%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE062+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n015.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 11:39:44 AM UTC 2026
% Result : Theorem 0.14s 0.89s
% Output : Proof 0.14s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE062+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.11/0.58 % Computer : n015.cluster.edu
% 0.11/0.58 % Model : x86_64 x86_64
% 0.11/0.58 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.58 % Memory : 8046.5625MB
% 0.11/0.58 % OS : Linux 6.8.0-71-generic
% 0.11/0.58 % CPULimit : 300
% 0.11/0.58 % WCLimit : 300
% 0.11/0.58 % DateTime : Sun Sep 27 13:12:00 UTC 2026
% 0.11/0.59 % CPUTime :
% 0.11/0.59 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.14/0.89 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.14/0.89
% 0.14/0.89 % SZS status Theorem
% 0.14/0.89
% 0.14/0.90 % SZS output start Proof
% 0.14/0.90 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.14/0.90 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.14/0.90 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.14/0.90 Axiom 4 (domain3): addition(domain(X), one) = one.
% 0.14/0.90 Axiom 5 (domain2): domain(multiplication(X, Y)) = domain(multiplication(X, domain(Y))).
% 0.14/0.90 Axiom 6 (domain5): domain(addition(X, Y)) = addition(domain(X), domain(Y)).
% 0.14/0.90 Axiom 7 (domain1): addition(X, multiplication(domain(X), X)) = multiplication(domain(X), X).
% 0.14/0.90 Axiom 8 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.14/0.90 Axiom 9 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.14/0.90
% 0.14/0.90 Lemma 10: domain(domain(X)) = domain(X).
% 0.14/0.91 Proof:
% 0.14/0.91 domain(domain(X))
% 0.14/0.91 = { by axiom 2 (multiplicative_left_identity) R->L }
% 0.14/0.91 domain(multiplication(one, domain(X)))
% 0.14/0.91 = { by axiom 5 (domain2) R->L }
% 0.14/0.91 domain(multiplication(one, X))
% 0.14/0.91 = { by axiom 2 (multiplicative_left_identity) }
% 0.14/0.91 domain(X)
% 0.14/0.91
% 0.14/0.91 Lemma 11: addition(one, domain(X)) = one.
% 0.14/0.91 Proof:
% 0.14/0.91 addition(one, domain(X))
% 0.14/0.91 = { by axiom 3 (additive_commutativity) R->L }
% 0.14/0.91 addition(domain(X), one)
% 0.14/0.91 = { by axiom 4 (domain3) }
% 0.14/0.91 one
% 0.14/0.91
% 0.14/0.91 Lemma 12: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
% 0.14/0.91 Proof:
% 0.14/0.91 addition(X, multiplication(Y, X))
% 0.14/0.91 = { by axiom 2 (multiplicative_left_identity) R->L }
% 0.14/0.91 addition(multiplication(one, X), multiplication(Y, X))
% 0.14/0.91 = { by axiom 9 (left_distributivity) R->L }
% 0.14/0.91 multiplication(addition(one, Y), X)
% 0.14/0.91 = { by axiom 3 (additive_commutativity) }
% 0.14/0.91 multiplication(addition(Y, one), X)
% 0.14/0.91
% 0.14/0.91 Lemma 13: addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Z, Y)).
% 0.14/0.91 Proof:
% 0.14/0.91 addition(multiplication(X, Y), multiplication(X, Z))
% 0.14/0.91 = { by axiom 8 (right_distributivity) R->L }
% 0.14/0.91 multiplication(X, addition(Y, Z))
% 0.14/0.91 = { by axiom 3 (additive_commutativity) }
% 0.14/0.91 multiplication(X, addition(Z, Y))
% 0.14/0.91
% 0.14/0.91 Lemma 14: addition(X, multiplication(X, domain(Y))) = X.
% 0.14/0.91 Proof:
% 0.14/0.91 addition(X, multiplication(X, domain(Y)))
% 0.14/0.91 = { by axiom 3 (additive_commutativity) R->L }
% 0.14/0.91 addition(multiplication(X, domain(Y)), X)
% 0.14/0.91 = { by axiom 1 (multiplicative_right_identity) R->L }
% 0.14/0.91 addition(multiplication(X, domain(Y)), multiplication(X, one))
% 0.14/0.91 = { by lemma 13 }
% 0.14/0.91 multiplication(X, addition(one, domain(Y)))
% 0.14/0.91 = { by lemma 11 }
% 0.14/0.91 multiplication(X, one)
% 0.14/0.91 = { by axiom 1 (multiplicative_right_identity) }
% 0.14/0.91 X
% 0.14/0.91
% 0.14/0.91 Lemma 15: multiplication(domain(X), multiplication(domain(Y), domain(X))) = multiplication(domain(Y), domain(X)).
% 0.14/0.91 Proof:
% 0.14/0.91 multiplication(domain(X), multiplication(domain(Y), domain(X)))
% 0.14/0.91 = { by lemma 10 R->L }
% 0.14/0.91 multiplication(domain(domain(X)), multiplication(domain(Y), domain(X)))
% 0.14/0.91 = { by axiom 2 (multiplicative_left_identity) R->L }
% 0.14/0.91 multiplication(domain(multiplication(one, domain(X))), multiplication(domain(Y), domain(X)))
% 0.14/0.91 = { by lemma 11 R->L }
% 0.14/0.91 multiplication(domain(multiplication(addition(one, domain(Y)), domain(X))), multiplication(domain(Y), domain(X)))
% 0.14/0.91 = { by axiom 3 (additive_commutativity) R->L }
% 0.14/0.91 multiplication(domain(multiplication(addition(domain(Y), one), domain(X))), multiplication(domain(Y), domain(X)))
% 0.14/0.91 = { by axiom 9 (left_distributivity) }
% 0.14/0.91 multiplication(domain(addition(multiplication(domain(Y), domain(X)), multiplication(one, domain(X)))), multiplication(domain(Y), domain(X)))
% 0.14/0.91 = { by axiom 2 (multiplicative_left_identity) }
% 0.14/0.91 multiplication(domain(addition(multiplication(domain(Y), domain(X)), domain(X))), multiplication(domain(Y), domain(X)))
% 0.14/0.91 = { by axiom 3 (additive_commutativity) }
% 0.14/0.91 multiplication(domain(addition(domain(X), multiplication(domain(Y), domain(X)))), multiplication(domain(Y), domain(X)))
% 0.14/0.91 = { by axiom 6 (domain5) }
% 0.14/0.91 multiplication(addition(domain(domain(X)), domain(multiplication(domain(Y), domain(X)))), multiplication(domain(Y), domain(X)))
% 0.14/0.91 = { by lemma 10 }
% 0.14/0.91 multiplication(addition(domain(X), domain(multiplication(domain(Y), domain(X)))), multiplication(domain(Y), domain(X)))
% 0.14/0.91 = { by axiom 9 (left_distributivity) }
% 0.14/0.91 addition(multiplication(domain(X), multiplication(domain(Y), domain(X))), multiplication(domain(multiplication(domain(Y), domain(X))), multiplication(domain(Y), domain(X))))
% 0.14/0.91 = { by axiom 7 (domain1) R->L }
% 0.14/0.91 addition(multiplication(domain(X), multiplication(domain(Y), domain(X))), addition(multiplication(domain(Y), domain(X)), multiplication(domain(multiplication(domain(Y), domain(X))), multiplication(domain(Y), domain(X)))))
% 0.14/0.91 = { by lemma 12 }
% 0.14/0.91 addition(multiplication(domain(X), multiplication(domain(Y), domain(X))), multiplication(addition(domain(multiplication(domain(Y), domain(X))), one), multiplication(domain(Y), domain(X))))
% 0.14/0.91 = { by axiom 3 (additive_commutativity) }
% 0.14/0.91 addition(multiplication(domain(X), multiplication(domain(Y), domain(X))), multiplication(addition(one, domain(multiplication(domain(Y), domain(X)))), multiplication(domain(Y), domain(X))))
% 0.14/0.91 = { by lemma 11 }
% 0.14/0.91 addition(multiplication(domain(X), multiplication(domain(Y), domain(X))), multiplication(one, multiplication(domain(Y), domain(X))))
% 0.14/0.91 = { by axiom 2 (multiplicative_left_identity) }
% 0.14/0.91 addition(multiplication(domain(X), multiplication(domain(Y), domain(X))), multiplication(domain(Y), domain(X)))
% 0.14/0.91 = { by axiom 3 (additive_commutativity) }
% 0.14/0.91 addition(multiplication(domain(Y), domain(X)), multiplication(domain(X), multiplication(domain(Y), domain(X))))
% 0.14/0.91 = { by lemma 12 }
% 0.14/0.91 multiplication(addition(domain(X), one), multiplication(domain(Y), domain(X)))
% 0.14/0.91 = { by axiom 3 (additive_commutativity) }
% 0.14/0.91 multiplication(addition(one, domain(X)), multiplication(domain(Y), domain(X)))
% 0.14/0.91 = { by lemma 11 }
% 0.14/0.91 multiplication(one, multiplication(domain(Y), domain(X)))
% 0.14/0.91 = { by axiom 2 (multiplicative_left_identity) }
% 0.14/0.91 multiplication(domain(Y), domain(X))
% 0.14/0.91
% 0.14/0.91 Goal 1 (goals): multiplication(domain(x0), domain(x1)) = multiplication(domain(x1), domain(x0)).
% 0.14/0.91 Proof:
% 0.14/0.91 multiplication(domain(x0), domain(x1))
% 0.14/0.91 = { by lemma 14 R->L }
% 0.14/0.91 multiplication(domain(x0), addition(domain(x1), multiplication(domain(x1), domain(x0))))
% 0.14/0.91 = { by axiom 3 (additive_commutativity) R->L }
% 0.14/0.91 multiplication(domain(x0), addition(multiplication(domain(x1), domain(x0)), domain(x1)))
% 0.14/0.91 = { by lemma 13 R->L }
% 0.14/0.91 addition(multiplication(domain(x0), domain(x1)), multiplication(domain(x0), multiplication(domain(x1), domain(x0))))
% 0.14/0.91 = { by lemma 15 }
% 0.14/0.91 addition(multiplication(domain(x0), domain(x1)), multiplication(domain(x1), domain(x0)))
% 0.14/0.91 = { by axiom 3 (additive_commutativity) R->L }
% 0.14/0.91 addition(multiplication(domain(x1), domain(x0)), multiplication(domain(x0), domain(x1)))
% 0.14/0.91 = { by lemma 15 R->L }
% 0.14/0.91 addition(multiplication(domain(x1), domain(x0)), multiplication(domain(x1), multiplication(domain(x0), domain(x1))))
% 0.14/0.91 = { by lemma 13 }
% 0.14/0.91 multiplication(domain(x1), addition(multiplication(domain(x0), domain(x1)), domain(x0)))
% 0.14/0.91 = { by axiom 3 (additive_commutativity) }
% 0.14/0.91 multiplication(domain(x1), addition(domain(x0), multiplication(domain(x0), domain(x1))))
% 0.14/0.91 = { by lemma 14 }
% 0.14/0.91 multiplication(domain(x1), domain(x0))
% 0.14/0.91 % SZS output end Proof
% 0.14/0.91
% 0.14/0.91 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------