%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE143+2 : TPTP v9.3.1. Released v4.0.0.
% 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 11:39:53 AM UTC 2026
% Result : Theorem 39.24s 5.41s
% Output : Proof 39.24s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : KLE143+2 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.36 % Computer : n014.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 : Sun Sep 27 13:12:16 UTC 2026
% 0.09/0.37 % CPUTime :
% 0.09/0.37 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 39.24/5.41 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
% 39.24/5.41
% 39.24/5.41 % SZS status Theorem
% 39.24/5.41
% 39.24/5.42 % SZS output start Proof
% 39.24/5.42 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 39.24/5.42 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 39.24/5.42 Axiom 3 (idempotence): addition(X, X) = X.
% 39.24/5.42 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 39.24/5.42 Axiom 5 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 39.24/5.42 Axiom 6 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
% 39.24/5.42 Axiom 7 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 39.24/5.42 Axiom 8 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 39.24/5.42 Axiom 9 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 39.24/5.42 Axiom 10 (distributivity1): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 39.24/5.42 Axiom 11 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 39.24/5.42 Axiom 12 (order): ifeq3(addition(X, Y), Y, leq(X, Y), true) = true.
% 39.24/5.42 Axiom 13 (infty_coinduction): ifeq(leq(X, addition(multiplication(Y, X), Z)), true, leq(X, multiplication(strong_iteration(Y), Z)), true) = true.
% 39.24/5.42
% 39.24/5.42 Lemma 14: addition(X, addition(X, Y)) = addition(X, Y).
% 39.24/5.42 Proof:
% 39.24/5.42 addition(X, addition(X, Y))
% 39.24/5.42 = { by axiom 7 (additive_associativity) }
% 39.24/5.42 addition(addition(X, X), Y)
% 39.24/5.42 = { by axiom 3 (idempotence) }
% 39.24/5.42 addition(X, Y)
% 39.24/5.42
% 39.24/5.42 Lemma 15: addition(one, multiplication(X, strong_iteration(X))) = strong_iteration(X).
% 39.24/5.42 Proof:
% 39.24/5.42 addition(one, multiplication(X, strong_iteration(X)))
% 39.24/5.42 = { by axiom 4 (additive_commutativity) R->L }
% 39.24/5.42 addition(multiplication(X, strong_iteration(X)), one)
% 39.24/5.42 = { by axiom 6 (infty_unfold1) R->L }
% 39.24/5.42 strong_iteration(X)
% 39.24/5.42
% 39.24/5.42 Goal 1 (goals): tuple(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), leq(strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)))) = tuple(true, true).
% 39.24/5.42 Proof:
% 39.24/5.42 tuple(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), leq(strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0))))
% 39.24/5.42 = { by axiom 1 (multiplicative_right_identity) R->L }
% 39.24/5.42 tuple(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), leq(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), strong_iteration(x0))))
% 39.24/5.42 = { by lemma 15 R->L }
% 39.24/5.42 tuple(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), leq(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), addition(one, multiplication(x0, strong_iteration(x0))))))
% 39.24/5.42 = { by axiom 10 (distributivity1) }
% 39.24/5.42 tuple(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), leq(multiplication(strong_iteration(x0), one), addition(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), multiplication(x0, strong_iteration(x0))))))
% 39.24/5.42 = { by axiom 9 (ifeq_axiom) R->L }
% 39.24/5.42 tuple(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), ifeq3(addition(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), multiplication(x0, strong_iteration(x0)))), addition(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), multiplication(x0, strong_iteration(x0)))), leq(multiplication(strong_iteration(x0), one), addition(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), multiplication(x0, strong_iteration(x0))))), true))
% 39.24/5.42 = { by lemma 14 R->L }
% 39.24/5.42 tuple(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), ifeq3(addition(multiplication(strong_iteration(x0), one), addition(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), multiplication(x0, strong_iteration(x0))))), addition(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), multiplication(x0, strong_iteration(x0)))), leq(multiplication(strong_iteration(x0), one), addition(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), multiplication(x0, strong_iteration(x0))))), true))
% 39.24/5.42 = { by axiom 12 (order) }
% 39.24/5.42 tuple(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true)
% 39.24/5.42 = { by axiom 8 (ifeq_axiom) R->L }
% 39.24/5.42 tuple(ifeq(true, true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.42 = { by axiom 12 (order) R->L }
% 39.24/5.42 tuple(ifeq(ifeq3(addition(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), multiplication(strong_iteration(x0_2), strong_iteration(x0_2))), multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), multiplication(strong_iteration(x0_2), strong_iteration(x0_2))), true), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.42 = { by axiom 3 (idempotence) }
% 39.24/5.42 tuple(ifeq(ifeq3(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), multiplication(strong_iteration(x0_2), strong_iteration(x0_2))), true), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.42 = { by axiom 9 (ifeq_axiom) }
% 39.24/5.42 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), multiplication(strong_iteration(x0_2), strong_iteration(x0_2))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.42 = { by lemma 15 R->L }
% 39.24/5.42 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), multiplication(addition(one, multiplication(x0_2, strong_iteration(x0_2))), strong_iteration(x0_2))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.42 = { by axiom 11 (distributivity2) }
% 39.24/5.42 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(multiplication(one, strong_iteration(x0_2)), multiplication(multiplication(x0_2, strong_iteration(x0_2)), strong_iteration(x0_2)))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.42 = { by axiom 2 (multiplicative_left_identity) }
% 39.24/5.42 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(strong_iteration(x0_2), multiplication(multiplication(x0_2, strong_iteration(x0_2)), strong_iteration(x0_2)))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.42 = { by axiom 5 (multiplicative_associativity) R->L }
% 39.24/5.42 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(strong_iteration(x0_2), multiplication(x0_2, multiplication(strong_iteration(x0_2), strong_iteration(x0_2))))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.42 = { by axiom 4 (additive_commutativity) R->L }
% 39.24/5.42 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(multiplication(x0_2, multiplication(strong_iteration(x0_2), strong_iteration(x0_2))), strong_iteration(x0_2))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.42 = { by lemma 15 R->L }
% 39.24/5.42 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(multiplication(x0_2, multiplication(strong_iteration(x0_2), strong_iteration(x0_2))), addition(one, multiplication(x0_2, strong_iteration(x0_2))))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.42 = { by axiom 4 (additive_commutativity) R->L }
% 39.24/5.42 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(multiplication(x0_2, multiplication(strong_iteration(x0_2), strong_iteration(x0_2))), addition(multiplication(x0_2, strong_iteration(x0_2)), one))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.42 = { by axiom 4 (additive_commutativity) R->L }
% 39.24/5.42 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(addition(multiplication(x0_2, strong_iteration(x0_2)), one), multiplication(x0_2, multiplication(strong_iteration(x0_2), strong_iteration(x0_2))))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.42 = { by axiom 7 (additive_associativity) R->L }
% 39.24/5.42 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(multiplication(x0_2, strong_iteration(x0_2)), addition(one, multiplication(x0_2, multiplication(strong_iteration(x0_2), strong_iteration(x0_2)))))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.42 = { by axiom 4 (additive_commutativity) }
% 39.24/5.42 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(multiplication(x0_2, strong_iteration(x0_2)), addition(multiplication(x0_2, multiplication(strong_iteration(x0_2), strong_iteration(x0_2))), one))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.42 = { by axiom 7 (additive_associativity) }
% 39.24/5.43 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(addition(multiplication(x0_2, strong_iteration(x0_2)), multiplication(x0_2, multiplication(strong_iteration(x0_2), strong_iteration(x0_2)))), one)), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.43 = { by axiom 10 (distributivity1) R->L }
% 39.24/5.43 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(multiplication(x0_2, addition(strong_iteration(x0_2), multiplication(strong_iteration(x0_2), strong_iteration(x0_2)))), one)), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.43 = { by axiom 4 (additive_commutativity) }
% 39.24/5.43 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(one, multiplication(x0_2, addition(strong_iteration(x0_2), multiplication(strong_iteration(x0_2), strong_iteration(x0_2)))))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.43 = { by axiom 4 (additive_commutativity) }
% 39.24/5.43 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(one, multiplication(x0_2, addition(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2))))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.43 = { by axiom 1 (multiplicative_right_identity) R->L }
% 39.24/5.43 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(one, multiplication(x0_2, addition(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), multiplication(strong_iteration(x0_2), one))))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.43 = { by axiom 10 (distributivity1) R->L }
% 39.24/5.43 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(one, multiplication(x0_2, multiplication(strong_iteration(x0_2), addition(strong_iteration(x0_2), one))))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.43 = { by axiom 4 (additive_commutativity) }
% 39.24/5.43 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(one, multiplication(x0_2, multiplication(strong_iteration(x0_2), addition(one, strong_iteration(x0_2)))))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.43 = { by lemma 15 R->L }
% 39.24/5.43 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(one, multiplication(x0_2, multiplication(strong_iteration(x0_2), addition(one, addition(one, multiplication(x0_2, strong_iteration(x0_2)))))))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.43 = { by lemma 14 }
% 39.24/5.43 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(one, multiplication(x0_2, multiplication(strong_iteration(x0_2), addition(one, multiplication(x0_2, strong_iteration(x0_2))))))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.43 = { by lemma 15 }
% 39.24/5.43 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(one, multiplication(x0_2, multiplication(strong_iteration(x0_2), strong_iteration(x0_2))))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.43 = { by axiom 4 (additive_commutativity) R->L }
% 39.24/5.43 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(multiplication(x0_2, multiplication(strong_iteration(x0_2), strong_iteration(x0_2))), one)), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2)), true), true)
% 39.24/5.43 = { by axiom 1 (multiplicative_right_identity) R->L }
% 39.24/5.43 tuple(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), addition(multiplication(x0_2, multiplication(strong_iteration(x0_2), strong_iteration(x0_2))), one)), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(x0_2)), multiplication(strong_iteration(x0_2), one)), true), true)
% 39.24/5.43 = { by axiom 13 (infty_coinduction) }
% 39.24/5.43 tuple(true, true)
% 39.24/5.43 % SZS output end Proof
% 39.24/5.43
% 39.24/5.43 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------