%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : COL078-2 : TPTP v9.3.1. Released v1.2.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n017.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 : Tue Sep 29 09:37:24 AM UTC 2026
% Result : Unsatisfiable 0.25s 0.32s
% Output : Proof 0.25s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : COL078-2 : TPTP v9.3.1. Released v1.2.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.11/0.19 % Computer : n017.cluster.edu
% 0.11/0.19 % Model : x86_64 x86_64
% 0.11/0.19 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.19 % Memory : 8046.5625MB
% 0.11/0.19 % OS : Linux 6.8.0-71-generic
% 0.11/0.19 % CPULimit : 300
% 0.11/0.19 % WCLimit : 300
% 0.11/0.19 % DateTime : Mon Sep 28 21:29:36 UTC 2026
% 0.11/0.20 % CPUTime :
% 0.11/0.20 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.25/0.32 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --complete-subsets --ground-joining-incomplete-limit 15 --flatten-every 2
% 0.25/0.32
% 0.25/0.32 % SZS status Unsatisfiable
% 0.25/0.32
% 0.25/0.33 % SZS output start Proof
% 0.25/0.33 Axiom 1 (identity_definition): apply(identity, X) = X.
% 0.25/0.33 Axiom 2 (k_definition): apply(k(X), Y) = X.
% 0.25/0.33 Axiom 3 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 0.25/0.33 Axiom 4 (abstraction): apply(apply(apply(abstraction, X), Y), Z) = apply(apply(X, k(Z)), apply(Y, Z)).
% 0.25/0.33 Axiom 5 (extensionality2): ifeq(apply(X, n(X, Y)), apply(Y, n(X, Y)), X, Y) = Y.
% 0.25/0.33
% 0.25/0.33 Lemma 6: apply(apply(apply(apply(abstraction, abstraction), Y), X), Z) = apply(X, apply(apply(Y, X), Z)).
% 0.25/0.33 Proof:
% 0.25/0.33 apply(apply(apply(apply(abstraction, abstraction), Y), X), Z)
% 0.25/0.33 = { by axiom 4 (abstraction) }
% 0.25/0.33 apply(apply(apply(abstraction, k(X)), apply(Y, X)), Z)
% 0.25/0.33 = { by axiom 4 (abstraction) }
% 0.25/0.33 apply(apply(k(X), k(Z)), apply(apply(Y, X), Z))
% 0.25/0.33 = { by axiom 2 (k_definition) }
% 0.25/0.33 apply(X, apply(apply(Y, X), Z))
% 0.25/0.33
% 0.25/0.33 Lemma 7: ifeq(X, apply(Y, n(k(X), Y)), k(X), Y) = Y.
% 0.25/0.33 Proof:
% 0.25/0.33 ifeq(X, apply(Y, n(k(X), Y)), k(X), Y)
% 0.25/0.33 = { by axiom 2 (k_definition) R->L }
% 0.25/0.33 ifeq(apply(k(X), n(k(X), Y)), apply(Y, n(k(X), Y)), k(X), Y)
% 0.25/0.33 = { by axiom 5 (extensionality2) }
% 0.25/0.33 Y
% 0.25/0.33
% 0.25/0.33 Goal 1 (prove_TRC1b): apply(apply(apply(abstraction, abstraction), abstraction), abstraction) = k(k(identity)).
% 0.25/0.33 Proof:
% 0.25/0.33 apply(apply(apply(abstraction, abstraction), abstraction), abstraction)
% 0.25/0.33 = { by lemma 7 R->L }
% 0.25/0.33 ifeq(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))
% 0.25/0.33 = { by lemma 7 R->L }
% 0.25/0.33 ifeq(k(identity), ifeq(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))))), k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))), k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))
% 0.25/0.33 = { by axiom 5 (extensionality2) R->L }
% 0.25/0.33 ifeq(k(identity), ifeq(identity, ifeq(apply(identity, n(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))))))), apply(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))))), n(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))))))), identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))))), k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))), k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))
% 0.25/0.33 = { by axiom 1 (identity_definition) }
% 0.25/0.33 ifeq(k(identity), ifeq(identity, ifeq(n(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))))), apply(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))))), n(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))))))), identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))))), k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))), k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))
% 0.25/0.33 = { by lemma 6 }
% 0.25/0.33 ifeq(k(identity), ifeq(identity, ifeq(n(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))))), apply(apply(apply(abstraction, apply(apply(abstraction, abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))))), n(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))))))), identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))))), k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))), k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))
% 0.25/0.34 = { by axiom 4 (abstraction) }
% 0.25/0.34 ifeq(k(identity), ifeq(identity, ifeq(n(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))))), apply(apply(apply(apply(abstraction, abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), k(n(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))))))), apply(n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))), n(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))))))), identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))))), k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))), k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))
% 0.25/0.34 = { by lemma 6 }
% 0.25/0.34 ifeq(k(identity), ifeq(identity, ifeq(n(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))))), apply(k(n(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))))))), apply(apply(n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)), k(n(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))))))), apply(n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))), n(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))))))))), identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))))), k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))), k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))
% 0.25/0.34 = { by axiom 2 (k_definition) }
% 0.25/0.34 ifeq(k(identity), ifeq(identity, ifeq(n(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))))), n(identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))))), identity, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))), n(k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))))), k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))), k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))
% 0.25/0.34 = { by axiom 3 (ifeq_axiom) }
% 0.25/0.34 ifeq(k(identity), ifeq(identity, identity, k(identity), apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), n(k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction)))), k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))
% 0.25/0.34 = { by axiom 3 (ifeq_axiom) }
% 0.25/0.34 ifeq(k(identity), k(identity), k(k(identity)), apply(apply(apply(abstraction, abstraction), abstraction), abstraction))
% 0.25/0.34 = { by axiom 3 (ifeq_axiom) }
% 0.25/0.34 k(k(identity))
% 0.25/0.34 % SZS output end Proof
% 0.25/0.34
% 0.25/0.34 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------