%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE040+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n026.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 11:39:41 AM UTC 2026
% Result : Theorem 3.98s 0.94s
% Output : Proof 3.98s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE040+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.06/0.34 % Computer : n026.cluster.edu
% 0.06/0.34 % Model : x86_64 x86_64
% 0.06/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.06/0.34 % Memory : 8046.5625MB
% 0.06/0.34 % OS : Linux 6.8.0-71-generic
% 0.06/0.35 % CPULimit : 300
% 0.06/0.35 % WCLimit : 300
% 0.06/0.35 % DateTime : Sun Sep 27 13:08:11 UTC 2026
% 0.06/0.35 % CPUTime :
% 0.06/0.35 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 3.98/0.94 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
% 3.98/0.94
% 3.98/0.94 % SZS status Theorem
% 3.98/0.94
% 3.98/0.95 % SZS output start Proof
% 3.98/0.95 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 3.98/0.95 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 3.98/0.95 Axiom 3 (additive_idempotence): addition(X, X) = X.
% 3.98/0.95 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 3.98/0.95 Axiom 5 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 3.98/0.95 Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 3.98/0.95 Axiom 7 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 3.98/0.95 Axiom 8 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 3.98/0.95 Axiom 9 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 3.98/0.95 Axiom 10 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 3.98/0.95 Axiom 11 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 3.98/0.95 Axiom 12 (star_unfold_left): leq(addition(one, multiplication(star(X), X)), star(X)) = true.
% 3.98/0.95 Axiom 13 (order): ifeq3(addition(X, Y), Y, leq(X, Y), true) = true.
% 3.98/0.95 Axiom 14 (order_1): ifeq2(leq(X, Y), true, addition(X, Y), Y) = Y.
% 3.98/0.95 Axiom 15 (star_induction_right): ifeq(leq(addition(multiplication(X, Y), Z), X), true, leq(multiplication(Z, star(Y)), X), true) = true.
% 3.98/0.95
% 3.98/0.95 Lemma 16: addition(X, addition(X, Y)) = addition(X, Y).
% 3.98/0.95 Proof:
% 3.98/0.95 addition(X, addition(X, Y))
% 3.98/0.95 = { by axiom 5 (additive_associativity) }
% 3.98/0.95 addition(addition(X, X), Y)
% 3.98/0.95 = { by axiom 3 (additive_idempotence) }
% 3.98/0.95 addition(X, Y)
% 3.98/0.95
% 3.98/0.95 Lemma 17: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
% 3.98/0.95 Proof:
% 3.98/0.95 addition(X, multiplication(X, Y))
% 3.98/0.95 = { by axiom 1 (multiplicative_right_identity) R->L }
% 3.98/0.95 addition(multiplication(X, one), multiplication(X, Y))
% 3.98/0.95 = { by axiom 9 (right_distributivity) R->L }
% 3.98/0.95 multiplication(X, addition(one, Y))
% 3.98/0.95 = { by axiom 4 (additive_commutativity) }
% 3.98/0.95 multiplication(X, addition(Y, one))
% 3.98/0.95
% 3.98/0.95 Lemma 18: addition(one, multiplication(star(X), addition(X, one))) = star(X).
% 3.98/0.95 Proof:
% 3.98/0.95 addition(one, multiplication(star(X), addition(X, one)))
% 3.98/0.95 = { by lemma 17 R->L }
% 3.98/0.95 addition(one, addition(star(X), multiplication(star(X), X)))
% 3.98/0.95 = { by axiom 4 (additive_commutativity) R->L }
% 3.98/0.95 addition(one, addition(multiplication(star(X), X), star(X)))
% 3.98/0.95 = { by axiom 5 (additive_associativity) }
% 3.98/0.95 addition(addition(one, multiplication(star(X), X)), star(X))
% 3.98/0.95 = { by axiom 8 (ifeq_axiom) R->L }
% 3.98/0.95 ifeq2(true, true, addition(addition(one, multiplication(star(X), X)), star(X)), star(X))
% 3.98/0.95 = { by axiom 12 (star_unfold_left) R->L }
% 3.98/0.95 ifeq2(leq(addition(one, multiplication(star(X), X)), star(X)), true, addition(addition(one, multiplication(star(X), X)), star(X)), star(X))
% 3.98/0.95 = { by axiom 14 (order_1) }
% 3.98/0.95 star(X)
% 3.98/0.95
% 3.98/0.95 Lemma 19: addition(one, multiplication(addition(X, one), star(X))) = star(X).
% 3.98/0.95 Proof:
% 3.98/0.95 addition(one, multiplication(addition(X, one), star(X)))
% 3.98/0.95 = { by axiom 4 (additive_commutativity) R->L }
% 3.98/0.95 addition(one, multiplication(addition(one, X), star(X)))
% 3.98/0.95 = { by axiom 10 (left_distributivity) }
% 3.98/0.96 addition(one, addition(multiplication(one, star(X)), multiplication(X, star(X))))
% 3.98/0.96 = { by axiom 2 (multiplicative_left_identity) }
% 3.98/0.96 addition(one, addition(star(X), multiplication(X, star(X))))
% 3.98/0.96 = { by axiom 4 (additive_commutativity) R->L }
% 3.98/0.96 addition(one, addition(multiplication(X, star(X)), star(X)))
% 3.98/0.96 = { by axiom 5 (additive_associativity) }
% 3.98/0.96 addition(addition(one, multiplication(X, star(X))), star(X))
% 3.98/0.96 = { by axiom 8 (ifeq_axiom) R->L }
% 3.98/0.96 ifeq2(true, true, addition(addition(one, multiplication(X, star(X))), star(X)), star(X))
% 3.98/0.96 = { by axiom 11 (star_unfold_right) R->L }
% 3.98/0.96 ifeq2(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(addition(one, multiplication(X, star(X))), star(X)), star(X))
% 3.98/0.96 = { by axiom 14 (order_1) }
% 3.98/0.96 star(X)
% 3.98/0.96
% 3.98/0.96 Goal 1 (goals): multiplication(star(x0), star(x0)) = star(x0).
% 3.98/0.96 Proof:
% 3.98/0.96 multiplication(star(x0), star(x0))
% 3.98/0.96 = { by lemma 19 R->L }
% 3.98/0.96 multiplication(star(x0), addition(one, multiplication(addition(x0, one), star(x0))))
% 3.98/0.96 = { by lemma 16 R->L }
% 3.98/0.96 multiplication(star(x0), addition(one, addition(one, multiplication(addition(x0, one), star(x0)))))
% 3.98/0.96 = { by lemma 19 }
% 3.98/0.96 multiplication(star(x0), addition(one, star(x0)))
% 3.98/0.96 = { by axiom 4 (additive_commutativity) R->L }
% 3.98/0.96 multiplication(star(x0), addition(star(x0), one))
% 3.98/0.96 = { by axiom 9 (right_distributivity) }
% 3.98/0.96 addition(multiplication(star(x0), star(x0)), multiplication(star(x0), one))
% 3.98/0.96 = { by axiom 1 (multiplicative_right_identity) }
% 3.98/0.96 addition(multiplication(star(x0), star(x0)), star(x0))
% 3.98/0.96 = { by axiom 8 (ifeq_axiom) R->L }
% 3.98/0.96 ifeq2(true, true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by axiom 15 (star_induction_right) R->L }
% 3.98/0.96 ifeq2(ifeq(leq(addition(multiplication(star(x0), x0), star(x0)), star(x0)), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by axiom 4 (additive_commutativity) }
% 3.98/0.96 ifeq2(ifeq(leq(addition(star(x0), multiplication(star(x0), x0)), star(x0)), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by lemma 17 }
% 3.98/0.96 ifeq2(ifeq(leq(multiplication(star(x0), addition(x0, one)), star(x0)), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by axiom 3 (additive_idempotence) R->L }
% 3.98/0.96 ifeq2(ifeq(leq(multiplication(star(x0), addition(x0, addition(one, one))), star(x0)), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by axiom 5 (additive_associativity) }
% 3.98/0.96 ifeq2(ifeq(leq(multiplication(star(x0), addition(addition(x0, one), one)), star(x0)), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by lemma 17 R->L }
% 3.98/0.96 ifeq2(ifeq(leq(addition(star(x0), multiplication(star(x0), addition(x0, one))), star(x0)), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by axiom 4 (additive_commutativity) R->L }
% 3.98/0.96 ifeq2(ifeq(leq(addition(multiplication(star(x0), addition(x0, one)), star(x0)), star(x0)), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by lemma 18 R->L }
% 3.98/0.96 ifeq2(ifeq(leq(addition(multiplication(star(x0), addition(x0, one)), addition(one, multiplication(star(x0), addition(x0, one)))), star(x0)), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by lemma 16 R->L }
% 3.98/0.96 ifeq2(ifeq(leq(addition(multiplication(star(x0), addition(x0, one)), addition(one, addition(one, multiplication(star(x0), addition(x0, one))))), star(x0)), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by axiom 4 (additive_commutativity) R->L }
% 3.98/0.96 ifeq2(ifeq(leq(addition(multiplication(star(x0), addition(x0, one)), addition(one, addition(multiplication(star(x0), addition(x0, one)), one))), star(x0)), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by axiom 5 (additive_associativity) }
% 3.98/0.96 ifeq2(ifeq(leq(addition(addition(multiplication(star(x0), addition(x0, one)), one), addition(multiplication(star(x0), addition(x0, one)), one)), star(x0)), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by axiom 3 (additive_idempotence) }
% 3.98/0.96 ifeq2(ifeq(leq(addition(multiplication(star(x0), addition(x0, one)), one), star(x0)), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by axiom 4 (additive_commutativity) }
% 3.98/0.96 ifeq2(ifeq(leq(addition(one, multiplication(star(x0), addition(x0, one))), star(x0)), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by lemma 18 }
% 3.98/0.96 ifeq2(ifeq(leq(star(x0), star(x0)), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by axiom 7 (ifeq_axiom) R->L }
% 3.98/0.96 ifeq2(ifeq(ifeq3(star(x0), star(x0), leq(star(x0), star(x0)), true), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by axiom 3 (additive_idempotence) R->L }
% 3.98/0.96 ifeq2(ifeq(ifeq3(addition(star(x0), star(x0)), star(x0), leq(star(x0), star(x0)), true), true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by axiom 13 (order) }
% 3.98/0.96 ifeq2(ifeq(true, true, leq(multiplication(star(x0), star(x0)), star(x0)), true), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by axiom 6 (ifeq_axiom) }
% 3.98/0.96 ifeq2(leq(multiplication(star(x0), star(x0)), star(x0)), true, addition(multiplication(star(x0), star(x0)), star(x0)), star(x0))
% 3.98/0.96 = { by axiom 14 (order_1) }
% 3.98/0.96 star(x0)
% 3.98/0.96 % SZS output end Proof
% 3.98/0.96
% 3.98/0.96 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------