%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE170+1.004 : TPTP v9.3.1. Released v5.2.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n014.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 11:39:58 AM UTC 2026
% Result : Theorem 3.79s 0.97s
% Output : Proof 3.79s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE170+1.004 : TPTP v9.3.1. Released v5.2.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.09/0.36 % Computer : n014.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 13:15:01 UTC 2026
% 0.09/0.36 % CPUTime :
% 0.09/0.36 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 3.79/0.97 Command-line arguments: --flatten --complete-subsets
% 3.79/0.97
% 3.79/0.97 % SZS status Theorem
% 3.79/0.97
% 3.79/0.98 % SZS output start Proof
% 3.79/0.98 Axiom 1 (additive_idempotence): addition(X, X) = X.
% 3.79/0.98 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 3.79/0.98 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 3.79/0.98 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 3.79/0.98 Axiom 5 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 3.79/0.98 Axiom 6 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 3.79/0.98 Axiom 7 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 3.79/0.98 Axiom 8 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 3.79/0.98 Axiom 9 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 3.79/0.98 Axiom 10 (star_unfold_left): leq(addition(one, multiplication(star(X), X)), star(X)) = true.
% 3.79/0.98 Axiom 11 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 3.79/0.98 Axiom 12 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 3.79/0.98 Axiom 13 (order): ifeq3(addition(X, Y), Y, leq(X, Y), true) = true.
% 3.79/0.98 Axiom 14 (order_1): ifeq2(leq(X, Y), true, addition(X, Y), Y) = Y.
% 3.79/0.98
% 3.79/0.98 Lemma 15: addition(one, addition(star(X), multiplication(X, star(X)))) = star(X).
% 3.79/0.98 Proof:
% 3.79/0.98 addition(one, addition(star(X), multiplication(X, star(X))))
% 3.79/0.98 = { by axiom 2 (additive_commutativity) R->L }
% 3.79/0.98 addition(one, addition(multiplication(X, star(X)), star(X)))
% 3.79/0.98 = { by axiom 7 (additive_associativity) }
% 3.79/0.98 addition(addition(one, multiplication(X, star(X))), star(X))
% 3.79/0.98 = { by axiom 6 (ifeq_axiom) R->L }
% 3.79/0.98 ifeq2(true, true, addition(addition(one, multiplication(X, star(X))), star(X)), star(X))
% 3.79/0.98 = { by axiom 9 (star_unfold_right) R->L }
% 3.79/0.98 ifeq2(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(addition(one, multiplication(X, star(X))), star(X)), star(X))
% 3.79/0.98 = { by axiom 14 (order_1) }
% 3.79/0.98 star(X)
% 3.79/0.98
% 3.79/0.98 Lemma 16: addition(one, star(X)) = star(X).
% 3.79/0.98 Proof:
% 3.79/0.98 addition(one, star(X))
% 3.79/0.98 = { by lemma 15 R->L }
% 3.79/0.98 addition(one, addition(one, addition(star(X), multiplication(X, star(X)))))
% 3.79/0.98 = { by axiom 7 (additive_associativity) }
% 3.79/0.98 addition(addition(one, one), addition(star(X), multiplication(X, star(X))))
% 3.79/0.98 = { by axiom 1 (additive_idempotence) }
% 3.79/0.98 addition(one, addition(star(X), multiplication(X, star(X))))
% 3.79/0.98 = { by lemma 15 }
% 3.79/0.98 star(X)
% 3.79/0.98
% 3.79/0.98 Lemma 17: addition(star(X), multiplication(star(X), X)) = star(X).
% 3.79/0.98 Proof:
% 3.79/0.98 addition(star(X), multiplication(star(X), X))
% 3.79/0.98 = { by lemma 16 R->L }
% 3.79/0.98 addition(addition(one, star(X)), multiplication(star(X), X))
% 3.79/0.98 = { by axiom 7 (additive_associativity) R->L }
% 3.79/0.98 addition(one, addition(star(X), multiplication(star(X), X)))
% 3.79/0.98 = { by axiom 2 (additive_commutativity) }
% 3.79/0.98 addition(one, addition(multiplication(star(X), X), star(X)))
% 3.79/0.98 = { by axiom 7 (additive_associativity) }
% 3.79/0.98 addition(addition(one, multiplication(star(X), X)), star(X))
% 3.79/0.98 = { by axiom 6 (ifeq_axiom) R->L }
% 3.79/0.98 ifeq2(true, true, addition(addition(one, multiplication(star(X), X)), star(X)), star(X))
% 3.79/0.98 = { by axiom 10 (star_unfold_left) R->L }
% 3.79/0.98 ifeq2(leq(addition(one, multiplication(star(X), X)), star(X)), true, addition(addition(one, multiplication(star(X), X)), star(X)), star(X))
% 3.79/0.98 = { by axiom 14 (order_1) }
% 3.79/0.98 star(X)
% 3.79/0.98
% 3.79/0.98 Lemma 18: multiplication(star(X), addition(X, one)) = star(X).
% 3.79/0.98 Proof:
% 3.79/0.98 multiplication(star(X), addition(X, one))
% 3.79/0.98 = { by axiom 2 (additive_commutativity) R->L }
% 3.79/0.98 multiplication(star(X), addition(one, X))
% 3.79/0.98 = { by axiom 11 (right_distributivity) }
% 3.79/0.98 addition(multiplication(star(X), one), multiplication(star(X), X))
% 3.79/0.98 = { by axiom 3 (multiplicative_right_identity) }
% 3.79/0.98 addition(star(X), multiplication(star(X), X))
% 3.79/0.98 = { by lemma 17 }
% 3.79/0.98 star(X)
% 3.79/0.98
% 3.79/0.98 Lemma 19: addition(star(X), addition(Y, multiplication(star(X), X))) = addition(Y, star(X)).
% 3.79/0.98 Proof:
% 3.79/0.98 addition(star(X), addition(Y, multiplication(star(X), X)))
% 3.79/0.98 = { by axiom 2 (additive_commutativity) R->L }
% 3.79/0.98 addition(star(X), addition(multiplication(star(X), X), Y))
% 3.79/0.98 = { by axiom 7 (additive_associativity) }
% 3.79/0.98 addition(addition(star(X), multiplication(star(X), X)), Y)
% 3.79/0.98 = { by lemma 17 }
% 3.79/0.98 addition(star(X), Y)
% 3.79/0.98 = { by axiom 2 (additive_commutativity) }
% 3.79/0.98 addition(Y, star(X))
% 3.79/0.98
% 3.79/0.98 Goal 1 (a): leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)) = true.
% 3.79/0.98 Proof:
% 3.79/0.98 leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a))
% 3.79/0.98 = { by axiom 5 (ifeq_axiom) R->L }
% 3.79/0.98 ifeq3(star(a), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by lemma 17 R->L }
% 3.79/0.98 ifeq3(addition(star(a), multiplication(star(a), a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by lemma 17 R->L }
% 3.79/0.98 ifeq3(addition(star(a), multiplication(addition(star(a), multiplication(star(a), a)), a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by lemma 17 R->L }
% 3.79/0.98 ifeq3(addition(star(a), multiplication(addition(star(a), multiplication(addition(star(a), multiplication(star(a), a)), a)), a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by lemma 18 R->L }
% 3.79/0.98 ifeq3(addition(star(a), multiplication(addition(star(a), multiplication(addition(star(a), multiplication(multiplication(star(a), addition(a, one)), a)), a)), a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by lemma 16 R->L }
% 3.79/0.98 ifeq3(addition(star(a), multiplication(addition(star(a), multiplication(addition(star(a), multiplication(multiplication(addition(one, star(a)), addition(a, one)), a)), a)), a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by axiom 12 (left_distributivity) }
% 3.79/0.98 ifeq3(addition(star(a), multiplication(addition(star(a), multiplication(addition(star(a), multiplication(addition(multiplication(one, addition(a, one)), multiplication(star(a), addition(a, one))), a)), a)), a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by axiom 4 (multiplicative_left_identity) }
% 3.79/0.98 ifeq3(addition(star(a), multiplication(addition(star(a), multiplication(addition(star(a), multiplication(addition(addition(a, one), multiplication(star(a), addition(a, one))), a)), a)), a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by lemma 18 }
% 3.79/0.98 ifeq3(addition(star(a), multiplication(addition(star(a), multiplication(addition(star(a), multiplication(addition(addition(a, one), star(a)), a)), a)), a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by axiom 7 (additive_associativity) R->L }
% 3.79/0.98 ifeq3(addition(star(a), multiplication(addition(star(a), multiplication(addition(star(a), multiplication(addition(a, addition(one, star(a))), a)), a)), a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by lemma 16 }
% 3.79/0.98 ifeq3(addition(star(a), multiplication(addition(star(a), multiplication(addition(star(a), multiplication(addition(a, star(a)), a)), a)), a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by axiom 12 (left_distributivity) }
% 3.79/0.98 ifeq3(addition(star(a), multiplication(addition(star(a), multiplication(addition(star(a), addition(multiplication(a, a), multiplication(star(a), a))), a)), a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by lemma 19 }
% 3.79/0.98 ifeq3(addition(star(a), multiplication(addition(star(a), multiplication(addition(multiplication(a, a), star(a)), a)), a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by axiom 12 (left_distributivity) }
% 3.79/0.98 ifeq3(addition(star(a), multiplication(addition(star(a), addition(multiplication(multiplication(a, a), a), multiplication(star(a), a))), a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by axiom 8 (multiplicative_associativity) R->L }
% 3.79/0.98 ifeq3(addition(star(a), multiplication(addition(star(a), addition(multiplication(a, multiplication(a, a)), multiplication(star(a), a))), a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by lemma 19 }
% 3.79/0.98 ifeq3(addition(star(a), multiplication(addition(multiplication(a, multiplication(a, a)), star(a)), a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by axiom 12 (left_distributivity) }
% 3.79/0.98 ifeq3(addition(star(a), addition(multiplication(multiplication(a, multiplication(a, a)), a), multiplication(star(a), a))), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by axiom 8 (multiplicative_associativity) R->L }
% 3.79/0.98 ifeq3(addition(star(a), addition(multiplication(a, multiplication(multiplication(a, a), a)), multiplication(star(a), a))), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by axiom 8 (multiplicative_associativity) R->L }
% 3.79/0.98 ifeq3(addition(star(a), addition(multiplication(a, multiplication(a, multiplication(a, a))), multiplication(star(a), a))), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by lemma 19 }
% 3.79/0.98 ifeq3(addition(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), star(a), leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), true)
% 3.79/0.98 = { by axiom 13 (order) }
% 3.79/0.98 true
% 3.79/0.98 % SZS output end Proof
% 3.79/0.98
% 3.79/0.98 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------