%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET794+4 : TPTP v9.3.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n018.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 12.53s 2.11s
% Output : CNFRefutation 12.53s
% Verified :
% SZS Type : Refutation
% Derivation depth : 18
% Number of leaves : 4
% Syntax : Number of formulae : 37 ( 9 unt; 0 def)
% Number of atoms : 114 ( 9 equ)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 120 ( 43 ~; 49 |; 20 &)
% ( 3 <=>; 5 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 8 ( 6 usr; 1 prp; 0-3 aty)
% Number of functors : 4 ( 4 usr; 3 con; 0-3 aty)
% Number of variables : 65 ( 1 sgn 39 !; 4 ?)
% Comments :
%------------------------------------------------------------------------------
fof(total_order,axiom,
! [R,E] :
( total_order(R,E)
<=> ( ! [X,Y] :
( ( member(Y,E)
& member(X,E) )
=> ( apply(R,Y,X)
| apply(R,X,Y) ) )
& order(R,E) ) ),
file('theBenchmark.p',total_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(thIV6,conjecture,
! [R,E,M] :
( ( min(M,R,E)
& total_order(R,E) )
=> least(M,R,E) ),
file('theBenchmark.p',thIV6) ).
fof(thIV6_neg,negated_conjecture,
~ ! [R,E,M] :
( ( min(M,R,E)
& total_order(R,E) )
=> least(M,R,E) ),
inference(assume_negation,[status(cth)],[thIV6]) ).
fof(nnf_11,plain,
? [R,E,M] :
( ~ least(M,R,E)
& min(M,R,E)
& total_order(R,E) ),
inference(fof_nnf,[status(thm)],[thIV6_neg]) ).
fof(sk_thIV6_neg,plain,
( ~ least(sK2,sK0,sK1)
& min(sK2,sK0,sK1)
& total_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_322,negated_conjecture,
total_order(sK0,sK1),
inference(split_conjunct,[status(thm)],[sk_thIV6_neg]) ).
cnf(c_323,plain,
( apply(A,D,C)
| apply(A,C,D)
| ~ member(D,B)
| ~ member(C,B)
| ~ total_order(A,B) ),
inference(clausify,[status(thm)],[total_order]) ).
cnf(c_334,plain,
( A = D
| ~ apply(B,D,A)
| ~ member(D,C)
| ~ min(A,B,C) ),
inference(clausify,[status(thm)],[min]) ).
cnf(c_335,plain,
( member(A,C)
| ~ min(A,B,C) ),
inference(clausify,[status(thm)],[min]) ).
cnf(c_338,negated_conjecture,
min(sK2,sK0,sK1),
inference(split_conjunct,[status(thm)],[sk_thIV6_neg]) ).
cnf(c_361,negated_conjecture,
member(sK2,sK1),
inference(resolve,[status(thm)],[c_338,c_335]) ).
cnf(c_362,negated_conjecture,
( sK2 = A
| ~ apply(sK0,A,sK2)
| ~ member(A,sK1) ),
inference(resolve,[status(thm)],[c_338,c_334]) ).
cnf(c_549,negated_conjecture,
~ least(sK2,sK0,sK1),
inference(split_conjunct,[status(thm)],[sk_thIV6_neg]) ).
fof(nnf_6,plain,
! [R,E,M] :
( ( ? [X] :
( ~ apply(R,M,X)
& member(X,E) )
| ~ member(M,E)
| least(M,R,E) )
& ( ( ! [X0] :
( apply(R,M,X0)
| ~ member(X0,E) )
& member(M,E) )
| ~ least(M,R,E) ) ),
inference(fof_nnf,[status(thm)],[least]) ).
fof(sk_least_sk,plain,
! [R,E,M] :
( ( ( ~ apply(R,M,sK3(R,E,M))
& member(sK3(R,E,M),E) )
| ~ member(M,E)
| least(M,R,E) )
& ( ( ! [X0] :
( apply(R,M,X0)
| ~ member(X0,E) )
& member(M,E) )
| ~ least(M,R,E) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK3]),skolemize(X,sK3(R,E,M))],[nnf_6]) ).
fof(sk_least,plain,
( ! [VAR_0,VAR_1,VAR_2] :
( ~ apply(VAR_1,VAR_0,sK3(VAR_1,VAR_2,VAR_0))
| ~ member(VAR_0,VAR_2)
| least(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)
| least(VAR_0,VAR_1,VAR_2) )
& ! [VAR_0,VAR_1,VAR_2,VAR_3] :
( apply(VAR_1,VAR_0,VAR_3)
| ~ member(VAR_3,VAR_2)
| ~ least(VAR_0,VAR_1,VAR_2) )
& ! [VAR_0,VAR_1,VAR_2] :
( member(VAR_0,VAR_2)
| ~ least(VAR_0,VAR_1,VAR_2) ) ),
inference(cnf_transformation,[status(thm)],[sk_least_sk]) ).
cnf(c_550,plain,
( ~ apply(B,A,sK3(B,C,A))
| ~ member(A,C)
| least(A,B,C) ),
inference(split_conjunct,[status(thm)],[sk_least]) ).
cnf(c_551,plain,
( member(sK3(B,C,A),C)
| ~ member(A,C)
| least(A,B,C) ),
inference(split_conjunct,[status(thm)],[sk_least]) ).
cnf(c_556,negated_conjecture,
( apply(sK0,B,A)
| apply(sK0,A,B)
| ~ member(B,sK1)
| ~ member(A,sK1) ),
inference(resolve,[status(thm)],[c_323,c_322]) ).
cnf(c_1036,plain,
( member(sK3(A,sK1,sK2),sK1)
| least(sK2,A,sK1) ),
inference(resolve,[status(thm)],[c_551,c_361]) ).
cnf(c_1037,plain,
( apply(sK0,A,sK2)
| apply(sK0,sK2,A)
| ~ member(A,sK1) ),
inference(resolve,[status(thm)],[c_556,c_361]) ).
cnf(c_3978,plain,
( apply(sK0,sK2,sK2)
| ~ member(sK2,sK1) ),
inference(factor,[status(thm)],[c_1037]) ).
cnf(c_1038,negated_conjecture,
apply(sK0,sK2,sK2),
inference(resolve,[status(thm)],[c_361,c_3978]) ).
cnf(c_1580,plain,
member(sK3(sK0,sK1,sK2),sK1),
inference(resolve,[status(thm)],[c_1036,c_549]) ).
cnf(c_1665,plain,
( apply(sK0,sK3(sK0,sK1,sK2),sK2)
| apply(sK0,sK2,sK3(sK0,sK1,sK2)) ),
inference(resolve,[status(thm)],[c_1037,c_1580]) ).
cnf(c_3979,negated_conjecture,
( sK2 = sK3(sK0,sK1,sK2)
| ~ member(sK3(sK0,sK1,sK2),sK1)
| apply(sK0,sK2,sK3(sK0,sK1,sK2)) ),
inference(resolve,[status(thm)],[c_1665,c_362]) ).
cnf(c_3980,negated_conjecture,
( sK3(sK0,sK1,sK2) = sK2
| ~ member(sK3(sK0,sK1,sK2),sK1)
| apply(sK0,sK2,sK3(sK0,sK1,sK2)) ),
inference(copy,[status(thm)],[c_3979]) ).
cnf(c_3969,negated_conjecture,
( sK3(sK0,sK1,sK2) = sK2
| apply(sK0,sK2,sK3(sK0,sK1,sK2)) ),
inference(resolve,[status(thm)],[c_1580,c_3980]) ).
cnf(c_3981,plain,
( ~ member(sK2,sK1)
| least(sK2,sK0,sK1)
| sK3(sK0,sK1,sK2) = sK2 ),
inference(resolve,[status(thm)],[c_3969,c_550]) ).
cnf(c_3982,negated_conjecture,
( ~ member(sK2,sK1)
| sK3(sK0,sK1,sK2) = sK2 ),
inference(resolve,[status(thm)],[c_549,c_3981]) ).
cnf(c_3976,negated_conjecture,
sK3(sK0,sK1,sK2) = sK2,
inference(resolve,[status(thm)],[c_361,c_3982]) ).
cnf(c_3983,plain,
( ~ apply(sK0,sK2,sK2)
| ~ member(sK2,sK1)
| least(sK2,sK0,sK1) ),
inference(paramod,[status(thm)],[c_3976,c_550]) ).
cnf(c_3984,negated_conjecture,
( ~ apply(sK0,sK2,sK2)
| ~ member(sK2,sK1) ),
inference(resolve,[status(thm)],[c_549,c_3983]) ).
cnf(c_3985,negated_conjecture,
~ apply(sK0,sK2,sK2),
inference(resolve,[status(thm)],[c_361,c_3984]) ).
cnf(c_3977,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_1038,c_3985]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET794+4 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.34 % Computer : n018.cluster.edu
% 0.09/0.34 % Model : x86_64 x86_64
% 0.09/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.34 % Memory : 8046.5625MB
% 0.09/0.34 % OS : Linux 6.8.0-71-generic
% 0.09/0.34 % CPULimit : 300
% 0.09/0.34 % WCLimit : 300
% 0.09/0.34 % DateTime : Sat Sep 26 08:18:51 UTC 2026
% 0.09/0.35 % CPUTime :
% 0.09/0.35 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.09/0.35 % Process 4168582 was started by sandbox2 on n018,
% 0.09/0.35 % Sat Sep 26 08:18:51 2026
% 0.09/0.35 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.14/0.36
% 0.14/0.36 % From the command line: assign(max_seconds, 300).
% 12.53/2.11
% 12.53/2.11 % given #500 (I,wt=46): 858 member(f1(A,B),B) | apply(A,f2(A,B),f3(A,B)) | -apply(A,f4(A,B),f6(A,B)) | -member(f4(A,B),B) | -member(f6(A,B),B) | -member(C,B) | -apply(A,f6(A,B),C) | apply(A,f4(A,B),C). [factor(497,c,g)].
% 12.53/2.11
% 12.53/2.11 % SZS status Theorem for theBenchmark
% 12.53/2.11
% 12.53/2.11 % Proof 1 at 1.48 (+ 0.17) seconds.
% 12.53/2.11 % Length of proof is 23.
% 12.53/2.11 % Level of proof is 8.
% 12.53/2.11 % Maximum clause weight is 14.000.
% 12.53/2.11 % Given clauses 681.
% 12.53/2.11
% 12.53/2.11 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------