%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE040+2 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n008.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 14.40s 2.34s
% Output : Proof 14.40s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE040+2 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.05 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.13/0.40 % Computer : n008.cluster.edu
% 0.13/0.40 % Model : x86_64 x86_64
% 0.13/0.40 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.40 % Memory : 8046.5625MB
% 0.13/0.40 % OS : Linux 6.8.0-71-generic
% 0.13/0.40 % CPULimit : 300
% 0.13/0.40 % WCLimit : 300
% 0.13/0.40 % DateTime : Sun Sep 27 13:05:54 UTC 2026
% 0.13/0.40 % CPUTime :
% 0.13/0.40 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 14.40/2.34 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
% 14.40/2.34
% 14.40/2.34 % SZS status Theorem
% 14.40/2.34
% 14.40/2.36 % SZS output start Proof
% 14.40/2.36 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 14.40/2.36 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 14.40/2.36 Axiom 3 (additive_idempotence): addition(X, X) = X.
% 14.40/2.36 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 14.40/2.36 Axiom 5 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 14.40/2.36 Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 14.40/2.36 Axiom 7 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 14.40/2.36 Axiom 8 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 14.40/2.36 Axiom 9 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 14.40/2.36 Axiom 10 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 14.40/2.36 Axiom 11 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 14.40/2.36 Axiom 12 (star_unfold_left): leq(addition(one, multiplication(star(X), X)), star(X)) = true.
% 14.40/2.36 Axiom 13 (order): ifeq3(addition(X, Y), Y, leq(X, Y), true) = true.
% 14.40/2.36 Axiom 14 (order_1): ifeq2(leq(X, Y), true, addition(X, Y), Y) = Y.
% 14.40/2.36 Axiom 15 (star_induction_right): ifeq(leq(addition(multiplication(X, Y), Z), X), true, leq(multiplication(Z, star(Y)), X), true) = true.
% 14.40/2.36
% 14.40/2.36 Lemma 16: addition(X, addition(X, Y)) = addition(X, Y).
% 14.40/2.36 Proof:
% 14.40/2.36 addition(X, addition(X, Y))
% 14.40/2.36 = { by axiom 5 (additive_associativity) }
% 14.40/2.36 addition(addition(X, X), Y)
% 14.40/2.36 = { by axiom 3 (additive_idempotence) }
% 14.40/2.36 addition(X, Y)
% 14.40/2.36
% 14.40/2.36 Lemma 17: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
% 14.40/2.36 Proof:
% 14.40/2.36 addition(X, multiplication(X, Y))
% 14.40/2.36 = { by axiom 1 (multiplicative_right_identity) R->L }
% 14.40/2.36 addition(multiplication(X, one), multiplication(X, Y))
% 14.40/2.36 = { by axiom 9 (right_distributivity) R->L }
% 14.40/2.36 multiplication(X, addition(one, Y))
% 14.40/2.36 = { by axiom 4 (additive_commutativity) }
% 14.40/2.36 multiplication(X, addition(Y, one))
% 14.40/2.36
% 14.40/2.36 Lemma 18: addition(one, multiplication(star(X), addition(X, one))) = star(X).
% 14.40/2.36 Proof:
% 14.40/2.36 addition(one, multiplication(star(X), addition(X, one)))
% 14.40/2.36 = { by lemma 17 R->L }
% 14.40/2.36 addition(one, addition(star(X), multiplication(star(X), X)))
% 14.40/2.36 = { by axiom 4 (additive_commutativity) R->L }
% 14.40/2.36 addition(one, addition(multiplication(star(X), X), star(X)))
% 14.40/2.36 = { by axiom 5 (additive_associativity) }
% 14.40/2.36 addition(addition(one, multiplication(star(X), X)), star(X))
% 14.40/2.36 = { by axiom 8 (ifeq_axiom) R->L }
% 14.40/2.36 ifeq2(true, true, addition(addition(one, multiplication(star(X), X)), star(X)), star(X))
% 14.40/2.36 = { by axiom 12 (star_unfold_left) R->L }
% 14.40/2.36 ifeq2(leq(addition(one, multiplication(star(X), X)), star(X)), true, addition(addition(one, multiplication(star(X), X)), star(X)), star(X))
% 14.40/2.36 = { by axiom 14 (order_1) }
% 14.40/2.36 star(X)
% 14.40/2.37
% 14.40/2.37 Goal 1 (goals): tuple(leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), leq(star(x0), multiplication(star(x0), star(x0)))) = tuple(true, true).
% 14.40/2.37 Proof:
% 14.40/2.37 tuple(leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), leq(star(x0), multiplication(star(x0), star(x0))))
% 14.40/2.37 = { by axiom 1 (multiplicative_right_identity) R->L }
% 14.40/2.37 tuple(leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), leq(multiplication(star(x0), one), multiplication(star(x0), star(x0))))
% 14.40/2.37 = { by axiom 14 (order_1) R->L }
% 14.40/2.37 tuple(leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), leq(multiplication(star(x0), one), multiplication(star(x0), ifeq2(leq(addition(one, multiplication(x0, star(x0))), star(x0)), true, addition(addition(one, multiplication(x0, star(x0))), star(x0)), star(x0)))))
% 14.40/2.37 = { by axiom 11 (star_unfold_right) }
% 14.40/2.37 tuple(leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), leq(multiplication(star(x0), one), multiplication(star(x0), ifeq2(true, true, addition(addition(one, multiplication(x0, star(x0))), star(x0)), star(x0)))))
% 14.40/2.37 = { by axiom 8 (ifeq_axiom) }
% 14.40/2.37 tuple(leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), leq(multiplication(star(x0), one), multiplication(star(x0), addition(addition(one, multiplication(x0, star(x0))), star(x0)))))
% 14.40/2.37 = { by axiom 5 (additive_associativity) R->L }
% 14.40/2.37 tuple(leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), leq(multiplication(star(x0), one), multiplication(star(x0), addition(one, addition(multiplication(x0, star(x0)), star(x0))))))
% 14.40/2.37 = { by axiom 4 (additive_commutativity) }
% 14.40/2.37 tuple(leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), leq(multiplication(star(x0), one), multiplication(star(x0), addition(one, addition(star(x0), multiplication(x0, star(x0)))))))
% 14.40/2.37 = { by axiom 2 (multiplicative_left_identity) R->L }
% 14.40/2.37 tuple(leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), leq(multiplication(star(x0), one), multiplication(star(x0), addition(one, addition(multiplication(one, star(x0)), multiplication(x0, star(x0)))))))
% 14.40/2.37 = { by axiom 10 (left_distributivity) R->L }
% 14.40/2.37 tuple(leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), leq(multiplication(star(x0), one), multiplication(star(x0), addition(one, multiplication(addition(one, x0), star(x0))))))
% 14.40/2.37 = { by axiom 4 (additive_commutativity) }
% 14.40/2.37 tuple(leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), leq(multiplication(star(x0), one), multiplication(star(x0), addition(one, multiplication(addition(x0, one), star(x0))))))
% 14.40/2.37 = { by axiom 9 (right_distributivity) }
% 14.40/2.37 tuple(leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), leq(multiplication(star(x0), one), addition(multiplication(star(x0), one), multiplication(star(x0), multiplication(addition(x0, one), star(x0))))))
% 14.40/2.37 = { by axiom 7 (ifeq_axiom) R->L }
% 14.40/2.37 tuple(leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), ifeq3(addition(multiplication(star(x0), one), multiplication(star(x0), multiplication(addition(x0, one), star(x0)))), addition(multiplication(star(x0), one), multiplication(star(x0), multiplication(addition(x0, one), star(x0)))), leq(multiplication(star(x0), one), addition(multiplication(star(x0), one), multiplication(star(x0), multiplication(addition(x0, one), star(x0))))), true))
% 14.40/2.37 = { by lemma 16 R->L }
% 14.40/2.37 tuple(leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), ifeq3(addition(multiplication(star(x0), one), addition(multiplication(star(x0), one), multiplication(star(x0), multiplication(addition(x0, one), star(x0))))), addition(multiplication(star(x0), one), multiplication(star(x0), multiplication(addition(x0, one), star(x0)))), leq(multiplication(star(x0), one), addition(multiplication(star(x0), one), multiplication(star(x0), multiplication(addition(x0, one), star(x0))))), true))
% 14.40/2.37 = { by axiom 13 (order) }
% 14.40/2.37 tuple(leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true)
% 14.40/2.37 = { by axiom 6 (ifeq_axiom) R->L }
% 14.40/2.37 tuple(ifeq(true, true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.37 = { by axiom 13 (order) R->L }
% 14.40/2.37 tuple(ifeq(ifeq3(addition(star(x0_2), star(x0_2)), star(x0_2), leq(star(x0_2), star(x0_2)), true), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.37 = { by axiom 3 (additive_idempotence) }
% 14.40/2.37 tuple(ifeq(ifeq3(star(x0_2), star(x0_2), leq(star(x0_2), star(x0_2)), true), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.37 = { by axiom 7 (ifeq_axiom) }
% 14.40/2.37 tuple(ifeq(leq(star(x0_2), star(x0_2)), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.37 = { by lemma 18 R->L }
% 14.40/2.37 tuple(ifeq(leq(addition(one, multiplication(star(x0_2), addition(x0_2, one))), star(x0_2)), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.37 = { by axiom 4 (additive_commutativity) R->L }
% 14.40/2.37 tuple(ifeq(leq(addition(multiplication(star(x0_2), addition(x0_2, one)), one), star(x0_2)), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.37 = { by axiom 3 (additive_idempotence) R->L }
% 14.40/2.37 tuple(ifeq(leq(addition(addition(multiplication(star(x0_2), addition(x0_2, one)), one), addition(multiplication(star(x0_2), addition(x0_2, one)), one)), star(x0_2)), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.38 = { by axiom 5 (additive_associativity) R->L }
% 14.40/2.38 tuple(ifeq(leq(addition(multiplication(star(x0_2), addition(x0_2, one)), addition(one, addition(multiplication(star(x0_2), addition(x0_2, one)), one))), star(x0_2)), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.38 = { by axiom 4 (additive_commutativity) }
% 14.40/2.38 tuple(ifeq(leq(addition(multiplication(star(x0_2), addition(x0_2, one)), addition(one, addition(one, multiplication(star(x0_2), addition(x0_2, one))))), star(x0_2)), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.38 = { by lemma 16 }
% 14.40/2.38 tuple(ifeq(leq(addition(multiplication(star(x0_2), addition(x0_2, one)), addition(one, multiplication(star(x0_2), addition(x0_2, one)))), star(x0_2)), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.38 = { by lemma 18 }
% 14.40/2.38 tuple(ifeq(leq(addition(multiplication(star(x0_2), addition(x0_2, one)), star(x0_2)), star(x0_2)), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.38 = { by axiom 4 (additive_commutativity) }
% 14.40/2.38 tuple(ifeq(leq(addition(star(x0_2), multiplication(star(x0_2), addition(x0_2, one))), star(x0_2)), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.38 = { by lemma 17 }
% 14.40/2.38 tuple(ifeq(leq(multiplication(star(x0_2), addition(addition(x0_2, one), one)), star(x0_2)), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.38 = { by axiom 5 (additive_associativity) R->L }
% 14.40/2.38 tuple(ifeq(leq(multiplication(star(x0_2), addition(x0_2, addition(one, one))), star(x0_2)), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.38 = { by axiom 3 (additive_idempotence) }
% 14.40/2.38 tuple(ifeq(leq(multiplication(star(x0_2), addition(x0_2, one)), star(x0_2)), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.38 = { by lemma 17 R->L }
% 14.40/2.38 tuple(ifeq(leq(addition(star(x0_2), multiplication(star(x0_2), x0_2)), star(x0_2)), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.38 = { by axiom 4 (additive_commutativity) R->L }
% 14.40/2.38 tuple(ifeq(leq(addition(multiplication(star(x0_2), x0_2), star(x0_2)), star(x0_2)), true, leq(multiplication(star(x0_2), star(x0_2)), star(x0_2)), true), true)
% 14.40/2.38 = { by axiom 15 (star_induction_right) }
% 14.40/2.38 tuple(true, true)
% 14.40/2.38 % SZS output end Proof
% 14.40/2.38
% 14.40/2.38 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------