%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : COL081-2 : TPTP v9.3.1. Released v1.2.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n005.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 06:57:39 AM UTC 2026
% Result : Unsatisfiable 0.07s 0.37s
% Output : CNFRefutation 0.07s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 4
% Syntax : Number of formulae : 19 ( 16 unt; 0 def)
% Number of atoms : 22 ( 21 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 16 ( 13 ~; 3 |; 0 &)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 5 ( 2 avg)
% Maximal term depth : 9 ( 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 : 19 ( 1 sgn 7 !; 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(prove_TRC2c,axiom,
k(k(b)) != apply(abstraction,k(k(b))),
file('theBenchmark.p',prove_TRC2c) ).
cnf(c_6,plain,
k(k(b)) != apply(abstraction,k(k(b))),
inference(clausify,[status(thm)],[prove_TRC2c]) ).
cnf(c_7,plain,
apply(abstraction,k(k(b))) != k(k(b)),
inference(copy,[status(thm)],[c_6]) ).
cnf(c_8,plain,
apply(apply(apply(abstraction,k(A)),B),C) = apply(A,apply(B,C)),
inference(paramod,[status(thm)],[c_1,c_2]) ).
cnf(c_32,plain,
apply(k(k(b)),n(apply(abstraction,k(k(b))),k(k(b)))) != apply(apply(abstraction,k(k(b))),n(apply(abstraction,k(k(b))),k(k(b)))),
inference(resolve,[status(thm)],[c_4,c_7]) ).
cnf(c_33,plain,
k(b) != apply(apply(abstraction,k(k(b))),n(apply(abstraction,k(k(b))),k(k(b)))),
inference(paramod,[status(thm)],[c_1,c_32]) ).
cnf(c_23,plain,
apply(apply(abstraction,k(k(b))),n(apply(abstraction,k(k(b))),k(k(b)))) != k(b),
inference(copy,[status(thm)],[c_33]) ).
cnf(c_34,plain,
apply(k(b),n(apply(apply(abstraction,k(k(b))),n(apply(abstraction,k(k(b))),k(k(b)))),k(b))) != apply(apply(apply(abstraction,k(k(b))),n(apply(abstraction,k(k(b))),k(k(b)))),n(apply(apply(abstraction,k(k(b))),n(apply(abstraction,k(k(b))),k(k(b)))),k(b))),
inference(resolve,[status(thm)],[c_4,c_23]) ).
cnf(c_35,plain,
b != apply(apply(apply(abstraction,k(k(b))),n(apply(abstraction,k(k(b))),k(k(b)))),n(apply(apply(abstraction,k(k(b))),n(apply(abstraction,k(k(b))),k(k(b)))),k(b))),
inference(paramod,[status(thm)],[c_1,c_34]) ).
cnf(c_36,plain,
b != apply(k(b),apply(n(apply(abstraction,k(k(b))),k(k(b))),n(apply(apply(abstraction,k(k(b))),n(apply(abstraction,k(k(b))),k(k(b)))),k(b)))),
inference(paramod,[status(thm)],[c_8,c_35]) ).
cnf(c_37,plain,
b != b,
inference(paramod,[status(thm)],[c_1,c_36]) ).
cnf(c_31,plain,
$false,
inference(copy,[status(thm)],[c_37]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : COL081-2 : TPTP v9.3.1. Released v1.2.0.
% 0.00/0.03 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.07/0.34 % Computer : n005.cluster.edu
% 0.07/0.34 % Model : x86_64 x86_64
% 0.07/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.34 % Memory : 8046.5625MB
% 0.07/0.34 % OS : Linux 6.8.0-71-generic
% 0.07/0.34 % CPULimit : 300
% 0.07/0.34 % WCLimit : 300
% 0.07/0.34 % DateTime : Sat Sep 26 23:13:01 UTC 2026
% 0.07/0.35 % CPUTime :
% 0.07/0.35 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.07/0.35 % Process 3345320 was started by sandbox on n005,
% 0.07/0.35 % Sat Sep 26 23:13:01 2026
% 0.07/0.35 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.07/0.35
% 0.07/0.35 % From the command line: assign(max_seconds, 300).
% 0.07/0.37
% 0.07/0.37 % SZS status Unsatisfiable for theBenchmark
% 0.07/0.37
% 0.07/0.37 % Proof 1 at 0.00 (+ 0.00) seconds.
% 0.07/0.37 % Length of proof is 9.
% 0.07/0.37 % Level of proof is 3.
% 0.07/0.37 % Maximum clause weight is 18.000.
% 0.07/0.37 % Given clauses 7.
% 0.07/0.37
% 0.07/0.37 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------