%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : COL078-1 : 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 : n016.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 17.93s 2.83s
% Output : CNFRefutation 17.93s
% Verified :
% SZS Type : Refutation
% Derivation depth : 19
% Number of leaves : 9
% Syntax : Number of formulae : 70 ( 59 unt; 0 def)
% Number of atoms : 81 ( 80 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 38 ( 27 ~; 11 |; 0 &)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 5 ( 2 avg)
% Maximal term depth : 12 ( 2 avg)
% Number of predicates : 2 ( 0 usr; 1 prp; 0-2 aty)
% Number of functors : 8 ( 8 usr; 4 con; 0-2 aty)
% Number of variables : 92 ( 14 sgn 16 !; 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(projection1,axiom,
! [VAR_0,VAR_1] : apply(projection1,pair(VAR_0,VAR_1)) = VAR_0,
file('theBenchmark.p',projection1) ).
cnf(c_2,plain,
apply(projection1,pair(A,B)) = A,
inference(clausify,[status(thm)],[projection1]) ).
fof(projection2,axiom,
! [VAR_0,VAR_1] : apply(projection2,pair(VAR_0,VAR_1)) = VAR_1,
file('theBenchmark.p',projection2) ).
cnf(c_3,plain,
apply(projection2,pair(A,B)) = B,
inference(clausify,[status(thm)],[projection2]) ).
fof(pairing,axiom,
! [VAR_0] : pair(apply(projection1,VAR_0),apply(projection2,VAR_0)) = VAR_0,
file('theBenchmark.p',pairing) ).
cnf(c_4,plain,
pair(apply(projection1,A),apply(projection2,A)) = A,
inference(clausify,[status(thm)],[pairing]) ).
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,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,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_10,plain,
( apply(A,n(A,B)) != apply(B,n(A,B))
| A = B ),
inference(clausify,[status(thm)],[extensionality2]) ).
cnf(c_11,plain,
( apply(B,n(A,B)) != apply(A,n(A,B))
| A = B ),
inference(copy,[status(thm)],[c_10]) ).
fof(identity_definition,axiom,
! [VAR_0] : apply(identity,VAR_0) = VAR_0,
file('theBenchmark.p',identity_definition) ).
cnf(c_14,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_15,plain,
apply(apply(apply(abstraction,abstraction),abstraction),abstraction) != k(k(identity)),
inference(clausify,[status(thm)],[prove_TRC1b]) ).
cnf(c_16,plain,
k(k(identity)) != apply(apply(apply(abstraction,abstraction),abstraction),abstraction),
inference(copy,[status(thm)],[c_15]) ).
cnf(c_17,plain,
apply(pair(projection1,projection2),A) = A,
inference(paramod,[status(thm)],[c_6,c_4]) ).
cnf(c_18,plain,
pair(A,apply(B,C)) = apply(pair(k(A),B),C),
inference(paramod,[status(thm)],[c_1,c_6]) ).
cnf(c_34,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_7,c_7]) ).
cnf(c_6164,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_7,c_7]) ).
cnf(c_37,plain,
apply(apply(apply(apply(abstraction,abstraction),A),B),C) = apply(B,apply(apply(A,B),C)),
inference(paramod,[status(thm)],[c_1,c_6164]) ).
cnf(c_6165,plain,
( B != apply(A,n(A,k(B)))
| A = k(B) ),
inference(paramod,[status(thm)],[c_1,c_11]) ).
cnf(c_6166,plain,
( B != apply(A,n(A,k(B)))
| k(B) = A ),
inference(copy,[status(thm)],[c_6165]) ).
cnf(c_47,plain,
( apply(B,n(B,k(A))) != A
| k(A) = B ),
inference(copy,[status(thm)],[c_6166]) ).
cnf(c_48,plain,
( apply(B,n(k(A),B)) != A
| k(A) = B ),
inference(paramod,[status(thm)],[c_1,c_11]) ).
cnf(c_6167,plain,
apply(apply(apply(abstraction,identity),A),B) = apply(k(B),apply(A,B)),
inference(paramod,[status(thm)],[c_14,c_7]) ).
cnf(c_55,plain,
apply(apply(apply(abstraction,identity),A),B) = B,
inference(paramod,[status(thm)],[c_1,c_6167]) ).
cnf(c_6168,plain,
( n(A,identity) != apply(A,n(A,identity))
| A = identity ),
inference(paramod,[status(thm)],[c_14,c_11]) ).
cnf(c_6169,plain,
( n(A,identity) != apply(A,n(A,identity))
| identity = A ),
inference(copy,[status(thm)],[c_6168]) ).
cnf(c_57,plain,
( apply(A,n(A,identity)) != n(A,identity)
| identity = A ),
inference(copy,[status(thm)],[c_6169]) ).
cnf(c_58,plain,
( apply(A,n(identity,A)) != n(identity,A)
| identity = A ),
inference(paramod,[status(thm)],[c_14,c_11]) ).
cnf(c_6170,plain,
apply(apply(apply(apply(abstraction,abstraction),abstraction),abstraction),n(k(k(identity)),apply(apply(apply(abstraction,abstraction),abstraction),abstraction))) != apply(k(k(identity)),n(k(k(identity)),apply(apply(apply(abstraction,abstraction),abstraction),abstraction))),
inference(resolve,[status(thm)],[c_11,c_16]) ).
cnf(c_6171,plain,
apply(abstraction,apply(apply(abstraction,abstraction),n(k(k(identity)),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)))) != apply(k(k(identity)),n(k(k(identity)),apply(apply(apply(abstraction,abstraction),abstraction),abstraction))),
inference(paramod,[status(thm)],[c_37,c_6170]) ).
cnf(c_59,plain,
apply(abstraction,apply(apply(abstraction,abstraction),n(k(k(identity)),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)))) != k(identity),
inference(paramod,[status(thm)],[c_1,c_6171]) ).
cnf(c_6172,plain,
pair(A,B) = apply(pair(k(A),k(B)),C),
inference(paramod,[status(thm)],[c_1,c_18]) ).
cnf(c_68,plain,
apply(pair(k(A),k(B)),C) = pair(A,B),
inference(copy,[status(thm)],[c_6172]) ).
cnf(c_84,plain,
pair(A,B) = apply(pair(k(A),identity),B),
inference(paramod,[status(thm)],[c_14,c_18]) ).
cnf(c_844,plain,
k(pair(A,B)) = pair(k(A),k(B)),
inference(resolve,[status(thm)],[c_47,c_68]) ).
cnf(c_6173,plain,
identity = apply(apply(abstraction,identity),A),
inference(resolve,[status(thm)],[c_57,c_55]) ).
cnf(c_1071,plain,
apply(apply(abstraction,identity),A) = identity,
inference(copy,[status(thm)],[c_6173]) ).
cnf(c_6174,plain,
identity = pair(projection1,projection2),
inference(resolve,[status(thm)],[c_57,c_17]) ).
cnf(c_1072,plain,
pair(projection1,projection2) = identity,
inference(copy,[status(thm)],[c_6174]) ).
cnf(c_1097,plain,
k(identity) = pair(k(projection1),k(projection2)),
inference(paramod,[status(thm)],[c_1072,c_844]) ).
cnf(c_6175,plain,
apply(abstraction,apply(apply(abstraction,abstraction),n(k(pair(k(projection1),k(projection2))),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)))) != k(identity),
inference(paramod,[status(thm)],[c_1097,c_59]) ).
cnf(c_6176,plain,
apply(abstraction,apply(apply(abstraction,abstraction),n(pair(k(k(projection1)),k(k(projection2))),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)))) != k(identity),
inference(paramod,[status(thm)],[c_844,c_6175]) ).
cnf(c_1125,plain,
apply(abstraction,apply(apply(abstraction,abstraction),n(pair(k(k(projection1)),k(k(projection2))),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)))) != pair(k(projection1),k(projection2)),
inference(paramod,[status(thm)],[c_1097,c_6176]) ).
cnf(c_6177,plain,
k(identity) = apply(abstraction,identity),
inference(resolve,[status(thm)],[c_1071,c_48]) ).
cnf(c_1257,plain,
pair(k(projection1),k(projection2)) = apply(abstraction,identity),
inference(paramod,[status(thm)],[c_1097,c_6177]) ).
cnf(c_1293,plain,
apply(abstraction,apply(apply(abstraction,abstraction),n(pair(k(k(projection1)),k(k(projection2))),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)))) != apply(abstraction,identity),
inference(paramod,[status(thm)],[c_1257,c_1125]) ).
cnf(c_6178,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_34,c_7]) ).
cnf(c_6179,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_6178]) ).
cnf(c_1873,plain,
apply(apply(apply(abstraction,apply(apply(abstraction,abstraction),A)),B),C) = C,
inference(paramod,[status(thm)],[c_1,c_6179]) ).
cnf(c_1982,plain,
apply(abstraction,apply(apply(abstraction,abstraction),n(apply(pair(k(k(k(projection1))),identity),k(k(projection2))),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)))) != apply(abstraction,identity),
inference(paramod,[status(thm)],[c_84,c_1293]) ).
cnf(c_6180,plain,
apply(projection1,apply(abstraction,identity)) = k(projection1),
inference(paramod,[status(thm)],[c_1257,c_2]) ).
cnf(c_1994,plain,
k(projection1) = apply(projection1,apply(abstraction,identity)),
inference(copy,[status(thm)],[c_6180]) ).
cnf(c_6181,plain,
apply(projection2,apply(abstraction,identity)) = k(projection2),
inference(paramod,[status(thm)],[c_1257,c_3]) ).
cnf(c_1995,plain,
k(projection2) = apply(projection2,apply(abstraction,identity)),
inference(copy,[status(thm)],[c_6181]) ).
cnf(c_6182,plain,
apply(abstraction,apply(apply(abstraction,abstraction),n(apply(pair(k(k(apply(projection1,apply(abstraction,identity)))),identity),k(k(projection2))),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)))) != apply(abstraction,identity),
inference(paramod,[status(thm)],[c_1994,c_1982]) ).
cnf(c_2004,plain,
apply(abstraction,apply(apply(abstraction,abstraction),n(apply(pair(k(k(apply(projection1,apply(abstraction,identity)))),identity),k(apply(projection2,apply(abstraction,identity)))),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)))) != apply(abstraction,identity),
inference(paramod,[status(thm)],[c_1995,c_6182]) ).
cnf(c_6183,plain,
apply(abstraction,identity) != apply(abstraction,apply(apply(abstraction,abstraction),n(apply(pair(k(k(apply(projection1,apply(abstraction,identity)))),identity),k(apply(projection2,apply(abstraction,identity)))),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)))),
inference(copy,[status(thm)],[c_2004]) ).
cnf(c_6184,plain,
apply(apply(abstraction,apply(apply(abstraction,abstraction),n(apply(pair(k(k(apply(projection1,apply(abstraction,identity)))),identity),k(apply(projection2,apply(abstraction,identity)))),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)))),n(apply(abstraction,identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(pair(k(k(apply(projection1,apply(abstraction,identity)))),identity),k(apply(projection2,apply(abstraction,identity)))),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)))))) != apply(apply(abstraction,identity),n(apply(abstraction,identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(pair(k(k(apply(projection1,apply(abstraction,identity)))),identity),k(apply(projection2,apply(abstraction,identity)))),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)))))),
inference(resolve,[status(thm)],[c_11,c_6183]) ).
cnf(c_4794,plain,
apply(apply(abstraction,apply(apply(abstraction,abstraction),n(apply(pair(k(k(apply(projection1,apply(abstraction,identity)))),identity),k(apply(projection2,apply(abstraction,identity)))),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)))),n(apply(abstraction,identity),apply(abstraction,apply(apply(abstraction,abstraction),n(apply(pair(k(k(apply(projection1,apply(abstraction,identity)))),identity),k(apply(projection2,apply(abstraction,identity)))),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)))))) != identity,
inference(paramod,[status(thm)],[c_1071,c_6184]) ).
cnf(c_6185,plain,
identity = apply(apply(abstraction,apply(apply(abstraction,abstraction),A)),B),
inference(resolve,[status(thm)],[c_1873,c_58]) ).
cnf(c_6162,plain,
apply(apply(abstraction,apply(apply(abstraction,abstraction),A)),B) = identity,
inference(copy,[status(thm)],[c_6185]) ).
cnf(c_6163,plain,
$false,
inference(resolve,[status(thm)],[c_6162,c_4794]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : COL078-1 : TPTP v9.3.1. Released v1.2.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.11/0.37 % Computer : n016.cluster.edu
% 0.11/0.37 % Model : x86_64 x86_64
% 0.11/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.37 % Memory : 8046.5625MB
% 0.11/0.37 % OS : Linux 6.8.0-71-generic
% 0.11/0.37 % CPULimit : 300
% 0.11/0.37 % WCLimit : 300
% 0.11/0.37 % DateTime : Sat Sep 26 23:16:01 UTC 2026
% 0.11/0.37 % CPUTime :
% 0.11/0.37 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.11/0.37 % Process 1982443 was started by sandbox on n016,
% 0.11/0.37 % Sat Sep 26 23:16:01 2026
% 0.11/0.37 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.11/0.38
% 0.11/0.38 % From the command line: assign(max_seconds, 300).
% 17.93/2.83
% 17.93/2.83 % SZS status Unsatisfiable for theBenchmark
% 17.93/2.83
% 17.93/2.83 % Proof 1 at 2.04 (+ 0.37) seconds.
% 17.93/2.83 % Length of proof is 39.
% 17.93/2.83 % Level of proof is 11.
% 17.93/2.83 % Maximum clause weight is 67.000.
% 17.93/2.83 % Given clauses 245.
% 17.93/2.83
% 17.93/2.83 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------