%------------------------------------------------------------------------------
% File : Drodi---4.1.1
% Problem : SET960+1 : TPTP v9.3.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : drodi -timeout(300) /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n012.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 02:11:03 PM UTC 2026
% Result : Theorem 0.13s 0.39s
% Output : CNFRefutation 0.13s
% Verified :
% SZS Type : Refutation
% Derivation depth : 25
% Number of leaves : 4
% Syntax : Number of formulae : 58 ( 9 unt; 1 def)
% Number of atoms : 188 ( 90 equ)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 210 ( 80 ~; 95 |; 26 &)
% ( 8 <=>; 0 =>; 0 <=; 1 <~>)
% Maximal formula depth : 12 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 4 ( 2 usr; 1 prp; 0-5 aty)
% Number of functors : 11 ( 11 usr; 3 con; 0-4 aty)
% Number of variables : 169 ( 149 !; 20 ?)
% Comments :
%------------------------------------------------------------------------------
fof(f3,axiom,
! [A] :
( A = empty_set
<=> ! [B] : ~ in(B,A) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p') ).
fof(f4,axiom,
! [A,B,C] :
( C = cartesian_product2(A,B)
<=> ! [D] :
( in(D,C)
<=> ? [E,F] :
( D = ordered_pair(E,F)
& in(F,B)
& in(E,A) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p') ).
fof(f10,conjecture,
! [A,B] :
( cartesian_product2(A,B) = empty_set
<=> ( B = empty_set
| A = empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p') ).
fof(f11,negated_conjecture,
~ ! [A,B] :
( cartesian_product2(A,B) = empty_set
<=> ( B = empty_set
| A = empty_set ) ),
inference(negated_conjecture,[status(cth)],[f10]) ).
fof(f15,plain,
! [A] :
( ( ? [B] : in(B,A)
| A = empty_set )
& ( ! [B] : ~ in(B,A)
| A != empty_set ) ),
inference(NNF_transformation,[status(thm)],[f3]) ).
fof(f16,plain,
( ! [A] :
( ? [B] : in(B,A)
| A = empty_set )
& ! [A] :
( ! [B] : ~ in(B,A)
| A != empty_set ) ),
inference(miniscoping,[status(thm)],[f15]) ).
fof(f17,plain,
( ! [A] :
( in(sK0_skl(A),A)
| A = empty_set )
& ! [A] :
( ! [B] : ~ in(B,A)
| A != empty_set ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0_skl]),skolemize(B,sK0_skl(A))],[f16]) ).
fof(f18,plain,
! [X0,X1] :
( ~ in(X1,X0)
| X0 != empty_set ),
inference(cnf_transformation,[status(thm)],[f17]) ).
fof(f19,plain,
! [X0] :
( in(sK0_skl(X0),X0)
| X0 = empty_set ),
inference(cnf_transformation,[status(thm)],[f17]) ).
fof(f20,definition,
! [A,B,D,E,F] :
( sP0_prd(F,E,D,B,A)
<=> ( D = ordered_pair(E,F)
& in(F,B)
& in(E,A) ) ),
introduced(definition,[new_symbols(definition,[sP0_prd])],[]) ).
fof(f21,plain,
! [A,B,C] :
( C = cartesian_product2(A,B)
<=> ! [D] :
( in(D,C)
<=> ? [E,F] : sP0_prd(F,E,D,B,A) ) ),
inference(formula_renaming,[status(thm)],[f4,f20]) ).
fof(f22,plain,
! [A,B,C] :
( ( ? [D] :
( ( ? [E,F] : sP0_prd(F,E,D,B,A)
| in(D,C) )
& ( ! [E,F] : ~ sP0_prd(F,E,D,B,A)
| ~ in(D,C) ) )
| C = cartesian_product2(A,B) )
& ( ! [D] :
( ( ! [E,F] : ~ sP0_prd(F,E,D,B,A)
| in(D,C) )
& ( ? [E,F] : sP0_prd(F,E,D,B,A)
| ~ in(D,C) ) )
| C != cartesian_product2(A,B) ) ),
inference(NNF_transformation,[status(thm)],[f21]) ).
fof(f23,plain,
( ! [A,B,C] :
( ? [D] :
( ( ? [E,F] : sP0_prd(F,E,D,B,A)
| in(D,C) )
& ( ! [E,F] : ~ sP0_prd(F,E,D,B,A)
| ~ in(D,C) ) )
| C = cartesian_product2(A,B) )
& ! [A,B,C] :
( ( ! [D] :
( ! [E,F] : ~ sP0_prd(F,E,D,B,A)
| in(D,C) )
& ! [D] :
( ? [E,F] : sP0_prd(F,E,D,B,A)
| ~ in(D,C) ) )
| C != cartesian_product2(A,B) ) ),
inference(miniscoping,[status(thm)],[f22]) ).
fof(f24,plain,
( ! [A,B,C] :
( ( ( sP0_prd(sK5_skl(C,B,A),sK4_skl(C,B,A),sK3_skl(C,B,A),B,A)
| in(sK3_skl(C,B,A),C) )
& ( ! [E,F] : ~ sP0_prd(F,E,sK3_skl(C,B,A),B,A)
| ~ in(sK3_skl(C,B,A),C) ) )
| C = cartesian_product2(A,B) )
& ! [A,B,C] :
( ( ! [D] :
( ! [E,F] : ~ sP0_prd(F,E,D,B,A)
| in(D,C) )
& ! [D] :
( sP0_prd(sK2_skl(D,C,B,A),sK1_skl(D,C,B,A),D,B,A)
| ~ in(D,C) ) )
| C != cartesian_product2(A,B) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1_skl,sK2_skl,sK3_skl,sK4_skl,sK5_skl]),skolemize(E,sK1_skl(D,C,B,A)),skolemize(F,sK2_skl(D,C,B,A)),skolemize(D,sK3_skl(C,B,A)),skolemize(E,sK4_skl(C,B,A)),skolemize(F,sK5_skl(C,B,A))],[f23]) ).
fof(f25,plain,
! [X0,X1,X2,X3] :
( sP0_prd(sK2_skl(X3,X0,X2,X1),sK1_skl(X3,X0,X2,X1),X3,X2,X1)
| ~ in(X3,X0)
| X0 != cartesian_product2(X1,X2) ),
inference(cnf_transformation,[status(thm)],[f24]) ).
fof(f26,plain,
! [X0,X1,X2,X3,X4,X5] :
( ~ sP0_prd(X4,X5,X3,X2,X1)
| in(X3,X0)
| X0 != cartesian_product2(X1,X2) ),
inference(cnf_transformation,[status(thm)],[f24]) ).
fof(f28,plain,
! [X0,X1,X2] :
( sP0_prd(sK5_skl(X0,X2,X1),sK4_skl(X0,X2,X1),sK3_skl(X0,X2,X1),X2,X1)
| in(sK3_skl(X0,X2,X1),X0)
| X0 = cartesian_product2(X1,X2) ),
inference(cnf_transformation,[status(thm)],[f24]) ).
fof(f36,plain,
? [A,B] :
( cartesian_product2(A,B) = empty_set
<~> ( B = empty_set
| A = empty_set ) ),
inference(pre_NNF_transformation,[status(thm)],[f11]) ).
fof(f37,plain,
? [A,B] :
( ( ( B != empty_set
& A != empty_set )
| cartesian_product2(A,B) != empty_set )
& ( B = empty_set
| A = empty_set
| cartesian_product2(A,B) = empty_set ) ),
inference(NNF_transformation,[status(thm)],[f36]) ).
fof(f38,plain,
( ( ( sK9_skl != empty_set
& sK8_skl != empty_set )
| cartesian_product2(sK8_skl,sK9_skl) != empty_set )
& ( sK9_skl = empty_set
| sK8_skl = empty_set
| cartesian_product2(sK8_skl,sK9_skl) = empty_set ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK8_skl,sK9_skl]),skolemize(A,sK8_skl),skolemize(B,sK9_skl)],[f37]) ).
fof(f39,plain,
( sK9_skl = empty_set
| sK8_skl = empty_set
| cartesian_product2(sK8_skl,sK9_skl) = empty_set ),
inference(cnf_transformation,[status(thm)],[f38]) ).
fof(f40,plain,
( sK8_skl != empty_set
| cartesian_product2(sK8_skl,sK9_skl) != empty_set ),
inference(cnf_transformation,[status(thm)],[f38]) ).
fof(f41,plain,
( sK9_skl != empty_set
| cartesian_product2(sK8_skl,sK9_skl) != empty_set ),
inference(cnf_transformation,[status(thm)],[f38]) ).
fof(f42,plain,
! [A,B,D,E,F] :
( ( D != ordered_pair(E,F)
| ~ in(F,B)
| ~ in(E,A)
| sP0_prd(F,E,D,B,A) )
& ( ( D = ordered_pair(E,F)
& in(F,B)
& in(E,A) )
| ~ sP0_prd(F,E,D,B,A) ) ),
inference(NNF_transformation,[status(thm)],[f20]) ).
fof(f43,plain,
( ! [A,B,D,E,F] :
( D != ordered_pair(E,F)
| ~ in(F,B)
| ~ in(E,A)
| sP0_prd(F,E,D,B,A) )
& ! [A,B,D,E,F] :
( ( D = ordered_pair(E,F)
& in(F,B)
& in(E,A) )
| ~ sP0_prd(F,E,D,B,A) ) ),
inference(miniscoping,[status(thm)],[f42]) ).
fof(f44,plain,
! [X0,X1,X2,X3,X4] :
( in(X1,X4)
| ~ sP0_prd(X0,X1,X2,X3,X4) ),
inference(cnf_transformation,[status(thm)],[f43]) ).
fof(f45,plain,
! [X0,X1,X2,X3,X4] :
( in(X0,X3)
| ~ sP0_prd(X0,X1,X2,X3,X4) ),
inference(cnf_transformation,[status(thm)],[f43]) ).
fof(f47,plain,
! [X0,X1,X2,X3,X4] :
( X2 != ordered_pair(X1,X0)
| ~ in(X0,X3)
| ~ in(X1,X4)
| sP0_prd(X0,X1,X2,X3,X4) ),
inference(cnf_transformation,[status(thm)],[f43]) ).
fof(f48,plain,
! [X0] : ~ in(X0,empty_set),
inference(destructive_equality_resolution,[status(thm)],[f18]) ).
fof(f49,plain,
! [X0,X1,X2] :
( sP0_prd(sK2_skl(X0,cartesian_product2(X1,X2),X2,X1),sK1_skl(X0,cartesian_product2(X1,X2),X2,X1),X0,X2,X1)
| ~ in(X0,cartesian_product2(X1,X2)) ),
inference(destructive_equality_resolution,[status(thm)],[f25]) ).
fof(f50,plain,
! [X0,X1,X2,X3,X4] :
( ~ sP0_prd(X3,X4,X0,X2,X1)
| in(X0,cartesian_product2(X1,X2)) ),
inference(destructive_equality_resolution,[status(thm)],[f26]) ).
fof(f51,plain,
! [X0,X1,X2,X3] :
( ~ in(X0,X2)
| ~ in(X1,X3)
| sP0_prd(X0,X1,ordered_pair(X1,X0),X2,X3) ),
inference(destructive_equality_resolution,[status(thm)],[f47]) ).
fof(f88,plain,
! [X0,X1,X2] :
( in(sK2_skl(X0,cartesian_product2(X1,X2),X2,X1),X2)
| ~ in(X0,cartesian_product2(X1,X2)) ),
inference(resolution,[status(thm)],[f49,f45]) ).
fof(f89,plain,
! [X0,X1,X2] :
( in(sK1_skl(X0,cartesian_product2(X1,X2),X2,X1),X1)
| ~ in(X0,cartesian_product2(X1,X2)) ),
inference(resolution,[status(thm)],[f49,f44]) ).
fof(f98,plain,
! [X0,X1,X2] :
( in(sK3_skl(X0,X2,X1),cartesian_product2(X1,X2))
| in(sK3_skl(X0,X2,X1),X0)
| X0 = cartesian_product2(X1,X2) ),
inference(resolution,[status(thm)],[f28,f50]) ).
fof(f143,plain,
! [X0,X1,X2,X3] :
( in(ordered_pair(X0,X2),cartesian_product2(X1,X3))
| ~ in(X2,X3)
| ~ in(X0,X1) ),
inference(resolution,[status(thm)],[f51,f50]) ).
fof(f162,plain,
! [X0,X1] : ~ in(X0,cartesian_product2(X1,empty_set)),
inference(resolution,[status(thm)],[f88,f48]) ).
fof(f178,plain,
! [X0] : cartesian_product2(X0,empty_set) = empty_set,
inference(resolution,[status(thm)],[f162,f19]) ).
fof(f184,plain,
! [X0,X1] :
( in(sK3_skl(X0,empty_set,X1),empty_set)
| in(sK3_skl(X0,empty_set,X1),X0)
| X0 = cartesian_product2(X1,empty_set) ),
inference(paramodulation,[status(thm)],[f178,f98]) ).
fof(f193,plain,
! [X0,X1] :
( in(sK3_skl(X0,empty_set,X1),empty_set)
| in(sK3_skl(X0,empty_set,X1),X0)
| X0 = empty_set ),
inference(forward_demodulation,[status(thm)],[f178,f184]) ).
fof(f194,plain,
! [X0,X1] :
( in(sK3_skl(X0,empty_set,X1),X0)
| X0 = empty_set ),
inference(forward_subsumption_resolution,[status(thm)],[f193,f48]) ).
fof(f205,plain,
! [X0,X1] :
( sK9_skl = empty_set
| sK8_skl = empty_set
| in(ordered_pair(X0,X1),empty_set)
| ~ in(X1,sK9_skl)
| ~ in(X0,sK8_skl) ),
inference(paramodulation,[status(thm)],[f39,f143]) ).
fof(f207,plain,
! [X0,X1] :
( sK9_skl = empty_set
| sK8_skl = empty_set
| ~ in(X1,sK9_skl)
| ~ in(X0,sK8_skl) ),
inference(forward_subsumption_resolution,[status(thm)],[f205,f48]) ).
fof(f210,plain,
! [X0] :
( sK9_skl = empty_set
| sK9_skl = empty_set
| sK8_skl = empty_set
| ~ in(X0,sK8_skl) ),
inference(resolution,[status(thm)],[f207,f194]) ).
fof(f215,plain,
! [X0] :
( sK9_skl = empty_set
| sK8_skl = empty_set
| ~ in(X0,sK8_skl) ),
inference(duplicate_literals_removal,[status(thm)],[f210]) ).
fof(f221,plain,
( sK8_skl = empty_set
| sK9_skl = empty_set
| sK8_skl = empty_set ),
inference(resolution,[status(thm)],[f215,f194]) ).
fof(f225,plain,
( sK9_skl = empty_set
| sK8_skl = empty_set ),
inference(duplicate_literals_removal,[status(thm)],[f221]) ).
fof(f228,plain,
( sK8_skl = empty_set
| sK9_skl != empty_set
| cartesian_product2(sK8_skl,empty_set) != empty_set ),
inference(paramodulation,[status(thm)],[f225,f41]) ).
fof(f230,plain,
( sK8_skl = empty_set
| sK9_skl != empty_set
| empty_set != empty_set ),
inference(forward_demodulation,[status(thm)],[f178,f228]) ).
fof(f231,plain,
( sK8_skl = empty_set
| sK9_skl != empty_set ),
inference(trivial_equality_resolution,[status(thm)],[f230]) ).
fof(f232,plain,
sK8_skl = empty_set,
inference(forward_subsumption_resolution,[status(thm)],[f231,f225]) ).
fof(f235,plain,
( empty_set != empty_set
| cartesian_product2(sK8_skl,sK9_skl) != empty_set ),
inference(backward_demodulation,[status(thm)],[f232,f40]) ).
fof(f236,plain,
( empty_set != empty_set
| cartesian_product2(empty_set,sK9_skl) != empty_set ),
inference(forward_demodulation,[status(thm)],[f232,f235]) ).
fof(f237,plain,
cartesian_product2(empty_set,sK9_skl) != empty_set,
inference(trivial_equality_resolution,[status(thm)],[f236]) ).
fof(f239,plain,
! [X0,X1] : ~ in(X0,cartesian_product2(empty_set,X1)),
inference(resolution,[status(thm)],[f89,f48]) ).
fof(f256,plain,
! [X0] : cartesian_product2(empty_set,X0) = empty_set,
inference(resolution,[status(thm)],[f239,f194]) ).
fof(f261,plain,
empty_set != empty_set,
inference(backward_demodulation,[status(thm)],[f256,f237]) ).
fof(f276,plain,
$false,
inference(trivial_equality_resolution,[status(thm)],[f261]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : SET960+1 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.03 % Command : drodi -timeout(300) /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.35 % Computer : n012.cluster.edu
% 0.09/0.35 % Model : x86_64 x86_64
% 0.09/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.35 % Memory : 8046.5625MB
% 0.09/0.35 % OS : Linux 6.8.0-71-generic
% 0.09/0.35 % CPULimit : 300
% 0.09/0.35 % WCLimit : 300
% 0.09/0.35 % DateTime : Mon Sep 21 05:45:22 UTC 2026
% 0.09/0.35 % CPUTime :
% 0.13/0.36 % Drodi V4.1.1
% 0.13/0.39 % Refutation found
% 0.13/0.39 % SZS status Theorem for theBenchmark: Theorem is valid
% 0.13/0.39 % SZS output start CNFRefutation for theBenchmark
% See solution above
% 0.13/0.41 % Elapsed time: 0.052151 seconds
% 0.13/0.41 % CPU time: 0.158193 seconds
% 0.13/0.41 % Total memory used: 103.366 MB
% 0.13/0.41 % Net memory used: 102.793 MB
%------------------------------------------------------------------------------