%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : COL006-2 : TPTP v9.3.1. Released v1.0.0.
% 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 06:57:22 AM UTC 2026
% Result : Unsatisfiable 1.26s 0.62s
% Output : CNFRefutation 1.26s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 4
% Syntax : Number of formulae : 33 ( 30 unt; 0 def)
% Number of atoms : 36 ( 30 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 26 ( 23 ~; 3 |; 0 &)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 4 ( 2 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of predicates : 3 ( 1 usr; 1 prp; 0-2 aty)
% Number of functors : 4 ( 4 usr; 3 con; 0-2 aty)
% Number of variables : 30 ( 2 sgn 6 !; 0 ?)
% Comments :
%------------------------------------------------------------------------------
fof(s_definition,axiom,
! [VAR_0,VAR_1,VAR_2] : apply(apply(apply(s,VAR_0),VAR_1),VAR_2) = apply(apply(VAR_0,VAR_2),apply(VAR_1,VAR_2)),
file('theBenchmark.p',s_definition) ).
cnf(c_1,plain,
apply(apply(apply(s,A),B),C) = apply(apply(A,C),apply(B,C)),
inference(clausify,[status(thm)],[s_definition]) ).
fof(k_definition,axiom,
! [VAR_0,VAR_1] : apply(apply(k,VAR_0),VAR_1) = VAR_0,
file('theBenchmark.p',k_definition) ).
cnf(c_2,plain,
apply(apply(k,A),B) = A,
inference(clausify,[status(thm)],[k_definition]) ).
fof(strong_fixed_point,axiom,
! [VAR_0] :
( fixed_point(VAR_0)
| apply(VAR_0,fixed_pt) != apply(fixed_pt,apply(VAR_0,fixed_pt)) ),
file('theBenchmark.p',strong_fixed_point) ).
cnf(c_3,plain,
( fixed_point(A)
| apply(A,fixed_pt) != apply(fixed_pt,apply(A,fixed_pt)) ),
inference(clausify,[status(thm)],[strong_fixed_point]) ).
cnf(c_4,plain,
( fixed_point(A)
| apply(fixed_pt,apply(A,fixed_pt)) != apply(A,fixed_pt) ),
inference(copy,[status(thm)],[c_3]) ).
fof(prove_strong_fixed_point,axiom,
~ fixed_point(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k)))),
file('theBenchmark.p',prove_strong_fixed_point) ).
cnf(c_5,plain,
~ fixed_point(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k)))),
inference(clausify,[status(thm)],[prove_strong_fixed_point]) ).
cnf(c_7,plain,
apply(apply(apply(s,A),apply(apply(s,B),C)),D) = apply(apply(A,D),apply(apply(B,D),apply(C,D))),
inference(paramod,[status(thm)],[c_1,c_1]) ).
cnf(c_12,plain,
apply(apply(apply(s,apply(k,A)),B),C) = apply(A,apply(B,C)),
inference(paramod,[status(thm)],[c_2,c_1]) ).
cnf(c_14,plain,
apply(apply(apply(s,k),A),B) = B,
inference(paramod,[status(thm)],[c_2,c_1]) ).
cnf(c_277,plain,
apply(fixed_pt,apply(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k))),fixed_pt)) != apply(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k))),fixed_pt),
inference(resolve,[status(thm)],[c_4,c_5]) ).
cnf(c_278,plain,
apply(fixed_pt,apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k)),fixed_pt))) != apply(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k))),fixed_pt),
inference(paramod,[status(thm)],[c_12,c_277]) ).
cnf(c_279,plain,
apply(fixed_pt,apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(s,s),apply(s,k)),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) != apply(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k))),fixed_pt),
inference(paramod,[status(thm)],[c_12,c_278]) ).
cnf(c_280,plain,
apply(fixed_pt,apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt))))) != apply(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k))),fixed_pt),
inference(paramod,[status(thm)],[c_12,c_279]) ).
cnf(c_281,plain,
apply(fixed_pt,apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt))))) != apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k)),fixed_pt)),
inference(paramod,[status(thm)],[c_12,c_280]) ).
cnf(c_282,plain,
apply(fixed_pt,apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt))))) != apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(s,s),apply(s,k)),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),
inference(paramod,[status(thm)],[c_12,c_281]) ).
cnf(c_18,plain,
apply(fixed_pt,apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt))))) != apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt)))),
inference(paramod,[status(thm)],[c_12,c_282]) ).
cnf(c_283,plain,
apply(fixed_pt,apply(apply(apply(apply(s,k),k),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt)))),apply(apply(apply(s,k),k),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt)))))) != apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt)))),
inference(paramod,[status(thm)],[c_1,c_18]) ).
cnf(c_284,plain,
apply(fixed_pt,apply(apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt))),apply(apply(apply(s,k),k),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt)))))) != apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt)))),
inference(paramod,[status(thm)],[c_14,c_283]) ).
cnf(c_50,plain,
apply(fixed_pt,apply(apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt))),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt))))) != apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt)))),
inference(paramod,[status(thm)],[c_14,c_284]) ).
cnf(c_57,plain,
apply(fixed_pt,apply(apply(apply(s,apply(apply(s,s),apply(s,k))),apply(apply(s,s),apply(s,k))),apply(s,apply(k,fixed_pt)))) != apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt)))),
inference(paramod,[status(thm)],[c_1,c_50]) ).
cnf(c_285,plain,
apply(fixed_pt,apply(apply(apply(s,apply(apply(s,s),apply(s,k))),apply(apply(s,s),apply(s,k))),apply(s,apply(k,fixed_pt)))) != apply(apply(apply(apply(s,k),k),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt)))),apply(apply(apply(s,k),k),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt))))),
inference(paramod,[status(thm)],[c_1,c_57]) ).
cnf(c_286,plain,
apply(fixed_pt,apply(apply(apply(s,apply(apply(s,s),apply(s,k))),apply(apply(s,s),apply(s,k))),apply(s,apply(k,fixed_pt)))) != apply(apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt))),apply(apply(apply(s,k),k),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt))))),
inference(paramod,[status(thm)],[c_14,c_285]) ).
cnf(c_59,plain,
apply(fixed_pt,apply(apply(apply(s,apply(apply(s,s),apply(s,k))),apply(apply(s,s),apply(s,k))),apply(s,apply(k,fixed_pt)))) != apply(apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt))),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt)))),
inference(paramod,[status(thm)],[c_14,c_286]) ).
cnf(c_60,plain,
apply(fixed_pt,apply(apply(apply(s,apply(apply(s,s),apply(s,k))),apply(apply(s,s),apply(s,k))),apply(s,apply(k,fixed_pt)))) != apply(apply(apply(s,apply(s,apply(k,fixed_pt))),apply(apply(s,k),apply(s,apply(k,fixed_pt)))),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt)))),
inference(paramod,[status(thm)],[c_1,c_59]) ).
cnf(c_287,plain,
apply(apply(apply(s,apply(k,A)),apply(apply(s,B),C)),D) = apply(A,apply(apply(B,D),apply(C,D))),
inference(paramod,[status(thm)],[c_2,c_7]) ).
cnf(c_94,plain,
apply(A,apply(apply(apply(s,B),C),D)) = apply(A,apply(apply(B,D),apply(C,D))),
inference(paramod,[status(thm)],[c_12,c_287]) ).
cnf(c_288,plain,
apply(fixed_pt,apply(apply(apply(s,apply(apply(s,s),apply(s,k))),apply(apply(s,s),apply(s,k))),apply(s,apply(k,fixed_pt)))) != apply(apply(apply(s,apply(k,fixed_pt)),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt)))),apply(apply(apply(s,k),apply(s,apply(k,fixed_pt))),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt))))),
inference(paramod,[status(thm)],[c_1,c_60]) ).
cnf(c_289,plain,
apply(fixed_pt,apply(apply(apply(s,apply(apply(s,s),apply(s,k))),apply(apply(s,s),apply(s,k))),apply(s,apply(k,fixed_pt)))) != apply(apply(apply(s,apply(k,fixed_pt)),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt)))),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt)))),
inference(paramod,[status(thm)],[c_14,c_288]) ).
cnf(c_275,plain,
apply(fixed_pt,apply(apply(apply(s,apply(apply(s,s),apply(s,k))),apply(apply(s,s),apply(s,k))),apply(s,apply(k,fixed_pt)))) != apply(fixed_pt,apply(apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt))),apply(apply(apply(s,s),apply(s,k)),apply(s,apply(k,fixed_pt))))),
inference(paramod,[status(thm)],[c_12,c_289]) ).
cnf(c_276,plain,
$false,
inference(resolve,[status(thm)],[c_275,c_94]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : COL006-2 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.06 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.17/0.43 % Computer : n018.cluster.edu
% 0.17/0.43 % Model : x86_64 x86_64
% 0.17/0.43 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.43 % Memory : 8046.5625MB
% 0.17/0.43 % OS : Linux 6.8.0-71-generic
% 0.17/0.43 % CPULimit : 300
% 0.17/0.43 % WCLimit : 300
% 0.17/0.43 % DateTime : Sat Sep 26 23:06:52 UTC 2026
% 0.17/0.44 % CPUTime :
% 0.17/0.44 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.17/0.44 % Process 1660898 was started by sandbox2 on n018,
% 0.17/0.44 % Sat Sep 26 23:06:52 2026
% 0.17/0.44 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.17/0.44
% 0.17/0.44 % From the command line: assign(max_seconds, 300).
% 1.26/0.62
% 1.26/0.62 % SZS status Unsatisfiable for theBenchmark
% 1.26/0.62
% 1.26/0.62 % Proof 1 at 0.07 (+ 0.01) seconds.
% 1.26/0.62 % Length of proof is 16.
% 1.26/0.62 % Level of proof is 8.
% 1.26/0.62 % Maximum clause weight is 57.000.
% 1.26/0.62 % Given clauses 26.
% 1.26/0.62
% 1.26/0.62 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------