%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE069+1 : 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 11:39:45 AM UTC 2026
% Result : Theorem 0.10s 0.46s
% Output : Proof 0.10s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE069+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 : n016.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 13:13:16 UTC 2026
% 0.10/0.36 % CPUTime :
% 0.10/0.36 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.10/0.46 Command-line arguments: --flatten --complete-subsets
% 0.10/0.46
% 0.10/0.46 % SZS status Theorem
% 0.10/0.46
% 0.10/0.46 % SZS output start Proof
% 0.10/0.47 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.10/0.47 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.10/0.47 Axiom 3 (additive_idempotence): addition(X, X) = X.
% 0.10/0.47 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.10/0.47 Axiom 5 (domain3): addition(domain(X), one) = one.
% 0.10/0.47 Axiom 6 (domain2): domain(multiplication(X, Y)) = domain(multiplication(X, domain(Y))).
% 0.10/0.47 Axiom 7 (domain5): domain(addition(X, Y)) = addition(domain(X), domain(Y)).
% 0.10/0.47 Axiom 8 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 0.10/0.47 Axiom 9 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 0.10/0.47 Axiom 10 (domain1): addition(X, multiplication(domain(X), X)) = multiplication(domain(X), X).
% 0.10/0.47 Axiom 11 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.10/0.47 Axiom 12 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.10/0.47 Axiom 13 (order_1): ifeq(leq(X, Y), true, addition(X, Y), Y) = Y.
% 0.10/0.47 Axiom 14 (order): ifeq2(addition(X, Y), Y, leq(X, Y), true) = true.
% 0.10/0.47
% 0.10/0.47 Lemma 15: domain(domain(X)) = domain(X).
% 0.10/0.47 Proof:
% 0.10/0.47 domain(domain(X))
% 0.10/0.47 = { by axiom 2 (multiplicative_left_identity) R->L }
% 0.10/0.47 domain(multiplication(one, domain(X)))
% 0.10/0.47 = { by axiom 6 (domain2) R->L }
% 0.10/0.47 domain(multiplication(one, X))
% 0.10/0.47 = { by axiom 2 (multiplicative_left_identity) }
% 0.10/0.47 domain(X)
% 0.10/0.47
% 0.10/0.47 Lemma 16: addition(one, domain(X)) = one.
% 0.10/0.47 Proof:
% 0.10/0.47 addition(one, domain(X))
% 0.10/0.47 = { by axiom 4 (additive_commutativity) R->L }
% 0.10/0.47 addition(domain(X), one)
% 0.10/0.47 = { by axiom 5 (domain3) }
% 0.10/0.47 one
% 0.10/0.47
% 0.10/0.47 Lemma 17: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
% 0.10/0.47 Proof:
% 0.10/0.47 addition(X, multiplication(X, Y))
% 0.10/0.47 = { by axiom 1 (multiplicative_right_identity) R->L }
% 0.10/0.47 addition(multiplication(X, one), multiplication(X, Y))
% 0.10/0.47 = { by axiom 12 (right_distributivity) R->L }
% 0.10/0.47 multiplication(X, addition(one, Y))
% 0.10/0.47 = { by axiom 4 (additive_commutativity) }
% 0.10/0.47 multiplication(X, addition(Y, one))
% 0.10/0.47
% 0.10/0.47 Goal 1 (goals): multiplication(domain(x0), addition(domain(x0), domain(x1))) = domain(x0).
% 0.10/0.47 Proof:
% 0.10/0.47 multiplication(domain(x0), addition(domain(x0), domain(x1)))
% 0.10/0.47 = { by axiom 4 (additive_commutativity) }
% 0.10/0.47 multiplication(domain(x0), addition(domain(x1), domain(x0)))
% 0.10/0.47 = { by axiom 3 (additive_idempotence) R->L }
% 0.10/0.47 multiplication(domain(x0), addition(domain(x1), addition(domain(x0), domain(x0))))
% 0.10/0.47 = { by axiom 11 (additive_associativity) }
% 0.10/0.47 multiplication(domain(x0), addition(addition(domain(x1), domain(x0)), domain(x0)))
% 0.10/0.47 = { by axiom 4 (additive_commutativity) R->L }
% 0.10/0.47 multiplication(domain(x0), addition(addition(domain(x0), domain(x1)), domain(x0)))
% 0.10/0.47 = { by axiom 12 (right_distributivity) }
% 0.10/0.47 addition(multiplication(domain(x0), addition(domain(x0), domain(x1))), multiplication(domain(x0), domain(x0)))
% 0.10/0.47 = { by axiom 13 (order_1) R->L }
% 0.10/0.47 addition(multiplication(domain(x0), addition(domain(x0), domain(x1))), ifeq(leq(domain(x0), multiplication(domain(x0), domain(x0))), true, addition(domain(x0), multiplication(domain(x0), domain(x0))), multiplication(domain(x0), domain(x0))))
% 0.10/0.47 = { by lemma 15 R->L }
% 0.10/0.47 addition(multiplication(domain(x0), addition(domain(x0), domain(x1))), ifeq(leq(domain(x0), multiplication(domain(domain(x0)), domain(x0))), true, addition(domain(x0), multiplication(domain(x0), domain(x0))), multiplication(domain(x0), domain(x0))))
% 0.10/0.47 = { by axiom 9 (ifeq_axiom) R->L }
% 0.10/0.47 addition(multiplication(domain(x0), addition(domain(x0), domain(x1))), ifeq(ifeq2(multiplication(domain(domain(x0)), domain(x0)), multiplication(domain(domain(x0)), domain(x0)), leq(domain(x0), multiplication(domain(domain(x0)), domain(x0))), true), true, addition(domain(x0), multiplication(domain(x0), domain(x0))), multiplication(domain(x0), domain(x0))))
% 0.10/0.47 = { by axiom 10 (domain1) R->L }
% 0.10/0.47 addition(multiplication(domain(x0), addition(domain(x0), domain(x1))), ifeq(ifeq2(addition(domain(x0), multiplication(domain(domain(x0)), domain(x0))), multiplication(domain(domain(x0)), domain(x0)), leq(domain(x0), multiplication(domain(domain(x0)), domain(x0))), true), true, addition(domain(x0), multiplication(domain(x0), domain(x0))), multiplication(domain(x0), domain(x0))))
% 0.10/0.47 = { by axiom 14 (order) }
% 0.10/0.47 addition(multiplication(domain(x0), addition(domain(x0), domain(x1))), ifeq(true, true, addition(domain(x0), multiplication(domain(x0), domain(x0))), multiplication(domain(x0), domain(x0))))
% 0.10/0.47 = { by axiom 8 (ifeq_axiom) }
% 0.10/0.47 addition(multiplication(domain(x0), addition(domain(x0), domain(x1))), addition(domain(x0), multiplication(domain(x0), domain(x0))))
% 0.10/0.47 = { by lemma 17 }
% 0.10/0.47 addition(multiplication(domain(x0), addition(domain(x0), domain(x1))), multiplication(domain(x0), addition(domain(x0), one)))
% 0.10/0.47 = { by axiom 4 (additive_commutativity) R->L }
% 0.10/0.47 addition(multiplication(domain(x0), addition(domain(x0), domain(x1))), multiplication(domain(x0), addition(one, domain(x0))))
% 0.10/0.47 = { by lemma 16 }
% 0.10/0.47 addition(multiplication(domain(x0), addition(domain(x0), domain(x1))), multiplication(domain(x0), one))
% 0.10/0.47 = { by axiom 1 (multiplicative_right_identity) }
% 0.10/0.47 addition(multiplication(domain(x0), addition(domain(x0), domain(x1))), domain(x0))
% 0.10/0.47 = { by axiom 4 (additive_commutativity) }
% 0.10/0.47 addition(domain(x0), multiplication(domain(x0), addition(domain(x0), domain(x1))))
% 0.10/0.47 = { by lemma 17 }
% 0.10/0.47 multiplication(domain(x0), addition(addition(domain(x0), domain(x1)), one))
% 0.10/0.47 = { by axiom 4 (additive_commutativity) R->L }
% 0.10/0.47 multiplication(domain(x0), addition(one, addition(domain(x0), domain(x1))))
% 0.10/0.47 = { by axiom 4 (additive_commutativity) }
% 0.10/0.47 multiplication(domain(x0), addition(one, addition(domain(x1), domain(x0))))
% 0.10/0.47 = { by lemma 15 R->L }
% 0.10/0.47 multiplication(domain(x0), addition(one, addition(domain(x1), domain(domain(x0)))))
% 0.10/0.47 = { by lemma 15 R->L }
% 0.10/0.47 multiplication(domain(x0), addition(one, addition(domain(domain(x1)), domain(domain(x0)))))
% 0.10/0.47 = { by axiom 7 (domain5) R->L }
% 0.10/0.47 multiplication(domain(x0), addition(one, domain(addition(domain(x1), domain(x0)))))
% 0.10/0.47 = { by axiom 4 (additive_commutativity) R->L }
% 0.10/0.47 multiplication(domain(x0), addition(one, domain(addition(domain(x0), domain(x1)))))
% 0.10/0.47 = { by lemma 16 }
% 0.10/0.47 multiplication(domain(x0), one)
% 0.10/0.47 = { by axiom 1 (multiplicative_right_identity) }
% 0.10/0.47 domain(x0)
% 0.10/0.47 % SZS output end Proof
% 0.10/0.47
% 0.10/0.47 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------