%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : COL076-2 : 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 : n010.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 2.99s 1.40s
% Output : CNFRefutation 2.99s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 6
% Syntax : Number of formulae : 36 ( 30 unt; 0 def)
% Number of atoms : 42 ( 41 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 22 ( 16 ~; 6 |; 0 &)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 5 ( 2 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of predicates : 2 ( 0 usr; 1 prp; 0-2 aty)
% Number of functors : 8 ( 8 usr; 5 con; 0-2 aty)
% Number of variables : 71 ( 2 sgn 13 !; 0 ?)
% Comments :
%------------------------------------------------------------------------------
fof(k_definition,axiom,
! [VAR_0,VAR_1] : apply(apply(k,VAR_0),VAR_1) = VAR_0,
file('theBenchmark.p',k_definition) ).
cnf(c_1,plain,
apply(apply(k,A),B) = A,
inference(clausify,[status(thm)],[k_definition]) ).
fof(pairwise_application,axiom,
! [VAR_0,VAR_1,VAR_2] : apply(pair(VAR_0,VAR_1),VAR_2) = pair(apply(VAR_0,VAR_2),apply(VAR_1,VAR_2)),
file('theBenchmark.p',pairwise_application) ).
cnf(c_5,plain,
apply(pair(A,B),C) = pair(apply(A,C),apply(B,C)),
inference(clausify,[status(thm)],[pairwise_application]) ).
cnf(c_6,plain,
pair(apply(A,B),apply(C,B)) = apply(pair(A,C),B),
inference(copy,[status(thm)],[c_5]) ).
fof(abstraction,axiom,
! [VAR_0,VAR_1,VAR_2] : apply(apply(apply(abstraction,VAR_0),VAR_1),VAR_2) = apply(apply(VAR_0,apply(k,VAR_2)),apply(VAR_1,VAR_2)),
file('theBenchmark.p',abstraction) ).
cnf(c_7,plain,
apply(apply(apply(abstraction,A),B),C) = apply(apply(A,apply(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_8,plain,
( apply(A,n(A,B)) != apply(B,n(A,B))
| A = B ),
inference(clausify,[status(thm)],[extensionality2]) ).
cnf(c_9,plain,
( apply(B,n(A,B)) != apply(A,n(A,B))
| A = B ),
inference(copy,[status(thm)],[c_8]) ).
fof(diagonal_combinator,axiom,
! [VAR_0,VAR_1] : apply(apply(f,VAR_0),VAR_1) = apply(VAR_0,VAR_0),
file('theBenchmark.p',diagonal_combinator) ).
cnf(c_10,plain,
apply(apply(f,A),B) = apply(A,A),
inference(clausify,[status(thm)],[diagonal_combinator]) ).
fof(prove_self_referential,axiom,
! [VAR_0] : VAR_0 != apply(eq,pair(apply(k,VAR_0),apply(k,projection2))),
file('theBenchmark.p',prove_self_referential) ).
cnf(c_11,plain,
A != apply(eq,pair(apply(k,A),apply(k,projection2))),
inference(clausify,[status(thm)],[prove_self_referential]) ).
cnf(c_12,plain,
apply(eq,pair(apply(k,A),apply(k,projection2))) != A,
inference(copy,[status(thm)],[c_11]) ).
cnf(c_14,plain,
pair(A,apply(B,C)) = apply(pair(apply(k,A),B),C),
inference(paramod,[status(thm)],[c_1,c_6]) ).
cnf(c_15,plain,
pair(apply(A,B),C) = apply(pair(A,apply(k,C)),B),
inference(paramod,[status(thm)],[c_1,c_6]) ).
cnf(c_23,plain,
pair(A,apply(B,C)) = apply(apply(pair(k,apply(k,B)),A),C),
inference(paramod,[status(thm)],[c_15,c_14]) ).
cnf(c_24,plain,
apply(eq,apply(pair(k,apply(k,apply(k,projection2))),A)) != A,
inference(paramod,[status(thm)],[c_15,c_12]) ).
cnf(c_27,plain,
apply(apply(apply(abstraction,apply(k,A)),B),C) = apply(A,apply(B,C)),
inference(paramod,[status(thm)],[c_1,c_7]) ).
cnf(c_28,plain,
apply(apply(apply(abstraction,A),apply(k,B)),C) = apply(apply(A,apply(k,C)),B),
inference(paramod,[status(thm)],[c_1,c_7]) ).
cnf(c_1912,plain,
( B != apply(A,n(A,apply(k,B)))
| A = apply(k,B) ),
inference(paramod,[status(thm)],[c_1,c_9]) ).
cnf(c_1913,plain,
( B != apply(A,n(A,apply(k,B)))
| apply(k,B) = A ),
inference(copy,[status(thm)],[c_1912]) ).
cnf(c_37,plain,
( apply(B,n(B,apply(k,A))) != A
| apply(k,A) = B ),
inference(copy,[status(thm)],[c_1913]) ).
cnf(c_1914,plain,
apply(k,apply(A,A)) = apply(f,A),
inference(resolve,[status(thm)],[c_37,c_10]) ).
cnf(c_1327,plain,
apply(f,A) = apply(k,apply(A,A)),
inference(copy,[status(thm)],[c_1914]) ).
cnf(c_1915,plain,
pair(apply(k,A),apply(B,C)) = apply(apply(apply(abstraction,pair(k,apply(k,B))),apply(k,C)),A),
inference(paramod,[status(thm)],[c_28,c_23]) ).
cnf(c_1647,plain,
apply(pair(k,apply(k,apply(A,B))),C) = apply(apply(apply(abstraction,pair(k,apply(k,A))),apply(k,B)),C),
inference(paramod,[status(thm)],[c_15,c_1915]) ).
cnf(c_1793,plain,
apply(eq,apply(apply(apply(abstraction,pair(k,apply(k,k))),apply(k,projection2)),A)) != A,
inference(paramod,[status(thm)],[c_1647,c_24]) ).
cnf(c_1916,plain,
apply(eq,apply(apply(pair(k,apply(k,k)),apply(k,A)),apply(apply(k,projection2),A))) != A,
inference(paramod,[status(thm)],[c_7,c_1793]) ).
cnf(c_1815,plain,
apply(eq,apply(apply(pair(k,apply(k,k)),apply(k,A)),projection2)) != A,
inference(paramod,[status(thm)],[c_1,c_1916]) ).
cnf(c_1824,plain,
apply(eq,apply(apply(pair(k,apply(k,k)),apply(f,A)),projection2)) != apply(A,A),
inference(paramod,[status(thm)],[c_1327,c_1815]) ).
cnf(c_1917,plain,
apply(eq,pair(apply(f,A),apply(k,projection2))) != apply(A,A),
inference(paramod,[status(thm)],[c_23,c_1824]) ).
cnf(c_1910,plain,
apply(eq,apply(pair(f,apply(k,apply(k,projection2))),A)) != apply(A,A),
inference(paramod,[status(thm)],[c_15,c_1917]) ).
cnf(c_1918,plain,
apply(A,apply(B,C)) = apply(apply(apply(abstraction,apply(k,A)),B),C),
inference(copy,[status(thm)],[c_27]) ).
cnf(c_1911,plain,
$false,
inference(resolve,[status(thm)],[c_1910,c_1918]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.23 % Problem : COL076-2 : TPTP v9.3.1. Released v1.2.0.
% 0.00/0.24 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.10/0.93 % Computer : n010.cluster.edu
% 0.10/0.93 % Model : x86_64 x86_64
% 0.10/0.93 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.93 % Memory : 8046.5625MB
% 0.10/0.93 % OS : Linux 6.8.0-71-generic
% 0.10/0.93 % CPULimit : 300
% 0.10/0.93 % WCLimit : 300
% 0.10/0.93 % DateTime : Sat Sep 26 23:11:49 UTC 2026
% 0.10/0.94 % CPUTime :
% 0.10/0.94 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.10/0.94 % Process 275954 was started by sandbox on n010,
% 0.10/0.94 % Sat Sep 26 23:11:49 2026
% 0.10/0.94 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.10/0.94
% 0.10/0.94 % From the command line: assign(max_seconds, 300).
% 2.99/1.40
% 2.99/1.40 % SZS status Unsatisfiable for theBenchmark
% 2.99/1.40
% 2.99/1.40 % Proof 1 at 0.30 (+ 0.05) seconds.
% 2.99/1.40 % Length of proof is 23.
% 2.99/1.40 % Level of proof is 9.
% 2.99/1.40 % Maximum clause weight is 23.000.
% 2.99/1.40 % Given clauses 100.
% 2.99/1.40
% 2.99/1.40 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------