%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : COL076-2 : TPTP v9.3.1. Released v1.2.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n026.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 98.71s 12.75s
% Output : Proof 98.71s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : COL076-2 : TPTP v9.3.1. Released v1.2.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.07/0.21 % Computer : n026.cluster.edu
% 0.07/0.21 % Model : x86_64 x86_64
% 0.07/0.21 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.21 % Memory : 8046.5625MB
% 0.07/0.21 % OS : Linux 6.8.0-71-generic
% 0.07/0.21 % CPULimit : 300
% 0.07/0.21 % WCLimit : 300
% 0.07/0.21 % DateTime : Mon Sep 28 21:36:56 UTC 2026
% 0.07/0.21 % CPUTime :
% 0.07/0.21 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 98.71/12.75 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --complete-subsets --ground-joining-incomplete-limit 15 --flatten-regeneralise
% 98.71/12.75
% 98.71/12.75 % SZS status Unsatisfiable
% 98.71/12.75
% 98.71/12.76 % SZS output start Proof
% 98.71/12.76 Axiom 1 (k_definition): apply(apply(k, X), Y) = X.
% 98.71/12.77 Axiom 2 (diagonal_combinator): apply(apply(f, X), Y) = apply(X, X).
% 98.71/12.77 Axiom 3 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 98.71/12.77 Axiom 4 (pairwise_application): apply(pair(X, Y), Z) = pair(apply(X, Z), apply(Y, Z)).
% 98.71/12.77 Axiom 5 (abstraction): apply(apply(apply(abstraction, X), Y), Z) = apply(apply(X, apply(k, Z)), apply(Y, Z)).
% 98.71/12.77 Axiom 6 (extensionality2): ifeq(apply(X, n(X, Y)), apply(Y, n(X, Y)), X, Y) = Y.
% 98.71/12.77
% 98.71/12.77 Lemma 7: ifeq(apply(X, X), Y, apply(f, X), apply(k, Y)) = apply(k, Y).
% 98.71/12.77 Proof:
% 98.71/12.77 ifeq(apply(X, X), Y, apply(f, X), apply(k, Y))
% 98.71/12.77 = { by axiom 1 (k_definition) R->L }
% 98.71/12.77 ifeq(apply(X, X), apply(apply(k, Y), n(apply(f, X), apply(k, Y))), apply(f, X), apply(k, Y))
% 98.71/12.77 = { by axiom 2 (diagonal_combinator) R->L }
% 98.71/12.77 ifeq(apply(apply(f, X), n(apply(f, X), apply(k, Y))), apply(apply(k, Y), n(apply(f, X), apply(k, Y))), apply(f, X), apply(k, Y))
% 98.71/12.77 = { by axiom 6 (extensionality2) }
% 98.71/12.77 apply(k, Y)
% 98.71/12.77
% 98.71/12.77 Goal 1 (prove_self_referential): X = apply(eq, pair(apply(k, X), apply(k, projection2))).
% 98.71/12.77 The goal is true when:
% 98.71/12.77 X = apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))
% 98.71/12.77
% 98.71/12.77 Proof:
% 98.71/12.77 apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))
% 98.71/12.77 = { by axiom 5 (abstraction) }
% 98.71/12.77 apply(apply(apply(k, eq), apply(k, pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))
% 98.71/12.77 = { by axiom 1 (k_definition) }
% 98.71/12.77 apply(eq, apply(f, pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))
% 98.71/12.77 = { by axiom 3 (ifeq_axiom) R->L }
% 98.71/12.77 apply(eq, ifeq(apply(pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(f, pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(k, apply(pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))))
% 98.71/12.77 = { by lemma 7 }
% 98.71/12.77 apply(eq, apply(k, apply(pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))
% 98.71/12.77 = { by axiom 4 (pairwise_application) }
% 98.71/12.77 apply(eq, apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))))
% 98.71/12.77 = { by axiom 1 (k_definition) }
% 98.71/12.77 apply(eq, apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), projection2)))
% 98.71/12.78 = { by axiom 1 (k_definition) R->L }
% 98.71/12.78 apply(eq, apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2)))))
% 98.71/12.78 = { by axiom 6 (extensionality2) R->L }
% 98.71/12.78 apply(eq, ifeq(apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, apply(k, projection2))), n(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, apply(k, projection2))), apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2)))))), apply(apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2)))), n(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, apply(k, projection2))), apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2)))))), pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, apply(k, projection2))), apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2))))))
% 98.71/12.78 = { by axiom 1 (k_definition) }
% 98.71/12.78 apply(eq, ifeq(apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, apply(k, projection2))), n(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, apply(k, projection2))), apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2)))))), pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2))), pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, apply(k, projection2))), apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2))))))
% 98.71/12.78 = { by axiom 4 (pairwise_application) }
% 98.71/12.78 apply(eq, ifeq(pair(apply(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), n(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, apply(k, projection2))), apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2)))))), apply(apply(f, apply(k, projection2)), n(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, apply(k, projection2))), apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2))))))), pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2))), pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, apply(k, projection2))), apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2))))))
% 98.71/12.79 = { by axiom 2 (diagonal_combinator) }
% 98.71/12.79 apply(eq, ifeq(pair(apply(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), n(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, apply(k, projection2))), apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2)))))), apply(apply(k, projection2), apply(k, projection2))), pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2))), pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, apply(k, projection2))), apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2))))))
% 98.71/12.79 = { by axiom 1 (k_definition) }
% 98.71/12.79 apply(eq, ifeq(pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2))), pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2))), pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, apply(k, projection2))), apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, projection2))))))
% 98.71/12.79 = { by axiom 3 (ifeq_axiom) }
% 98.71/12.79 apply(eq, pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, apply(k, projection2))))
% 98.71/12.79 = { by axiom 3 (ifeq_axiom) R->L }
% 98.71/12.79 apply(eq, pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), ifeq(projection2, projection2, apply(f, apply(k, projection2)), apply(k, projection2))))
% 98.71/12.79 = { by axiom 1 (k_definition) R->L }
% 98.71/12.79 apply(eq, pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), ifeq(apply(apply(k, projection2), apply(k, projection2)), projection2, apply(f, apply(k, projection2)), apply(k, projection2))))
% 98.71/12.79 = { by lemma 7 }
% 98.71/12.79 apply(eq, pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)))
% 98.71/12.79 % SZS output end Proof
% 98.71/12.79
% 98.71/12.79 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------