%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE169+1 : TPTP v9.3.1. Released v5.2.0.
% 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 11:39:57 AM UTC 2026
% Result : Theorem 1.55s 0.77s
% Output : Proof 1.55s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : KLE169+1 : TPTP v9.3.1. Released v5.2.0.
% 0.00/0.07 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.19/0.45 % Computer : n010.cluster.edu
% 0.19/0.45 % Model : x86_64 x86_64
% 0.19/0.45 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.19/0.45 % Memory : 8046.5625MB
% 0.19/0.45 % OS : Linux 6.8.0-71-generic
% 0.19/0.45 % CPULimit : 300
% 0.19/0.45 % WCLimit : 300
% 0.19/0.45 % DateTime : Sun Sep 27 13:14:46 UTC 2026
% 0.19/0.45 % CPUTime :
% 0.19/0.45 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 1.55/0.77 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 1.55/0.77
% 1.55/0.77 % SZS status Theorem
% 1.55/0.77
% 1.55/0.78 % SZS output start Proof
% 1.55/0.78 Axiom 1 (multiplicative_left_identity): multiplication(one, X) = X.
% 1.55/0.78 Axiom 2 (additive_idempotence): addition(X, X) = X.
% 1.55/0.78 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 1.55/0.79 Axiom 4 (an): sigma = addition(a, b).
% 1.55/0.79 Axiom 5 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 1.55/0.79 Axiom 6 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 1.55/0.79 Axiom 7 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 1.55/0.79 Axiom 8 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 1.55/0.79 Axiom 9 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 1.55/0.79 Axiom 10 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 1.55/0.79 Axiom 11 (order): ifeq3(addition(X, Y), Y, leq(X, Y), true) = true.
% 1.55/0.79 Axiom 12 (order_1): ifeq2(leq(X, Y), true, addition(X, Y), Y) = Y.
% 1.55/0.79
% 1.55/0.79 Lemma 13: addition(X, addition(X, Y)) = addition(X, Y).
% 1.55/0.79 Proof:
% 1.55/0.79 addition(X, addition(X, Y))
% 1.55/0.79 = { by axiom 5 (additive_associativity) }
% 1.55/0.79 addition(addition(X, X), Y)
% 1.55/0.79 = { by axiom 2 (additive_idempotence) }
% 1.55/0.79 addition(X, Y)
% 1.55/0.79
% 1.55/0.79 Lemma 14: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
% 1.55/0.79 Proof:
% 1.55/0.79 addition(X, multiplication(Y, X))
% 1.55/0.79 = { by axiom 1 (multiplicative_left_identity) R->L }
% 1.55/0.79 addition(multiplication(one, X), multiplication(Y, X))
% 1.55/0.79 = { by axiom 9 (left_distributivity) R->L }
% 1.55/0.79 multiplication(addition(one, Y), X)
% 1.55/0.79 = { by axiom 3 (additive_commutativity) }
% 1.55/0.79 multiplication(addition(Y, one), X)
% 1.55/0.79
% 1.55/0.79 Lemma 15: addition(one, multiplication(addition(X, one), star(X))) = star(X).
% 1.55/0.79 Proof:
% 1.55/0.79 addition(one, multiplication(addition(X, one), star(X)))
% 1.55/0.79 = { by lemma 14 R->L }
% 1.55/0.79 addition(one, addition(star(X), multiplication(X, star(X))))
% 1.55/0.79 = { by axiom 3 (additive_commutativity) R->L }
% 1.55/0.79 addition(one, addition(multiplication(X, star(X)), star(X)))
% 1.55/0.79 = { by axiom 5 (additive_associativity) }
% 1.55/0.79 addition(addition(one, multiplication(X, star(X))), star(X))
% 1.55/0.79 = { by axiom 7 (ifeq_axiom) R->L }
% 1.55/0.79 ifeq2(true, true, addition(addition(one, multiplication(X, star(X))), star(X)), star(X))
% 1.55/0.79 = { by axiom 10 (star_unfold_right) R->L }
% 1.55/0.79 ifeq2(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(addition(one, multiplication(X, star(X))), star(X)), star(X))
% 1.55/0.79 = { by axiom 12 (order_1) }
% 1.55/0.79 star(X)
% 1.55/0.79
% 1.55/0.79 Goal 1 (a): leq(multiplication(a, multiplication(b, a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a)))) = true.
% 1.55/0.79 Proof:
% 1.55/0.79 leq(multiplication(a, multiplication(b, a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a))))
% 1.55/0.79 = { by lemma 15 R->L }
% 1.55/0.79 leq(multiplication(a, multiplication(b, a)), multiplication(addition(one, multiplication(addition(sigma, one), star(sigma))), multiplication(a, multiplication(sigma, a))))
% 1.55/0.79 = { by lemma 13 R->L }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), multiplication(addition(one, addition(one, multiplication(addition(sigma, one), star(sigma)))), multiplication(a, multiplication(sigma, a))))
% 1.55/0.80 = { by lemma 15 }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), multiplication(addition(one, star(sigma)), multiplication(a, multiplication(sigma, a))))
% 1.55/0.80 = { by axiom 3 (additive_commutativity) R->L }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), multiplication(addition(star(sigma), one), multiplication(a, multiplication(sigma, a))))
% 1.55/0.80 = { by lemma 14 R->L }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(sigma, a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a)))))
% 1.55/0.80 = { by axiom 4 (an) }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(addition(a, b), a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a)))))
% 1.55/0.80 = { by axiom 3 (additive_commutativity) R->L }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(addition(b, a), a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a)))))
% 1.55/0.80 = { by lemma 13 R->L }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(addition(b, addition(b, a)), a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a)))))
% 1.55/0.80 = { by axiom 3 (additive_commutativity) }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(addition(b, addition(a, b)), a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a)))))
% 1.55/0.80 = { by axiom 4 (an) R->L }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(addition(b, sigma), a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a)))))
% 1.55/0.80 = { by axiom 9 (left_distributivity) }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), addition(multiplication(a, addition(multiplication(b, a), multiplication(sigma, a))), multiplication(star(sigma), multiplication(a, multiplication(sigma, a)))))
% 1.55/0.80 = { by axiom 3 (additive_commutativity) R->L }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), addition(multiplication(a, addition(multiplication(sigma, a), multiplication(b, a))), multiplication(star(sigma), multiplication(a, multiplication(sigma, a)))))
% 1.55/0.80 = { by axiom 8 (right_distributivity) }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), addition(addition(multiplication(a, multiplication(sigma, a)), multiplication(a, multiplication(b, a))), multiplication(star(sigma), multiplication(a, multiplication(sigma, a)))))
% 1.55/0.80 = { by axiom 5 (additive_associativity) R->L }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(sigma, a)), addition(multiplication(a, multiplication(b, a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a))))))
% 1.55/0.80 = { by axiom 3 (additive_commutativity) R->L }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), addition(addition(multiplication(a, multiplication(b, a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a)))), multiplication(a, multiplication(sigma, a))))
% 1.55/0.80 = { by axiom 5 (additive_associativity) R->L }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(b, a)), addition(multiplication(star(sigma), multiplication(a, multiplication(sigma, a))), multiplication(a, multiplication(sigma, a)))))
% 1.55/0.80 = { by axiom 3 (additive_commutativity) }
% 1.55/0.80 leq(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(sigma, a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a))))))
% 1.55/0.80 = { by axiom 6 (ifeq_axiom) R->L }
% 1.55/0.80 ifeq3(addition(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(sigma, a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a))))), addition(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(sigma, a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a))))), leq(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(sigma, a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a)))))), true)
% 1.55/0.80 = { by lemma 13 R->L }
% 1.55/0.80 ifeq3(addition(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(sigma, a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a)))))), addition(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(sigma, a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a))))), leq(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(b, a)), addition(multiplication(a, multiplication(sigma, a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a)))))), true)
% 1.55/0.80 = { by axiom 11 (order) }
% 1.55/0.80 true
% 1.55/0.80 % SZS output end Proof
% 1.55/0.80
% 1.55/0.80 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------