%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE082+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 11:39:47 AM UTC 2026
% Result : Theorem 0.26s 0.54s
% Output : Proof 0.26s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : KLE082+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.11/0.36 % Computer : n015.cluster.edu
% 0.11/0.36 % Model : x86_64 x86_64
% 0.11/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.36 % Memory : 8046.5625MB
% 0.11/0.36 % OS : Linux 6.8.0-71-generic
% 0.11/0.36 % CPULimit : 300
% 0.11/0.36 % WCLimit : 300
% 0.11/0.36 % DateTime : Sun Sep 27 13:12:30 UTC 2026
% 0.11/0.36 % CPUTime :
% 0.11/0.36 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.26/0.54 Command-line arguments: --flatten --complete-subsets
% 0.26/0.54
% 0.26/0.54 % SZS status Theorem
% 0.26/0.54
% 0.26/0.54 % SZS output start Proof
% 0.26/0.54 Axiom 1 (domain4): domain(zero) = zero.
% 0.26/0.54 Axiom 2 (additive_idempotence): addition(X, X) = X.
% 0.26/0.54 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.26/0.54 Axiom 4 (additive_identity): addition(X, zero) = X.
% 0.26/0.54 Axiom 5 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.26/0.54 Axiom 6 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.26/0.54 Axiom 7 (domain2): domain(multiplication(X, Y)) = domain(multiplication(X, domain(Y))).
% 0.26/0.54 Axiom 8 (goals): addition(domain(X), antidomain(X)) = one.
% 0.26/0.54 Axiom 9 (goals_1): multiplication(domain(X), antidomain(X)) = zero.
% 0.26/0.54 Axiom 10 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.26/0.54 Axiom 11 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.26/0.54 Axiom 12 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.26/0.55
% 0.26/0.55 Lemma 13: addition(antidomain(X), domain(X)) = one.
% 0.26/0.55 Proof:
% 0.26/0.55 addition(antidomain(X), domain(X))
% 0.26/0.55 = { by axiom 3 (additive_commutativity) R->L }
% 0.26/0.55 addition(domain(X), antidomain(X))
% 0.26/0.55 = { by axiom 8 (goals) }
% 0.26/0.55 one
% 0.26/0.55
% 0.26/0.55 Goal 1 (goals_2): addition(antidomain(multiplication(x0, x1)), antidomain(multiplication(x0, domain(x1)))) = antidomain(multiplication(x0, domain(x1))).
% 0.26/0.55 Proof:
% 0.26/0.55 addition(antidomain(multiplication(x0, x1)), antidomain(multiplication(x0, domain(x1))))
% 0.26/0.55 = { by axiom 3 (additive_commutativity) }
% 0.26/0.55 addition(antidomain(multiplication(x0, domain(x1))), antidomain(multiplication(x0, x1)))
% 0.26/0.55 = { by axiom 6 (multiplicative_left_identity) R->L }
% 0.26/0.55 addition(antidomain(multiplication(x0, domain(x1))), multiplication(one, antidomain(multiplication(x0, x1))))
% 0.26/0.55 = { by lemma 13 R->L }
% 0.26/0.55 addition(antidomain(multiplication(x0, domain(x1))), multiplication(addition(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, domain(x1)))), antidomain(multiplication(x0, x1))))
% 0.26/0.55 = { by axiom 3 (additive_commutativity) R->L }
% 0.26/0.55 addition(antidomain(multiplication(x0, domain(x1))), multiplication(addition(domain(multiplication(x0, domain(x1))), antidomain(multiplication(x0, domain(x1)))), antidomain(multiplication(x0, x1))))
% 0.26/0.55 = { by axiom 7 (domain2) R->L }
% 0.26/0.55 addition(antidomain(multiplication(x0, domain(x1))), multiplication(addition(domain(multiplication(x0, x1)), antidomain(multiplication(x0, domain(x1)))), antidomain(multiplication(x0, x1))))
% 0.26/0.55 = { by axiom 12 (left_distributivity) }
% 0.26/0.55 addition(antidomain(multiplication(x0, domain(x1))), addition(multiplication(domain(multiplication(x0, x1)), antidomain(multiplication(x0, x1))), multiplication(antidomain(multiplication(x0, domain(x1))), antidomain(multiplication(x0, x1)))))
% 0.26/0.55 = { by axiom 9 (goals_1) }
% 0.26/0.55 addition(antidomain(multiplication(x0, domain(x1))), addition(zero, multiplication(antidomain(multiplication(x0, domain(x1))), antidomain(multiplication(x0, x1)))))
% 0.26/0.55 = { by axiom 1 (domain4) R->L }
% 0.26/0.55 addition(antidomain(multiplication(x0, domain(x1))), addition(domain(zero), multiplication(antidomain(multiplication(x0, domain(x1))), antidomain(multiplication(x0, x1)))))
% 0.26/0.55 = { by axiom 3 (additive_commutativity) R->L }
% 0.26/0.55 addition(antidomain(multiplication(x0, domain(x1))), addition(multiplication(antidomain(multiplication(x0, domain(x1))), antidomain(multiplication(x0, x1))), domain(zero)))
% 0.26/0.55 = { by axiom 1 (domain4) }
% 0.26/0.55 addition(antidomain(multiplication(x0, domain(x1))), addition(multiplication(antidomain(multiplication(x0, domain(x1))), antidomain(multiplication(x0, x1))), zero))
% 0.26/0.55 = { by axiom 4 (additive_identity) }
% 0.26/0.55 addition(antidomain(multiplication(x0, domain(x1))), multiplication(antidomain(multiplication(x0, domain(x1))), antidomain(multiplication(x0, x1))))
% 0.26/0.55 = { by axiom 5 (multiplicative_right_identity) R->L }
% 0.26/0.55 addition(multiplication(antidomain(multiplication(x0, domain(x1))), one), multiplication(antidomain(multiplication(x0, domain(x1))), antidomain(multiplication(x0, x1))))
% 0.26/0.55 = { by axiom 11 (right_distributivity) R->L }
% 0.26/0.55 multiplication(antidomain(multiplication(x0, domain(x1))), addition(one, antidomain(multiplication(x0, x1))))
% 0.26/0.55 = { by axiom 3 (additive_commutativity) }
% 0.26/0.55 multiplication(antidomain(multiplication(x0, domain(x1))), addition(antidomain(multiplication(x0, x1)), one))
% 0.26/0.55 = { by lemma 13 R->L }
% 0.26/0.55 multiplication(antidomain(multiplication(x0, domain(x1))), addition(antidomain(multiplication(x0, x1)), addition(antidomain(multiplication(x0, x1)), domain(multiplication(x0, x1)))))
% 0.26/0.55 = { by axiom 10 (additive_associativity) }
% 0.26/0.55 multiplication(antidomain(multiplication(x0, domain(x1))), addition(addition(antidomain(multiplication(x0, x1)), antidomain(multiplication(x0, x1))), domain(multiplication(x0, x1))))
% 0.26/0.55 = { by axiom 2 (additive_idempotence) }
% 0.26/0.55 multiplication(antidomain(multiplication(x0, domain(x1))), addition(antidomain(multiplication(x0, x1)), domain(multiplication(x0, x1))))
% 0.26/0.55 = { by lemma 13 }
% 0.26/0.55 multiplication(antidomain(multiplication(x0, domain(x1))), one)
% 0.26/0.55 = { by axiom 5 (multiplicative_right_identity) }
% 0.26/0.55 antidomain(multiplication(x0, domain(x1)))
% 0.26/0.55 % SZS output end Proof
% 0.26/0.55
% 0.26/0.55 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------