%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE044+2 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n016.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 11:39:42 AM UTC 2026
% Result : Theorem 66.60s 8.83s
% Output : Proof 66.60s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : KLE044+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 : n016.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 13:11:57 UTC 2026
% 0.09/0.36 % CPUTime :
% 0.09/0.36 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 66.60/8.83 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
% 66.60/8.83
% 66.60/8.83 % SZS status Theorem
% 66.60/8.83
% 66.60/8.85 % SZS output start Proof
% 66.60/8.85 Axiom 1 (additive_idempotence): addition(X, X) = X.
% 66.60/8.85 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 66.60/8.85 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 66.60/8.85 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 66.60/8.85 Axiom 5 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 66.60/8.85 Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 66.60/8.85 Axiom 7 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 66.60/8.85 Axiom 8 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 66.60/8.85 Axiom 9 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 66.60/8.85 Axiom 10 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 66.60/8.85 Axiom 11 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 66.60/8.85 Axiom 12 (star_unfold_left): leq(addition(one, multiplication(star(X), X)), star(X)) = true.
% 66.60/8.85 Axiom 13 (order): ifeq3(addition(X, Y), Y, leq(X, Y), true) = true.
% 66.60/8.85 Axiom 14 (order_1): ifeq2(leq(X, Y), true, addition(X, Y), Y) = Y.
% 66.60/8.85 Axiom 15 (star_induction_right): ifeq(leq(addition(multiplication(X, Y), Z), X), true, leq(multiplication(Z, star(Y)), X), true) = true.
% 66.60/8.85 Axiom 16 (star_induction_left): ifeq(leq(addition(multiplication(X, Y), Z), Y), true, leq(multiplication(star(X), Z), Y), true) = true.
% 66.60/8.85
% 66.60/8.85 Lemma 17: addition(X, addition(Y, X)) = addition(Y, X).
% 66.60/8.85 Proof:
% 66.60/8.85 addition(X, addition(Y, X))
% 66.60/8.85 = { by axiom 1 (additive_idempotence) R->L }
% 66.60/8.85 addition(X, addition(addition(Y, Y), X))
% 66.60/8.85 = { by axiom 5 (additive_associativity) R->L }
% 66.60/8.85 addition(X, addition(Y, addition(Y, X)))
% 66.60/8.85 = { by axiom 2 (additive_commutativity) R->L }
% 66.60/8.85 addition(X, addition(Y, addition(X, Y)))
% 66.60/8.85 = { by axiom 5 (additive_associativity) }
% 66.60/8.85 addition(addition(X, Y), addition(X, Y))
% 66.60/8.85 = { by axiom 1 (additive_idempotence) }
% 66.60/8.85 addition(X, Y)
% 66.60/8.85 = { by axiom 2 (additive_commutativity) }
% 66.60/8.85 addition(Y, X)
% 66.60/8.85
% 66.60/8.85 Lemma 18: multiplication(X, addition(Y, one)) = addition(X, multiplication(X, Y)).
% 66.60/8.85 Proof:
% 66.60/8.85 multiplication(X, addition(Y, one))
% 66.60/8.85 = { by axiom 2 (additive_commutativity) R->L }
% 66.60/8.85 multiplication(X, addition(one, Y))
% 66.60/8.85 = { by axiom 9 (right_distributivity) }
% 66.60/8.85 addition(multiplication(X, one), multiplication(X, Y))
% 66.60/8.85 = { by axiom 3 (multiplicative_right_identity) }
% 66.60/8.85 addition(X, multiplication(X, Y))
% 66.60/8.85
% 66.60/8.85 Lemma 19: multiplication(addition(one, Y), X) = addition(X, multiplication(Y, X)).
% 66.60/8.85 Proof:
% 66.60/8.85 multiplication(addition(one, Y), X)
% 66.60/8.85 = { by axiom 10 (left_distributivity) }
% 66.60/8.85 addition(multiplication(one, X), multiplication(Y, X))
% 66.60/8.85 = { by axiom 4 (multiplicative_left_identity) }
% 66.60/8.86 addition(X, multiplication(Y, X))
% 66.60/8.86
% 66.60/8.86 Lemma 20: addition(one, addition(star(X), multiplication(X, star(X)))) = star(X).
% 66.60/8.86 Proof:
% 66.60/8.86 addition(one, addition(star(X), multiplication(X, star(X))))
% 66.60/8.86 = { by axiom 2 (additive_commutativity) R->L }
% 66.60/8.86 addition(one, addition(multiplication(X, star(X)), star(X)))
% 66.60/8.86 = { by axiom 5 (additive_associativity) }
% 66.60/8.86 addition(addition(one, multiplication(X, star(X))), star(X))
% 66.60/8.86 = { by axiom 8 (ifeq_axiom) R->L }
% 66.60/8.86 ifeq2(true, true, addition(addition(one, multiplication(X, star(X))), star(X)), star(X))
% 66.60/8.86 = { by axiom 11 (star_unfold_right) R->L }
% 66.60/8.86 ifeq2(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(addition(one, multiplication(X, star(X))), star(X)), star(X))
% 66.60/8.86 = { by axiom 14 (order_1) }
% 66.60/8.86 star(X)
% 66.60/8.86
% 66.60/8.86 Lemma 21: addition(star(X), multiplication(X, star(X))) = star(X).
% 66.60/8.86 Proof:
% 66.60/8.86 addition(star(X), multiplication(X, star(X)))
% 66.60/8.86 = { by axiom 2 (additive_commutativity) R->L }
% 66.60/8.86 addition(multiplication(X, star(X)), star(X))
% 66.60/8.86 = { by lemma 17 R->L }
% 66.60/8.86 addition(star(X), addition(multiplication(X, star(X)), star(X)))
% 66.60/8.86 = { by axiom 5 (additive_associativity) }
% 66.60/8.86 addition(addition(star(X), multiplication(X, star(X))), star(X))
% 66.60/8.86 = { by lemma 20 R->L }
% 66.60/8.86 addition(addition(star(X), multiplication(X, star(X))), addition(one, addition(star(X), multiplication(X, star(X)))))
% 66.60/8.86 = { by lemma 17 }
% 66.60/8.86 addition(one, addition(star(X), multiplication(X, star(X))))
% 66.60/8.86 = { by lemma 20 }
% 66.60/8.86 star(X)
% 66.60/8.86
% 66.60/8.86 Lemma 22: multiplication(addition(one, X), star(X)) = star(X).
% 66.60/8.86 Proof:
% 66.60/8.86 multiplication(addition(one, X), star(X))
% 66.60/8.86 = { by lemma 19 }
% 66.60/8.86 addition(star(X), multiplication(X, star(X)))
% 66.60/8.86 = { by lemma 21 }
% 66.60/8.86 star(X)
% 66.60/8.86
% 66.60/8.86 Lemma 23: addition(one, addition(star(X), multiplication(star(X), X))) = star(X).
% 66.60/8.86 Proof:
% 66.60/8.86 addition(one, addition(star(X), multiplication(star(X), X)))
% 66.60/8.86 = { by axiom 2 (additive_commutativity) R->L }
% 66.60/8.86 addition(one, addition(multiplication(star(X), X), star(X)))
% 66.60/8.86 = { by axiom 5 (additive_associativity) }
% 66.60/8.86 addition(addition(one, multiplication(star(X), X)), star(X))
% 66.60/8.86 = { by axiom 8 (ifeq_axiom) R->L }
% 66.60/8.86 ifeq2(true, true, addition(addition(one, multiplication(star(X), X)), star(X)), star(X))
% 66.60/8.86 = { by axiom 12 (star_unfold_left) R->L }
% 66.60/8.86 ifeq2(leq(addition(one, multiplication(star(X), X)), star(X)), true, addition(addition(one, multiplication(star(X), X)), star(X)), star(X))
% 66.60/8.86 = { by axiom 14 (order_1) }
% 66.60/8.86 star(X)
% 66.60/8.86
% 66.60/8.86 Goal 1 (goals): tuple(leq(star(addition(one, x0_2)), star(x0_2)), leq(star(x0), star(addition(one, x0)))) = tuple(true, true).
% 66.60/8.86 Proof:
% 66.60/8.86 tuple(leq(star(addition(one, x0_2)), star(x0_2)), leq(star(x0), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 2 (additive_commutativity) R->L }
% 66.60/8.86 tuple(leq(star(addition(x0_2, one)), star(x0_2)), leq(star(x0), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 6 (ifeq_axiom) R->L }
% 66.60/8.86 tuple(ifeq(true, true, leq(star(addition(x0_2, one)), star(x0_2)), true), leq(star(x0), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 13 (order) R->L }
% 66.60/8.86 tuple(ifeq(ifeq3(addition(star(x0_2), star(x0_2)), star(x0_2), leq(star(x0_2), star(x0_2)), true), true, leq(star(addition(x0_2, one)), star(x0_2)), true), leq(star(x0), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 1 (additive_idempotence) }
% 66.60/8.86 tuple(ifeq(ifeq3(star(x0_2), star(x0_2), leq(star(x0_2), star(x0_2)), true), true, leq(star(addition(x0_2, one)), star(x0_2)), true), leq(star(x0), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 7 (ifeq_axiom) }
% 66.60/8.86 tuple(ifeq(leq(star(x0_2), star(x0_2)), true, leq(star(addition(x0_2, one)), star(x0_2)), true), leq(star(x0), star(addition(one, x0))))
% 66.60/8.86 = { by lemma 23 R->L }
% 66.60/8.86 tuple(ifeq(leq(addition(one, addition(star(x0_2), multiplication(star(x0_2), x0_2))), star(x0_2)), true, leq(star(addition(x0_2, one)), star(x0_2)), true), leq(star(x0), star(addition(one, x0))))
% 66.60/8.86 = { by lemma 18 R->L }
% 66.60/8.86 tuple(ifeq(leq(addition(one, multiplication(star(x0_2), addition(x0_2, one))), star(x0_2)), true, leq(star(addition(x0_2, one)), star(x0_2)), true), leq(star(x0), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 2 (additive_commutativity) R->L }
% 66.60/8.86 tuple(ifeq(leq(addition(multiplication(star(x0_2), addition(x0_2, one)), one), star(x0_2)), true, leq(star(addition(x0_2, one)), star(x0_2)), true), leq(star(x0), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 4 (multiplicative_left_identity) R->L }
% 66.60/8.86 tuple(ifeq(leq(addition(multiplication(star(x0_2), addition(x0_2, one)), one), star(x0_2)), true, leq(multiplication(one, star(addition(x0_2, one))), star(x0_2)), true), leq(star(x0), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 15 (star_induction_right) }
% 66.60/8.86 tuple(true, leq(star(x0), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 14 (order_1) R->L }
% 66.60/8.86 tuple(true, leq(ifeq2(leq(multiplication(star(addition(one, x0)), star(x0)), star(x0)), true, addition(multiplication(star(addition(one, x0)), star(x0)), star(x0)), star(x0)), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 2 (additive_commutativity) }
% 66.60/8.86 tuple(true, leq(ifeq2(leq(multiplication(star(addition(one, x0)), star(x0)), star(x0)), true, addition(star(x0), multiplication(star(addition(one, x0)), star(x0))), star(x0)), star(addition(one, x0))))
% 66.60/8.86 = { by lemma 22 R->L }
% 66.60/8.86 tuple(true, leq(ifeq2(leq(multiplication(star(addition(one, x0)), multiplication(addition(one, x0), star(x0))), star(x0)), true, addition(star(x0), multiplication(star(addition(one, x0)), star(x0))), star(x0)), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 6 (ifeq_axiom) R->L }
% 66.60/8.86 tuple(true, leq(ifeq2(ifeq(true, true, leq(multiplication(star(addition(one, x0)), multiplication(addition(one, x0), star(x0))), star(x0)), true), true, addition(star(x0), multiplication(star(addition(one, x0)), star(x0))), star(x0)), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 13 (order) R->L }
% 66.60/8.86 tuple(true, leq(ifeq2(ifeq(ifeq3(addition(star(x0), star(x0)), star(x0), leq(star(x0), star(x0)), true), true, leq(multiplication(star(addition(one, x0)), multiplication(addition(one, x0), star(x0))), star(x0)), true), true, addition(star(x0), multiplication(star(addition(one, x0)), star(x0))), star(x0)), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 1 (additive_idempotence) }
% 66.60/8.86 tuple(true, leq(ifeq2(ifeq(ifeq3(star(x0), star(x0), leq(star(x0), star(x0)), true), true, leq(multiplication(star(addition(one, x0)), multiplication(addition(one, x0), star(x0))), star(x0)), true), true, addition(star(x0), multiplication(star(addition(one, x0)), star(x0))), star(x0)), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 7 (ifeq_axiom) }
% 66.60/8.86 tuple(true, leq(ifeq2(ifeq(leq(star(x0), star(x0)), true, leq(multiplication(star(addition(one, x0)), multiplication(addition(one, x0), star(x0))), star(x0)), true), true, addition(star(x0), multiplication(star(addition(one, x0)), star(x0))), star(x0)), star(addition(one, x0))))
% 66.60/8.86 = { by lemma 22 R->L }
% 66.60/8.86 tuple(true, leq(ifeq2(ifeq(leq(multiplication(addition(one, x0), star(x0)), star(x0)), true, leq(multiplication(star(addition(one, x0)), multiplication(addition(one, x0), star(x0))), star(x0)), true), true, addition(star(x0), multiplication(star(addition(one, x0)), star(x0))), star(x0)), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 1 (additive_idempotence) R->L }
% 66.60/8.86 tuple(true, leq(ifeq2(ifeq(leq(addition(multiplication(addition(one, x0), star(x0)), multiplication(addition(one, x0), star(x0))), star(x0)), true, leq(multiplication(star(addition(one, x0)), multiplication(addition(one, x0), star(x0))), star(x0)), true), true, addition(star(x0), multiplication(star(addition(one, x0)), star(x0))), star(x0)), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 16 (star_induction_left) }
% 66.60/8.86 tuple(true, leq(ifeq2(true, true, addition(star(x0), multiplication(star(addition(one, x0)), star(x0))), star(x0)), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 8 (ifeq_axiom) }
% 66.60/8.86 tuple(true, leq(addition(star(x0), multiplication(star(addition(one, x0)), star(x0))), star(addition(one, x0))))
% 66.60/8.86 = { by lemma 21 R->L }
% 66.60/8.86 tuple(true, leq(addition(star(x0), multiplication(addition(star(addition(one, x0)), multiplication(addition(one, x0), star(addition(one, x0)))), star(x0))), star(addition(one, x0))))
% 66.60/8.86 = { by lemma 19 R->L }
% 66.60/8.86 tuple(true, leq(multiplication(addition(one, addition(star(addition(one, x0)), multiplication(addition(one, x0), star(addition(one, x0))))), star(x0)), star(addition(one, x0))))
% 66.60/8.86 = { by lemma 20 }
% 66.60/8.86 tuple(true, leq(multiplication(star(addition(one, x0)), star(x0)), star(addition(one, x0))))
% 66.60/8.86 = { by axiom 6 (ifeq_axiom) R->L }
% 66.60/8.86 tuple(true, ifeq(true, true, leq(multiplication(star(addition(one, x0)), star(x0)), star(addition(one, x0))), true))
% 66.60/8.86 = { by axiom 13 (order) R->L }
% 66.60/8.86 tuple(true, ifeq(ifeq3(addition(multiplication(star(addition(one, x0)), addition(one, x0)), star(addition(one, x0))), star(addition(one, x0)), leq(multiplication(star(addition(one, x0)), addition(one, x0)), star(addition(one, x0))), true), true, leq(multiplication(star(addition(one, x0)), star(x0)), star(addition(one, x0))), true))
% 66.60/8.86 = { by lemma 17 R->L }
% 66.60/8.86 tuple(true, ifeq(ifeq3(addition(star(addition(one, x0)), addition(multiplication(star(addition(one, x0)), addition(one, x0)), star(addition(one, x0)))), star(addition(one, x0)), leq(multiplication(star(addition(one, x0)), addition(one, x0)), star(addition(one, x0))), true), true, leq(multiplication(star(addition(one, x0)), star(x0)), star(addition(one, x0))), true))
% 66.60/8.86 = { by axiom 5 (additive_associativity) }
% 66.60/8.87 tuple(true, ifeq(ifeq3(addition(addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0))), star(addition(one, x0))), star(addition(one, x0)), leq(multiplication(star(addition(one, x0)), addition(one, x0)), star(addition(one, x0))), true), true, leq(multiplication(star(addition(one, x0)), star(x0)), star(addition(one, x0))), true))
% 66.60/8.87 = { by lemma 23 R->L }
% 66.60/8.87 tuple(true, ifeq(ifeq3(addition(addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0))), addition(one, addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0))))), star(addition(one, x0)), leq(multiplication(star(addition(one, x0)), addition(one, x0)), star(addition(one, x0))), true), true, leq(multiplication(star(addition(one, x0)), star(x0)), star(addition(one, x0))), true))
% 66.60/8.87 = { by lemma 17 }
% 66.60/8.87 tuple(true, ifeq(ifeq3(addition(one, addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0)))), star(addition(one, x0)), leq(multiplication(star(addition(one, x0)), addition(one, x0)), star(addition(one, x0))), true), true, leq(multiplication(star(addition(one, x0)), star(x0)), star(addition(one, x0))), true))
% 66.60/8.87 = { by lemma 23 }
% 66.60/8.87 tuple(true, ifeq(ifeq3(star(addition(one, x0)), star(addition(one, x0)), leq(multiplication(star(addition(one, x0)), addition(one, x0)), star(addition(one, x0))), true), true, leq(multiplication(star(addition(one, x0)), star(x0)), star(addition(one, x0))), true))
% 66.60/8.87 = { by axiom 7 (ifeq_axiom) }
% 66.60/8.87 tuple(true, ifeq(leq(multiplication(star(addition(one, x0)), addition(one, x0)), star(addition(one, x0))), true, leq(multiplication(star(addition(one, x0)), star(x0)), star(addition(one, x0))), true))
% 66.60/8.87 = { by axiom 2 (additive_commutativity) R->L }
% 66.60/8.87 tuple(true, ifeq(leq(multiplication(star(addition(one, x0)), addition(x0, one)), star(addition(one, x0))), true, leq(multiplication(star(addition(one, x0)), star(x0)), star(addition(one, x0))), true))
% 66.60/8.87 = { by lemma 18 }
% 66.60/8.87 tuple(true, ifeq(leq(addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), x0)), star(addition(one, x0))), true, leq(multiplication(star(addition(one, x0)), star(x0)), star(addition(one, x0))), true))
% 66.60/8.87 = { by axiom 2 (additive_commutativity) R->L }
% 66.60/8.87 tuple(true, ifeq(leq(addition(multiplication(star(addition(one, x0)), x0), star(addition(one, x0))), star(addition(one, x0))), true, leq(multiplication(star(addition(one, x0)), star(x0)), star(addition(one, x0))), true))
% 66.60/8.87 = { by axiom 15 (star_induction_right) }
% 66.60/8.87 tuple(true, true)
% 66.60/8.87 % SZS output end Proof
% 66.60/8.87
% 66.60/8.87 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------