%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SWC098+1 : TPTP v9.3.1. Released v2.4.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n007.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 : Sun Sep 27 08:56:49 AM UTC 2026
% Result : Theorem 33.46s 4.68s
% Output : CNFRefutation 33.46s
% Verified :
% SZS Type : Refutation
% Derivation depth : 26
% Number of leaves : 3
% Syntax : Number of formulae : 81 ( 16 unt; 0 def)
% Number of atoms : 324 ( 141 equ)
% Maximal formula atoms : 46 ( 4 avg)
% Number of connectives : 379 ( 136 ~; 164 |; 65 &)
% ( 1 <=>; 13 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 4 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 7 ( 5 usr; 1 prp; 0-2 aty)
% Number of functors : 8 ( 8 usr; 6 con; 0-2 aty)
% Number of variables : 45 ( 0 sgn 23 !; 11 ?)
% Comments :
%------------------------------------------------------------------------------
fof(ax4,axiom,
! [U] :
( ssList(U)
=> ( singletonP(U)
<=> ? [V] :
( cons(V,nil) = U
& ssItem(V) ) ) ),
file('theBenchmark.p',ax4) ).
fof(co1,conjecture,
! [U] :
( ssList(U)
=> ! [V] :
( ssList(V)
=> ! [W] :
( ssList(W)
=> ! [X] :
( ssList(X)
=> ( ( ( nil != W
| nil != X )
& ! [X1] :
( ssItem(X1)
=> ( ? [X2] :
( leq(X1,X2)
& memberP(X,X2)
& X1 != X2
& ssItem(X2) )
| ~ memberP(X,X1)
| cons(X1,nil) != W ) ) )
| ( nil = U
& nil = V )
| ? [Y] :
( memberP(V,Y)
& ! [Z] :
( ssItem(Z)
=> ( Y = Z
| ~ leq(Y,Z)
| ~ memberP(V,Z) ) )
& cons(Y,nil) = U
& ssItem(Y) )
| U != W
| V != X ) ) ) ) ),
file('theBenchmark.p',co1) ).
fof(co1_neg,negated_conjecture,
~ ! [U] :
( ssList(U)
=> ! [V] :
( ssList(V)
=> ! [W] :
( ssList(W)
=> ! [X] :
( ssList(X)
=> ( ( ( nil != W
| nil != X )
& ! [X1] :
( ssItem(X1)
=> ( ? [X2] :
( leq(X1,X2)
& memberP(X,X2)
& X1 != X2
& ssItem(X2) )
| ~ memberP(X,X1)
| cons(X1,nil) != W ) ) )
| ( nil = U
& nil = V )
| ? [Y] :
( memberP(V,Y)
& ! [Z] :
( ssItem(Z)
=> ( Y = Z
| ~ leq(Y,Z)
| ~ memberP(V,Z) ) )
& cons(Y,nil) = U
& ssItem(Y) )
| U != W
| V != X ) ) ) ) ),
inference(assume_negation,[status(cth)],[co1]) ).
cnf(c_203,plain,
( cons(B,nil) != A
| ~ ssItem(B)
| singletonP(A)
| ~ ssList(A) ),
inference(clausify,[status(thm)],[ax4]) ).
fof(ax39,axiom,
~ singletonP(nil),
file('theBenchmark.p',ax39) ).
cnf(c_262,plain,
~ singletonP(nil),
inference(clausify,[status(thm)],[ax39]) ).
fof(nnf_87,plain,
? [U] :
( ? [V] :
( ? [W] :
( ? [X] :
( ( ( nil = W
& nil = X )
| ? [X1] :
( ! [X2] :
( ~ leq(X1,X2)
| ~ memberP(X,X2)
| X2 = X1
| ~ ssItem(X2) )
& memberP(X,X1)
& cons(X1,nil) = W
& ssItem(X1) ) )
& ( nil != U
| nil != V )
& ! [Y] :
( ~ memberP(V,Y)
| ? [Z] :
( Z != Y
& leq(Y,Z)
& memberP(V,Z)
& ssItem(Z) )
| cons(Y,nil) != U
| ~ ssItem(Y) )
& W = U
& X = V
& ssList(X) )
& ssList(W) )
& ssList(V) )
& ssList(U) ),
inference(fof_nnf,[status(thm)],[co1_neg]) ).
fof(sk_co1_neg_sk,plain,
( ( ( nil = sK3
& nil = sK1 )
| ( ! [X2] :
( ~ leq(sK5,X2)
| ~ memberP(sK1,X2)
| X2 = sK5
| ~ ssItem(X2) )
& memberP(sK1,sK5)
& cons(sK5,nil) = sK3
& ssItem(sK5) ) )
& ( nil != sK0
| nil != sK2 )
& ! [Y] :
( ~ memberP(sK2,Y)
| ( sK4(Y) != Y
& leq(Y,sK4(Y))
& memberP(sK2,sK4(Y))
& ssItem(sK4(Y)) )
| cons(Y,nil) != sK0
| ~ ssItem(Y) )
& sK3 = sK0
& sK1 = sK2
& ssList(sK1)
& ssList(sK3)
& ssList(sK2)
& ssList(sK0) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK2,sK3,sK1,sK4,sK5]),skolemize(U,sK0),skolemize(V,sK2),skolemize(W,sK3),skolemize(X,sK1),skolemize(Z,sK4(Y)),skolemize(X1,sK5)],[nnf_87]) ).
fof(sk_co1_neg,plain,
( ! [VAR_0] :
( nil = sK3
| ~ leq(sK5,VAR_0)
| ~ memberP(sK1,VAR_0)
| VAR_0 = sK5
| ~ ssItem(VAR_0) )
& ! [VAR_0] :
( nil = sK1
| ~ leq(sK5,VAR_0)
| ~ memberP(sK1,VAR_0)
| VAR_0 = sK5
| ~ ssItem(VAR_0) )
& ( nil = sK3
| memberP(sK1,sK5) )
& ( nil = sK1
| memberP(sK1,sK5) )
& ( nil = sK3
| cons(sK5,nil) = sK3 )
& ( nil = sK1
| cons(sK5,nil) = sK3 )
& ( nil = sK3
| ssItem(sK5) )
& ( nil = sK1
| ssItem(sK5) )
& ( nil != sK0
| nil != sK2 )
& ! [VAR_0] :
( ~ memberP(sK2,VAR_0)
| sK4(VAR_0) != VAR_0
| cons(VAR_0,nil) != sK0
| ~ ssItem(VAR_0) )
& ! [VAR_0] :
( ~ memberP(sK2,VAR_0)
| leq(VAR_0,sK4(VAR_0))
| cons(VAR_0,nil) != sK0
| ~ ssItem(VAR_0) )
& ! [VAR_0] :
( ~ memberP(sK2,VAR_0)
| memberP(sK2,sK4(VAR_0))
| cons(VAR_0,nil) != sK0
| ~ ssItem(VAR_0) )
& ! [VAR_0] :
( ~ memberP(sK2,VAR_0)
| ssItem(sK4(VAR_0))
| cons(VAR_0,nil) != sK0
| ~ ssItem(VAR_0) )
& sK3 = sK0
& sK1 = sK2
& ssList(sK1)
& ssList(sK3)
& ssList(sK2)
& ssList(sK0) ),
inference(cnf_transformation,[status(thm)],[sk_co1_neg_sk]) ).
cnf(c_327,negated_conjecture,
ssList(sK0),
inference(split_conjunct,[status(thm)],[sk_co1_neg]) ).
cnf(c_331,negated_conjecture,
sK1 = sK2,
inference(split_conjunct,[status(thm)],[sk_co1_neg]) ).
cnf(c_332,negated_conjecture,
sK3 = sK0,
inference(split_conjunct,[status(thm)],[sk_co1_neg]) ).
cnf(c_333,negated_conjecture,
( ~ memberP(sK2,A)
| ssItem(sK4(A))
| cons(A,nil) != sK0
| ~ ssItem(A) ),
inference(split_conjunct,[status(thm)],[sk_co1_neg]) ).
cnf(c_334,negated_conjecture,
( ~ memberP(sK2,A)
| memberP(sK2,sK4(A))
| cons(A,nil) != sK0
| ~ ssItem(A) ),
inference(split_conjunct,[status(thm)],[sk_co1_neg]) ).
cnf(c_335,negated_conjecture,
( ~ memberP(sK2,A)
| leq(A,sK4(A))
| cons(A,nil) != sK0
| ~ ssItem(A) ),
inference(split_conjunct,[status(thm)],[sk_co1_neg]) ).
cnf(c_336,negated_conjecture,
( ~ memberP(sK2,A)
| sK4(A) != A
| cons(A,nil) != sK0
| ~ ssItem(A) ),
inference(split_conjunct,[status(thm)],[sk_co1_neg]) ).
cnf(c_337,negated_conjecture,
( nil != sK0
| nil != sK2 ),
inference(split_conjunct,[status(thm)],[sk_co1_neg]) ).
cnf(c_6625,negated_conjecture,
( nil != sK0
| sK2 != nil ),
inference(copy,[status(thm)],[c_337]) ).
cnf(c_338,negated_conjecture,
( sK0 != nil
| sK2 != nil ),
inference(copy,[status(thm)],[c_6625]) ).
cnf(c_339,negated_conjecture,
( nil = sK1
| ssItem(sK5) ),
inference(split_conjunct,[status(thm)],[sk_co1_neg]) ).
cnf(c_6626,negated_conjecture,
( nil = sK2
| ssItem(sK5) ),
inference(paramod,[status(thm)],[c_331,c_339]) ).
cnf(c_340,negated_conjecture,
( sK2 = nil
| ssItem(sK5) ),
inference(copy,[status(thm)],[c_6626]) ).
cnf(c_341,negated_conjecture,
( nil = sK3
| ssItem(sK5) ),
inference(split_conjunct,[status(thm)],[sk_co1_neg]) ).
cnf(c_6627,negated_conjecture,
( nil = sK0
| ssItem(sK5) ),
inference(paramod,[status(thm)],[c_332,c_341]) ).
cnf(c_342,negated_conjecture,
( sK0 = nil
| ssItem(sK5) ),
inference(copy,[status(thm)],[c_6627]) ).
cnf(c_343,negated_conjecture,
( nil = sK1
| cons(sK5,nil) = sK3 ),
inference(split_conjunct,[status(thm)],[sk_co1_neg]) ).
cnf(c_6628,negated_conjecture,
( nil = sK1
| cons(sK5,nil) = sK0 ),
inference(paramod,[status(thm)],[c_332,c_343]) ).
cnf(c_6629,negated_conjecture,
( nil = sK2
| cons(sK5,nil) = sK0 ),
inference(paramod,[status(thm)],[c_331,c_6628]) ).
cnf(c_344,negated_conjecture,
( sK2 = nil
| cons(sK5,nil) = sK0 ),
inference(copy,[status(thm)],[c_6629]) ).
cnf(c_345,negated_conjecture,
( nil = sK3
| cons(sK5,nil) = sK3 ),
inference(split_conjunct,[status(thm)],[sk_co1_neg]) ).
cnf(c_6630,negated_conjecture,
( nil = sK3
| cons(sK5,nil) = sK0 ),
inference(paramod,[status(thm)],[c_332,c_345]) ).
cnf(c_6631,negated_conjecture,
( nil = sK0
| cons(sK5,nil) = sK0 ),
inference(paramod,[status(thm)],[c_332,c_6630]) ).
cnf(c_346,negated_conjecture,
( sK0 = nil
| cons(sK5,nil) = sK0 ),
inference(copy,[status(thm)],[c_6631]) ).
cnf(c_349,negated_conjecture,
( nil = sK3
| memberP(sK1,sK5) ),
inference(split_conjunct,[status(thm)],[sk_co1_neg]) ).
cnf(c_6632,negated_conjecture,
( nil = sK3
| memberP(sK2,sK5) ),
inference(paramod,[status(thm)],[c_331,c_349]) ).
cnf(c_6633,negated_conjecture,
( nil = sK0
| memberP(sK2,sK5) ),
inference(paramod,[status(thm)],[c_332,c_6632]) ).
cnf(c_350,negated_conjecture,
( sK0 = nil
| memberP(sK2,sK5) ),
inference(copy,[status(thm)],[c_6633]) ).
cnf(c_353,negated_conjecture,
( nil = sK3
| ~ leq(sK5,A)
| ~ memberP(sK1,A)
| A = sK5
| ~ ssItem(A) ),
inference(split_conjunct,[status(thm)],[sk_co1_neg]) ).
cnf(c_6634,negated_conjecture,
( nil = sK3
| ~ leq(sK5,A)
| ~ memberP(sK2,A)
| A = sK5
| ~ ssItem(A) ),
inference(paramod,[status(thm)],[c_331,c_353]) ).
cnf(c_6635,negated_conjecture,
( nil = sK0
| ~ leq(sK5,A)
| ~ memberP(sK2,A)
| A = sK5
| ~ ssItem(A) ),
inference(paramod,[status(thm)],[c_332,c_6634]) ).
cnf(c_6636,negated_conjecture,
( nil = sK0
| ~ leq(sK5,A)
| ~ memberP(sK2,A)
| sK5 = A
| ~ ssItem(A) ),
inference(copy,[status(thm)],[c_6635]) ).
cnf(c_354,negated_conjecture,
( sK0 = nil
| ~ leq(sK5,A)
| ~ memberP(sK2,A)
| sK5 = A
| ~ ssItem(A) ),
inference(copy,[status(thm)],[c_6636]) ).
cnf(c_2466,plain,
( sK0 != nil
| ssItem(sK5) ),
inference(resolve,[status(thm)],[c_340,c_338]) ).
cnf(c_6637,negated_conjecture,
( ~ ssItem(sK5)
| singletonP(sK0)
| ~ ssList(sK0)
| sK2 = nil ),
inference(resolve,[status(thm)],[c_344,c_203]) ).
cnf(c_2615,negated_conjecture,
( ~ ssItem(sK5)
| singletonP(sK0)
| sK2 = nil ),
inference(resolve,[status(thm)],[c_327,c_6637]) ).
cnf(c_6638,negated_conjecture,
( ~ ssItem(sK5)
| singletonP(sK0)
| ~ ssList(sK0)
| sK0 = nil ),
inference(resolve,[status(thm)],[c_346,c_203]) ).
cnf(c_2660,negated_conjecture,
( ~ ssItem(sK5)
| singletonP(sK0)
| sK0 = nil ),
inference(resolve,[status(thm)],[c_327,c_6638]) ).
cnf(c_6639,negated_conjecture,
( ssItem(sK5)
| ssItem(sK5) ),
inference(resolve,[status(thm)],[c_2466,c_342]) ).
cnf(c_4854,negated_conjecture,
ssItem(sK5),
inference(copy,[status(thm)],[c_6639]) ).
cnf(c_5321,negated_conjecture,
( singletonP(sK0)
| sK0 = nil ),
inference(resolve,[status(thm)],[c_4854,c_2660]) ).
cnf(c_5346,negated_conjecture,
( singletonP(sK0)
| sK2 = nil ),
inference(resolve,[status(thm)],[c_4854,c_2615]) ).
cnf(c_5549,plain,
( ~ memberP(sK2,sK5)
| sK4(sK5) != sK5
| cons(sK5,nil) != sK0 ),
inference(resolve,[status(thm)],[c_4854,c_336]) ).
cnf(c_5550,plain,
( ~ memberP(sK2,sK5)
| leq(sK5,sK4(sK5))
| cons(sK5,nil) != sK0 ),
inference(resolve,[status(thm)],[c_4854,c_335]) ).
cnf(c_5551,plain,
( ~ memberP(sK2,sK5)
| memberP(sK2,sK4(sK5))
| cons(sK5,nil) != sK0 ),
inference(resolve,[status(thm)],[c_4854,c_334]) ).
cnf(c_5552,plain,
( ~ memberP(sK2,sK5)
| ssItem(sK4(sK5))
| cons(sK5,nil) != sK0 ),
inference(resolve,[status(thm)],[c_4854,c_333]) ).
cnf(c_5886,plain,
( sK2 != nil
| singletonP(sK0) ),
inference(resolve,[status(thm)],[c_5321,c_338]) ).
cnf(c_6640,plain,
( singletonP(sK0)
| singletonP(sK0) ),
inference(resolve,[status(thm)],[c_5886,c_5346]) ).
cnf(c_5926,plain,
singletonP(sK0),
inference(copy,[status(thm)],[c_6640]) ).
cnf(c_6641,plain,
( singletonP(nil)
| cons(sK5,nil) = sK0 ),
inference(paramod,[status(thm)],[c_346,c_5926]) ).
cnf(c_5937,plain,
cons(sK5,nil) = sK0,
inference(resolve,[status(thm)],[c_262,c_6641]) ).
cnf(c_6642,plain,
( singletonP(nil)
| memberP(sK2,sK5) ),
inference(paramod,[status(thm)],[c_350,c_5926]) ).
cnf(c_5938,plain,
memberP(sK2,sK5),
inference(resolve,[status(thm)],[c_262,c_6642]) ).
cnf(c_6643,plain,
( ~ memberP(sK2,sK5)
| ssItem(sK4(sK5))
| sK0 != sK0 ),
inference(paramod,[status(thm)],[c_5937,c_5552]) ).
cnf(c_6644,plain,
( ~ memberP(sK2,sK5)
| ssItem(sK4(sK5)) ),
inference(copy,[status(thm)],[c_6643]) ).
cnf(c_6001,plain,
ssItem(sK4(sK5)),
inference(resolve,[status(thm)],[c_5938,c_6644]) ).
cnf(c_6645,plain,
( ~ memberP(sK2,sK5)
| memberP(sK2,sK4(sK5))
| sK0 != sK0 ),
inference(paramod,[status(thm)],[c_5937,c_5551]) ).
cnf(c_6646,plain,
( ~ memberP(sK2,sK5)
| memberP(sK2,sK4(sK5)) ),
inference(copy,[status(thm)],[c_6645]) ).
cnf(c_6002,plain,
memberP(sK2,sK4(sK5)),
inference(resolve,[status(thm)],[c_5938,c_6646]) ).
cnf(c_6647,plain,
( ~ memberP(sK2,sK5)
| leq(sK5,sK4(sK5))
| sK0 != sK0 ),
inference(paramod,[status(thm)],[c_5937,c_5550]) ).
cnf(c_6648,plain,
( ~ memberP(sK2,sK5)
| leq(sK5,sK4(sK5)) ),
inference(copy,[status(thm)],[c_6647]) ).
cnf(c_6003,plain,
leq(sK5,sK4(sK5)),
inference(resolve,[status(thm)],[c_5938,c_6648]) ).
cnf(c_6649,plain,
( ~ memberP(sK2,sK5)
| sK4(sK5) != sK5
| sK0 != sK0 ),
inference(paramod,[status(thm)],[c_5937,c_5549]) ).
cnf(c_6650,plain,
( ~ memberP(sK2,sK5)
| sK4(sK5) != sK5 ),
inference(copy,[status(thm)],[c_6649]) ).
cnf(c_6004,plain,
sK4(sK5) != sK5,
inference(resolve,[status(thm)],[c_5938,c_6650]) ).
cnf(c_6651,negated_conjecture,
( sK0 = nil
| ~ leq(sK5,sK4(sK5))
| ~ memberP(sK2,sK4(sK5))
| sK5 = sK4(sK5) ),
inference(resolve,[status(thm)],[c_6001,c_354]) ).
cnf(c_6652,negated_conjecture,
( sK0 = nil
| ~ leq(sK5,sK4(sK5))
| ~ memberP(sK2,sK4(sK5))
| sK4(sK5) = sK5 ),
inference(copy,[status(thm)],[c_6651]) ).
cnf(c_6653,negated_conjecture,
( sK0 = nil
| ~ leq(sK5,sK4(sK5))
| ~ memberP(sK2,sK4(sK5)) ),
inference(resolve,[status(thm)],[c_6004,c_6652]) ).
cnf(c_6654,negated_conjecture,
( sK0 = nil
| ~ leq(sK5,sK4(sK5)) ),
inference(resolve,[status(thm)],[c_6002,c_6653]) ).
cnf(c_6377,negated_conjecture,
sK0 = nil,
inference(resolve,[status(thm)],[c_6003,c_6654]) ).
cnf(c_6655,plain,
singletonP(nil),
inference(paramod,[status(thm)],[c_6377,c_5926]) ).
cnf(c_6624,plain,
$false,
inference(resolve,[status(thm)],[c_262,c_6655]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SWC098+1 : TPTP v9.3.1. Released v2.4.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.09/0.36 % Computer : n007.cluster.edu
% 0.09/0.36 % Model : x86_64 x86_64
% 0.09/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.36 % Memory : 8046.5625MB
% 0.09/0.36 % OS : Linux 6.8.0-71-generic
% 0.09/0.36 % CPULimit : 300
% 0.09/0.36 % WCLimit : 300
% 0.09/0.36 % DateTime : Sat Sep 26 12:12:55 UTC 2026
% 0.09/0.36 % CPUTime :
% 0.09/0.37 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.09/0.37 % Process 3687749 was started by sandbox on n007,
% 0.09/0.37 % Sat Sep 26 12:12:55 2026
% 0.09/0.37 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.09/0.39
% 0.09/0.39 % From the command line: assign(max_seconds, 300).
% 33.46/4.68
% 33.46/4.68 % given #500 (I,wt=34): 703 -ssList(A) | ssItem(f34(A)) | -ssItem(B) | -ssList(C) | -ssList(D) | -ssList(E) | app(app(C,cons(B,D)),cons(B,E)) != A. [factor(377,c,d),xx(h)].
% 33.46/4.68
% 33.46/4.68 % given #1000 (I,wt=35): 1223 -ssList(A) | ssList(f37(A)) | -ssItem(B) | -ssItem(C) | -ssList(D) | app(app(D,cons(B,D)),cons(C,A)) != A | C != B. [factor(723,e,f)].
% 33.46/4.68
% 33.46/4.68 % SZS status Theorem for theBenchmark
% 33.46/4.68
% 33.46/4.68 % Proof 1 at 3.50 (+ 0.75) seconds.
% 33.46/4.68 % Length of proof is 45.
% 33.46/4.68 % Level of proof is 11.
% 33.46/4.68 % Maximum clause weight is 18.000.
% 33.46/4.68 % Given clauses 1494.
% 33.46/4.68
% 33.46/4.68 % SZS output start CNFRefutation for theBenchmark
% See solution above
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