%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE068+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n013.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 0.60s 0.56s
% Output : Proof 0.60s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE068+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.10/0.36 % Computer : n013.cluster.edu
% 0.10/0.36 % Model : x86_64 x86_64
% 0.10/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.36 % Memory : 8046.5625MB
% 0.10/0.36 % OS : Linux 6.8.0-71-generic
% 0.10/0.36 % CPULimit : 300
% 0.10/0.36 % WCLimit : 300
% 0.10/0.36 % DateTime : Sun Sep 27 13:07:51 UTC 2026
% 0.10/0.37 % CPUTime :
% 0.10/0.37 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.60/0.56 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.60/0.56
% 0.60/0.56 % SZS status Theorem
% 0.60/0.56
% 0.60/0.57 % SZS output start Proof
% 0.60/0.57 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.60/0.57 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.60/0.57 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.60/0.57 Axiom 4 (domain3): addition(domain(X), one) = one.
% 0.60/0.57 Axiom 5 (domain2): domain(multiplication(X, Y)) = domain(multiplication(X, domain(Y))).
% 0.60/0.57 Axiom 6 (domain5): domain(addition(X, Y)) = addition(domain(X), domain(Y)).
% 0.60/0.57 Axiom 7 (domain1): addition(X, multiplication(domain(X), X)) = multiplication(domain(X), X).
% 0.60/0.57 Axiom 8 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.60/0.57 Axiom 9 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.60/0.57
% 0.60/0.57 Lemma 10: domain(domain(X)) = domain(X).
% 0.60/0.57 Proof:
% 0.60/0.57 domain(domain(X))
% 0.60/0.57 = { by axiom 2 (multiplicative_left_identity) R->L }
% 0.60/0.57 domain(multiplication(one, domain(X)))
% 0.60/0.57 = { by axiom 5 (domain2) R->L }
% 0.60/0.57 domain(multiplication(one, X))
% 0.60/0.57 = { by axiom 2 (multiplicative_left_identity) }
% 0.60/0.57 domain(X)
% 0.60/0.57
% 0.60/0.57 Lemma 11: addition(one, domain(X)) = one.
% 0.60/0.57 Proof:
% 0.60/0.57 addition(one, domain(X))
% 0.60/0.57 = { by axiom 3 (additive_commutativity) R->L }
% 0.60/0.57 addition(domain(X), one)
% 0.60/0.57 = { by axiom 4 (domain3) }
% 0.60/0.57 one
% 0.60/0.57
% 0.60/0.57 Lemma 12: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
% 0.60/0.57 Proof:
% 0.60/0.57 addition(X, multiplication(X, Y))
% 0.60/0.57 = { by axiom 1 (multiplicative_right_identity) R->L }
% 0.60/0.57 addition(multiplication(X, one), multiplication(X, Y))
% 0.60/0.57 = { by axiom 8 (right_distributivity) R->L }
% 0.60/0.57 multiplication(X, addition(one, Y))
% 0.60/0.57 = { by axiom 3 (additive_commutativity) }
% 0.60/0.57 multiplication(X, addition(Y, one))
% 0.60/0.57
% 0.60/0.57 Lemma 13: addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(Z, X), Y).
% 0.60/0.57 Proof:
% 0.60/0.57 addition(multiplication(X, Y), multiplication(Z, Y))
% 0.60/0.57 = { by axiom 9 (left_distributivity) R->L }
% 0.60/0.57 multiplication(addition(X, Z), Y)
% 0.60/0.57 = { by axiom 3 (additive_commutativity) }
% 0.60/0.57 multiplication(addition(Z, X), Y)
% 0.60/0.57
% 0.60/0.57 Lemma 14: addition(X, multiplication(domain(Y), X)) = X.
% 0.60/0.57 Proof:
% 0.60/0.57 addition(X, multiplication(domain(Y), X))
% 0.60/0.57 = { by axiom 3 (additive_commutativity) R->L }
% 0.60/0.57 addition(multiplication(domain(Y), X), X)
% 0.60/0.57 = { by axiom 2 (multiplicative_left_identity) R->L }
% 0.60/0.57 addition(multiplication(domain(Y), X), multiplication(one, X))
% 0.60/0.57 = { by lemma 13 }
% 0.60/0.57 multiplication(addition(one, domain(Y)), X)
% 0.60/0.57 = { by lemma 11 }
% 0.60/0.57 multiplication(one, X)
% 0.60/0.57 = { by axiom 2 (multiplicative_left_identity) }
% 0.60/0.57 X
% 0.60/0.57
% 0.60/0.57 Goal 1 (goals): domain(multiplication(domain(x0), domain(x1))) = multiplication(domain(x0), domain(x1)).
% 0.60/0.57 Proof:
% 0.60/0.57 domain(multiplication(domain(x0), domain(x1)))
% 0.60/0.57 = { by axiom 1 (multiplicative_right_identity) R->L }
% 0.60/0.57 multiplication(domain(multiplication(domain(x0), domain(x1))), one)
% 0.60/0.57 = { by lemma 11 R->L }
% 0.60/0.57 multiplication(domain(multiplication(domain(x0), domain(x1))), addition(one, domain(x1)))
% 0.60/0.57 = { by axiom 3 (additive_commutativity) R->L }
% 0.60/0.57 multiplication(domain(multiplication(domain(x0), domain(x1))), addition(domain(x1), one))
% 0.60/0.57 = { by lemma 12 R->L }
% 0.60/0.57 addition(domain(multiplication(domain(x0), domain(x1))), multiplication(domain(multiplication(domain(x0), domain(x1))), domain(x1)))
% 0.60/0.57 = { by axiom 1 (multiplicative_right_identity) R->L }
% 0.60/0.57 addition(multiplication(domain(multiplication(domain(x0), domain(x1))), one), multiplication(domain(multiplication(domain(x0), domain(x1))), domain(x1)))
% 0.60/0.57 = { by lemma 11 R->L }
% 0.60/0.57 addition(multiplication(domain(multiplication(domain(x0), domain(x1))), addition(one, domain(multiplication(domain(x0), domain(x1))))), multiplication(domain(multiplication(domain(x0), domain(x1))), domain(x1)))
% 0.60/0.57 = { by axiom 3 (additive_commutativity) R->L }
% 0.60/0.57 addition(multiplication(domain(multiplication(domain(x0), domain(x1))), addition(domain(multiplication(domain(x0), domain(x1))), one)), multiplication(domain(multiplication(domain(x0), domain(x1))), domain(x1)))
% 0.60/0.57 = { by lemma 12 R->L }
% 0.60/0.57 addition(addition(domain(multiplication(domain(x0), domain(x1))), multiplication(domain(multiplication(domain(x0), domain(x1))), domain(multiplication(domain(x0), domain(x1))))), multiplication(domain(multiplication(domain(x0), domain(x1))), domain(x1)))
% 0.60/0.57 = { by lemma 10 R->L }
% 0.60/0.57 addition(addition(domain(multiplication(domain(x0), domain(x1))), multiplication(domain(domain(multiplication(domain(x0), domain(x1)))), domain(multiplication(domain(x0), domain(x1))))), multiplication(domain(multiplication(domain(x0), domain(x1))), domain(x1)))
% 0.60/0.57 = { by axiom 7 (domain1) }
% 0.60/0.57 addition(multiplication(domain(domain(multiplication(domain(x0), domain(x1)))), domain(multiplication(domain(x0), domain(x1)))), multiplication(domain(multiplication(domain(x0), domain(x1))), domain(x1)))
% 0.60/0.57 = { by lemma 10 }
% 0.60/0.57 addition(multiplication(domain(multiplication(domain(x0), domain(x1))), domain(multiplication(domain(x0), domain(x1)))), multiplication(domain(multiplication(domain(x0), domain(x1))), domain(x1)))
% 0.60/0.57 = { by axiom 8 (right_distributivity) R->L }
% 0.60/0.57 multiplication(domain(multiplication(domain(x0), domain(x1))), addition(domain(multiplication(domain(x0), domain(x1))), domain(x1)))
% 0.60/0.57 = { by axiom 3 (additive_commutativity) }
% 0.60/0.57 multiplication(domain(multiplication(domain(x0), domain(x1))), addition(domain(x1), domain(multiplication(domain(x0), domain(x1)))))
% 0.60/0.57 = { by lemma 10 R->L }
% 0.60/0.57 multiplication(domain(multiplication(domain(x0), domain(x1))), addition(domain(domain(x1)), domain(multiplication(domain(x0), domain(x1)))))
% 0.60/0.57 = { by axiom 6 (domain5) R->L }
% 0.60/0.57 multiplication(domain(multiplication(domain(x0), domain(x1))), domain(addition(domain(x1), multiplication(domain(x0), domain(x1)))))
% 0.60/0.57 = { by lemma 14 }
% 0.60/0.57 multiplication(domain(multiplication(domain(x0), domain(x1))), domain(domain(x1)))
% 0.60/0.57 = { by lemma 10 }
% 0.60/0.57 multiplication(domain(multiplication(domain(x0), domain(x1))), domain(x1))
% 0.60/0.57 = { by lemma 14 R->L }
% 0.60/0.57 multiplication(domain(multiplication(domain(x0), domain(x1))), addition(domain(x1), multiplication(domain(x0), domain(x1))))
% 0.60/0.58 = { by axiom 3 (additive_commutativity) R->L }
% 0.60/0.58 multiplication(domain(multiplication(domain(x0), domain(x1))), addition(multiplication(domain(x0), domain(x1)), domain(x1)))
% 0.60/0.58 = { by axiom 8 (right_distributivity) }
% 0.60/0.58 addition(multiplication(domain(multiplication(domain(x0), domain(x1))), multiplication(domain(x0), domain(x1))), multiplication(domain(multiplication(domain(x0), domain(x1))), domain(x1)))
% 0.60/0.58 = { by axiom 7 (domain1) R->L }
% 0.60/0.58 addition(addition(multiplication(domain(x0), domain(x1)), multiplication(domain(multiplication(domain(x0), domain(x1))), multiplication(domain(x0), domain(x1)))), multiplication(domain(multiplication(domain(x0), domain(x1))), domain(x1)))
% 0.60/0.58 = { by axiom 2 (multiplicative_left_identity) R->L }
% 0.60/0.58 addition(addition(multiplication(one, multiplication(domain(x0), domain(x1))), multiplication(domain(multiplication(domain(x0), domain(x1))), multiplication(domain(x0), domain(x1)))), multiplication(domain(multiplication(domain(x0), domain(x1))), domain(x1)))
% 0.60/0.58 = { by axiom 9 (left_distributivity) R->L }
% 0.60/0.58 addition(multiplication(addition(one, domain(multiplication(domain(x0), domain(x1)))), multiplication(domain(x0), domain(x1))), multiplication(domain(multiplication(domain(x0), domain(x1))), domain(x1)))
% 0.60/0.58 = { by lemma 11 }
% 0.60/0.58 addition(multiplication(one, multiplication(domain(x0), domain(x1))), multiplication(domain(multiplication(domain(x0), domain(x1))), domain(x1)))
% 0.60/0.58 = { by axiom 2 (multiplicative_left_identity) }
% 0.60/0.58 addition(multiplication(domain(x0), domain(x1)), multiplication(domain(multiplication(domain(x0), domain(x1))), domain(x1)))
% 0.60/0.58 = { by lemma 13 }
% 0.60/0.58 multiplication(addition(domain(multiplication(domain(x0), domain(x1))), domain(x0)), domain(x1))
% 0.60/0.58 = { by axiom 3 (additive_commutativity) }
% 0.60/0.58 multiplication(addition(domain(x0), domain(multiplication(domain(x0), domain(x1)))), domain(x1))
% 0.60/0.58 = { by lemma 10 R->L }
% 0.60/0.58 multiplication(addition(domain(domain(x0)), domain(multiplication(domain(x0), domain(x1)))), domain(x1))
% 0.60/0.58 = { by axiom 6 (domain5) R->L }
% 0.60/0.58 multiplication(domain(addition(domain(x0), multiplication(domain(x0), domain(x1)))), domain(x1))
% 0.60/0.58 = { by axiom 3 (additive_commutativity) R->L }
% 0.60/0.58 multiplication(domain(addition(multiplication(domain(x0), domain(x1)), domain(x0))), domain(x1))
% 0.60/0.58 = { by axiom 1 (multiplicative_right_identity) R->L }
% 0.60/0.58 multiplication(domain(addition(multiplication(domain(x0), domain(x1)), multiplication(domain(x0), one))), domain(x1))
% 0.60/0.58 = { by axiom 8 (right_distributivity) R->L }
% 0.60/0.58 multiplication(domain(multiplication(domain(x0), addition(domain(x1), one))), domain(x1))
% 0.60/0.58 = { by axiom 3 (additive_commutativity) }
% 0.60/0.58 multiplication(domain(multiplication(domain(x0), addition(one, domain(x1)))), domain(x1))
% 0.60/0.58 = { by lemma 11 }
% 0.60/0.58 multiplication(domain(multiplication(domain(x0), one)), domain(x1))
% 0.60/0.58 = { by axiom 1 (multiplicative_right_identity) }
% 0.60/0.58 multiplication(domain(domain(x0)), domain(x1))
% 0.60/0.58 = { by lemma 10 }
% 0.60/0.58 multiplication(domain(x0), domain(x1))
% 0.60/0.58 % SZS output end Proof
% 0.60/0.58
% 0.60/0.58 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------