%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE139+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n010.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 11:39:52 AM UTC 2026
% Result : Theorem 18.29s 2.88s
% Output : Proof 18.29s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE139+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.05 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.15/0.43 % Computer : n010.cluster.edu
% 0.15/0.43 % Model : x86_64 x86_64
% 0.15/0.43 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.43 % Memory : 8046.5625MB
% 0.15/0.43 % OS : Linux 6.8.0-71-generic
% 0.15/0.43 % CPULimit : 300
% 0.15/0.43 % WCLimit : 300
% 0.15/0.43 % DateTime : Sun Sep 27 13:12:31 UTC 2026
% 0.15/0.43 % CPUTime :
% 0.15/0.43 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 18.29/2.88 Command-line arguments: --no-flatten-goal
% 18.29/2.88
% 18.29/2.88 % SZS status Theorem
% 18.29/2.88
% 18.29/2.88 % SZS output start Proof
% 18.29/2.88 Axiom 1 (idempotence): addition(X, X) = X.
% 18.29/2.88 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 18.29/2.88 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 18.29/2.88 Axiom 4 (left_annihilation): multiplication(zero, X) = zero.
% 18.29/2.88 Axiom 5 (multiplicative_left_identity): multiplication(one, X) = X.
% 18.29/2.88 Axiom 6 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 18.29/2.88 Axiom 7 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
% 18.29/2.88 Axiom 8 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 18.29/2.88 Axiom 9 (star_unfold2): addition(one, multiplication(star(X), X)) = star(X).
% 18.29/2.88 Axiom 10 (isolation): strong_iteration(X) = addition(star(X), multiplication(strong_iteration(X), zero)).
% 18.29/2.88 Axiom 11 (distributivity1): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 18.29/2.88 Axiom 12 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 18.29/2.88
% 18.29/2.88 Lemma 13: addition(one, multiplication(X, strong_iteration(X))) = strong_iteration(X).
% 18.29/2.88 Proof:
% 18.29/2.88 addition(one, multiplication(X, strong_iteration(X)))
% 18.29/2.88 = { by axiom 2 (additive_commutativity) R->L }
% 18.29/2.88 addition(multiplication(X, strong_iteration(X)), one)
% 18.29/2.88 = { by axiom 7 (infty_unfold1) R->L }
% 18.29/2.88 strong_iteration(X)
% 18.29/2.88
% 18.29/2.88 Lemma 14: addition(X, addition(X, Y)) = addition(X, Y).
% 18.29/2.88 Proof:
% 18.29/2.88 addition(X, addition(X, Y))
% 18.29/2.88 = { by axiom 6 (additive_associativity) }
% 18.29/2.88 addition(addition(X, X), Y)
% 18.29/2.88 = { by axiom 1 (idempotence) }
% 18.29/2.88 addition(X, Y)
% 18.29/2.88
% 18.29/2.88 Lemma 15: addition(one, strong_iteration(X)) = strong_iteration(X).
% 18.29/2.88 Proof:
% 18.29/2.88 addition(one, strong_iteration(X))
% 18.29/2.88 = { by lemma 13 R->L }
% 18.29/2.89 addition(one, addition(one, multiplication(X, strong_iteration(X))))
% 18.29/2.89 = { by lemma 14 }
% 18.29/2.89 addition(one, multiplication(X, strong_iteration(X)))
% 18.29/2.89 = { by lemma 13 }
% 18.29/2.89 strong_iteration(X)
% 18.29/2.89
% 18.29/2.89 Lemma 16: addition(one, multiplication(X, zero)) = strong_iteration(multiplication(X, zero)).
% 18.29/2.89 Proof:
% 18.29/2.89 addition(one, multiplication(X, zero))
% 18.29/2.89 = { by axiom 4 (left_annihilation) R->L }
% 18.29/2.89 addition(one, multiplication(X, multiplication(zero, strong_iteration(multiplication(X, zero)))))
% 18.29/2.89 = { by axiom 8 (multiplicative_associativity) }
% 18.29/2.89 addition(one, multiplication(multiplication(X, zero), strong_iteration(multiplication(X, zero))))
% 18.29/2.89 = { by lemma 13 }
% 18.29/2.89 strong_iteration(multiplication(X, zero))
% 18.29/2.89
% 18.29/2.89 Lemma 17: multiplication(addition(X, one), Y) = addition(Y, multiplication(X, Y)).
% 18.29/2.89 Proof:
% 18.29/2.89 multiplication(addition(X, one), Y)
% 18.29/2.89 = { by axiom 2 (additive_commutativity) R->L }
% 18.29/2.89 multiplication(addition(one, X), Y)
% 18.29/2.89 = { by axiom 12 (distributivity2) }
% 18.29/2.89 addition(multiplication(one, Y), multiplication(X, Y))
% 18.29/2.89 = { by axiom 5 (multiplicative_left_identity) }
% 18.29/2.89 addition(Y, multiplication(X, Y))
% 18.29/2.89
% 18.29/2.89 Lemma 18: multiplication(addition(X, one), strong_iteration(X)) = strong_iteration(X).
% 18.29/2.89 Proof:
% 18.29/2.89 multiplication(addition(X, one), strong_iteration(X))
% 18.29/2.89 = { by lemma 17 }
% 18.29/2.89 addition(strong_iteration(X), multiplication(X, strong_iteration(X)))
% 18.29/2.89 = { by axiom 2 (additive_commutativity) R->L }
% 18.29/2.89 addition(multiplication(X, strong_iteration(X)), strong_iteration(X))
% 18.29/2.89 = { by lemma 13 R->L }
% 18.29/2.89 addition(multiplication(X, strong_iteration(X)), addition(one, multiplication(X, strong_iteration(X))))
% 18.29/2.89 = { by axiom 2 (additive_commutativity) R->L }
% 18.29/2.89 addition(multiplication(X, strong_iteration(X)), addition(multiplication(X, strong_iteration(X)), one))
% 18.29/2.89 = { by lemma 14 }
% 18.29/2.89 addition(multiplication(X, strong_iteration(X)), one)
% 18.29/2.89 = { by axiom 2 (additive_commutativity) }
% 18.29/2.89 addition(one, multiplication(X, strong_iteration(X)))
% 18.29/2.89 = { by lemma 13 }
% 18.29/2.89 strong_iteration(X)
% 18.29/2.89
% 18.29/2.89 Lemma 19: addition(star(X), addition(Y, multiplication(strong_iteration(X), zero))) = addition(Y, strong_iteration(X)).
% 18.29/2.89 Proof:
% 18.29/2.89 addition(star(X), addition(Y, multiplication(strong_iteration(X), zero)))
% 18.29/2.89 = { by axiom 2 (additive_commutativity) R->L }
% 18.29/2.89 addition(star(X), addition(multiplication(strong_iteration(X), zero), Y))
% 18.29/2.89 = { by axiom 6 (additive_associativity) }
% 18.29/2.89 addition(addition(star(X), multiplication(strong_iteration(X), zero)), Y)
% 18.29/2.89 = { by axiom 10 (isolation) R->L }
% 18.29/2.89 addition(strong_iteration(X), Y)
% 18.29/2.89 = { by axiom 2 (additive_commutativity) }
% 18.29/2.89 addition(Y, strong_iteration(X))
% 18.29/2.89
% 18.29/2.89 Goal 1 (goals): strong_iteration(x0) = addition(multiplication(strong_iteration(x0), x0), one).
% 18.29/2.89 Proof:
% 18.29/2.89 strong_iteration(x0)
% 18.29/2.89 = { by lemma 18 R->L }
% 18.29/2.89 multiplication(addition(x0, one), strong_iteration(x0))
% 18.29/2.89 = { by lemma 15 R->L }
% 18.29/2.89 multiplication(addition(x0, one), addition(one, strong_iteration(x0)))
% 18.29/2.89 = { by axiom 11 (distributivity1) }
% 18.29/2.89 addition(multiplication(addition(x0, one), one), multiplication(addition(x0, one), strong_iteration(x0)))
% 18.29/2.89 = { by axiom 3 (multiplicative_right_identity) }
% 18.29/2.89 addition(addition(x0, one), multiplication(addition(x0, one), strong_iteration(x0)))
% 18.29/2.89 = { by lemma 18 }
% 18.29/2.89 addition(addition(x0, one), strong_iteration(x0))
% 18.29/2.89 = { by axiom 6 (additive_associativity) R->L }
% 18.29/2.89 addition(x0, addition(one, strong_iteration(x0)))
% 18.29/2.89 = { by lemma 15 }
% 18.29/2.89 addition(x0, strong_iteration(x0))
% 18.29/2.89 = { by lemma 19 R->L }
% 18.29/2.89 addition(star(x0), addition(x0, multiplication(strong_iteration(x0), zero)))
% 18.29/2.89 = { by axiom 4 (left_annihilation) R->L }
% 18.29/2.89 addition(star(x0), addition(x0, multiplication(strong_iteration(x0), multiplication(zero, x0))))
% 18.29/2.89 = { by axiom 8 (multiplicative_associativity) }
% 18.29/2.89 addition(star(x0), addition(x0, multiplication(multiplication(strong_iteration(x0), zero), x0)))
% 18.29/2.89 = { by lemma 17 R->L }
% 18.29/2.89 addition(star(x0), multiplication(addition(multiplication(strong_iteration(x0), zero), one), x0))
% 18.29/2.89 = { by axiom 2 (additive_commutativity) }
% 18.29/2.89 addition(star(x0), multiplication(addition(one, multiplication(strong_iteration(x0), zero)), x0))
% 18.29/2.89 = { by lemma 16 }
% 18.29/2.89 addition(star(x0), multiplication(strong_iteration(multiplication(strong_iteration(x0), zero)), x0))
% 18.29/2.89 = { by axiom 9 (star_unfold2) R->L }
% 18.29/2.89 addition(addition(one, multiplication(star(x0), x0)), multiplication(strong_iteration(multiplication(strong_iteration(x0), zero)), x0))
% 18.29/2.89 = { by axiom 6 (additive_associativity) R->L }
% 18.29/2.89 addition(one, addition(multiplication(star(x0), x0), multiplication(strong_iteration(multiplication(strong_iteration(x0), zero)), x0)))
% 18.29/2.89 = { by axiom 2 (additive_commutativity) }
% 18.29/2.89 addition(one, addition(multiplication(strong_iteration(multiplication(strong_iteration(x0), zero)), x0), multiplication(star(x0), x0)))
% 18.29/2.89 = { by axiom 12 (distributivity2) R->L }
% 18.29/2.89 addition(one, multiplication(addition(strong_iteration(multiplication(strong_iteration(x0), zero)), star(x0)), x0))
% 18.29/2.89 = { by axiom 2 (additive_commutativity) }
% 18.29/2.89 addition(one, multiplication(addition(star(x0), strong_iteration(multiplication(strong_iteration(x0), zero))), x0))
% 18.29/2.89 = { by lemma 16 R->L }
% 18.29/2.89 addition(one, multiplication(addition(star(x0), addition(one, multiplication(strong_iteration(x0), zero))), x0))
% 18.29/2.89 = { by lemma 19 }
% 18.29/2.89 addition(one, multiplication(addition(one, strong_iteration(x0)), x0))
% 18.29/2.89 = { by lemma 15 }
% 18.29/2.89 addition(one, multiplication(strong_iteration(x0), x0))
% 18.29/2.89 = { by axiom 2 (additive_commutativity) R->L }
% 18.29/2.89 addition(multiplication(strong_iteration(x0), x0), one)
% 18.29/2.89 % SZS output end Proof
% 18.29/2.89
% 18.29/2.89 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------