%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE071+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n009.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 11:39:45 AM UTC 2026
% Result : Theorem 7.03s 1.38s
% Output : Proof 7.03s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : KLE071+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.03 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.08/0.35 % Computer : n009.cluster.edu
% 0.08/0.35 % Model : x86_64 x86_64
% 0.08/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.35 % Memory : 8046.5625MB
% 0.08/0.35 % OS : Linux 6.8.0-71-generic
% 0.08/0.35 % CPULimit : 300
% 0.08/0.35 % WCLimit : 300
% 0.08/0.35 % DateTime : Sun Sep 27 13:08:59 UTC 2026
% 0.08/0.36 % CPUTime :
% 0.08/0.36 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 7.03/1.38 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 7.03/1.38
% 7.03/1.38 % SZS status Theorem
% 7.03/1.38
% 7.03/1.39 % SZS output start Proof
% 7.03/1.39 Axiom 1 (additive_idempotence): addition(X, X) = X.
% 7.03/1.39 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 7.03/1.39 Axiom 3 (domain3): addition(domain(X), one) = one.
% 7.03/1.39 Axiom 4 (multiplicative_right_identity): multiplication(X, one) = X.
% 7.03/1.39 Axiom 5 (multiplicative_left_identity): multiplication(one, X) = X.
% 7.03/1.39 Axiom 6 (domain2): domain(multiplication(X, Y)) = domain(multiplication(X, domain(Y))).
% 7.03/1.39 Axiom 7 (domain1): addition(X, multiplication(domain(X), X)) = multiplication(domain(X), X).
% 7.03/1.39 Axiom 8 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 7.03/1.39 Axiom 9 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 7.03/1.39 Axiom 10 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 7.03/1.39 Axiom 11 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 7.03/1.39
% 7.03/1.39 Lemma 12: domain(domain(X)) = domain(X).
% 7.03/1.39 Proof:
% 7.03/1.39 domain(domain(X))
% 7.03/1.39 = { by axiom 5 (multiplicative_left_identity) R->L }
% 7.03/1.39 domain(multiplication(one, domain(X)))
% 7.03/1.39 = { by axiom 6 (domain2) R->L }
% 7.03/1.39 domain(multiplication(one, X))
% 7.03/1.39 = { by axiom 5 (multiplicative_left_identity) }
% 7.03/1.39 domain(X)
% 7.03/1.39
% 7.03/1.39 Lemma 13: addition(one, domain(X)) = one.
% 7.03/1.39 Proof:
% 7.03/1.39 addition(one, domain(X))
% 7.03/1.39 = { by axiom 2 (additive_commutativity) R->L }
% 7.03/1.39 addition(domain(X), one)
% 7.03/1.39 = { by axiom 3 (domain3) }
% 7.03/1.39 one
% 7.03/1.39
% 7.03/1.39 Lemma 14: multiplication(addition(one, Y), X) = addition(X, multiplication(Y, X)).
% 7.03/1.39 Proof:
% 7.03/1.39 multiplication(addition(one, Y), X)
% 7.03/1.39 = { by axiom 11 (left_distributivity) }
% 7.03/1.39 addition(multiplication(one, X), multiplication(Y, X))
% 7.03/1.39 = { by axiom 5 (multiplicative_left_identity) }
% 7.03/1.39 addition(X, multiplication(Y, X))
% 7.03/1.39
% 7.03/1.39 Lemma 15: addition(X, multiplication(domain(Y), X)) = X.
% 7.03/1.39 Proof:
% 7.03/1.39 addition(X, multiplication(domain(Y), X))
% 7.03/1.39 = { by lemma 14 R->L }
% 7.03/1.39 multiplication(addition(one, domain(Y)), X)
% 7.03/1.39 = { by lemma 13 }
% 7.03/1.39 multiplication(one, X)
% 7.03/1.39 = { by axiom 5 (multiplicative_left_identity) }
% 7.03/1.39 X
% 7.03/1.39
% 7.03/1.39 Goal 1 (goals): addition(domain(x0), multiplication(domain(x1), domain(x2))) = multiplication(addition(domain(x0), domain(x1)), addition(domain(x0), domain(x2))).
% 7.03/1.39 Proof:
% 7.03/1.39 addition(domain(x0), multiplication(domain(x1), domain(x2)))
% 7.03/1.39 = { by lemma 15 R->L }
% 7.03/1.39 addition(addition(domain(x0), multiplication(domain(x1), domain(x2))), multiplication(domain(x1), addition(domain(x0), multiplication(domain(x1), domain(x2)))))
% 7.03/1.39 = { by axiom 2 (additive_commutativity) R->L }
% 7.03/1.39 addition(multiplication(domain(x1), addition(domain(x0), multiplication(domain(x1), domain(x2)))), addition(domain(x0), multiplication(domain(x1), domain(x2))))
% 7.03/1.39 = { by axiom 2 (additive_commutativity) R->L }
% 7.03/1.39 addition(multiplication(domain(x1), addition(multiplication(domain(x1), domain(x2)), domain(x0))), addition(domain(x0), multiplication(domain(x1), domain(x2))))
% 7.03/1.39 = { by axiom 10 (right_distributivity) }
% 7.03/1.39 addition(addition(multiplication(domain(x1), multiplication(domain(x1), domain(x2))), multiplication(domain(x1), domain(x0))), addition(domain(x0), multiplication(domain(x1), domain(x2))))
% 7.03/1.39 = { by lemma 12 R->L }
% 7.03/1.39 addition(addition(multiplication(domain(domain(x1)), multiplication(domain(x1), domain(x2))), multiplication(domain(x1), domain(x0))), addition(domain(x0), multiplication(domain(x1), domain(x2))))
% 7.03/1.39 = { by axiom 9 (multiplicative_associativity) }
% 7.03/1.39 addition(addition(multiplication(multiplication(domain(domain(x1)), domain(x1)), domain(x2)), multiplication(domain(x1), domain(x0))), addition(domain(x0), multiplication(domain(x1), domain(x2))))
% 7.03/1.39 = { by axiom 7 (domain1) R->L }
% 7.03/1.39 addition(addition(multiplication(addition(domain(x1), multiplication(domain(domain(x1)), domain(x1))), domain(x2)), multiplication(domain(x1), domain(x0))), addition(domain(x0), multiplication(domain(x1), domain(x2))))
% 7.03/1.39 = { by axiom 11 (left_distributivity) }
% 7.03/1.39 addition(addition(addition(multiplication(domain(x1), domain(x2)), multiplication(multiplication(domain(domain(x1)), domain(x1)), domain(x2))), multiplication(domain(x1), domain(x0))), addition(domain(x0), multiplication(domain(x1), domain(x2))))
% 7.03/1.39 = { by axiom 9 (multiplicative_associativity) R->L }
% 7.03/1.39 addition(addition(addition(multiplication(domain(x1), domain(x2)), multiplication(domain(domain(x1)), multiplication(domain(x1), domain(x2)))), multiplication(domain(x1), domain(x0))), addition(domain(x0), multiplication(domain(x1), domain(x2))))
% 7.03/1.39 = { by lemma 12 }
% 7.03/1.39 addition(addition(addition(multiplication(domain(x1), domain(x2)), multiplication(domain(x1), multiplication(domain(x1), domain(x2)))), multiplication(domain(x1), domain(x0))), addition(domain(x0), multiplication(domain(x1), domain(x2))))
% 7.03/1.39 = { by lemma 15 }
% 7.03/1.39 addition(addition(multiplication(domain(x1), domain(x2)), multiplication(domain(x1), domain(x0))), addition(domain(x0), multiplication(domain(x1), domain(x2))))
% 7.03/1.40 = { by axiom 10 (right_distributivity) R->L }
% 7.03/1.40 addition(multiplication(domain(x1), addition(domain(x2), domain(x0))), addition(domain(x0), multiplication(domain(x1), domain(x2))))
% 7.03/1.40 = { by axiom 2 (additive_commutativity) }
% 7.03/1.40 addition(multiplication(domain(x1), addition(domain(x0), domain(x2))), addition(domain(x0), multiplication(domain(x1), domain(x2))))
% 7.03/1.40 = { by axiom 2 (additive_commutativity) R->L }
% 7.03/1.40 addition(addition(domain(x0), multiplication(domain(x1), domain(x2))), multiplication(domain(x1), addition(domain(x0), domain(x2))))
% 7.03/1.40 = { by axiom 8 (additive_associativity) R->L }
% 7.03/1.40 addition(domain(x0), addition(multiplication(domain(x1), domain(x2)), multiplication(domain(x1), addition(domain(x0), domain(x2)))))
% 7.03/1.40 = { by axiom 10 (right_distributivity) R->L }
% 7.03/1.40 addition(domain(x0), multiplication(domain(x1), addition(domain(x2), addition(domain(x0), domain(x2)))))
% 7.03/1.40 = { by axiom 2 (additive_commutativity) }
% 7.03/1.40 addition(domain(x0), multiplication(domain(x1), addition(domain(x2), addition(domain(x2), domain(x0)))))
% 7.03/1.40 = { by axiom 8 (additive_associativity) }
% 7.03/1.40 addition(domain(x0), multiplication(domain(x1), addition(addition(domain(x2), domain(x2)), domain(x0))))
% 7.03/1.40 = { by axiom 1 (additive_idempotence) }
% 7.03/1.40 addition(domain(x0), multiplication(domain(x1), addition(domain(x2), domain(x0))))
% 7.03/1.40 = { by axiom 2 (additive_commutativity) R->L }
% 7.03/1.40 addition(domain(x0), multiplication(domain(x1), addition(domain(x0), domain(x2))))
% 7.03/1.40 = { by axiom 4 (multiplicative_right_identity) R->L }
% 7.03/1.40 addition(multiplication(domain(x0), one), multiplication(domain(x1), addition(domain(x0), domain(x2))))
% 7.03/1.40 = { by lemma 13 R->L }
% 7.03/1.40 addition(multiplication(domain(x0), addition(one, domain(x2))), multiplication(domain(x1), addition(domain(x0), domain(x2))))
% 7.03/1.40 = { by axiom 10 (right_distributivity) }
% 7.03/1.40 addition(addition(multiplication(domain(x0), one), multiplication(domain(x0), domain(x2))), multiplication(domain(x1), addition(domain(x0), domain(x2))))
% 7.03/1.40 = { by axiom 4 (multiplicative_right_identity) }
% 7.03/1.40 addition(addition(domain(x0), multiplication(domain(x0), domain(x2))), multiplication(domain(x1), addition(domain(x0), domain(x2))))
% 7.03/1.40 = { by axiom 5 (multiplicative_left_identity) R->L }
% 7.03/1.40 addition(addition(multiplication(one, domain(x0)), multiplication(domain(x0), domain(x2))), multiplication(domain(x1), addition(domain(x0), domain(x2))))
% 7.03/1.40 = { by lemma 13 R->L }
% 7.03/1.40 addition(addition(multiplication(addition(one, domain(domain(x0))), domain(x0)), multiplication(domain(x0), domain(x2))), multiplication(domain(x1), addition(domain(x0), domain(x2))))
% 7.03/1.40 = { by lemma 14 }
% 7.03/1.40 addition(addition(addition(domain(x0), multiplication(domain(domain(x0)), domain(x0))), multiplication(domain(x0), domain(x2))), multiplication(domain(x1), addition(domain(x0), domain(x2))))
% 7.03/1.40 = { by axiom 7 (domain1) }
% 7.03/1.40 addition(addition(multiplication(domain(domain(x0)), domain(x0)), multiplication(domain(x0), domain(x2))), multiplication(domain(x1), addition(domain(x0), domain(x2))))
% 7.03/1.40 = { by lemma 12 }
% 7.03/1.40 addition(addition(multiplication(domain(x0), domain(x0)), multiplication(domain(x0), domain(x2))), multiplication(domain(x1), addition(domain(x0), domain(x2))))
% 7.03/1.40 = { by axiom 10 (right_distributivity) R->L }
% 7.03/1.40 addition(multiplication(domain(x0), addition(domain(x0), domain(x2))), multiplication(domain(x1), addition(domain(x0), domain(x2))))
% 7.03/1.40 = { by axiom 11 (left_distributivity) R->L }
% 7.03/1.40 multiplication(addition(domain(x0), domain(x1)), addition(domain(x0), domain(x2)))
% 7.03/1.40 % SZS output end Proof
% 7.03/1.40
% 7.03/1.40 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------