%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE088+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n026.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 11:39:48 AM UTC 2026
% Result : Theorem 3.75s 0.98s
% Output : Proof 3.75s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE088+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.11/0.38 % Computer : n026.cluster.edu
% 0.11/0.38 % Model : x86_64 x86_64
% 0.11/0.38 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.38 % Memory : 8046.5625MB
% 0.11/0.38 % OS : Linux 6.8.0-71-generic
% 0.11/0.38 % CPULimit : 300
% 0.11/0.38 % WCLimit : 300
% 0.11/0.38 % DateTime : Sun Sep 27 13:12:11 UTC 2026
% 0.11/0.38 % CPUTime :
% 0.11/0.38 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 3.75/0.98 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 3.75/0.98
% 3.75/0.98 % SZS status Theorem
% 3.75/0.98
% 3.75/0.98 % SZS output start Proof
% 3.75/0.98 Axiom 1 (domain4): domain(X) = antidomain(antidomain(X)).
% 3.75/0.98 Axiom 2 (multiplicative_right_identity): multiplication(X, one) = X.
% 3.75/0.98 Axiom 3 (multiplicative_left_identity): multiplication(one, X) = X.
% 3.75/0.98 Axiom 4 (domain1): multiplication(antidomain(X), X) = zero.
% 3.75/0.98 Axiom 5 (additive_idempotence): addition(X, X) = X.
% 3.75/0.98 Axiom 6 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 3.75/0.98 Axiom 7 (additive_identity): addition(X, zero) = X.
% 3.75/0.98 Axiom 8 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 3.75/0.98 Axiom 9 (goals): multiplication(domain(x0), x1) = zero.
% 3.75/0.98 Axiom 10 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 3.75/0.98 Axiom 11 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 3.75/0.98 Axiom 12 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 3.75/0.98 Axiom 13 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 3.75/0.98
% 3.75/0.98 Lemma 14: antidomain(one) = zero.
% 3.75/0.99 Proof:
% 3.75/0.99 antidomain(one)
% 3.75/0.99 = { by axiom 2 (multiplicative_right_identity) R->L }
% 3.75/0.99 multiplication(antidomain(one), one)
% 3.75/0.99 = { by axiom 4 (domain1) }
% 3.75/0.99 zero
% 3.75/0.99
% 3.75/0.99 Lemma 15: addition(zero, X) = X.
% 3.75/0.99 Proof:
% 3.75/0.99 addition(zero, X)
% 3.75/0.99 = { by axiom 6 (additive_commutativity) R->L }
% 3.75/0.99 addition(X, zero)
% 3.75/0.99 = { by axiom 7 (additive_identity) }
% 3.75/0.99 X
% 3.75/0.99
% 3.75/0.99 Lemma 16: addition(X, addition(X, Y)) = addition(X, Y).
% 3.75/0.99 Proof:
% 3.75/0.99 addition(X, addition(X, Y))
% 3.75/0.99 = { by axiom 10 (additive_associativity) }
% 3.75/0.99 addition(addition(X, X), Y)
% 3.75/0.99 = { by axiom 5 (additive_idempotence) }
% 3.75/0.99 addition(X, Y)
% 3.75/0.99
% 3.75/0.99 Lemma 17: addition(antidomain(X), antidomain(antidomain(X))) = one.
% 3.75/0.99 Proof:
% 3.75/0.99 addition(antidomain(X), antidomain(antidomain(X)))
% 3.75/0.99 = { by axiom 6 (additive_commutativity) R->L }
% 3.75/0.99 addition(antidomain(antidomain(X)), antidomain(X))
% 3.75/0.99 = { by axiom 8 (domain3) }
% 3.75/0.99 one
% 3.75/0.99
% 3.75/0.99 Lemma 18: addition(domain(X), antidomain(domain(X))) = one.
% 3.75/0.99 Proof:
% 3.75/0.99 addition(domain(X), antidomain(domain(X)))
% 3.75/0.99 = { by axiom 1 (domain4) }
% 3.75/0.99 addition(domain(X), antidomain(antidomain(antidomain(X))))
% 3.75/0.99 = { by axiom 1 (domain4) }
% 3.75/0.99 addition(antidomain(antidomain(X)), antidomain(antidomain(antidomain(X))))
% 3.75/0.99 = { by lemma 17 }
% 3.75/0.99 one
% 3.75/0.99
% 3.75/0.99 Lemma 19: multiplication(addition(X, antidomain(Y)), Y) = multiplication(X, Y).
% 3.75/0.99 Proof:
% 3.75/0.99 multiplication(addition(X, antidomain(Y)), Y)
% 3.75/0.99 = { by axiom 6 (additive_commutativity) R->L }
% 3.75/0.99 multiplication(addition(antidomain(Y), X), Y)
% 3.75/0.99 = { by axiom 12 (left_distributivity) }
% 3.75/0.99 addition(multiplication(antidomain(Y), Y), multiplication(X, Y))
% 3.75/0.99 = { by axiom 4 (domain1) }
% 3.75/0.99 addition(zero, multiplication(X, Y))
% 3.75/0.99 = { by lemma 15 }
% 3.75/0.99 multiplication(X, Y)
% 3.75/0.99
% 3.75/0.99 Goal 1 (goals_1): addition(domain(x0), antidomain(x1)) = antidomain(x1).
% 3.75/0.99 Proof:
% 3.75/0.99 addition(domain(x0), antidomain(x1))
% 3.75/0.99 = { by axiom 2 (multiplicative_right_identity) R->L }
% 3.75/0.99 multiplication(addition(domain(x0), antidomain(x1)), one)
% 3.75/0.99 = { by lemma 17 R->L }
% 3.75/0.99 multiplication(addition(domain(x0), antidomain(x1)), addition(antidomain(x1), antidomain(antidomain(x1))))
% 3.75/0.99 = { by axiom 11 (right_distributivity) }
% 3.75/0.99 addition(multiplication(addition(domain(x0), antidomain(x1)), antidomain(x1)), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))
% 3.75/0.99 = { by lemma 19 R->L }
% 3.75/0.99 addition(multiplication(addition(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))), antidomain(x1)), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))
% 3.75/0.99 = { by axiom 10 (additive_associativity) R->L }
% 3.75/0.99 addition(multiplication(addition(domain(x0), addition(antidomain(x1), antidomain(antidomain(x1)))), antidomain(x1)), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))
% 3.75/0.99 = { by lemma 17 }
% 3.75/0.99 addition(multiplication(addition(domain(x0), one), antidomain(x1)), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))
% 3.75/0.99 = { by lemma 18 R->L }
% 3.75/0.99 addition(multiplication(addition(domain(x0), addition(domain(x0), antidomain(domain(x0)))), antidomain(x1)), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))
% 3.75/0.99 = { by lemma 16 }
% 3.75/0.99 addition(multiplication(addition(domain(x0), antidomain(domain(x0))), antidomain(x1)), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))
% 3.75/0.99 = { by lemma 18 }
% 3.75/0.99 addition(multiplication(one, antidomain(x1)), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))
% 3.75/0.99 = { by axiom 3 (multiplicative_left_identity) }
% 3.75/0.99 addition(antidomain(x1), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))
% 3.75/0.99 = { by axiom 3 (multiplicative_left_identity) R->L }
% 3.75/0.99 addition(antidomain(x1), multiplication(one, multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))))
% 3.75/0.99 = { by lemma 17 R->L }
% 3.75/0.99 addition(antidomain(x1), multiplication(addition(antidomain(multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))), antidomain(antidomain(multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))))), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))))
% 3.75/0.99 = { by lemma 16 R->L }
% 3.75/0.99 addition(antidomain(x1), multiplication(addition(antidomain(multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))), addition(antidomain(multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))), antidomain(antidomain(multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))))), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))))
% 3.75/0.99 = { by lemma 17 }
% 3.75/0.99 addition(antidomain(x1), multiplication(addition(antidomain(multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))), one), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))))
% 3.75/0.99 = { by axiom 6 (additive_commutativity) }
% 3.75/0.99 addition(antidomain(x1), multiplication(addition(one, antidomain(multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))))
% 3.75/0.99 = { by lemma 17 R->L }
% 3.75/0.99 addition(antidomain(x1), multiplication(addition(addition(antidomain(one), antidomain(antidomain(one))), antidomain(multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))))
% 3.75/0.99 = { by lemma 14 }
% 3.75/0.99 addition(antidomain(x1), multiplication(addition(addition(zero, antidomain(antidomain(one))), antidomain(multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))))
% 3.75/1.00 = { by lemma 15 }
% 3.75/1.00 addition(antidomain(x1), multiplication(addition(antidomain(antidomain(one)), antidomain(multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))))
% 3.75/1.00 = { by lemma 14 }
% 3.75/1.00 addition(antidomain(x1), multiplication(addition(antidomain(zero), antidomain(multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))))
% 3.75/1.00 = { by axiom 9 (goals) R->L }
% 3.75/1.00 addition(antidomain(x1), multiplication(addition(antidomain(multiplication(domain(x0), x1)), antidomain(multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))))
% 3.75/1.00 = { by lemma 19 R->L }
% 3.75/1.00 addition(antidomain(x1), multiplication(addition(antidomain(multiplication(addition(domain(x0), antidomain(x1)), x1)), antidomain(multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1))))), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))))
% 3.75/1.00 = { by axiom 13 (domain2) }
% 3.75/1.00 addition(antidomain(x1), multiplication(antidomain(multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))), multiplication(addition(domain(x0), antidomain(x1)), antidomain(antidomain(x1)))))
% 3.75/1.00 = { by axiom 4 (domain1) }
% 3.75/1.00 addition(antidomain(x1), zero)
% 3.75/1.00 = { by axiom 7 (additive_identity) }
% 3.75/1.00 antidomain(x1)
% 3.75/1.00 % SZS output end Proof
% 3.75/1.00
% 3.75/1.00 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------