%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : COL006-3 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n014.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 09:37:09 AM UTC 2026
% Result : Unsatisfiable 0.21s 0.32s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : COL006-3 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.07/0.18 % Computer : n014.cluster.edu
% 0.07/0.18 % Model : x86_64 x86_64
% 0.07/0.18 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.18 % Memory : 8046.5625MB
% 0.07/0.18 % OS : Linux 6.8.0-71-generic
% 0.07/0.18 % CPULimit : 300
% 0.07/0.18 % WCLimit : 300
% 0.07/0.18 % DateTime : Mon Sep 28 21:29:15 UTC 2026
% 0.07/0.18 % CPUTime :
% 0.07/0.18 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.21/0.32 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
% 0.21/0.32
% 0.21/0.32 % SZS status Unsatisfiable
% 0.21/0.32
% 0.21/0.33 % SZS output start Proof
% 0.21/0.33 Axiom 1 (k_definition): apply(apply(k, X), Y) = X.
% 0.21/0.33 Axiom 2 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 0.21/0.33 Axiom 3 (s_definition): apply(apply(apply(s, X), Y), Z) = apply(apply(X, Z), apply(Y, Z)).
% 0.21/0.33 Axiom 4 (strong_fixed_point): ifeq(apply(X, fixed_pt), apply(fixed_pt, apply(X, fixed_pt)), fixed_point(X), true) = true.
% 0.21/0.33
% 0.21/0.33 Lemma 5: apply(apply(apply(s, k), X), Y) = Y.
% 0.21/0.33 Proof:
% 0.21/0.33 apply(apply(apply(s, k), X), Y)
% 0.21/0.33 = { by axiom 3 (s_definition) }
% 0.21/0.33 apply(apply(k, Y), apply(X, Y))
% 0.21/0.33 = { by axiom 1 (k_definition) }
% 0.21/0.33 Y
% 0.21/0.33
% 0.21/0.33 Lemma 6: apply(apply(apply(s, apply(k, X)), Y), Z) = apply(X, apply(Y, Z)).
% 0.21/0.33 Proof:
% 0.21/0.33 apply(apply(apply(s, apply(k, X)), Y), Z)
% 0.21/0.33 = { by axiom 3 (s_definition) }
% 0.21/0.33 apply(apply(apply(k, X), Z), apply(Y, Z))
% 0.21/0.33 = { by axiom 1 (k_definition) }
% 0.21/0.34 apply(X, apply(Y, Z))
% 0.21/0.34
% 0.21/0.34 Goal 1 (prove_strong_fixed_point): fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))))) = true.
% 0.21/0.34 Proof:
% 0.21/0.34 fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))))
% 0.21/0.34 = { by axiom 2 (ifeq_axiom) R->L }
% 0.21/0.34 ifeq(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))))), true)
% 0.21/0.34 = { by lemma 6 }
% 0.21/0.34 ifeq(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt)), fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))))), true)
% 0.21/0.35 = { by axiom 3 (s_definition) }
% 0.21/0.35 ifeq(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(apply(s, k), k), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt)), apply(apply(apply(s, k), k), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt))), fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))))), true)
% 0.21/0.35 = { by lemma 5 }
% 0.21/0.35 ifeq(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt), apply(apply(apply(s, k), k), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt))), fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))))), true)
% 0.21/0.35 = { by lemma 5 }
% 0.21/0.36 ifeq(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt)), fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))))), true)
% 0.21/0.36 = { by axiom 3 (s_definition) }
% 0.21/0.36 ifeq(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(apply(apply(s, apply(k, s)), k), fixed_pt), apply(apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))), fixed_pt)), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt)), fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))))), true)
% 0.21/0.36 = { by axiom 1 (k_definition) }
% 0.21/0.36 ifeq(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(apply(apply(s, apply(k, s)), k), fixed_pt), apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt)), fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))))), true)
% 0.21/0.36 = { by lemma 6 }
% 0.21/0.36 ifeq(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(s, apply(k, fixed_pt)), apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt)), fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))))), true)
% 0.21/0.36 = { by axiom 3 (s_definition) }
% 0.21/0.36 ifeq(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(k, fixed_pt), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt)), apply(apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt))), fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))))), true)
% 0.21/0.36 = { by axiom 1 (k_definition) }
% 0.21/0.36 ifeq(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(fixed_pt, apply(apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt))), fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))))), true)
% 0.21/0.36 = { by lemma 6 R->L }
% 0.21/0.36 ifeq(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(fixed_pt, apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt)), fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))))), true)
% 0.21/0.36 = { by axiom 4 (strong_fixed_point) }
% 0.21/0.36 true
% 0.21/0.36 % SZS output end Proof
% 0.21/0.36
% 0.21/0.36 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------