( ( ? [E3] :
( precedes(E3,E2,P)
& sequential(E1,E3) )
| sequential(E1,E2) )
& on_path(E2,P)
& on_path(E1,P) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p') ).
fof(f10,axiom,
! [P,V1,V2] :
( path(V1,V2,P)
=> ! [E1,E2] :
( precedes(E1,E2,P)
=> ( ( sequential(E1,E2)
<~> ? [E3] :
( precedes(E3,E2,P)
& sequential(E1,E3) ) )
& on_path(E2,P)
& on_path(E1,P) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p') ).
fof(f11,axiom,
! [V1,V2,SP] :
( shortest_path(V1,V2,SP)
<=> ( ! [P] :
( path(V1,V2,P)
=> less_or_equal(length_of(SP),length_of(P)) )
& V1 != V2
& path(V1,V2,SP) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p') ).
fof(f12,axiom,
! [V1,V2,E1,E2,P] :
( ( precedes(E1,E2,P)
& shortest_path(V1,V2,P) )
=> ( ~ precedes(E2,E1,P)
& ~ ? [E3] :
( head_of(E3) = head_of(E2)
& tail_of(E3) = tail_of(E1) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p') ).
fof(f19,conjecture,
( complete
=> ! [V1,V2,E1,E2,P] :
( ( precedes(E1,E2,P)
& shortest_path(V1,V2,P) )
=> ? [E3] :
( head_of(E3) = tail_of(E1)
& tail_of(E3) = head_of(E2)
& edge(E3) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p') ).
fof(f20,negated_conjecture,
~ ( complete
=> ! [V1,V2,E1,E2,P] :
( ( precedes(E1,E2,P)
& shortest_path(V1,V2,P) )
=> ? [E3] :
( head_of(E3) = tail_of(E1)
& tail_of(E3) = head_of(E2)
& edge(E3) ) ) ),
inference(negated_conjecture,[status(cth)],[f19]) ).
fof(f26,plain,
( ! [V1,V2] :
( ? [E] :
( ( ( V2 = tail_of(E)
& V1 = head_of(E) )
<~> ( V1 = tail_of(E)
& V2 = head_of(E) ) )
& edge(E) )
| V1 = V2
| ~ vertex(V2)
| ~ vertex(V1) )
| ~ complete ),
inference(pre_NNF_transformation,[status(thm)],[f3]) ).
fof(f27,definition,
! [V1,V2,E] :
( sP0_prd(E,V2,V1)
<=> ( V2 = tail_of(E)
& V1 = head_of(E) ) ),
introduced(definition,[new_symbols(definition,[sP0_prd])],[]) ).
fof(f28,plain,
( ! [V1,V2] :
( ? [E] :
( ( sP0_prd(E,V2,V1)
<~> ( V1 = tail_of(E)
& V2 = head_of(E) ) )
& edge(E) )
| V1 = V2
| ~ vertex(V2)
| ~ vertex(V1) )
| ~ complete ),
inference(formula_renaming,[status(thm)],[f26,f27]) ).
fof(f29,plain,
( ! [V1,V2] :
( ? [E] :
( ( V1 != tail_of(E)
| V2 != head_of(E)
| ~ sP0_prd(E,V2,V1) )
& ( ( V1 = tail_of(E)
& V2 = head_of(E) )
| sP0_prd(E,V2,V1) )
& edge(E) )
| V1 = V2
| ~ vertex(V2)
| ~ vertex(V1) )
| ~ complete ),
inference(NNF_transformation,[status(thm)],[f28]) ).
fof(f30,plain,
( ! [V1,V2] :
( ( ( V1 != tail_of(sK0_skl(V2,V1))
| V2 != head_of(sK0_skl(V2,V1))
| ~ sP0_prd(sK0_skl(V2,V1),V2,V1) )
& ( ( V1 = tail_of(sK0_skl(V2,V1))
& V2 = head_of(sK0_skl(V2,V1)) )
| sP0_prd(sK0_skl(V2,V1),V2,V1) )
& edge(sK0_skl(V2,V1)) )
| V1 = V2
| ~ vertex(V2)
| ~ vertex(V1) )
| ~ complete ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0_skl]),skolemize(E,sK0_skl(V2,V1))],[f29]) ).
fof(f31,plain,
! [X0,X1] :
( edge(sK0_skl(X1,X0))
| X0 = X1
| ~ vertex(X1)
| ~ vertex(X0)
| ~ complete ),
inference(cnf_transformation,[status(thm)],[f30]) ).
fof(f32,plain,
! [X0,X1] :
( X1 = head_of(sK0_skl(X1,X0))
| sP0_prd(sK0_skl(X1,X0),X1,X0)
| X0 = X1
| ~ vertex(X1)
| ~ vertex(X0)
| ~ complete ),
inference(cnf_transformation,[status(thm)],[f30]) ).
fof(f33,plain,
! [X0,X1] :
( X0 = tail_of(sK0_skl(X1,X0))
| sP0_prd(sK0_skl(X1,X0),X1,X0)
| X0 = X1
| ~ vertex(X1)
| ~ vertex(X0)
| ~ complete ),
inference(cnf_transformation,[status(thm)],[f30]) ).
fof(f50,plain,
! [V1,V2,P,E] :
( ( in_path(tail_of(E),P)
& in_path(head_of(E),P)
& edge(E) )
| ~ on_path(E,P)
| ~ path(V1,V2,P) ),
inference(pre_NNF_transformation,[status(thm)],[f6]) ).
fof(f51,plain,
! [P,E] :
( ( in_path(tail_of(E),P)
& in_path(head_of(E),P)
& edge(E) )
| ~ on_path(E,P)
| ! [V1,V2] : ~ path(V1,V2,P) ),
inference(miniscoping,[status(thm)],[f50]) ).
fof(f52,plain,
! [X0,X1,X2,X3] :
( edge(X3)
| ~ on_path(X3,X2)
| ~ path(X0,X1,X2) ),
inference(cnf_transformation,[status(thm)],[f51]) ).
fof(f53,plain,
! [X0,X1,X2,X3] :
( in_path(head_of(X3),X2)
| ~ on_path(X3,X2)
| ~ path(X0,X1,X2) ),
inference(cnf_transformation,[status(thm)],[f51]) ).
fof(f54,plain,
! [X0,X1,X2,X3] :
( in_path(tail_of(X3),X2)
| ~ on_path(X3,X2)
| ~ path(X0,X1,X2) ),
inference(cnf_transformation,[status(thm)],[f51]) ).
fof(f55,plain,
! [V1,V2,P,V] :
( ( ? [E] :
( ( V = tail_of(E)
| V = head_of(E) )
& on_path(E,P) )
& vertex(V) )
| ~ in_path(V,P)
| ~ path(V1,V2,P) ),
inference(pre_NNF_transformation,[status(thm)],[f7]) ).
fof(f56,plain,
! [P,V] :
( ( ? [E] :
( ( V = tail_of(E)
| V = head_of(E) )
& on_path(E,P) )
& vertex(V) )
| ~ in_path(V,P)
| ! [V1,V2] : ~ path(V1,V2,P) ),
inference(miniscoping,[status(thm)],[f55]) ).
fof(f57,plain,
! [P,V] :
( ( ( V = tail_of(sK3_skl(V,P))
| V = head_of(sK3_skl(V,P)) )
& on_path(sK3_skl(V,P),P)
& vertex(V) )
| ~ in_path(V,P)
| ! [V1,V2] : ~ path(V1,V2,P) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK3_skl]),skolemize(E,sK3_skl(V,P))],[f56]) ).
fof(f58,plain,
! [X0,X1,X2,X3] :
( vertex(X3)
| ~ in_path(X3,X2)
| ~ path(X0,X1,X2) ),
inference(cnf_transformation,[status(thm)],[f57]) ).
fof(f61,plain,
! [E1,E2] :
( ( head_of(E1) != tail_of(E2)
| E1 = E2
| ~ edge(E2)
| ~ edge(E1)
| sequential(E1,E2) )
& ( ( head_of(E1) = tail_of(E2)
& E1 != E2
& edge(E2)
& edge(E1) )
| ~ sequential(E1,E2) ) ),
inference(NNF_transformation,[status(thm)],[f8]) ).
fof(f62,plain,
( ! [E1,E2] :
( head_of(E1) != tail_of(E2)
| E1 = E2
| ~ edge(E2)
| ~ edge(E1)
| sequential(E1,E2) )
& ! [E1,E2] :
( ( head_of(E1) = tail_of(E2)
& E1 != E2
& edge(E2)
& edge(E1) )
| ~ sequential(E1,E2) ) ),
inference(miniscoping,[status(thm)],[f61]) ).
fof(f67,plain,
! [X0,X1] :
( head_of(X0) != tail_of(X1)
| X0 = X1
| ~ edge(X1)
| ~ edge(X0)
| sequential(X0,X1) ),
inference(cnf_transformation,[status(thm)],[f62]) ).
fof(f68,plain,
! [P,V1,V2] :
( ! [E1,E2] :
( ( ! [E3] :
( ~ precedes(E3,E2,P)
| ~ sequential(E1,E3) )
& ~ sequential(E1,E2) )
| ~ on_path(E2,P)
| ~ on_path(E1,P)
| precedes(E1,E2,P) )
| ~ path(V1,V2,P) ),
inference(pre_NNF_transformation,[status(thm)],[f9]) ).
fof(f69,plain,
! [P] :
( ! [E1,E2] :
( ( ! [E3] :
( ~ precedes(E3,E2,P)
| ~ sequential(E1,E3) )
& ~ sequential(E1,E2) )
| ~ on_path(E2,P)
| ~ on_path(E1,P)
| precedes(E1,E2,P) )
| ! [V1,V2] : ~ path(V1,V2,P) ),
inference(miniscoping,[status(thm)],[f68]) ).
fof(f70,plain,
! [X0,X1,X2,X3,X4] :
( ~ sequential(X3,X4)
| ~ on_path(X4,X2)
| ~ on_path(X3,X2)
| precedes(X3,X4,X2)
| ~ path(X0,X1,X2) ),
inference(cnf_transformation,[status(thm)],[f69]) ).
fof(f72,plain,
! [P,V1,V2] :
( ! [E1,E2] :
( ( ( sequential(E1,E2)
<~> ? [E3] :
( precedes(E3,E2,P)
& sequential(E1,E3) ) )
& on_path(E2,P)
& on_path(E1,P) )
| ~ precedes(E1,E2,P) )
| ~ path(V1,V2,P) ),
inference(pre_NNF_transformation,[status(thm)],[f10]) ).
fof(f73,plain,
! [P,V1,V2] :
( ! [E1,E2] :
( ( ( ! [E3] :
( ~ precedes(E3,E2,P)
| ~ sequential(E1,E3) )
| ~ sequential(E1,E2) )
& ( ? [E3] :
( precedes(E3,E2,P)
& sequential(E1,E3) )
| sequential(E1,E2) )
& on_path(E2,P)
& on_path(E1,P) )
| ~ precedes(E1,E2,P) )
| ~ path(V1,V2,P) ),
inference(NNF_transformation,[status(thm)],[f72]) ).
fof(f74,plain,
! [P] :
( ! [E1,E2] :
( ( ( ! [E3] :
( ~ precedes(E3,E2,P)
| ~ sequential(E1,E3) )
| ~ sequential(E1,E2) )
& ( ? [E3] :
( precedes(E3,E2,P)
& sequential(E1,E3) )
| sequential(E1,E2) )
& on_path(E2,P)
& on_path(E1,P) )
| ~ precedes(E1,E2,P) )
| ! [V1,V2] : ~ path(V1,V2,P) ),
inference(miniscoping,[status(thm)],[f73]) ).
fof(f75,plain,
! [P] :
( ! [E1,E2] :
( ( ( ! [E3] :
( ~ precedes(E3,E2,P)
| ~ sequential(E1,E3) )
| ~ sequential(E1,E2) )
& ( ( precedes(sK4_skl(E2,E1,P),E2,P)
& sequential(E1,sK4_skl(E2,E1,P)) )
| sequential(E1,E2) )
& on_path(E2,P)
& on_path(E1,P) )
| ~ precedes(E1,E2,P) )
| ! [V1,V2] : ~ path(V1,V2,P) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK4_skl]),skolemize(E3,sK4_skl(E2,E1,P))],[f74]) ).
fof(f76,plain,
! [X0,X1,X2,X3,X4] :
( on_path(X3,X2)
| ~ precedes(X3,X4,X2)
| ~ path(X0,X1,X2) ),
inference(cnf_transformation,[status(thm)],[f75]) ).
fof(f77,plain,
! [X0,X1,X2,X3,X4] :
( on_path(X4,X2)
| ~ precedes(X3,X4,X2)
| ~ path(X0,X1,X2) ),
inference(cnf_transformation,[status(thm)],[f75]) ).
fof(f81,plain,
! [V1,V2,SP] :
( shortest_path(V1,V2,SP)
<=> ( ! [P] :
( less_or_equal(length_of(SP),length_of(P))
| ~ path(V1,V2,P) )
& V1 != V2
& path(V1,V2,SP) ) ),
inference(pre_NNF_transformation,[status(thm)],[f11]) ).
fof(f82,plain,
! [V1,V2,SP] :
( ( ? [P] :
( ~ less_or_equal(length_of(SP),length_of(P))
& path(V1,V2,P) )
| V1 = V2
| ~ path(V1,V2,SP)
| shortest_path(V1,V2,SP) )
& ( ( ! [P] :
( less_or_equal(length_of(SP),length_of(P))
| ~ path(V1,V2,P) )
& V1 != V2
& path(V1,V2,SP) )
| ~ shortest_path(V1,V2,SP) ) ),
inference(NNF_transformation,[status(thm)],[f81]) ).
fof(f83,plain,
( ! [V1,V2,SP] :
( ? [P] :
( ~ less_or_equal(length_of(SP),length_of(P))
& path(V1,V2,P) )
| V1 = V2
| ~ path(V1,V2,SP)
| shortest_path(V1,V2,SP) )
& ! [V1,V2,SP] :
( ( ! [P] :
( less_or_equal(length_of(SP),length_of(P))
| ~ path(V1,V2,P) )
& V1 != V2
& path(V1,V2,SP) )
| ~ shortest_path(V1,V2,SP) ) ),
inference(miniscoping,[status(thm)],[f82]) ).
fof(f84,plain,
( ! [V1,V2,SP] :
( ( ~ less_or_equal(length_of(SP),length_of(sK5_skl(SP,V2,V1)))
& path(V1,V2,sK5_skl(SP,V2,V1)) )
| V1 = V2
| ~ path(V1,V2,SP)
| shortest_path(V1,V2,SP) )
& ! [V1,V2,SP] :
( ( ! [P] :
( less_or_equal(length_of(SP),length_of(P))
| ~ path(V1,V2,P) )
& V1 != V2
& path(V1,V2,SP) )
| ~ shortest_path(V1,V2,SP) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK5_skl]),skolemize(P,sK5_skl(SP,V2,V1))],[f83]) ).
fof(f85,plain,
! [X0,X1,X2] :
( path(X0,X1,X2)
| ~ shortest_path(X0,X1,X2) ),
inference(cnf_transformation,[status(thm)],[f84]) ).
fof(f90,plain,
! [V1,V2,E1,E2,P] :
( ( ~ precedes(E2,E1,P)
& ! [E3] :
( head_of(E3) != head_of(E2)
| tail_of(E3) != tail_of(E1) ) )
| ~ precedes(E1,E2,P)
| ~ shortest_path(V1,V2,P) ),
inference(pre_NNF_transformation,[status(thm)],[f12]) ).
fof(f91,plain,
! [E1,E2,P] :
( ( ~ precedes(E2,E1,P)
& ! [E3] :
( head_of(E3) != head_of(E2)
| tail_of(E3) != tail_of(E1) ) )
| ~ precedes(E1,E2,P)
| ! [V1,V2] : ~ shortest_path(V1,V2,P) ),
inference(miniscoping,[status(thm)],[f90]) ).
fof(f92,plain,
! [X0,X1,X2,X3,X4,X5] :
( head_of(X5) != head_of(X4)
| tail_of(X5) != tail_of(X3)
| ~ precedes(X3,X4,X2)
| ~ shortest_path(X0,X1,X2) ),
inference(cnf_transformation,[status(thm)],[f91]) ).
fof(f93,plain,
! [X0,X1,X2,X3,X4] :
( ~ precedes(X4,X3,X2)
| ~ precedes(X3,X4,X2)
| ~ shortest_path(X0,X1,X2) ),
inference(cnf_transformation,[status(thm)],[f91]) ).
fof(f120,plain,
( ? [V1,V2,E1,E2,P] :
( ! [E3] :
( head_of(E3) != tail_of(E1)
| tail_of(E3) != head_of(E2)
| ~ edge(E3) )
& precedes(E1,E2,P)
& shortest_path(V1,V2,P) )
& complete ),
inference(pre_NNF_transformation,[status(thm)],[f20]) ).
fof(f121,plain,
( ? [E1,E2] :
( ! [E3] :
( head_of(E3) != tail_of(E1)
| tail_of(E3) != head_of(E2)
| ~ edge(E3) )
& ? [P] :
( precedes(E1,E2,P)
& ? [V1,V2] : shortest_path(V1,V2,P) ) )
& complete ),
inference(miniscoping,[status(thm)],[f120]) ).
fof(f122,plain,
( ! [E3] :
( head_of(E3) != tail_of(sK8_skl)
| tail_of(E3) != head_of(sK9_skl)
| ~ edge(E3) )
& precedes(sK8_skl,sK9_skl,sK10_skl)
& shortest_path(sK11_skl,sK12_skl,sK10_skl)
& complete ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK8_skl,sK9_skl,sK10_skl,sK11_skl,sK12_skl]),skolemize(E1,sK8_skl),skolemize(E2,sK9_skl),skolemize(P,sK10_skl),skolemize(V1,sK11_skl),skolemize(V2,sK12_skl)],[f121]) ).
fof(f123,plain,
complete,
inference(cnf_transformation,[status(thm)],[f122]) ).
fof(f124,plain,
shortest_path(sK11_skl,sK12_skl,sK10_skl),
inference(cnf_transformation,[status(thm)],[f122]) ).
fof(f125,plain,
precedes(sK8_skl,sK9_skl,sK10_skl),
inference(cnf_transformation,[status(thm)],[f122]) ).
fof(f126,plain,
! [X0] :
( head_of(X0) != tail_of(sK8_skl)
| tail_of(X0) != head_of(sK9_skl)
| ~ edge(X0) ),
inference(cnf_transformation,[status(thm)],[f122]) ).
fof(f127,plain,
! [V1,V2,E] :
( ( V2 != tail_of(E)
| V1 != head_of(E)
| sP0_prd(E,V2,V1) )
& ( ( V2 = tail_of(E)
& V1 = head_of(E) )
| ~ sP0_prd(E,V2,V1) ) ),
inference(NNF_transformation,[status(thm)],[f27]) ).
fof(f128,plain,
( ! [V1,V2,E] :
( V2 != tail_of(E)
| V1 != head_of(E)
| sP0_prd(E,V2,V1) )
& ! [V1,V2,E] :
( ( V2 = tail_of(E)
& V1 = head_of(E) )
| ~ sP0_prd(E,V2,V1) ) ),
inference(miniscoping,[status(thm)],[f127]) ).
fof(f129,plain,
! [X0,X1,X2] :
( X2 = head_of(X0)
| ~ sP0_prd(X0,X1,X2) ),
inference(cnf_transformation,[status(thm)],[f128]) ).
fof(f130,plain,
! [X0,X1,X2] :
( X1 = tail_of(X0)
| ~ sP0_prd(X0,X1,X2) ),
inference(cnf_transformation,[status(thm)],[f128]) ).
fof(f137,definition,
( sQ0_spl
<=> complete ),
introduced(definition,[new_symbols(definition,[sQ0_spl])],[split_symbol_definition]) ).
fof(f139,plain,
( sQ0_spl
| ~ complete ),
inference(component_clause,[status(thm)],[f137]) ).
fof(f140,definition,
! [X0,X1] :
( sQ1_spl
<=> ( edge(sK0_skl(X1,X0))
| X0 = X1
| ~ vertex(X1)
| ~ vertex(X0) ) ),
introduced(definition,[new_symbols(definition,[sQ1_spl])],[split_symbol_definition]) ).
fof(f141,plain,
! [X0,X1] :
( ~ sQ1_spl
| edge(sK0_skl(X1,X0))
| X0 = X1
| ~ vertex(X1)
| ~ vertex(X0) ),
inference(component_clause,[status(thm)],[f140]) ).
fof(f143,plain,
( sQ1_spl
| ~ sQ0_spl ),
inference(split_clause,[status(thm)],[f31,f137,f140]) ).
fof(f144,definition,
! [X0,X1] :
( sQ2_spl
<=> ( X1 = head_of(sK0_skl(X1,X0))
| sP0_prd(sK0_skl(X1,X0),X1,X0)
| X0 = X1
| ~ vertex(X1)
| ~ vertex(X0) ) ),
introduced(definition,[new_symbols(definition,[sQ2_spl])],[split_symbol_definition]) ).
fof(f145,plain,
! [X0,X1] :
( ~ sQ2_spl
| X1 = head_of(sK0_skl(X1,X0))
| sP0_prd(sK0_skl(X1,X0),X1,X0)
| X0 = X1
| ~ vertex(X1)
| ~ vertex(X0) ),
inference(component_clause,[status(thm)],[f144]) ).
fof(f147,plain,
( sQ2_spl
| ~ sQ0_spl ),
inference(split_clause,[status(thm)],[f32,f137,f144]) ).
fof(f148,definition,
! [X0,X1] :
( sQ3_spl
<=> ( X0 = tail_of(sK0_skl(X1,X0))
| sP0_prd(sK0_skl(X1,X0),X1,X0)
| X0 = X1
| ~ vertex(X1)
| ~ vertex(X0) ) ),
introduced(definition,[new_symbols(definition,[sQ3_spl])],[split_symbol_definition]) ).
fof(f149,plain,
! [X0,X1] :
( ~ sQ3_spl
| X0 = tail_of(sK0_skl(X1,X0))
| sP0_prd(sK0_skl(X1,X0),X1,X0)
| X0 = X1
| ~ vertex(X1)
| ~ vertex(X0) ),
inference(component_clause,[status(thm)],[f148]) ).
fof(f151,plain,
( sQ3_spl
| ~ sQ0_spl ),
inference(split_clause,[status(thm)],[f33,f137,f148]) ).
fof(f162,plain,
( sQ0_spl
| $false ),
inference(forward_subsumption_resolution,[status(thm)],[f139,f123]) ).
fof(f163,plain,
sQ0_spl,
inference(contradiction_clause,[status(thm)],[f162]) ).
fof(f170,definition,
! [X0,X1] :
( sQ5_spl
<=> ~ path(X0,X1,sK10_skl) ),
introduced(definition,[new_symbols(definition,[sQ5_spl])],[split_symbol_definition]) ).
fof(f171,plain,
! [X0,X1] :
( ~ sQ5_spl
| ~ path(X0,X1,sK10_skl) ),
inference(component_clause,[status(thm)],[f170]) ).
fof(f193,plain,
path(sK11_skl,sK12_skl,sK10_skl),
inference(resolution,[status(thm)],[f85,f124]) ).
fof(f194,plain,
( ~ sQ5_spl
| $false ),
inference(forward_subsumption_resolution,[status(thm)],[f193,f171]) ).
fof(f195,plain,
~ sQ5_spl,
inference(contradiction_clause,[status(thm)],[f194]) ).
fof(f196,plain,
! [X0,X1,X2,X3,X4] :
( head_of(X3) != head_of(X4)
| ~ precedes(X3,X4,X2)
| ~ shortest_path(X0,X1,X2) ),
inference(equality_resolution,[status(thm)],[f92]) ).
fof(f197,plain,
! [X0,X1] :
( ~ precedes(X1,X0,sK10_skl)
| ~ precedes(X0,X1,sK10_skl) ),
inference(resolution,[status(thm)],[f93,f124]) ).
fof(f211,plain,
~ precedes(sK9_skl,sK8_skl,sK10_skl),
inference(resolution,[status(thm)],[f197,f125]) ).
fof(f214,plain,
! [X0,X1,X2,X3] :
( ~ precedes(X3,X3,X2)
| ~ shortest_path(X0,X1,X2) ),
inference(equality_resolution,[status(thm)],[f196]) ).
fof(f215,plain,
! [X0] : ~ precedes(X0,X0,sK10_skl),
inference(resolution,[status(thm)],[f214,f124]) ).
fof(f216,plain,
! [X0,X1] :
( ~ sQ2_spl
| X1 = head_of(sK0_skl(X1,X0))
| X0 = X1
| ~ vertex(X1)
| ~ vertex(X0)
| X0 = head_of(sK0_skl(X1,X0)) ),
inference(resolution,[status(thm)],[f129,f145]) ).
fof(f217,plain,
! [X0,X1] :
( ~ sQ2_spl
| X0 = head_of(sK0_skl(X0,X1))
| X1 = X0
| ~ vertex(X0)
| ~ vertex(X1)
| X0 = tail_of(sK0_skl(X0,X1)) ),
inference(resolution,[status(thm)],[f130,f145]) ).
fof(f220,plain,
! [X0] :
( edge(X0)
| ~ on_path(X0,sK10_skl) ),
inference(resolution,[status(thm)],[f52,f193]) ).
fof(f224,plain,
! [X0,X1] :
( ~ sQ3_spl
| X0 = head_of(sK0_skl(X1,X0))
| X0 = tail_of(sK0_skl(X1,X0))
| X0 = X1
| ~ vertex(X1)
| ~ vertex(X0) ),
inference(resolution,[status(thm)],[f149,f129]) ).
fof(f227,plain,
! [X0] :
( ~ sQ2_spl
| tail_of(sK0_skl(tail_of(sK8_skl),X0)) != head_of(sK9_skl)
| ~ edge(sK0_skl(tail_of(sK8_skl),X0))
| X0 = tail_of(sK8_skl)
| ~ vertex(tail_of(sK8_skl))
| ~ vertex(X0)
| X0 = head_of(sK0_skl(tail_of(sK8_skl),X0)) ),
inference(resolution,[status(thm)],[f216,f126]) ).
fof(f230,definition,
! [X0] :
( sQ10_spl
<=> ( tail_of(sK0_skl(tail_of(sK8_skl),X0)) != head_of(sK9_skl)
| ~ edge(sK0_skl(tail_of(sK8_skl),X0))
| X0 = tail_of(sK8_skl)
| ~ vertex(X0)
| X0 = head_of(sK0_skl(tail_of(sK8_skl),X0)) ) ),
introduced(definition,[new_symbols(definition,[sQ10_spl])],[split_symbol_definition]) ).
fof(f231,plain,
! [X0] :
( ~ sQ10_spl
| tail_of(sK0_skl(tail_of(sK8_skl),X0)) != head_of(sK9_skl)
| ~ edge(sK0_skl(tail_of(sK8_skl),X0))
| X0 = tail_of(sK8_skl)
| ~ vertex(X0)
| X0 = head_of(sK0_skl(tail_of(sK8_skl),X0)) ),
inference(component_clause,[status(thm)],[f230]) ).
fof(f233,definition,
( sQ11_spl
<=> vertex(tail_of(sK8_skl)) ),
introduced(definition,[new_symbols(definition,[sQ11_spl])],[split_symbol_definition]) ).
fof(f235,plain,
( sQ11_spl
| ~ vertex(tail_of(sK8_skl)) ),
inference(component_clause,[status(thm)],[f233]) ).
fof(f236,plain,
( ~ sQ2_spl
| ~ sQ11_spl
| sQ10_spl ),
inference(split_clause,[status(thm)],[f227,f230,f233,f144]) ).
fof(f240,plain,
! [X0,X1,X2,X3,X4,X5] :
( ~ sQ2_spl
| head_of(sK0_skl(tail_of(X1),X0)) != head_of(X5)
| ~ precedes(X1,X5,X4)
| ~ shortest_path(X2,X3,X4)
| tail_of(X1) = head_of(sK0_skl(tail_of(X1),X0))
| X0 = tail_of(X1)
| ~ vertex(tail_of(X1))
| ~ vertex(X0) ),
inference(resolution,[status(thm)],[f217,f92]) ).
fof(f246,definition,
( sQ13_spl
<=> vertex(head_of(sK9_skl)) ),
introduced(definition,[new_symbols(definition,[sQ13_spl])],[split_symbol_definition]) ).
fof(f248,plain,
( sQ13_spl
| ~ vertex(head_of(sK9_skl)) ),
inference(component_clause,[status(thm)],[f246]) ).
fof(f272,plain,
! [X0] :
( in_path(head_of(X0),sK10_skl)
| ~ on_path(X0,sK10_skl) ),
inference(resolution,[status(thm)],[f53,f193]) ).
fof(f276,plain,
! [X0] :
( in_path(tail_of(X0),sK10_skl)
| ~ on_path(X0,sK10_skl) ),
inference(resolution,[status(thm)],[f54,f193]) ).
fof(f280,plain,
! [X0] :
( vertex(X0)
| ~ in_path(X0,sK10_skl) ),
inference(resolution,[status(thm)],[f58,f193]) ).
fof(f283,plain,
! [X0] :
( vertex(head_of(X0))
| ~ on_path(X0,sK10_skl) ),
inference(resolution,[status(thm)],[f272,f280]) ).
fof(f286,plain,
! [X0] :
( vertex(tail_of(X0))
| ~ on_path(X0,sK10_skl) ),
inference(resolution,[status(thm)],[f276,f280]) ).
fof(f290,plain,
( ~ sQ10_spl
| ~ sQ3_spl
| ~ edge(sK0_skl(tail_of(sK8_skl),head_of(sK9_skl)))
| head_of(sK9_skl) = tail_of(sK8_skl)
| ~ vertex(head_of(sK9_skl))
| head_of(sK9_skl) = head_of(sK0_skl(tail_of(sK8_skl),head_of(sK9_skl)))
| head_of(sK9_skl) = head_of(sK0_skl(tail_of(sK8_skl),head_of(sK9_skl)))
| head_of(sK9_skl) = tail_of(sK8_skl)
| ~ vertex(tail_of(sK8_skl))
| ~ vertex(head_of(sK9_skl)) ),
inference(resolution,[status(thm)],[f224,f231]) ).
fof(f295,definition,
( sQ15_spl
<=> head_of(sK9_skl) = tail_of(sK8_skl) ),
introduced(definition,[new_symbols(definition,[sQ15_spl])],[split_symbol_definition]) ).
fof(f296,plain,
( ~ sQ15_spl
| head_of(sK9_skl) = tail_of(sK8_skl) ),
inference(component_clause,[status(thm)],[f295]) ).
fof(f298,definition,
( sQ16_spl
<=> head_of(sK9_skl) = head_of(sK0_skl(tail_of(sK8_skl),head_of(sK9_skl))) ),
introduced(definition,[new_symbols(definition,[sQ16_spl])],[split_symbol_definition]) ).
fof(f299,plain,
( ~ sQ16_spl
| head_of(sK9_skl) = head_of(sK0_skl(tail_of(sK8_skl),head_of(sK9_skl))) ),
inference(component_clause,[status(thm)],[f298]) ).
fof(f301,definition,
( sQ17_spl
<=> edge(sK0_skl(tail_of(sK8_skl),head_of(sK9_skl))) ),
introduced(definition,[new_symbols(definition,[sQ17_spl])],[split_symbol_definition]) ).
fof(f303,plain,
( sQ17_spl
| ~ edge(sK0_skl(tail_of(sK8_skl),head_of(sK9_skl))) ),
inference(component_clause,[status(thm)],[f301]) ).
fof(f304,plain,
( ~ sQ10_spl
| ~ sQ3_spl
| ~ sQ17_spl
| sQ16_spl
| sQ15_spl
| ~ sQ11_spl
| ~ sQ13_spl ),
inference(split_clause,[status(thm)],[f290,f246,f233,f295,f298,f301,f148,f230]) ).
fof(f315,definition,
( sQ20_spl
<=> edge(sK9_skl) ),
introduced(definition,[new_symbols(definition,[sQ20_spl])],[split_symbol_definition]) ).
fof(f317,plain,
( sQ20_spl
| ~ edge(sK9_skl) ),
inference(component_clause,[status(thm)],[f315]) ).
fof(f322,plain,
! [X0,X1] :
( on_path(sK8_skl,sK10_skl)
| ~ path(X0,X1,sK10_skl) ),
inference(resolution,[status(thm)],[f76,f125]) ).
fof(f323,definition,
( sQ21_spl
<=> on_path(sK8_skl,sK10_skl) ),
introduced(definition,[new_symbols(definition,[sQ21_spl])],[split_symbol_definition]) ).
fof(f324,plain,
( ~ sQ21_spl
| on_path(sK8_skl,sK10_skl) ),
inference(component_clause,[status(thm)],[f323]) ).
fof(f326,plain,
( sQ21_spl
| sQ5_spl ),
inference(split_clause,[status(thm)],[f322,f170,f323]) ).
fof(f327,plain,
( ~ sQ21_spl
| vertex(tail_of(sK8_skl)) ),
inference(resolution,[status(thm)],[f324,f286]) ).
fof(f329,plain,
( ~ sQ21_spl
| edge(sK8_skl) ),
inference(resolution,[status(thm)],[f324,f220]) ).
fof(f330,plain,
( ~ sQ21_spl
| sQ11_spl
| $false ),
inference(forward_subsumption_resolution,[status(thm)],[f327,f235]) ).
fof(f331,plain,
( ~ sQ21_spl
| sQ11_spl ),
inference(contradiction_clause,[status(thm)],[f330]) ).
fof(f352,plain,
! [X0,X1] :
( ~ sequential(X0,X1)
| ~ on_path(X1,sK10_skl)
| ~ on_path(X0,sK10_skl)
| precedes(X0,X1,sK10_skl) ),
inference(resolution,[status(thm)],[f70,f193]) ).
fof(f355,plain,
( ~ sequential(sK9_skl,sK8_skl)
| ~ on_path(sK8_skl,sK10_skl)
| ~ on_path(sK9_skl,sK10_skl) ),
inference(resolution,[status(thm)],[f352,f211]) ).
fof(f364,definition,
( sQ25_spl
<=> on_path(sK9_skl,sK10_skl) ),
introduced(definition,[new_symbols(definition,[sQ25_spl])],[split_symbol_definition]) ).
fof(f365,plain,
( ~ sQ25_spl
| on_path(sK9_skl,sK10_skl) ),
inference(component_clause,[status(thm)],[f364]) ).
fof(f367,definition,
( sQ26_spl
<=> sequential(sK9_skl,sK8_skl) ),
introduced(definition,[new_symbols(definition,[sQ26_spl])],[split_symbol_definition]) ).
fof(f370,plain,
( ~ sQ26_spl
| ~ sQ21_spl
| ~ sQ25_spl ),
inference(split_clause,[status(thm)],[f355,f364,f323,f367]) ).
fof(f400,plain,
! [X0,X1] :
( on_path(sK9_skl,sK10_skl)
| ~ path(X0,X1,sK10_skl) ),
inference(resolution,[status(thm)],[f77,f125]) ).
fof(f402,plain,
( sQ25_spl
| sQ5_spl ),
inference(split_clause,[status(thm)],[f400,f170,f364]) ).
fof(f409,plain,
( ~ sQ25_spl
| vertex(head_of(sK9_skl)) ),
inference(resolution,[status(thm)],[f365,f283]) ).
fof(f410,plain,
( ~ sQ25_spl
| edge(sK9_skl) ),
inference(resolution,[status(thm)],[f365,f220]) ).
fof(f411,plain,
( ~ sQ25_spl
| sQ13_spl
| $false ),
inference(forward_subsumption_resolution,[status(thm)],[f409,f248]) ).
fof(f412,plain,
( ~ sQ25_spl
| sQ13_spl ),
inference(contradiction_clause,[status(thm)],[f411]) ).
fof(f413,plain,
( sQ20_spl
| ~ sQ25_spl
| $false ),
inference(forward_subsumption_resolution,[status(thm)],[f317,f410]) ).
fof(f414,plain,
( sQ20_spl
| ~ sQ25_spl ),
inference(contradiction_clause,[status(thm)],[f413]) ).
fof(f484,plain,
( ~ sQ1_spl
| sQ17_spl
| head_of(sK9_skl) = tail_of(sK8_skl)
| ~ vertex(tail_of(sK8_skl))
| ~ vertex(head_of(sK9_skl)) ),
inference(resolution,[status(thm)],[f303,f141]) ).
fof(f485,plain,
( ~ sQ1_spl
| sQ17_spl
| sQ15_spl
| ~ sQ11_spl
| ~ sQ13_spl ),
inference(split_clause,[status(thm)],[f484,f246,f233,f295,f301,f140]) ).
fof(f488,plain,
( ~ sQ15_spl
| sK9_skl = sK8_skl
| ~ edge(sK8_skl)
| ~ edge(sK9_skl)
| sequential(sK9_skl,sK8_skl) ),
inference(resolution,[status(thm)],[f296,f67]) ).
fof(f502,definition,
( sQ43_spl
<=> edge(sK8_skl) ),
introduced(definition,[new_symbols(definition,[sQ43_spl])],[split_symbol_definition]) ).
fof(f504,plain,
( sQ43_spl
| ~ edge(sK8_skl) ),
inference(component_clause,[status(thm)],[f502]) ).
fof(f505,definition,
( sQ44_spl
<=> sK9_skl = sK8_skl ),
introduced(definition,[new_symbols(definition,[sQ44_spl])],[split_symbol_definition]) ).
fof(f506,plain,
( ~ sQ44_spl
| sK9_skl = sK8_skl ),
inference(component_clause,[status(thm)],[f505]) ).
fof(f508,plain,
( ~ sQ15_spl
| sQ44_spl
| ~ sQ43_spl
| ~ sQ20_spl
| sQ26_spl ),
inference(split_clause,[status(thm)],[f488,f367,f315,f502,f505,f295]) ).
fof(f537,plain,
( sQ43_spl
| ~ sQ21_spl
| $false ),
inference(forward_subsumption_resolution,[status(thm)],[f504,f329]) ).
fof(f538,plain,
( sQ43_spl
| ~ sQ21_spl ),
inference(contradiction_clause,[status(thm)],[f537]) ).
fof(f1251,plain,
! [X0,X1,X2,X3] :
( ~ sQ16_spl
| ~ sQ2_spl
| head_of(sK0_skl(tail_of(sK8_skl),head_of(sK9_skl))) != head_of(X3)
| ~ precedes(sK8_skl,X3,X2)
| ~ shortest_path(X0,X1,X2)
| tail_of(sK8_skl) = head_of(sK9_skl)
| head_of(sK9_skl) = tail_of(sK8_skl)
| ~ vertex(tail_of(sK8_skl))
| ~ vertex(head_of(sK9_skl)) ),
inference(paramodulation,[status(thm)],[f299,f240]) ).
fof(f1258,definition,
! [X0,X1,X2,X3] :
( sQ124_spl
<=> ( head_of(sK0_skl(tail_of(sK8_skl),head_of(sK9_skl))) != head_of(X3)
| ~ precedes(sK8_skl,X3,X2)
| ~ shortest_path(X0,X1,X2) ) ),
introduced(definition,[new_symbols(definition,[sQ124_spl])],[split_symbol_definition]) ).
fof(f1259,plain,
! [X0,X1,X2,X3] :
( ~ sQ124_spl
| head_of(sK0_skl(tail_of(sK8_skl),head_of(sK9_skl))) != head_of(X3)
| ~ precedes(sK8_skl,X3,X2)
| ~ shortest_path(X0,X1,X2) ),
inference(component_clause,[status(thm)],[f1258]) ).
fof(f1261,plain,
( ~ sQ16_spl
| ~ sQ2_spl
| sQ124_spl
| sQ15_spl
| ~ sQ11_spl
| ~ sQ13_spl ),
inference(split_clause,[status(thm)],[f1251,f246,f233,f295,f1258,f144,f298]) ).
fof(f1262,plain,
! [X0,X1,X2,X3] :
( ~ sQ124_spl
| ~ sQ16_spl
| head_of(sK9_skl) != head_of(X3)
| ~ precedes(sK8_skl,X3,X2)
| ~ shortest_path(X0,X1,X2) ),
inference(forward_demodulation,[status(thm)],[f299,f1259]) ).
fof(f1286,plain,
! [X0,X1,X2] :
( ~ sQ124_spl
| ~ sQ16_spl
| ~ precedes(sK8_skl,sK9_skl,X2)
| ~ shortest_path(X0,X1,X2) ),
inference(equality_resolution,[status(thm)],[f1262]) ).
fof(f1309,plain,
( ~ sQ124_spl
| ~ sQ16_spl
| ~ precedes(sK8_skl,sK9_skl,sK10_skl) ),
inference(resolution,[status(thm)],[f1286,f124]) ).
fof(f1310,plain,
( ~ sQ124_spl
| ~ sQ16_spl
| $false ),
inference(forward_subsumption_resolution,[status(thm)],[f1309,f125]) ).
fof(f1311,plain,
( ~ sQ124_spl
| ~ sQ16_spl ),
inference(contradiction_clause,[status(thm)],[f1310]) ).
fof(f1338,plain,
( ~ sQ44_spl
| precedes(sK8_skl,sK8_skl,sK10_skl) ),
inference(backward_demodulation,[status(thm)],[f506,f125]) ).
fof(f1353,plain,
( ~ sQ44_spl
| $false ),
inference(forward_subsumption_resolution,[status(thm)],[f1338,f215]) ).
fof(f1354,plain,
~ sQ44_spl,
inference(contradiction_clause,[status(thm)],[f1353]) ).
fof(f1355,plain,
$false,
inference(sat_refutation,[status(thm)],[f143,f147,f151,f163,f195,f236,f304,f326,f331,f370,f402,f412,f414,f485,f508,f538,f1261,f1311,f1354]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : GRA007+2 : TPTP v9.3.1. Bugfixed v3.2.0.
% 0.00/0.04 % Command : drodi -timeout(300) /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.14/0.40 % Computer : n016.cluster.edu
% 0.14/0.40 % Model : x86_64 x86_64
% 0.14/0.40 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.40 % Memory : 8046.5625MB
% 0.14/0.40 % OS : Linux 6.8.0-71-generic
% 0.14/0.40 % CPULimit : 300
% 0.14/0.40 % WCLimit : 300
% 0.14/0.40 % DateTime : Sun Sep 20 18:25:07 UTC 2026
% 0.14/0.40 % CPUTime :
% 0.14/0.41 % Drodi V4.1.1
% 0.14/0.56 % Refutation found
% 0.14/0.56 % SZS status Theorem for theBenchmark: Theorem is valid
% 0.14/0.56 % SZS output start CNFRefutation for theBenchmark
% See solution above
% 0.14/0.59 % Elapsed time: 0.176486 seconds
% 0.14/0.59 % CPU time: 1.198143 seconds
% 0.14/0.59 % Total memory used: 99.509 MB
% 0.14/0.59 % Net memory used: 98.188 MB
%------------------------------------------------------------------------------