%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE078+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n005.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 11:39:46 AM UTC 2026
% Result : Theorem 1.91s 0.76s
% Output : Proof 1.91s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE078+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.10/0.37 % Computer : n005.cluster.edu
% 0.10/0.37 % Model : x86_64 x86_64
% 0.10/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.37 % Memory : 8046.5625MB
% 0.10/0.37 % OS : Linux 6.8.0-71-generic
% 0.10/0.37 % CPULimit : 300
% 0.10/0.37 % WCLimit : 300
% 0.10/0.37 % DateTime : Sun Sep 27 13:09:01 UTC 2026
% 0.10/0.38 % CPUTime :
% 0.10/0.38 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 1.91/0.76 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 1.91/0.76
% 1.91/0.76 % SZS status Theorem
% 1.91/0.76
% 1.91/0.78 % SZS output start Proof
% 1.91/0.78 Axiom 1 (domain4): domain(zero) = zero.
% 1.91/0.78 Axiom 2 (additive_idempotence): addition(X, X) = X.
% 1.91/0.78 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 1.91/0.78 Axiom 4 (additive_identity): addition(X, zero) = X.
% 1.91/0.78 Axiom 5 (domain3): addition(domain(X), one) = one.
% 1.91/0.78 Axiom 6 (multiplicative_right_identity): multiplication(X, one) = X.
% 1.91/0.78 Axiom 7 (left_annihilation): multiplication(zero, X) = zero.
% 1.91/0.78 Axiom 8 (multiplicative_left_identity): multiplication(one, X) = X.
% 1.91/0.78 Axiom 9 (domain2): domain(multiplication(X, Y)) = domain(multiplication(X, domain(Y))).
% 1.91/0.78 Axiom 10 (goals): addition(domain(X), antidomain(X)) = one.
% 1.91/0.78 Axiom 11 (goals_1): multiplication(domain(X), antidomain(X)) = zero.
% 1.91/0.78 Axiom 12 (domain1): addition(X, multiplication(domain(X), X)) = multiplication(domain(X), X).
% 1.91/0.78 Axiom 13 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 1.91/0.78 Axiom 14 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 1.91/0.78 Axiom 15 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 1.91/0.78
% 1.91/0.78 Lemma 16: addition(one, domain(X)) = one.
% 1.91/0.78 Proof:
% 1.91/0.78 addition(one, domain(X))
% 1.91/0.78 = { by axiom 3 (additive_commutativity) R->L }
% 1.91/0.78 addition(domain(X), one)
% 1.91/0.78 = { by axiom 5 (domain3) }
% 1.91/0.78 one
% 1.91/0.78
% 1.91/0.78 Lemma 17: addition(antidomain(X), domain(X)) = one.
% 1.91/0.78 Proof:
% 1.91/0.78 addition(antidomain(X), domain(X))
% 1.91/0.78 = { by axiom 3 (additive_commutativity) R->L }
% 1.91/0.78 addition(domain(X), antidomain(X))
% 1.91/0.78 = { by axiom 10 (goals) }
% 1.91/0.78 one
% 1.91/0.78
% 1.91/0.78 Lemma 18: multiplication(X, addition(one, Y)) = addition(X, multiplication(X, Y)).
% 1.91/0.78 Proof:
% 1.91/0.78 multiplication(X, addition(one, Y))
% 1.91/0.78 = { by axiom 14 (right_distributivity) }
% 1.91/0.78 addition(multiplication(X, one), multiplication(X, Y))
% 1.91/0.78 = { by axiom 6 (multiplicative_right_identity) }
% 1.91/0.78 addition(X, multiplication(X, Y))
% 1.91/0.78
% 1.91/0.78 Lemma 19: domain(multiplication(domain(X), domain(antidomain(X)))) = zero.
% 1.91/0.78 Proof:
% 1.91/0.78 domain(multiplication(domain(X), domain(antidomain(X))))
% 1.91/0.78 = { by axiom 9 (domain2) R->L }
% 1.91/0.78 domain(multiplication(domain(X), antidomain(X)))
% 1.91/0.78 = { by axiom 11 (goals_1) }
% 1.91/0.78 domain(zero)
% 1.91/0.78 = { by axiom 1 (domain4) }
% 1.91/0.78 zero
% 1.91/0.78
% 1.91/0.78 Goal 1 (goals_2): domain(antidomain(x0)) = antidomain(x0).
% 1.91/0.78 Proof:
% 1.91/0.78 domain(antidomain(x0))
% 1.91/0.78 = { by axiom 8 (multiplicative_left_identity) R->L }
% 1.91/0.78 multiplication(one, domain(antidomain(x0)))
% 1.91/0.78 = { by lemma 17 R->L }
% 1.91/0.78 multiplication(addition(antidomain(x0), domain(x0)), domain(antidomain(x0)))
% 1.91/0.78 = { by axiom 15 (left_distributivity) }
% 1.91/0.78 addition(multiplication(antidomain(x0), domain(antidomain(x0))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 6 (multiplicative_right_identity) R->L }
% 1.91/0.78 addition(multiplication(antidomain(x0), multiplication(domain(antidomain(x0)), one)), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by lemma 17 R->L }
% 1.91/0.78 addition(multiplication(antidomain(x0), multiplication(domain(antidomain(x0)), addition(antidomain(x0), domain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 2 (additive_idempotence) R->L }
% 1.91/0.78 addition(multiplication(antidomain(x0), multiplication(domain(antidomain(x0)), addition(addition(antidomain(x0), antidomain(x0)), domain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 13 (additive_associativity) R->L }
% 1.91/0.78 addition(multiplication(antidomain(x0), multiplication(domain(antidomain(x0)), addition(antidomain(x0), addition(antidomain(x0), domain(x0))))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by lemma 17 }
% 1.91/0.78 addition(multiplication(antidomain(x0), multiplication(domain(antidomain(x0)), addition(antidomain(x0), one))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 3 (additive_commutativity) }
% 1.91/0.78 addition(multiplication(antidomain(x0), multiplication(domain(antidomain(x0)), addition(one, antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by lemma 18 }
% 1.91/0.78 addition(multiplication(antidomain(x0), addition(domain(antidomain(x0)), multiplication(domain(antidomain(x0)), antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 12 (domain1) R->L }
% 1.91/0.78 addition(multiplication(antidomain(x0), addition(domain(antidomain(x0)), addition(antidomain(x0), multiplication(domain(antidomain(x0)), antidomain(x0))))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 8 (multiplicative_left_identity) R->L }
% 1.91/0.78 addition(multiplication(antidomain(x0), addition(domain(antidomain(x0)), addition(multiplication(one, antidomain(x0)), multiplication(domain(antidomain(x0)), antidomain(x0))))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 15 (left_distributivity) R->L }
% 1.91/0.78 addition(multiplication(antidomain(x0), addition(domain(antidomain(x0)), multiplication(addition(one, domain(antidomain(x0))), antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by lemma 16 }
% 1.91/0.78 addition(multiplication(antidomain(x0), addition(domain(antidomain(x0)), multiplication(one, antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 8 (multiplicative_left_identity) }
% 1.91/0.78 addition(multiplication(antidomain(x0), addition(domain(antidomain(x0)), antidomain(x0))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 3 (additive_commutativity) R->L }
% 1.91/0.78 addition(multiplication(antidomain(x0), addition(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 14 (right_distributivity) }
% 1.91/0.78 addition(addition(multiplication(antidomain(x0), antidomain(x0)), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 4 (additive_identity) R->L }
% 1.91/0.78 addition(addition(addition(multiplication(antidomain(x0), antidomain(x0)), zero), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 3 (additive_commutativity) }
% 1.91/0.78 addition(addition(addition(zero, multiplication(antidomain(x0), antidomain(x0))), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 11 (goals_1) R->L }
% 1.91/0.78 addition(addition(addition(multiplication(domain(x0), antidomain(x0)), multiplication(antidomain(x0), antidomain(x0))), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 15 (left_distributivity) R->L }
% 1.91/0.78 addition(addition(multiplication(addition(domain(x0), antidomain(x0)), antidomain(x0)), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 3 (additive_commutativity) }
% 1.91/0.78 addition(addition(multiplication(addition(antidomain(x0), domain(x0)), antidomain(x0)), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by lemma 17 }
% 1.91/0.78 addition(addition(multiplication(one, antidomain(x0)), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 8 (multiplicative_left_identity) }
% 1.91/0.78 addition(addition(antidomain(x0), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by lemma 18 R->L }
% 1.91/0.78 addition(multiplication(antidomain(x0), addition(one, domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by lemma 16 }
% 1.91/0.78 addition(multiplication(antidomain(x0), one), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 6 (multiplicative_right_identity) }
% 1.91/0.78 addition(antidomain(x0), multiplication(domain(x0), domain(antidomain(x0))))
% 1.91/0.78 = { by axiom 4 (additive_identity) R->L }
% 1.91/0.78 addition(antidomain(x0), addition(multiplication(domain(x0), domain(antidomain(x0))), zero))
% 1.91/0.78 = { by axiom 7 (left_annihilation) R->L }
% 1.91/0.78 addition(antidomain(x0), addition(multiplication(domain(x0), domain(antidomain(x0))), multiplication(zero, multiplication(domain(x0), domain(antidomain(x0))))))
% 1.91/0.78 = { by lemma 19 R->L }
% 1.91/0.78 addition(antidomain(x0), addition(multiplication(domain(x0), domain(antidomain(x0))), multiplication(domain(multiplication(domain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))))
% 1.91/0.78 = { by axiom 12 (domain1) }
% 1.91/0.78 addition(antidomain(x0), multiplication(domain(multiplication(domain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0)))))
% 1.91/0.78 = { by lemma 19 }
% 1.91/0.78 addition(antidomain(x0), multiplication(zero, multiplication(domain(x0), domain(antidomain(x0)))))
% 1.91/0.78 = { by axiom 7 (left_annihilation) }
% 1.91/0.78 addition(antidomain(x0), zero)
% 1.91/0.78 = { by axiom 4 (additive_identity) }
% 1.91/0.78 antidomain(x0)
% 1.91/0.78 % SZS output end Proof
% 1.91/0.78
% 1.91/0.78 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------