%------------------------------------------------------------------------------
% File : FindProof---0.1
% Problem : COL078-2 : TPTP v9.3.1. Released v1.2.0.
% Transfm : none
% Format : tptp:raw
% Command : run_findproof /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n012.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 : Fri Sep 25 01:02:00 PM UTC 2026
% Result : Unsatisfiable 13.35s 2.08s
% Output : Proof 13.35s
% Verified :
% SZS Type : Refutation
% Derivation depth : 28
% Number of leaves : 6
% Syntax : Number of formulae : 60 ( 56 unt; 0 def)
% Number of atoms : 64 ( 63 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 14 ( 10 ~; 4 |; 0 &)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 5 ( 2 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 2 ( 0 usr; 1 prp; 0-2 aty)
% Number of functors : 6 ( 6 usr; 2 con; 0-4 aty)
% Number of variables : 106 ( 16 sgn 16 !; 0 ?)
% Comments :
%------------------------------------------------------------------------------
cnf(f2,axiom,
( apply(X,n(X,Y)) != apply(Y,n(X,Y))
| X = Y ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',extensionality2) ).
fof(f2_nnf,plain,
! [X,Y] :
( apply(X,n(X,Y)) != apply(Y,n(X,Y))
| X = Y ),
inference(nnf_transformation,[status(thm)],[f2]) ).
fof(f2_sk,plain,
! [X,Y] :
( apply(X,n(X,Y)) != apply(Y,n(X,Y))
| X = Y ),
inference(skolemisation,[status(esa)],[f2_nnf]) ).
cnf(c2,plain,
( apply(X0,n(X0,X1)) != apply(X1,n(X0,X1))
| X0 = X1 ),
inference(cnf_transformation,[status(esa)],[f2_sk]) ).
cnf(t3,plain,
ifeq(apply(X1,n(X1,X2)),apply(X2,n(X1,X2)),X1,X2) = X2,
inference(equality_encoding,[status(esa)],[c2]) ).
cnf(t39,plain,
ifeq(apply(X1,n(X1,X2)),apply(X2,n(X1,X2)),X1,X2) = X2,
inference(orient,[status(thm)],[t3]) ).
cnf(f0,axiom,
apply(k(X),Y) = X,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',k_definition) ).
fof(f0_nnf,plain,
! [X,Y] : apply(k(X),Y) = X,
inference(nnf_transformation,[status(thm)],[f0]) ).
fof(f0_sk,plain,
! [X,Y] : apply(k(X),Y) = X,
inference(skolemisation,[status(esa)],[f0_nnf]) ).
cnf(c0,plain,
apply(k(X0),X1) = X0,
inference(cnf_transformation,[status(esa)],[f0_sk]) ).
cnf(t1,plain,
apply(k(X1),X2) = X1,
inference(equality_encoding,[status(esa)],[c0]) ).
cnf(t6,plain,
apply(k(X1),X2) = X1,
inference(orient,[status(thm)],[t1]) ).
cnf(t42,plain,
X1 = ifeq(X2,apply(X1,n(k(X2),X1)),k(X2),X1),
inference(cp,[status(thm)],[t39,t6]) ).
cnf(t117,plain,
ifeq(X1,apply(X2,n(k(X1),X2)),k(X1),X2) = X2,
inference(orient,[status(thm)],[t42]) ).
cnf(f1,axiom,
apply(apply(apply(abstraction,X),Y),Z) = apply(apply(X,k(Z)),apply(Y,Z)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',abstraction) ).
fof(f1_nnf,plain,
! [X,Y,Z] : apply(apply(apply(abstraction,X),Y),Z) = apply(apply(X,k(Z)),apply(Y,Z)),
inference(nnf_transformation,[status(thm)],[f1]) ).
fof(f1_sk,plain,
! [X,Y,Z] : apply(apply(apply(abstraction,X),Y),Z) = apply(apply(X,k(Z)),apply(Y,Z)),
inference(skolemisation,[status(esa)],[f1_nnf]) ).
cnf(c1,plain,
apply(apply(apply(abstraction,X0),X1),X2) = apply(apply(X0,k(X2)),apply(X1,X2)),
inference(cnf_transformation,[status(esa)],[f1_sk]) ).
cnf(t4,plain,
apply(apply(X1,k(X2)),apply(X3,X2)) = apply(apply(apply(abstraction,X1),X3),X2),
inference(equality_encoding,[status(esa)],[c1]) ).
cnf(t8,plain,
apply(apply(X1,k(X2)),apply(X3,X2)) = apply(apply(apply(abstraction,X1),X3),X2),
inference(orient,[status(thm)],[t4]) ).
cnf(t11,plain,
apply(apply(apply(abstraction,k(X1)),X2),X3) = apply(X1,apply(X2,X3)),
inference(cp,[status(thm)],[t8,t6]) ).
cnf(t18,plain,
apply(apply(apply(abstraction,k(X1)),X2),X3) = apply(X1,apply(X2,X3)),
inference(orient,[status(thm)],[t11]) ).
cnf(t119,plain,
apply(apply(abstraction,k(X1)),X2) = ifeq(X3,apply(X1,apply(X2,n(k(X3),apply(apply(abstraction,k(X1)),X2)))),k(X3),apply(apply(abstraction,k(X1)),X2)),
inference(cp,[status(thm)],[t117,t18]) ).
cnf(t4278,plain,
ifeq(X1,apply(X2,apply(X3,n(k(X1),apply(apply(abstraction,k(X2)),X3)))),k(X1),apply(apply(abstraction,k(X2)),X3)) = apply(apply(abstraction,k(X2)),X3),
inference(orient,[status(thm)],[t119]) ).
cnf(f3,axiom,
apply(identity,X) = X,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_definition) ).
fof(f3_nnf,plain,
! [X] : apply(identity,X) = X,
inference(nnf_transformation,[status(thm)],[f3]) ).
fof(f3_sk,plain,
! [X] : apply(identity,X) = X,
inference(skolemisation,[status(esa)],[f3_nnf]) ).
cnf(c3,plain,
apply(identity,X0) = X0,
inference(cnf_transformation,[status(esa)],[f3_sk]) ).
cnf(t0,plain,
apply(identity,X1) = X1,
inference(equality_encoding,[status(esa)],[c3]) ).
cnf(t5,plain,
apply(identity,X1) = X1,
inference(orient,[status(thm)],[t0]) ).
cnf(t40,plain,
X1 = ifeq(n(identity,X1),apply(X1,n(identity,X1)),identity,X1),
inference(cp,[status(thm)],[t39,t5]) ).
cnf(t62,plain,
ifeq(n(identity,X1),apply(X1,n(identity,X1)),identity,X1) = X1,
inference(orient,[status(thm)],[t40]) ).
cnf(t20,plain,
apply(X1,apply(apply(X2,X1),X3)) = apply(apply(apply(apply(abstraction,abstraction),X2),X1),X3),
inference(cp,[status(thm)],[t18,t8]) ).
cnf(t207,plain,
apply(X1,apply(apply(X2,X1),X3)) = apply(apply(apply(apply(abstraction,abstraction),X2),X1),X3),
inference(orient,[status(thm)],[t20]) ).
cnf(t212,plain,
apply(apply(apply(apply(abstraction,abstraction),X1),k(X2)),apply(X3,X2)) = apply(k(X2),apply(apply(apply(abstraction,X1),X3),X2)),
inference(cp,[status(thm)],[t207,t8]) ).
cnf(t4384,plain,
apply(apply(apply(abstraction,apply(apply(abstraction,abstraction),X1)),X3),X2) = apply(k(X2),apply(apply(apply(abstraction,X1),X3),X2)),
inference(step,[status(thm)],[t212,t8]) ).
cnf(t4385,plain,
apply(apply(apply(abstraction,apply(apply(abstraction,abstraction),X1)),X3),X2) = X2,
inference(step,[status(thm)],[t4384,t6]) ).
cnf(t258,plain,
apply(apply(apply(abstraction,apply(apply(abstraction,abstraction),X1)),X2),X3) = X3,
inference(orient,[status(thm)],[t4385]) ).
cnf(t270,plain,
apply(apply(abstraction,apply(apply(abstraction,abstraction),X1)),X2) = ifeq(n(identity,apply(apply(abstraction,apply(apply(abstraction,abstraction),X1)),X2)),n(identity,apply(apply(abstraction,apply(apply(abstraction,abstraction),X1)),X2)),identity,apply(apply(abstraction,apply(apply(abstraction,abstraction),X1)),X2)),
inference(cp,[status(thm)],[t62,t258]) ).
cnf(t2,plain,
ifeq(X1,X1,X2,X3) = X2,
introduced(definition) ).
cnf(t7,plain,
ifeq(X1,X1,X2,X3) = X2,
inference(orient,[status(thm)],[t2]) ).
cnf(t4386,plain,
apply(apply(abstraction,apply(apply(abstraction,abstraction),X1)),X2) = identity,
inference(step,[status(thm)],[t270,t7]) ).
cnf(t276,plain,
apply(apply(abstraction,apply(apply(abstraction,abstraction),X1)),X2) = identity,
inference(orient,[status(thm)],[t4386]) ).
cnf(t287,plain,
apply(abstraction,apply(apply(abstraction,abstraction),X1)) = ifeq(X2,identity,k(X2),apply(abstraction,apply(apply(abstraction,abstraction),X1))),
inference(cp,[status(thm)],[t117,t276]) ).
cnf(t369,plain,
ifeq(X1,identity,k(X1),apply(abstraction,apply(apply(abstraction,abstraction),X2))) = apply(abstraction,apply(apply(abstraction,abstraction),X2)),
inference(orient,[status(thm)],[t287]) ).
cnf(t370,plain,
apply(abstraction,apply(apply(abstraction,abstraction),X1)) = k(identity),
inference(cp,[status(thm)],[t369,t7]) ).
cnf(t373,plain,
apply(abstraction,apply(apply(abstraction,abstraction),X1)) = k(identity),
inference(orient,[status(thm)],[t370]) ).
cnf(t4297,plain,
apply(apply(abstraction,k(abstraction)),apply(abstraction,abstraction)) = ifeq(X1,k(identity),k(X1),apply(apply(abstraction,k(abstraction)),apply(abstraction,abstraction))),
inference(cp,[status(thm)],[t4278,t373]) ).
cnf(t4497,plain,
apply(apply(apply(abstraction,abstraction),abstraction),abstraction) = ifeq(X1,k(identity),k(X1),apply(apply(abstraction,k(abstraction)),apply(abstraction,abstraction))),
inference(step,[status(thm)],[t4297,t8]) ).
cnf(t4498,plain,
apply(apply(apply(abstraction,abstraction),abstraction),abstraction) = ifeq(X1,k(identity),k(X1),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)),
inference(step,[status(thm)],[t4497,t8]) ).
cnf(t4355,plain,
ifeq(X1,k(identity),k(X1),apply(apply(apply(abstraction,abstraction),abstraction),abstraction)) = apply(apply(apply(abstraction,abstraction),abstraction),abstraction),
inference(orient,[status(thm)],[t4498]) ).
cnf(t4356,plain,
apply(apply(apply(abstraction,abstraction),abstraction),abstraction) = k(k(identity)),
inference(cp,[status(thm)],[t4355,t7]) ).
cnf(t4357,plain,
apply(apply(apply(abstraction,abstraction),abstraction),abstraction) = k(k(identity)),
inference(orient,[status(thm)],[t4356]) ).
cnf(f4,negated_conjecture,
apply(apply(apply(abstraction,abstraction),abstraction),abstraction) != k(k(identity)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_TRC1b) ).
fof(f4_nnf,plain,
apply(apply(apply(abstraction,abstraction),abstraction),abstraction) != k(k(identity)),
inference(nnf_transformation,[status(thm)],[f4]) ).
fof(f4_sk,plain,
apply(apply(apply(abstraction,abstraction),abstraction),abstraction) != k(k(identity)),
inference(skolemisation,[status(esa)],[f4_nnf]) ).
cnf(c4,plain,
apply(apply(apply(abstraction,abstraction),abstraction),abstraction) != k(k(identity)),
inference(cnf_transformation,[status(esa)],[f4_sk]) ).
cnf(goal_0,negated_conjecture,
apply(apply(apply(abstraction,abstraction),abstraction),abstraction) != k(k(identity)),
inference(equality_encoding,[status(esa)],[c4]) ).
cnf(g0_0,plain,
k(k(identity)) != k(k(identity)),
inference(rw,[status(thm)],[goal_0,t4357]) ).
cnf(contradiction_0,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[g0_0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : COL078-2 : TPTP v9.3.1. Released v1.2.0.
% 0.00/0.04 % Command : run_findproof /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.09/0.36 % Computer : n012.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 : Fri Sep 25 07:29:36 UTC 2026
% 0.09/0.36 % CPUTime :
% 0.09/0.36 Running run_findproof /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 13.35/2.08 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p
% 13.35/2.08 % SZS output start Proof for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
%------------------------------------------------------------------------------