%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET810+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 : 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 08:37:00 AM UTC 2026
% Result : Theorem 3.49s 0.97s
% Output : CNFRefutation 3.49s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 6
% Syntax : Number of formulae : 31 ( 10 unt; 0 def)
% Number of atoms : 87 ( 0 equ)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 89 ( 33 ~; 23 |; 20 &)
% ( 5 <=>; 8 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 1 ( 1 avg)
% Number of predicates : 8 ( 7 usr; 1 prp; 0-3 aty)
% Number of functors : 4 ( 4 usr; 4 con; 0-0 aty)
% Number of variables : 46 ( 0 sgn 21 !; 4 ?)
% Comments :
%------------------------------------------------------------------------------
fof(subset,axiom,
! [A,B] :
( subset(A,B)
<=> ! [X] :
( member(X,A)
=> member(X,B) ) ),
file('theBenchmark.p',subset) ).
fof(ordinal_number,axiom,
! [A] :
( member(A,on)
<=> ( ! [X] :
( member(X,A)
=> subset(X,A) )
& strict_well_order(member_predicate,A)
& set(A) ) ),
file('theBenchmark.p',ordinal_number) ).
fof(strict_well_order,axiom,
! [R,E] :
( strict_well_order(R,E)
<=> ( ! [A] :
( ( ? [X] : member(X,A)
& subset(A,E) )
=> ? [Y] : least(Y,R,A) )
& strict_order(R,E) ) ),
file('theBenchmark.p',strict_well_order) ).
fof(rel_member,axiom,
! [X,Y] :
( apply(member_predicate,X,Y)
<=> member(X,Y) ),
file('theBenchmark.p',rel_member) ).
fof(strict_order,axiom,
! [R,E] :
( strict_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) ) ) ) ),
file('theBenchmark.p',strict_order) ).
fof(thV3,conjecture,
! [A,B] :
( ( member(B,on)
& member(A,on) )
=> ~ ( member(B,A)
& member(A,B) ) ),
file('theBenchmark.p',thV3) ).
fof(thV3_neg,negated_conjecture,
~ ! [A,B] :
( ( member(B,on)
& member(A,on) )
=> ~ ( member(B,A)
& member(A,B) ) ),
inference(assume_negation,[status(cth)],[thV3]) ).
cnf(c_49,plain,
( ~ apply(A,D,C)
| ~ apply(A,C,D)
| ~ member(D,B)
| ~ member(C,B)
| ~ strict_order(A,B) ),
inference(clausify,[status(thm)],[strict_order]) ).
cnf(c_53,plain,
( strict_order(A,B)
| ~ strict_well_order(A,B) ),
inference(clausify,[status(thm)],[strict_well_order]) ).
cnf(c_55,plain,
( ~ apply(A,D,C)
| ~ apply(A,C,D)
| ~ member(D,B)
| ~ member(C,B)
| ~ strict_well_order(A,B) ),
inference(resolve,[status(thm)],[c_53,c_49]) ).
cnf(c_176,plain,
( member(C,B)
| ~ member(C,A)
| ~ subset(A,B) ),
inference(clausify,[status(thm)],[subset]) ).
cnf(c_203,plain,
( strict_well_order(member_predicate,A)
| ~ member(A,on) ),
inference(clausify,[status(thm)],[ordinal_number]) ).
cnf(c_204,plain,
( subset(B,A)
| ~ member(B,A)
| ~ member(A,on) ),
inference(clausify,[status(thm)],[ordinal_number]) ).
cnf(c_215,plain,
( ~ member(A,B)
| apply(member_predicate,A,B) ),
inference(clausify,[status(thm)],[rel_member]) ).
fof(nnf_20,plain,
? [A,B] :
( member(B,A)
& member(A,B)
& member(B,on)
& member(A,on) ),
inference(fof_nnf,[status(thm)],[thV3_neg]) ).
fof(sk_thV3_neg,plain,
( member(sK0,sK1)
& member(sK1,sK0)
& member(sK0,on)
& member(sK1,on) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1,sK0]),skolemize(A,sK1),skolemize(B,sK0)],[nnf_20]) ).
cnf(c_223,negated_conjecture,
member(sK0,on),
inference(split_conjunct,[status(thm)],[sk_thV3_neg]) ).
cnf(c_224,negated_conjecture,
member(sK1,sK0),
inference(split_conjunct,[status(thm)],[sk_thV3_neg]) ).
cnf(c_225,negated_conjecture,
member(sK0,sK1),
inference(split_conjunct,[status(thm)],[sk_thV3_neg]) ).
cnf(c_2543,plain,
strict_well_order(member_predicate,sK0),
inference(resolve,[status(thm)],[c_223,c_203]) ).
cnf(c_2553,plain,
apply(member_predicate,sK1,sK0),
inference(resolve,[status(thm)],[c_224,c_215]) ).
cnf(c_2840,negated_conjecture,
( subset(sK1,sK0)
| ~ member(sK0,on) ),
inference(resolve,[status(thm)],[c_224,c_204]) ).
cnf(c_2556,negated_conjecture,
subset(sK1,sK0),
inference(resolve,[status(thm)],[c_223,c_2840]) ).
cnf(c_2565,plain,
apply(member_predicate,sK0,sK1),
inference(resolve,[status(thm)],[c_225,c_215]) ).
cnf(c_2592,plain,
( member(A,sK0)
| ~ member(A,sK1) ),
inference(resolve,[status(thm)],[c_2556,c_176]) ).
cnf(c_2841,plain,
( ~ apply(member_predicate,sK0,sK1)
| ~ member(sK1,A)
| ~ member(sK0,A)
| ~ strict_well_order(member_predicate,A) ),
inference(resolve,[status(thm)],[c_2553,c_55]) ).
cnf(c_2821,plain,
( ~ member(sK1,A)
| ~ member(sK0,A)
| ~ strict_well_order(member_predicate,A) ),
inference(resolve,[status(thm)],[c_2565,c_2841]) ).
cnf(c_2842,plain,
( ~ member(sK1,sK0)
| ~ member(sK0,sK0) ),
inference(resolve,[status(thm)],[c_2821,c_2543]) ).
cnf(c_2836,negated_conjecture,
~ member(sK0,sK0),
inference(resolve,[status(thm)],[c_224,c_2842]) ).
cnf(c_2843,negated_conjecture,
member(sK0,sK0),
inference(resolve,[status(thm)],[c_2592,c_225]) ).
cnf(c_2839,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_2836,c_2843]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : SET810+4 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.07 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.17/0.45 % Computer : n018.cluster.edu
% 0.17/0.45 % Model : x86_64 x86_64
% 0.17/0.45 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.45 % Memory : 8046.5625MB
% 0.17/0.45 % OS : Linux 6.8.0-71-generic
% 0.17/0.45 % CPULimit : 300
% 0.17/0.45 % WCLimit : 300
% 0.17/0.45 % DateTime : Sat Sep 26 08:19:38 UTC 2026
% 0.17/0.45 % CPUTime :
% 0.17/0.45 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.17/0.45 % Process 4169418 was started by sandbox on n018,
% 0.17/0.45 % Sat Sep 26 08:19:38 2026
% 0.17/0.45 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.17/0.47
% 0.17/0.47 % From the command line: assign(max_seconds, 300).
% 3.49/0.97
% 3.49/0.97 % SZS status Theorem for theBenchmark
% 3.49/0.97
% 3.49/0.97 % Proof 1 at 0.41 (+ 0.08) seconds.
% 3.49/0.97 % Length of proof is 24.
% 3.49/0.97 % Level of proof is 5.
% 3.49/0.97 % Maximum clause weight is 22.000.
% 3.49/0.97 % Given clauses 261.
% 3.49/0.97
% 3.49/0.97 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------