%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : COL006-4 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n015.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 1.27s 0.41s
% Output : Proof 1.27s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : COL006-4 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.07/0.17 % Computer : n015.cluster.edu
% 0.07/0.17 % Model : x86_64 x86_64
% 0.07/0.17 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.17 % Memory : 8046.5625MB
% 0.07/0.17 % 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 21:32:46 UTC 2026
% 0.07/0.18 % CPUTime :
% 0.07/0.18 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 1.27/0.41 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
% 1.27/0.41
% 1.27/0.41 % SZS status Unsatisfiable
% 1.27/0.41
% 1.27/0.42 % SZS output start Proof
% 1.27/0.42 Axiom 1 (k_definition): apply(apply(k, X), Y) = X.
% 1.27/0.42 Axiom 2 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 1.27/0.42 Axiom 3 (s_definition): apply(apply(apply(s, X), Y), Z) = apply(apply(X, Z), apply(Y, Z)).
% 1.27/0.42 Axiom 4 (strong_fixed_point): ifeq(apply(X, fixed_pt), apply(fixed_pt, apply(X, fixed_pt)), fixed_point(X), true) = true.
% 1.27/0.42
% 1.27/0.42 Lemma 5: apply(apply(apply(s, k), X), Y) = Y.
% 1.27/0.42 Proof:
% 1.27/0.42 apply(apply(apply(s, k), X), Y)
% 1.27/0.42 = { by axiom 3 (s_definition) }
% 1.27/0.42 apply(apply(k, Y), apply(X, Y))
% 1.27/0.42 = { by axiom 1 (k_definition) }
% 1.27/0.42 Y
% 1.27/0.42
% 1.27/0.42 Lemma 6: apply(apply(apply(s, apply(k, X)), Y), Z) = apply(X, apply(Y, Z)).
% 1.27/0.42 Proof:
% 1.27/0.42 apply(apply(apply(s, apply(k, X)), Y), Z)
% 1.27/0.42 = { by axiom 3 (s_definition) }
% 1.27/0.42 apply(apply(apply(k, X), Z), apply(Y, Z))
% 1.27/0.42 = { by axiom 1 (k_definition) }
% 1.27/0.42 apply(X, apply(Y, Z))
% 1.27/0.42
% 1.27/0.42 Lemma 7: apply(apply(apply(apply(s, s), Z), X), Y) = apply(apply(X, Y), apply(apply(Z, X), Y)).
% 1.27/0.42 Proof:
% 1.27/0.42 apply(apply(apply(apply(s, s), Z), X), Y)
% 1.27/0.42 = { by axiom 3 (s_definition) }
% 1.27/0.42 apply(apply(apply(s, X), apply(Z, X)), Y)
% 1.27/0.42 = { by axiom 3 (s_definition) }
% 1.27/0.42 apply(apply(X, Y), apply(apply(Z, X), Y))
% 1.27/0.42
% 1.27/0.42 Lemma 8: apply(apply(apply(apply(s, s), apply(apply(s, k), Z)), X), Y) = apply(apply(X, Y), apply(X, Y)).
% 1.27/0.42 Proof:
% 1.27/0.42 apply(apply(apply(apply(s, s), apply(apply(s, k), Z)), X), Y)
% 1.27/0.42 = { by lemma 7 }
% 1.27/0.42 apply(apply(X, Y), apply(apply(apply(apply(s, k), Z), X), Y))
% 1.27/0.42 = { by lemma 5 }
% 1.27/0.42 apply(apply(X, Y), apply(X, Y))
% 1.27/0.42
% 1.27/0.42 Goal 1 (prove_strong_fixed_point): fixed_point(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k))) = true.
% 1.27/0.42 Proof:
% 1.27/0.42 fixed_point(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k)))
% 1.27/0.42 = { by axiom 2 (ifeq_axiom) R->L }
% 1.27/0.42 ifeq(apply(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k)), fixed_pt), apply(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k)), fixed_pt), fixed_point(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k))), true)
% 1.27/0.42 = { by lemma 6 }
% 1.27/0.42 ifeq(apply(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k)), fixed_pt), apply(apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))), apply(apply(apply(s, apply(k, s)), k), fixed_pt)), fixed_point(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k))), true)
% 1.27/0.43 = { by lemma 8 }
% 1.27/0.43 ifeq(apply(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k)), fixed_pt), apply(apply(apply(apply(s, s), apply(s, k)), apply(apply(apply(s, apply(k, s)), k), fixed_pt)), apply(apply(apply(s, s), apply(s, k)), apply(apply(apply(s, apply(k, s)), k), fixed_pt))), fixed_point(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k))), true)
% 1.27/0.43 = { by lemma 7 }
% 1.27/0.43 ifeq(apply(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k)), fixed_pt), apply(apply(apply(apply(apply(s, apply(k, s)), k), fixed_pt), apply(apply(apply(s, s), apply(s, k)), apply(apply(apply(s, apply(k, s)), k), fixed_pt))), apply(apply(apply(s, k), apply(apply(apply(s, apply(k, s)), k), fixed_pt)), apply(apply(apply(s, s), apply(s, k)), apply(apply(apply(s, apply(k, s)), k), fixed_pt)))), fixed_point(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k))), true)
% 1.27/0.43 = { by lemma 5 }
% 1.27/0.43 ifeq(apply(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k)), fixed_pt), apply(apply(apply(apply(apply(s, apply(k, s)), k), fixed_pt), apply(apply(apply(s, s), apply(s, k)), apply(apply(apply(s, apply(k, s)), k), fixed_pt))), apply(apply(apply(s, s), apply(s, k)), apply(apply(apply(s, apply(k, s)), k), fixed_pt))), fixed_point(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k))), true)
% 1.27/0.43 = { by lemma 6 }
% 1.27/0.43 ifeq(apply(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k)), fixed_pt), apply(apply(apply(s, apply(k, fixed_pt)), apply(apply(apply(s, s), apply(s, k)), apply(apply(apply(s, apply(k, s)), k), fixed_pt))), apply(apply(apply(s, s), apply(s, k)), apply(apply(apply(s, apply(k, s)), k), fixed_pt))), fixed_point(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k))), true)
% 1.27/0.43 = { by axiom 3 (s_definition) }
% 1.27/0.43 ifeq(apply(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k)), fixed_pt), apply(apply(apply(k, fixed_pt), apply(apply(apply(s, s), apply(s, k)), apply(apply(apply(s, apply(k, s)), k), fixed_pt))), apply(apply(apply(apply(s, s), apply(s, k)), apply(apply(apply(s, apply(k, s)), k), fixed_pt)), apply(apply(apply(s, s), apply(s, k)), apply(apply(apply(s, apply(k, s)), k), fixed_pt)))), fixed_point(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k))), true)
% 1.27/0.43 = { by axiom 1 (k_definition) }
% 1.27/0.43 ifeq(apply(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k)), fixed_pt), apply(fixed_pt, apply(apply(apply(apply(s, s), apply(s, k)), apply(apply(apply(s, apply(k, s)), k), fixed_pt)), apply(apply(apply(s, s), apply(s, k)), apply(apply(apply(s, apply(k, s)), k), fixed_pt)))), fixed_point(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k))), true)
% 1.27/0.43 = { by lemma 8 R->L }
% 1.27/0.43 ifeq(apply(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k)), fixed_pt), apply(fixed_pt, apply(apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))), apply(apply(apply(s, apply(k, s)), k), fixed_pt))), fixed_point(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k))), true)
% 1.27/0.43 = { by lemma 6 R->L }
% 1.27/0.43 ifeq(apply(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k)), fixed_pt), apply(fixed_pt, apply(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k)), fixed_pt)), fixed_point(apply(apply(s, apply(k, apply(apply(apply(s, s), apply(apply(s, k), k)), apply(apply(s, s), apply(s, k))))), apply(apply(s, apply(k, s)), k))), true)
% 1.27/0.43 = { by axiom 4 (strong_fixed_point) }
% 1.27/0.43 true
% 1.27/0.43 % SZS output end Proof
% 1.27/0.43
% 1.27/0.43 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------