%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : COL078-2 : TPTP v9.3.1. Released v1.2.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n008.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 06:57:38 AM UTC 2026
% Result : Unsatisfiable 0.09s 0.41s
% Output : CNFRefutation 0.09s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 5
% Syntax : Number of formulae : 42 ( 35 unt; 0 def)
% Number of atoms : 49 ( 48 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 31 ( 24 ~; 7 |; 0 &)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 5 ( 2 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of predicates : 2 ( 0 usr; 1 prp; 0-2 aty)
% Number of functors : 5 ( 5 usr; 2 con; 0-2 aty)
% Number of variables : 48 ( 6 sgn 8 !; 0 ?)
% Comments :
%------------------------------------------------------------------------------
fof(k_definition,axiom,
! [VAR_0,VAR_1] : apply(k(VAR_0),VAR_1) = VAR_0,
file('theBenchmark.p',k_definition) ).
cnf(c_1,plain,
apply(k(A),B) = A,
inference(clausify,[status(thm)],[k_definition]) ).
fof(abstraction,axiom,
! [VAR_0,VAR_1,VAR_2] : apply(apply(apply(abstraction,VAR_0),VAR_1),VAR_2) = apply(apply(VAR_0,k(VAR_2)),apply(VAR_1,VAR_2)),
file('theBenchmark.p',abstraction) ).
cnf(c_2,plain,
apply(apply(apply(abstraction,A),B),C) = apply(apply(A,k(C)),apply(B,C)),
inference(clausify,[status(thm)],[abstraction]) ).
fof(extensionality2,axiom,
! [VAR_0,VAR_1] :
( apply(VAR_0,n(VAR_0,VAR_1)) != apply(VAR_1,n(VAR_0,VAR_1))
| VAR_0 = VAR_1 ),
file('theBenchmark.p',extensionality2) ).
cnf(c_3,plain,
( apply(A,n(A,B)) != apply(B,n(A,B))
| A = B ),
inference(clausify,[status(thm)],[extensionality2]) ).
cnf(c_4,plain,
( apply(B,n(A,B)) != apply(A,n(A,B))
| A = B ),
inference(copy,[status(thm)],[c_3]) ).
fof(identity_definition,axiom,
! [VAR_0] : apply(identity,VAR_0) = VAR_0,
file('theBenchmark.p',identity_definition) ).
cnf(c_5,plain,
apply(identity,A) = A,
inference(clausify,[status(thm)],[identity_definition]) ).
fof(prove_TRC1b,axiom,
apply(apply(apply(abstraction,abstraction),abstraction),abstraction) != k(k(identity)),
file('theBenchmark.p',prove_TRC1b) ).
cnf(c_6,plain,
apply(apply(apply(abstraction,abstraction),abstraction),abstraction) != k(k(identity)),
inference(clausify,[status(thm)],[prove_TRC1b]) ).
cnf(c_7,plain,
k(k(identity)) != apply(apply(apply(abstraction,abstraction),abstraction),abstraction),
inference(copy,[status(thm)],[c_6]) ).
cnf(c_10,plain,
apply(apply(apply(abstraction,apply(apply(abstraction,A),B)),C),D) = apply(apply(apply(A,k(k(D))),apply(B,k(D))),apply(C,D)),
inference(paramod,[status(thm)],[c_2,c_2]) ).
cnf(c_213,plain,
apply(apply(apply(apply(abstraction,abstraction),A),B),C) = apply(apply(k(B),k(C)),apply(apply(A,B),C)),
inference(paramod,[status(thm)],[c_2,c_2]) ).
cnf(c_13,plain,
apply(apply(apply(apply(abstraction,abstraction),A),B),C) = apply(B,apply(apply(A,B),C)),
inference(paramod,[status(thm)],[c_1,c_213]) ).
cnf(c_16,plain,
( apply(B,n(k(A),B)) != A
| k(A) = B ),
inference(paramod,[status(thm)],[c_1,c_4]) ).
cnf(c_214,plain,
apply(apply(apply(abstraction,identity),A),B) = apply(k(B),apply(A,B)),
inference(paramod,[status(thm)],[c_5,c_2]) ).
cnf(c_19,plain,
apply(apply(apply(abstraction,identity),A),B) = B,
inference(paramod,[status(thm)],[c_1,c_214]) ).
cnf(c_215,plain,
( n(A,identity) != apply(A,n(A,identity))
| A = identity ),
inference(paramod,[status(thm)],[c_5,c_4]) ).
cnf(c_216,plain,
( n(A,identity) != apply(A,n(A,identity))
| identity = A ),
inference(copy,[status(thm)],[c_215]) ).
cnf(c_21,plain,
( apply(A,n(A,identity)) != n(A,identity)
| identity = A ),
inference(copy,[status(thm)],[c_216]) ).
cnf(c_217,plain,
apply(apply(apply(abstraction,abstraction),abstraction),abstraction) != k(k(identity)),
inference(copy,[status(thm)],[c_7]) ).
cnf(c_218,plain,
apply(k(k(identity)),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(k(identity)))) != apply(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(k(identity)))),
inference(resolve,[status(thm)],[c_4,c_217]) ).
cnf(c_219,plain,
k(identity) != apply(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(k(identity)))),
inference(paramod,[status(thm)],[c_1,c_218]) ).
cnf(c_220,plain,
k(identity) != apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(k(identity))))),
inference(paramod,[status(thm)],[c_13,c_219]) ).
cnf(c_24,plain,
apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(k(identity))))) != k(identity),
inference(copy,[status(thm)],[c_220]) ).
cnf(c_221,plain,
k(identity) != apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(k(identity))))),
inference(copy,[status(thm)],[c_24]) ).
cnf(c_222,plain,
apply(apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(k(identity))))),n(k(identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(k(identity))))))) != apply(k(identity),n(k(identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(k(identity))))))),
inference(resolve,[status(thm)],[c_4,c_221]) ).
cnf(c_37,plain,
apply(apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(k(identity))))),n(k(identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(k(identity))))))) != identity,
inference(paramod,[status(thm)],[c_1,c_222]) ).
cnf(c_223,plain,
identity = apply(apply(abstraction,identity),A),
inference(resolve,[status(thm)],[c_21,c_19]) ).
cnf(c_83,plain,
apply(apply(abstraction,identity),A) = identity,
inference(copy,[status(thm)],[c_223]) ).
cnf(c_88,plain,
k(identity) = apply(abstraction,identity),
inference(resolve,[status(thm)],[c_16,c_83]) ).
cnf(c_224,plain,
apply(apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))),n(k(identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(k(identity))))))) != identity,
inference(paramod,[status(thm)],[c_88,c_37]) ).
cnf(c_225,plain,
apply(apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))),n(apply(abstraction,identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(k(identity))))))) != identity,
inference(paramod,[status(thm)],[c_88,c_224]) ).
cnf(c_104,plain,
apply(apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))),n(apply(abstraction,identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))))) != identity,
inference(paramod,[status(thm)],[c_88,c_225]) ).
cnf(c_226,plain,
apply(apply(apply(abstraction,apply(apply(abstraction,abstraction),A)),B),C) = apply(apply(k(k(C)),k(apply(B,C))),apply(apply(A,k(C)),apply(B,C))),
inference(paramod,[status(thm)],[c_10,c_2]) ).
cnf(c_227,plain,
apply(apply(apply(abstraction,apply(apply(abstraction,abstraction),A)),B),C) = apply(k(C),apply(apply(A,k(C)),apply(B,C))),
inference(paramod,[status(thm)],[c_1,c_226]) ).
cnf(c_119,plain,
apply(apply(apply(abstraction,apply(apply(abstraction,abstraction),A)),B),C) = C,
inference(paramod,[status(thm)],[c_1,c_227]) ).
cnf(c_228,plain,
apply(identity,n(apply(apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))),n(apply(abstraction,identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))))),identity)) != apply(apply(apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))),n(apply(abstraction,identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))))),n(apply(apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))),n(apply(abstraction,identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))))),identity)),
inference(resolve,[status(thm)],[c_4,c_104]) ).
cnf(c_229,plain,
n(apply(apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))),n(apply(abstraction,identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))))),identity) != apply(apply(apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))),n(apply(abstraction,identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))))),n(apply(apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))),n(apply(abstraction,identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))))),identity)),
inference(paramod,[status(thm)],[c_5,c_228]) ).
cnf(c_230,plain,
n(apply(apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))),n(apply(abstraction,identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))))),identity) != n(apply(apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))),n(apply(abstraction,identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),k(apply(abstraction,identity))))))),identity),
inference(paramod,[status(thm)],[c_119,c_229]) ).
cnf(c_212,plain,
$false,
inference(copy,[status(thm)],[c_230]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : COL078-2 : TPTP v9.3.1. Released v1.2.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.36 % Computer : n008.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 23:13:10 UTC 2026
% 0.09/0.36 % CPUTime :
% 0.09/0.36 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.09/0.36 % Process 596660 was started by sandbox2 on n008,
% 0.09/0.36 % Sat Sep 26 23:13:10 2026
% 0.09/0.36 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.09/0.37
% 0.09/0.37 % From the command line: assign(max_seconds, 300).
% 0.09/0.41
% 0.09/0.41 % SZS status Unsatisfiable for theBenchmark
% 0.09/0.41
% 0.09/0.41 % Proof 1 at 0.03 (+ 0.01) seconds.
% 0.09/0.41 % Length of proof is 19.
% 0.09/0.41 % Level of proof is 6.
% 0.09/0.41 % Maximum clause weight is 43.000.
% 0.09/0.41 % Given clauses 36.
% 0.09/0.41
% 0.09/0.41 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------