%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET792+4 : TPTP v9.3.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n017.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 08:36:58 AM UTC 2026
% Result : Theorem 6.00s 1.95s
% Output : CNFRefutation 6.00s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 4
% Syntax : Number of formulae : 36 ( 10 unt; 0 def)
% Number of atoms : 129 ( 18 equ)
% Maximal formula atoms : 15 ( 3 avg)
% Number of connectives : 144 ( 51 ~; 53 |; 28 &)
% ( 3 <=>; 9 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 4 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 7 ( 5 usr; 1 prp; 0-3 aty)
% Number of functors : 4 ( 4 usr; 3 con; 0-3 aty)
% Number of variables : 79 ( 1 sgn 46 !; 4 ?)
% Comments :
%------------------------------------------------------------------------------
fof(order,axiom,
! [R,E] :
( order(R,E)
<=> ( ! [X,Y,Z] :
( ( member(Z,E)
& member(Y,E)
& member(X,E) )
=> ( ( apply(R,Y,Z)
& apply(R,X,Y) )
=> apply(R,X,Z) ) )
& ! [X,Y] :
( ( member(Y,E)
& member(X,E) )
=> ( ( apply(R,Y,X)
& apply(R,X,Y) )
=> X = Y ) )
& ! [X] :
( member(X,E)
=> apply(R,X,X) ) ) ),
file('theBenchmark.p',order) ).
fof(least,axiom,
! [R,E,M] :
( least(M,R,E)
<=> ( ! [X] :
( member(X,E)
=> apply(R,M,X) )
& member(M,E) ) ),
file('theBenchmark.p',least) ).
fof(min,axiom,
! [R,E,M] :
( min(M,R,E)
<=> ( ! [X] :
( ( apply(R,X,M)
& member(X,E) )
=> M = X )
& member(M,E) ) ),
file('theBenchmark.p',min) ).
fof(thIV4,conjecture,
! [R,E,M] :
( ( least(M,R,E)
& order(R,E) )
=> min(M,R,E) ),
file('theBenchmark.p',thIV4) ).
fof(thIV4_neg,negated_conjecture,
~ ! [R,E,M] :
( ( least(M,R,E)
& order(R,E) )
=> min(M,R,E) ),
inference(assume_negation,[status(cth)],[thIV4]) ).
cnf(c_14,plain,
( D = C
| ~ apply(A,D,C)
| ~ apply(A,C,D)
| ~ member(D,B)
| ~ member(C,B)
| ~ order(A,B) ),
inference(clausify,[status(thm)],[order]) ).
fof(nnf_11,plain,
? [R,E,M] :
( ~ min(M,R,E)
& least(M,R,E)
& order(R,E) ),
inference(fof_nnf,[status(thm)],[thIV4_neg]) ).
fof(sk_thIV4_neg,plain,
( ~ min(sK2,sK0,sK1)
& least(sK2,sK0,sK1)
& order(sK0,sK1) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK1,sK2]),skolemize(R,sK0),skolemize(E,sK1),skolemize(M,sK2)],[nnf_11]) ).
cnf(c_80,negated_conjecture,
order(sK0,sK1),
inference(split_conjunct,[status(thm)],[sk_thIV4_neg]) ).
cnf(c_332,plain,
( apply(B,A,D)
| ~ member(D,C)
| ~ least(A,B,C) ),
inference(clausify,[status(thm)],[least]) ).
cnf(c_335,negated_conjecture,
least(sK2,sK0,sK1),
inference(split_conjunct,[status(thm)],[sk_thIV4_neg]) ).
cnf(c_336,plain,
( member(A,C)
| ~ least(A,B,C) ),
inference(clausify,[status(thm)],[least]) ).
cnf(c_371,negated_conjecture,
( B = A
| ~ apply(sK0,B,A)
| ~ apply(sK0,A,B)
| ~ member(B,sK1)
| ~ member(A,sK1) ),
inference(resolve,[status(thm)],[c_80,c_14]) ).
cnf(c_553,negated_conjecture,
~ min(sK2,sK0,sK1),
inference(split_conjunct,[status(thm)],[sk_thIV4_neg]) ).
fof(nnf_8,plain,
! [R,E,M] :
( ( ? [X] :
( M != X
& apply(R,X,M)
& member(X,E) )
| ~ member(M,E)
| min(M,R,E) )
& ( ( ! [X7] :
( M = X7
| ~ apply(R,X7,M)
| ~ member(X7,E) )
& member(M,E) )
| ~ min(M,R,E) ) ),
inference(fof_nnf,[status(thm)],[min]) ).
fof(sk_min_sk,plain,
! [R,E,M] :
( ( ( M != sK3(R,E,M)
& apply(R,sK3(R,E,M),M)
& member(sK3(R,E,M),E) )
| ~ member(M,E)
| min(M,R,E) )
& ( ( ! [X7] :
( M = X7
| ~ apply(R,X7,M)
| ~ member(X7,E) )
& member(M,E) )
| ~ min(M,R,E) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK3]),skolemize(X,sK3(R,E,M))],[nnf_8]) ).
fof(sk_min,plain,
( ! [VAR_0,VAR_1,VAR_2] :
( VAR_0 != sK3(VAR_1,VAR_2,VAR_0)
| ~ member(VAR_0,VAR_2)
| min(VAR_0,VAR_1,VAR_2) )
& ! [VAR_0,VAR_1,VAR_2] :
( apply(VAR_1,sK3(VAR_1,VAR_2,VAR_0),VAR_0)
| ~ member(VAR_0,VAR_2)
| min(VAR_0,VAR_1,VAR_2) )
& ! [VAR_0,VAR_1,VAR_2] :
( member(sK3(VAR_1,VAR_2,VAR_0),VAR_2)
| ~ member(VAR_0,VAR_2)
| min(VAR_0,VAR_1,VAR_2) )
& ! [VAR_0,VAR_1,VAR_2,VAR_3] :
( VAR_0 = VAR_3
| ~ apply(VAR_1,VAR_3,VAR_0)
| ~ member(VAR_3,VAR_2)
| ~ min(VAR_0,VAR_1,VAR_2) )
& ! [VAR_0,VAR_1,VAR_2] :
( member(VAR_0,VAR_2)
| ~ min(VAR_0,VAR_1,VAR_2) ) ),
inference(cnf_transformation,[status(thm)],[sk_min_sk]) ).
cnf(c_554,plain,
( A != sK3(B,C,A)
| ~ member(A,C)
| min(A,B,C) ),
inference(split_conjunct,[status(thm)],[sk_min]) ).
cnf(c_555,plain,
( sK3(B,C,A) != A
| ~ member(A,C)
| min(A,B,C) ),
inference(copy,[status(thm)],[c_554]) ).
cnf(c_556,plain,
( apply(B,sK3(B,C,A),A)
| ~ member(A,C)
| min(A,B,C) ),
inference(split_conjunct,[status(thm)],[sk_min]) ).
cnf(c_557,plain,
( member(sK3(B,C,A),C)
| ~ member(A,C)
| min(A,B,C) ),
inference(split_conjunct,[status(thm)],[sk_min]) ).
cnf(c_806,negated_conjecture,
( apply(sK0,sK2,A)
| ~ member(A,sK1) ),
inference(resolve,[status(thm)],[c_332,c_335]) ).
cnf(c_807,negated_conjecture,
member(sK2,sK1),
inference(resolve,[status(thm)],[c_336,c_335]) ).
cnf(c_1067,plain,
( member(sK3(A,sK1,sK2),sK1)
| min(sK2,A,sK1) ),
inference(resolve,[status(thm)],[c_807,c_557]) ).
cnf(c_1068,plain,
( apply(A,sK3(A,sK1,sK2),sK2)
| min(sK2,A,sK1) ),
inference(resolve,[status(thm)],[c_807,c_556]) ).
cnf(c_1069,plain,
( sK3(A,sK1,sK2) != sK2
| min(sK2,A,sK1) ),
inference(resolve,[status(thm)],[c_807,c_555]) ).
cnf(c_1102,plain,
member(sK3(sK0,sK1,sK2),sK1),
inference(resolve,[status(thm)],[c_1067,c_553]) ).
cnf(c_1105,plain,
apply(sK0,sK2,sK3(sK0,sK1,sK2)),
inference(resolve,[status(thm)],[c_1102,c_806]) ).
cnf(c_1867,negated_conjecture,
( sK2 = sK3(sK0,sK1,sK2)
| ~ apply(sK0,sK3(sK0,sK1,sK2),sK2)
| ~ member(sK2,sK1)
| ~ member(sK3(sK0,sK1,sK2),sK1) ),
inference(resolve,[status(thm)],[c_1105,c_371]) ).
cnf(c_1868,negated_conjecture,
( sK3(sK0,sK1,sK2) = sK2
| ~ apply(sK0,sK3(sK0,sK1,sK2),sK2)
| ~ member(sK2,sK1)
| ~ member(sK3(sK0,sK1,sK2),sK1) ),
inference(copy,[status(thm)],[c_1867]) ).
cnf(c_1869,negated_conjecture,
( sK3(sK0,sK1,sK2) = sK2
| ~ apply(sK0,sK3(sK0,sK1,sK2),sK2)
| ~ member(sK2,sK1) ),
inference(resolve,[status(thm)],[c_1102,c_1868]) ).
cnf(c_1374,negated_conjecture,
( sK3(sK0,sK1,sK2) = sK2
| ~ apply(sK0,sK3(sK0,sK1,sK2),sK2) ),
inference(resolve,[status(thm)],[c_807,c_1869]) ).
cnf(c_1864,plain,
apply(sK0,sK3(sK0,sK1,sK2),sK2),
inference(resolve,[status(thm)],[c_1068,c_553]) ).
cnf(c_1865,negated_conjecture,
sK3(sK0,sK1,sK2) = sK2,
inference(resolve,[status(thm)],[c_1864,c_1374]) ).
cnf(c_1870,plain,
min(sK2,sK0,sK1),
inference(resolve,[status(thm)],[c_1865,c_1069]) ).
cnf(c_1866,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_553,c_1870]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET792+4 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.08/0.35 % Computer : n017.cluster.edu
% 0.08/0.35 % Model : x86_64 x86_64
% 0.08/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.35 % Memory : 8046.5625MB
% 0.08/0.35 % OS : Linux 6.8.0-71-generic
% 0.08/0.35 % CPULimit : 300
% 0.08/0.35 % WCLimit : 300
% 0.08/0.35 % DateTime : Sat Sep 26 08:11:48 UTC 2026
% 0.08/0.35 % CPUTime :
% 0.08/0.35 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.08/0.35 % Process 241283 was started by sandbox on n017,
% 0.08/0.35 % Sat Sep 26 08:11:48 2026
% 0.08/0.35 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.08/0.37
% 0.08/0.37 % From the command line: assign(max_seconds, 300).
% 6.00/1.95
% 6.00/1.95 % given #500 (I,wt=43): 861 member(f1(A,B),B) | apply(A,f3(A,B),f2(A,B)) | member(f5(A,B),B) | -member(f3(A,B),B) | -member(C,B) | -member(f2(A,B),B) | -apply(A,f3(A,B),C) | -apply(A,C,f2(A,B)). [factor(495,b,i)].
% 6.00/1.95
% 6.00/1.95 % SZS status Theorem for theBenchmark
% 6.00/1.95
% 6.00/1.95 % Proof 1 at 0.71 (+ 0.04) seconds.
% 6.00/1.95 % Length of proof is 26.
% 6.00/1.95 % Level of proof is 8.
% 6.00/1.95 % Maximum clause weight is 17.000.
% 6.00/1.95 % Given clauses 666.
% 6.00/1.95
% 6.00/1.95 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------