%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE098+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n017.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 11:39:49 AM UTC 2026
% Result : Theorem 3.05s 0.82s
% Output : Proof 3.05s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : KLE098+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.09/0.36 % Computer : n017.cluster.edu
% 0.09/0.36 % Model : x86_64 x86_64
% 0.09/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.36 % Memory : 8046.5625MB
% 0.09/0.36 % OS : Linux 6.8.0-71-generic
% 0.09/0.36 % CPULimit : 300
% 0.09/0.36 % WCLimit : 300
% 0.09/0.36 % DateTime : Sun Sep 27 13:05:20 UTC 2026
% 0.09/0.36 % CPUTime :
% 0.09/0.36 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 3.05/0.82 Command-line arguments: --flatten --complete-subsets
% 3.05/0.82
% 3.05/0.82 % SZS status Theorem
% 3.05/0.82
% 3.05/0.84 % SZS output start Proof
% 3.05/0.84 Axiom 1 (complement): c(X) = antidomain(domain(X)).
% 3.05/0.84 Axiom 2 (domain4): domain(X) = antidomain(antidomain(X)).
% 3.05/0.84 Axiom 3 (additive_idempotence): addition(X, X) = X.
% 3.05/0.84 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 3.05/0.84 Axiom 5 (additive_identity): addition(X, zero) = X.
% 3.05/0.84 Axiom 6 (multiplicative_right_identity): multiplication(X, one) = X.
% 3.05/0.84 Axiom 7 (multiplicative_left_identity): multiplication(one, X) = X.
% 3.05/0.84 Axiom 8 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 3.05/0.84 Axiom 9 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 3.05/0.84 Axiom 10 (domain1): multiplication(antidomain(X), X) = zero.
% 3.05/0.84 Axiom 11 (goals): addition(forward_diamond(x0, domain(x1)), domain(x2)) = domain(x2).
% 3.05/0.84 Axiom 12 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 3.05/0.84 Axiom 13 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 3.05/0.84 Axiom 14 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 3.05/0.84 Axiom 15 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 3.05/0.84 Axiom 16 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 3.05/0.84
% 3.05/0.84 Lemma 17: addition(domain(X), antidomain(X)) = one.
% 3.05/0.84 Proof:
% 3.05/0.84 addition(domain(X), antidomain(X))
% 3.05/0.84 = { by axiom 2 (domain4) }
% 3.05/0.84 addition(antidomain(antidomain(X)), antidomain(X))
% 3.05/0.84 = { by axiom 14 (domain3) }
% 3.05/0.84 one
% 3.05/0.84
% 3.05/0.84 Lemma 18: multiplication(domain(X), X) = X.
% 3.05/0.84 Proof:
% 3.05/0.84 multiplication(domain(X), X)
% 3.05/0.84 = { by axiom 5 (additive_identity) R->L }
% 3.05/0.84 addition(multiplication(domain(X), X), zero)
% 3.05/0.84 = { by axiom 10 (domain1) R->L }
% 3.05/0.84 addition(multiplication(domain(X), X), multiplication(antidomain(X), X))
% 3.05/0.84 = { by axiom 16 (left_distributivity) R->L }
% 3.05/0.84 multiplication(addition(domain(X), antidomain(X)), X)
% 3.05/0.84 = { by lemma 17 }
% 3.05/0.84 multiplication(one, X)
% 3.05/0.84 = { by axiom 7 (multiplicative_left_identity) }
% 3.05/0.84 X
% 3.05/0.84
% 3.05/0.84 Lemma 19: domain(antidomain(X)) = c(X).
% 3.05/0.84 Proof:
% 3.05/0.84 domain(antidomain(X))
% 3.05/0.84 = { by axiom 2 (domain4) }
% 3.05/0.84 antidomain(antidomain(antidomain(X)))
% 3.05/0.84 = { by axiom 2 (domain4) R->L }
% 3.05/0.84 antidomain(domain(X))
% 3.05/0.84 = { by axiom 1 (complement) R->L }
% 3.05/0.84 c(X)
% 3.05/0.84
% 3.05/0.84 Lemma 20: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 3.05/0.84 Proof:
% 3.05/0.84 multiplication(antidomain(X), addition(X, Y))
% 3.05/0.84 = { by axiom 4 (additive_commutativity) R->L }
% 3.05/0.84 multiplication(antidomain(X), addition(Y, X))
% 3.05/0.84 = { by axiom 15 (right_distributivity) }
% 3.05/0.84 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 3.05/0.84 = { by axiom 10 (domain1) }
% 3.05/0.84 addition(multiplication(antidomain(X), Y), zero)
% 3.05/0.84 = { by axiom 5 (additive_identity) }
% 3.05/0.84 multiplication(antidomain(X), Y)
% 3.05/0.84
% 3.05/0.84 Lemma 21: antidomain(X) = c(X).
% 3.05/0.84 Proof:
% 3.05/0.84 antidomain(X)
% 3.05/0.84 = { by lemma 18 R->L }
% 3.05/0.84 multiplication(domain(antidomain(X)), antidomain(X))
% 3.05/0.84 = { by lemma 19 }
% 3.05/0.84 multiplication(c(X), antidomain(X))
% 3.05/0.84 = { by axiom 1 (complement) }
% 3.05/0.84 multiplication(antidomain(domain(X)), antidomain(X))
% 3.05/0.84 = { by lemma 20 R->L }
% 3.05/0.84 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 3.05/0.84 = { by lemma 17 }
% 3.05/0.84 multiplication(antidomain(domain(X)), one)
% 3.05/0.84 = { by axiom 6 (multiplicative_right_identity) }
% 3.05/0.84 antidomain(domain(X))
% 3.05/0.84 = { by axiom 1 (complement) R->L }
% 3.05/0.84 c(X)
% 3.05/0.84
% 3.05/0.84 Lemma 22: antidomain(c(X)) = domain(domain(X)).
% 3.05/0.84 Proof:
% 3.05/0.84 antidomain(c(X))
% 3.05/0.84 = { by axiom 1 (complement) }
% 3.05/0.84 antidomain(antidomain(domain(X)))
% 3.05/0.84 = { by axiom 2 (domain4) R->L }
% 3.05/0.84 domain(domain(X))
% 3.05/0.84
% 3.05/0.84 Lemma 23: addition(domain(X), one) = one.
% 3.05/0.84 Proof:
% 3.05/0.84 addition(domain(X), one)
% 3.05/0.84 = { by lemma 17 R->L }
% 3.05/0.84 addition(domain(X), addition(domain(X), antidomain(X)))
% 3.05/0.84 = { by axiom 12 (additive_associativity) }
% 3.05/0.84 addition(addition(domain(X), domain(X)), antidomain(X))
% 3.05/0.84 = { by axiom 3 (additive_idempotence) }
% 3.05/0.84 addition(domain(X), antidomain(X))
% 3.05/0.84 = { by lemma 17 }
% 3.05/0.84 one
% 3.05/0.84
% 3.05/0.84 Lemma 24: antidomain(forward_diamond(X, Y)) = c(multiplication(X, domain(Y))).
% 3.05/0.84 Proof:
% 3.05/0.84 antidomain(forward_diamond(X, Y))
% 3.05/0.84 = { by axiom 8 (forward_diamond) }
% 3.05/0.84 antidomain(domain(multiplication(X, domain(Y))))
% 3.05/0.84 = { by axiom 1 (complement) R->L }
% 3.05/0.84 c(multiplication(X, domain(Y)))
% 3.05/0.84
% 3.05/0.84 Lemma 25: multiplication(domain(X), c(Y)) = domain_difference(X, domain(Y)).
% 3.05/0.84 Proof:
% 3.05/0.84 multiplication(domain(X), c(Y))
% 3.05/0.84 = { by axiom 1 (complement) }
% 3.05/0.84 multiplication(domain(X), antidomain(domain(Y)))
% 3.05/0.84 = { by axiom 9 (domain_difference) R->L }
% 3.05/0.84 domain_difference(X, domain(Y))
% 3.05/0.84
% 3.05/0.84 Goal 1 (goals_1): addition(multiplication(x0, domain(x1)), multiplication(domain(x2), x0)) = multiplication(domain(x2), x0).
% 3.05/0.85 Proof:
% 3.05/0.85 addition(multiplication(x0, domain(x1)), multiplication(domain(x2), x0))
% 3.05/0.85 = { by axiom 4 (additive_commutativity) }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(x0, domain(x1)))
% 3.05/0.85 = { by axiom 7 (multiplicative_left_identity) R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(one, multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 23 R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), one), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 17 R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), addition(domain(antidomain(multiplication(x0, domain(domain(x1))))), antidomain(antidomain(multiplication(x0, domain(domain(x1))))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 19 }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), addition(c(multiplication(x0, domain(domain(x1)))), antidomain(antidomain(multiplication(x0, domain(domain(x1))))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by axiom 2 (domain4) R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), addition(c(multiplication(x0, domain(domain(x1)))), domain(multiplication(x0, domain(domain(x1)))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by axiom 4 (additive_commutativity) }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), addition(domain(multiplication(x0, domain(domain(x1)))), c(multiplication(x0, domain(domain(x1)))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 24 R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), addition(domain(multiplication(x0, domain(domain(x1)))), antidomain(forward_diamond(x0, domain(x1))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by axiom 8 (forward_diamond) R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), addition(forward_diamond(x0, domain(x1)), antidomain(forward_diamond(x0, domain(x1))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by axiom 12 (additive_associativity) }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(addition(domain(x2), forward_diamond(x0, domain(x1))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by axiom 4 (additive_commutativity) R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(addition(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by axiom 11 (goals) }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), antidomain(forward_diamond(x0, domain(x1)))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 24 }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, domain(domain(x1))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 22 R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, antidomain(c(x1))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 18 R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, antidomain(multiplication(domain(c(x1)), c(x1)))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 25 }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, antidomain(domain_difference(c(x1), domain(x1)))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by axiom 1 (complement) }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, antidomain(domain_difference(antidomain(domain(x1)), domain(x1)))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 25 R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, antidomain(multiplication(domain(antidomain(domain(x1))), c(x1)))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 19 }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, antidomain(multiplication(c(domain(x1)), c(x1)))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by axiom 1 (complement) }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, antidomain(multiplication(antidomain(domain(domain(x1))), c(x1)))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 20 R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, antidomain(multiplication(antidomain(domain(domain(x1))), addition(domain(domain(x1)), c(x1))))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by axiom 1 (complement) }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, antidomain(multiplication(antidomain(domain(domain(x1))), addition(domain(domain(x1)), antidomain(domain(x1)))))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 17 }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, antidomain(multiplication(antidomain(domain(domain(x1))), one))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by axiom 6 (multiplicative_right_identity) }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, antidomain(antidomain(domain(domain(x1))))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by axiom 1 (complement) R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, antidomain(c(domain(x1)))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 22 }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, domain(domain(domain(x1)))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 22 R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, domain(antidomain(c(x1)))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 19 }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, c(c(x1))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 21 R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, c(antidomain(x1))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 21 R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, antidomain(antidomain(x1))))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by axiom 2 (domain4) R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), c(multiplication(x0, domain(x1)))), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by axiom 4 (additive_commutativity) R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(c(multiplication(x0, domain(x1))), domain(x2)), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by lemma 21 R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(addition(antidomain(multiplication(x0, domain(x1))), domain(x2)), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by axiom 16 (left_distributivity) }
% 3.05/0.85 addition(multiplication(domain(x2), x0), addition(multiplication(antidomain(multiplication(x0, domain(x1))), multiplication(x0, domain(x1))), multiplication(domain(x2), multiplication(x0, domain(x1)))))
% 3.05/0.85 = { by axiom 10 (domain1) }
% 3.05/0.85 addition(multiplication(domain(x2), x0), addition(zero, multiplication(domain(x2), multiplication(x0, domain(x1)))))
% 3.05/0.85 = { by axiom 4 (additive_commutativity) R->L }
% 3.05/0.85 addition(multiplication(domain(x2), x0), addition(multiplication(domain(x2), multiplication(x0, domain(x1))), zero))
% 3.05/0.85 = { by axiom 5 (additive_identity) }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(domain(x2), multiplication(x0, domain(x1))))
% 3.05/0.85 = { by axiom 13 (multiplicative_associativity) }
% 3.05/0.85 addition(multiplication(domain(x2), x0), multiplication(multiplication(domain(x2), x0), domain(x1)))
% 3.05/0.85 = { by axiom 6 (multiplicative_right_identity) R->L }
% 3.05/0.85 addition(multiplication(multiplication(domain(x2), x0), one), multiplication(multiplication(domain(x2), x0), domain(x1)))
% 3.05/0.86 = { by axiom 15 (right_distributivity) R->L }
% 3.05/0.86 multiplication(multiplication(domain(x2), x0), addition(one, domain(x1)))
% 3.05/0.86 = { by axiom 4 (additive_commutativity) }
% 3.05/0.86 multiplication(multiplication(domain(x2), x0), addition(domain(x1), one))
% 3.05/0.86 = { by lemma 23 }
% 3.05/0.86 multiplication(multiplication(domain(x2), x0), one)
% 3.05/0.86 = { by axiom 6 (multiplicative_right_identity) }
% 3.05/0.86 multiplication(domain(x2), x0)
% 3.05/0.86 % SZS output end Proof
% 3.05/0.86
% 3.05/0.86 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------