%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE139+2 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n019.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 19.37s 2.90s
% Output : Proof 19.37s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE139+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 : n019.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 13:12:18 UTC 2026
% 0.09/0.36 % CPUTime :
% 0.09/0.36 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 19.37/2.90 Command-line arguments: --no-flatten-goal
% 19.37/2.90
% 19.37/2.90 % SZS status Theorem
% 19.37/2.90
% 19.37/2.91 % SZS output start Proof
% 19.37/2.91 Axiom 1 (idempotence): addition(X, X) = X.
% 19.37/2.91 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 19.37/2.91 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 19.37/2.91 Axiom 4 (left_annihilation): multiplication(zero, X) = zero.
% 19.37/2.91 Axiom 5 (multiplicative_left_identity): multiplication(one, X) = X.
% 19.37/2.91 Axiom 6 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 19.37/2.91 Axiom 7 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 19.37/2.91 Axiom 8 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
% 19.37/2.91 Axiom 9 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 19.37/2.91 Axiom 10 (star_unfold2): addition(one, multiplication(star(X), X)) = star(X).
% 19.37/2.91 Axiom 11 (isolation): strong_iteration(X) = addition(star(X), multiplication(strong_iteration(X), zero)).
% 19.37/2.91 Axiom 12 (distributivity1): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 19.37/2.91 Axiom 13 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 19.37/2.91 Axiom 14 (order): ifeq3(addition(X, Y), Y, leq(X, Y), true) = true.
% 19.37/2.91
% 19.37/2.91 Lemma 15: addition(one, multiplication(X, strong_iteration(X))) = strong_iteration(X).
% 19.37/2.91 Proof:
% 19.37/2.91 addition(one, multiplication(X, strong_iteration(X)))
% 19.37/2.91 = { by axiom 2 (additive_commutativity) R->L }
% 19.37/2.91 addition(multiplication(X, strong_iteration(X)), one)
% 19.37/2.91 = { by axiom 8 (infty_unfold1) R->L }
% 19.37/2.91 strong_iteration(X)
% 19.37/2.91
% 19.37/2.91 Lemma 16: addition(X, addition(X, Y)) = addition(X, Y).
% 19.37/2.91 Proof:
% 19.37/2.91 addition(X, addition(X, Y))
% 19.37/2.91 = { by axiom 7 (additive_associativity) }
% 19.37/2.91 addition(addition(X, X), Y)
% 19.37/2.91 = { by axiom 1 (idempotence) }
% 19.37/2.91 addition(X, Y)
% 19.37/2.91
% 19.37/2.91 Lemma 17: addition(one, strong_iteration(X)) = strong_iteration(X).
% 19.37/2.91 Proof:
% 19.37/2.91 addition(one, strong_iteration(X))
% 19.37/2.91 = { by lemma 15 R->L }
% 19.37/2.91 addition(one, addition(one, multiplication(X, strong_iteration(X))))
% 19.37/2.91 = { by lemma 16 }
% 19.37/2.91 addition(one, multiplication(X, strong_iteration(X)))
% 19.37/2.91 = { by lemma 15 }
% 19.37/2.91 strong_iteration(X)
% 19.37/2.91
% 19.37/2.92 Lemma 18: addition(one, multiplication(X, zero)) = strong_iteration(multiplication(X, zero)).
% 19.37/2.92 Proof:
% 19.37/2.92 addition(one, multiplication(X, zero))
% 19.37/2.92 = { by axiom 4 (left_annihilation) R->L }
% 19.37/2.92 addition(one, multiplication(X, multiplication(zero, strong_iteration(multiplication(X, zero)))))
% 19.37/2.92 = { by axiom 9 (multiplicative_associativity) }
% 19.37/2.92 addition(one, multiplication(multiplication(X, zero), strong_iteration(multiplication(X, zero))))
% 19.37/2.92 = { by lemma 15 }
% 19.37/2.92 strong_iteration(multiplication(X, zero))
% 19.37/2.92
% 19.37/2.92 Lemma 19: multiplication(addition(X, one), Y) = addition(Y, multiplication(X, Y)).
% 19.37/2.92 Proof:
% 19.37/2.92 multiplication(addition(X, one), Y)
% 19.37/2.92 = { by axiom 2 (additive_commutativity) R->L }
% 19.37/2.92 multiplication(addition(one, X), Y)
% 19.37/2.92 = { by axiom 13 (distributivity2) }
% 19.37/2.92 addition(multiplication(one, Y), multiplication(X, Y))
% 19.37/2.92 = { by axiom 5 (multiplicative_left_identity) }
% 19.37/2.92 addition(Y, multiplication(X, Y))
% 19.37/2.92
% 19.37/2.92 Lemma 20: addition(star(X), addition(Y, multiplication(strong_iteration(X), zero))) = addition(Y, strong_iteration(X)).
% 19.37/2.92 Proof:
% 19.37/2.92 addition(star(X), addition(Y, multiplication(strong_iteration(X), zero)))
% 19.37/2.92 = { by axiom 2 (additive_commutativity) R->L }
% 19.37/2.92 addition(star(X), addition(multiplication(strong_iteration(X), zero), Y))
% 19.37/2.92 = { by axiom 7 (additive_associativity) }
% 19.37/2.92 addition(addition(star(X), multiplication(strong_iteration(X), zero)), Y)
% 19.37/2.92 = { by axiom 11 (isolation) R->L }
% 19.37/2.92 addition(strong_iteration(X), Y)
% 19.37/2.92 = { by axiom 2 (additive_commutativity) }
% 19.37/2.92 addition(Y, strong_iteration(X))
% 19.37/2.92
% 19.37/2.92 Lemma 21: multiplication(addition(X, one), strong_iteration(X)) = strong_iteration(X).
% 19.37/2.92 Proof:
% 19.37/2.92 multiplication(addition(X, one), strong_iteration(X))
% 19.37/2.92 = { by lemma 19 }
% 19.37/2.92 addition(strong_iteration(X), multiplication(X, strong_iteration(X)))
% 19.37/2.92 = { by axiom 2 (additive_commutativity) R->L }
% 19.37/2.92 addition(multiplication(X, strong_iteration(X)), strong_iteration(X))
% 19.37/2.92 = { by lemma 15 R->L }
% 19.37/2.92 addition(multiplication(X, strong_iteration(X)), addition(one, multiplication(X, strong_iteration(X))))
% 19.37/2.92 = { by axiom 2 (additive_commutativity) R->L }
% 19.37/2.92 addition(multiplication(X, strong_iteration(X)), addition(multiplication(X, strong_iteration(X)), one))
% 19.37/2.92 = { by lemma 16 }
% 19.37/2.92 addition(multiplication(X, strong_iteration(X)), one)
% 19.37/2.92 = { by axiom 2 (additive_commutativity) }
% 19.37/2.92 addition(one, multiplication(X, strong_iteration(X)))
% 19.37/2.92 = { by lemma 15 }
% 19.37/2.92 strong_iteration(X)
% 19.37/2.92
% 19.37/2.92 Lemma 22: addition(one, multiplication(strong_iteration(X), X)) = strong_iteration(X).
% 19.37/2.92 Proof:
% 19.37/2.92 addition(one, multiplication(strong_iteration(X), X))
% 19.37/2.92 = { by lemma 17 R->L }
% 19.37/2.92 addition(one, multiplication(addition(one, strong_iteration(X)), X))
% 19.37/2.92 = { by lemma 20 R->L }
% 19.37/2.92 addition(one, multiplication(addition(star(X), addition(one, multiplication(strong_iteration(X), zero))), X))
% 19.37/2.92 = { by lemma 18 }
% 19.37/2.92 addition(one, multiplication(addition(star(X), strong_iteration(multiplication(strong_iteration(X), zero))), X))
% 19.37/2.92 = { by axiom 2 (additive_commutativity) R->L }
% 19.37/2.92 addition(one, multiplication(addition(strong_iteration(multiplication(strong_iteration(X), zero)), star(X)), X))
% 19.37/2.92 = { by axiom 13 (distributivity2) }
% 19.37/2.92 addition(one, addition(multiplication(strong_iteration(multiplication(strong_iteration(X), zero)), X), multiplication(star(X), X)))
% 19.37/2.92 = { by axiom 2 (additive_commutativity) R->L }
% 19.37/2.92 addition(one, addition(multiplication(star(X), X), multiplication(strong_iteration(multiplication(strong_iteration(X), zero)), X)))
% 19.37/2.92 = { by axiom 7 (additive_associativity) }
% 19.37/2.92 addition(addition(one, multiplication(star(X), X)), multiplication(strong_iteration(multiplication(strong_iteration(X), zero)), X))
% 19.37/2.92 = { by axiom 10 (star_unfold2) }
% 19.37/2.92 addition(star(X), multiplication(strong_iteration(multiplication(strong_iteration(X), zero)), X))
% 19.37/2.92 = { by lemma 18 R->L }
% 19.37/2.92 addition(star(X), multiplication(addition(one, multiplication(strong_iteration(X), zero)), X))
% 19.37/2.92 = { by axiom 2 (additive_commutativity) R->L }
% 19.37/2.92 addition(star(X), multiplication(addition(multiplication(strong_iteration(X), zero), one), X))
% 19.37/2.92 = { by lemma 19 }
% 19.37/2.92 addition(star(X), addition(X, multiplication(multiplication(strong_iteration(X), zero), X)))
% 19.37/2.92 = { by axiom 9 (multiplicative_associativity) R->L }
% 19.37/2.92 addition(star(X), addition(X, multiplication(strong_iteration(X), multiplication(zero, X))))
% 19.37/2.92 = { by axiom 4 (left_annihilation) }
% 19.37/2.92 addition(star(X), addition(X, multiplication(strong_iteration(X), zero)))
% 19.37/2.92 = { by lemma 20 }
% 19.37/2.92 addition(X, strong_iteration(X))
% 19.37/2.92 = { by lemma 17 R->L }
% 19.37/2.92 addition(X, addition(one, strong_iteration(X)))
% 19.37/2.92 = { by axiom 7 (additive_associativity) }
% 19.37/2.92 addition(addition(X, one), strong_iteration(X))
% 19.37/2.92 = { by lemma 21 R->L }
% 19.37/2.92 addition(addition(X, one), multiplication(addition(X, one), strong_iteration(X)))
% 19.37/2.92 = { by axiom 3 (multiplicative_right_identity) R->L }
% 19.37/2.92 addition(multiplication(addition(X, one), one), multiplication(addition(X, one), strong_iteration(X)))
% 19.37/2.92 = { by axiom 12 (distributivity1) R->L }
% 19.37/2.92 multiplication(addition(X, one), addition(one, strong_iteration(X)))
% 19.37/2.92 = { by lemma 17 }
% 19.37/2.92 multiplication(addition(X, one), strong_iteration(X))
% 19.37/2.92 = { by lemma 21 }
% 19.37/2.92 strong_iteration(X)
% 19.37/2.92
% 19.37/2.92 Goal 1 (goals): tuple(leq(addition(multiplication(strong_iteration(x0), x0), one), strong_iteration(x0)), leq(strong_iteration(x0_2), addition(multiplication(strong_iteration(x0_2), x0_2), one))) = tuple(true, true).
% 19.37/2.92 Proof:
% 19.37/2.92 tuple(leq(addition(multiplication(strong_iteration(x0), x0), one), strong_iteration(x0)), leq(strong_iteration(x0_2), addition(multiplication(strong_iteration(x0_2), x0_2), one)))
% 19.37/2.92 = { by axiom 2 (additive_commutativity) }
% 19.37/2.92 tuple(leq(addition(one, multiplication(strong_iteration(x0), x0)), strong_iteration(x0)), leq(strong_iteration(x0_2), addition(multiplication(strong_iteration(x0_2), x0_2), one)))
% 19.37/2.92 = { by axiom 2 (additive_commutativity) }
% 19.37/2.92 tuple(leq(addition(one, multiplication(strong_iteration(x0), x0)), strong_iteration(x0)), leq(strong_iteration(x0_2), addition(one, multiplication(strong_iteration(x0_2), x0_2))))
% 19.37/2.92 = { by lemma 22 }
% 19.37/2.92 tuple(leq(strong_iteration(x0), strong_iteration(x0)), leq(strong_iteration(x0_2), addition(one, multiplication(strong_iteration(x0_2), x0_2))))
% 19.37/2.92 = { by axiom 6 (ifeq_axiom) R->L }
% 19.37/2.92 tuple(ifeq3(strong_iteration(x0), strong_iteration(x0), leq(strong_iteration(x0), strong_iteration(x0)), true), leq(strong_iteration(x0_2), addition(one, multiplication(strong_iteration(x0_2), x0_2))))
% 19.37/2.92 = { by axiom 1 (idempotence) R->L }
% 19.37/2.92 tuple(ifeq3(addition(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0), leq(strong_iteration(x0), strong_iteration(x0)), true), leq(strong_iteration(x0_2), addition(one, multiplication(strong_iteration(x0_2), x0_2))))
% 19.37/2.92 = { by axiom 14 (order) }
% 19.37/2.92 tuple(true, leq(strong_iteration(x0_2), addition(one, multiplication(strong_iteration(x0_2), x0_2))))
% 19.37/2.92 = { by lemma 22 }
% 19.37/2.92 tuple(true, leq(strong_iteration(x0_2), strong_iteration(x0_2)))
% 19.37/2.92 = { by axiom 6 (ifeq_axiom) R->L }
% 19.37/2.92 tuple(true, ifeq3(strong_iteration(x0_2), strong_iteration(x0_2), leq(strong_iteration(x0_2), strong_iteration(x0_2)), true))
% 19.37/2.92 = { by axiom 1 (idempotence) R->L }
% 19.37/2.92 tuple(true, ifeq3(addition(strong_iteration(x0_2), strong_iteration(x0_2)), strong_iteration(x0_2), leq(strong_iteration(x0_2), strong_iteration(x0_2)), true))
% 19.37/2.92 = { by axiom 14 (order) }
% 19.37/2.92 tuple(true, true)
% 19.37/2.92 % SZS output end Proof
% 19.37/2.92
% 19.37/2.92 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------