%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE084+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n019.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 3.48s 0.97s
% Output : Proof 3.48s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE084+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 : n019.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
% 0.10/0.38 % DateTime : Sun Sep 27 13:09:34 UTC 2026
% 0.10/0.38 % CPUTime :
% 0.10/0.38 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 3.48/0.97 Command-line arguments: --flatten --complete-subsets
% 3.48/0.97
% 3.48/0.97 % SZS status Theorem
% 3.48/0.97
% 3.48/0.99 % SZS output start Proof
% 3.48/0.99 Axiom 1 (domain4): domain(X) = antidomain(antidomain(X)).
% 3.48/0.99 Axiom 2 (additive_idempotence): addition(X, X) = X.
% 3.48/0.99 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 3.48/0.99 Axiom 4 (additive_identity): addition(X, zero) = X.
% 3.48/0.99 Axiom 5 (multiplicative_right_identity): multiplication(X, one) = X.
% 3.48/0.99 Axiom 6 (multiplicative_left_identity): multiplication(one, X) = X.
% 3.48/0.99 Axiom 7 (left_annihilation): multiplication(zero, X) = zero.
% 3.48/0.99 Axiom 8 (domain1): multiplication(antidomain(X), X) = zero.
% 3.48/0.99 Axiom 9 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 3.48/0.99 Axiom 10 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 3.48/0.99 Axiom 11 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 3.48/0.99 Axiom 12 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 3.48/0.99 Axiom 13 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 3.48/0.99 Axiom 14 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 3.48/0.99 Axiom 15 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 3.48/0.99 Axiom 16 (order_1): ifeq(leq(X, Y), true, addition(X, Y), Y) = Y.
% 3.48/0.99 Axiom 17 (order): ifeq2(addition(X, Y), Y, leq(X, Y), true) = true.
% 3.48/0.99 Axiom 18 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 3.48/0.99
% 3.48/0.99 Lemma 19: antidomain(one) = zero.
% 3.48/0.99 Proof:
% 3.48/0.99 antidomain(one)
% 3.48/0.99 = { by axiom 5 (multiplicative_right_identity) R->L }
% 3.48/0.99 multiplication(antidomain(one), one)
% 3.48/0.99 = { by axiom 8 (domain1) }
% 3.48/0.99 zero
% 3.48/0.99
% 3.48/0.99 Lemma 20: addition(domain(X), antidomain(X)) = one.
% 3.48/0.99 Proof:
% 3.48/0.99 addition(domain(X), antidomain(X))
% 3.48/0.99 = { by axiom 1 (domain4) }
% 3.48/0.99 addition(antidomain(antidomain(X)), antidomain(X))
% 3.48/0.99 = { by axiom 13 (domain3) }
% 3.48/0.99 one
% 3.48/0.99
% 3.48/0.99 Lemma 21: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 3.48/0.99 Proof:
% 3.48/0.99 multiplication(antidomain(X), addition(X, Y))
% 3.48/0.99 = { by axiom 3 (additive_commutativity) R->L }
% 3.48/0.99 multiplication(antidomain(X), addition(Y, X))
% 3.48/0.99 = { by axiom 14 (right_distributivity) }
% 3.48/0.99 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 3.48/0.99 = { by axiom 8 (domain1) }
% 3.48/0.99 addition(multiplication(antidomain(X), Y), zero)
% 3.48/0.99 = { by axiom 4 (additive_identity) }
% 3.48/0.99 multiplication(antidomain(X), Y)
% 3.48/0.99
% 3.48/0.99 Lemma 22: multiplication(domain(X), X) = X.
% 3.48/0.99 Proof:
% 3.48/0.99 multiplication(domain(X), X)
% 3.48/0.99 = { by axiom 4 (additive_identity) R->L }
% 3.48/0.99 addition(multiplication(domain(X), X), zero)
% 3.48/0.99 = { by axiom 8 (domain1) R->L }
% 3.48/0.99 addition(multiplication(domain(X), X), multiplication(antidomain(X), X))
% 3.48/0.99 = { by axiom 15 (left_distributivity) R->L }
% 3.48/0.99 multiplication(addition(domain(X), antidomain(X)), X)
% 3.48/0.99 = { by lemma 20 }
% 3.48/0.99 multiplication(one, X)
% 3.48/0.99 = { by axiom 6 (multiplicative_left_identity) }
% 3.48/0.99 X
% 3.48/0.99
% 3.48/0.99 Lemma 23: antidomain(domain(X)) = antidomain(X).
% 3.48/0.99 Proof:
% 3.48/0.99 antidomain(domain(X))
% 3.48/0.99 = { by axiom 5 (multiplicative_right_identity) R->L }
% 3.48/0.99 multiplication(antidomain(domain(X)), one)
% 3.48/0.99 = { by lemma 20 R->L }
% 3.48/0.99 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 3.48/0.99 = { by lemma 21 }
% 3.48/0.99 multiplication(antidomain(domain(X)), antidomain(X))
% 3.48/0.99 = { by axiom 1 (domain4) }
% 3.48/0.99 multiplication(antidomain(antidomain(antidomain(X))), antidomain(X))
% 3.48/0.99 = { by axiom 1 (domain4) R->L }
% 3.48/0.99 multiplication(domain(antidomain(X)), antidomain(X))
% 3.48/0.99 = { by lemma 22 }
% 3.48/0.99 antidomain(X)
% 3.48/0.99
% 3.48/0.99 Lemma 24: domain(domain(X)) = domain(X).
% 3.48/0.99 Proof:
% 3.48/0.99 domain(domain(X))
% 3.48/0.99 = { by axiom 1 (domain4) }
% 3.48/0.99 antidomain(antidomain(domain(X)))
% 3.48/0.99 = { by lemma 23 }
% 3.48/0.99 antidomain(antidomain(X))
% 3.48/0.99 = { by axiom 1 (domain4) R->L }
% 3.48/0.99 domain(X)
% 3.48/0.99
% 3.48/0.99 Lemma 25: addition(one, antidomain(X)) = one.
% 3.48/0.99 Proof:
% 3.48/0.99 addition(one, antidomain(X))
% 3.48/0.99 = { by axiom 3 (additive_commutativity) R->L }
% 3.48/0.99 addition(antidomain(X), one)
% 3.48/0.99 = { by axiom 9 (ifeq_axiom) R->L }
% 3.48/0.99 ifeq(true, true, addition(antidomain(X), one), one)
% 3.48/0.99 = { by axiom 17 (order) R->L }
% 3.48/0.99 ifeq(ifeq2(addition(antidomain(X), addition(antidomain(X), domain(X))), addition(antidomain(X), domain(X)), leq(antidomain(X), addition(antidomain(X), domain(X))), true), true, addition(antidomain(X), one), one)
% 3.48/0.99 = { by axiom 11 (additive_associativity) }
% 3.48/0.99 ifeq(ifeq2(addition(addition(antidomain(X), antidomain(X)), domain(X)), addition(antidomain(X), domain(X)), leq(antidomain(X), addition(antidomain(X), domain(X))), true), true, addition(antidomain(X), one), one)
% 3.48/0.99 = { by axiom 2 (additive_idempotence) }
% 3.48/0.99 ifeq(ifeq2(addition(antidomain(X), domain(X)), addition(antidomain(X), domain(X)), leq(antidomain(X), addition(antidomain(X), domain(X))), true), true, addition(antidomain(X), one), one)
% 3.48/0.99 = { by axiom 10 (ifeq_axiom) }
% 3.48/0.99 ifeq(leq(antidomain(X), addition(antidomain(X), domain(X))), true, addition(antidomain(X), one), one)
% 3.48/0.99 = { by axiom 3 (additive_commutativity) }
% 3.48/0.99 ifeq(leq(antidomain(X), addition(domain(X), antidomain(X))), true, addition(antidomain(X), one), one)
% 3.48/0.99 = { by lemma 20 }
% 3.48/0.99 ifeq(leq(antidomain(X), one), true, addition(antidomain(X), one), one)
% 3.48/0.99 = { by axiom 16 (order_1) }
% 3.48/0.99 one
% 3.48/0.99
% 3.48/0.99 Lemma 26: addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))) = antidomain(multiplication(X, domain(Y))).
% 3.48/0.99 Proof:
% 3.48/0.99 addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y))))
% 3.48/0.99 = { by axiom 1 (domain4) }
% 3.48/0.99 addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y)))))
% 3.48/0.99 = { by axiom 18 (domain2) }
% 3.48/0.99 antidomain(multiplication(X, antidomain(antidomain(Y))))
% 3.48/0.99 = { by axiom 1 (domain4) R->L }
% 3.48/0.99 antidomain(multiplication(X, domain(Y)))
% 3.48/0.99
% 3.48/0.99 Goal 1 (goals): domain(multiplication(x0, x1)) = domain(multiplication(x0, domain(x1))).
% 3.48/0.99 Proof:
% 3.48/0.99 domain(multiplication(x0, x1))
% 3.48/0.99 = { by axiom 6 (multiplicative_left_identity) R->L }
% 3.48/0.99 multiplication(one, domain(multiplication(x0, x1)))
% 3.48/0.99 = { by lemma 20 R->L }
% 3.48/0.99 multiplication(addition(domain(multiplication(x0, domain(x1))), antidomain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))
% 3.48/1.00 = { by axiom 15 (left_distributivity) }
% 3.48/1.00 addition(multiplication(domain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by lemma 24 R->L }
% 3.48/1.00 addition(multiplication(domain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1))), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by axiom 1 (domain4) }
% 3.48/1.00 addition(multiplication(antidomain(antidomain(domain(multiplication(x0, domain(x1))))), domain(multiplication(x0, x1))), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by lemma 21 R->L }
% 3.48/1.00 addition(multiplication(antidomain(antidomain(domain(multiplication(x0, domain(x1))))), addition(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by axiom 3 (additive_commutativity) }
% 3.48/1.00 addition(multiplication(antidomain(antidomain(domain(multiplication(x0, domain(x1))))), addition(domain(multiplication(x0, x1)), antidomain(domain(multiplication(x0, domain(x1)))))), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by lemma 23 }
% 3.48/1.00 addition(multiplication(antidomain(antidomain(domain(multiplication(x0, domain(x1))))), addition(domain(multiplication(x0, x1)), antidomain(multiplication(x0, domain(x1))))), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by lemma 26 R->L }
% 3.48/1.00 addition(multiplication(antidomain(antidomain(domain(multiplication(x0, domain(x1))))), addition(domain(multiplication(x0, x1)), addition(antidomain(multiplication(x0, x1)), antidomain(multiplication(x0, domain(x1)))))), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by axiom 11 (additive_associativity) }
% 3.48/1.00 addition(multiplication(antidomain(antidomain(domain(multiplication(x0, domain(x1))))), addition(addition(domain(multiplication(x0, x1)), antidomain(multiplication(x0, x1))), antidomain(multiplication(x0, domain(x1))))), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by lemma 20 }
% 3.48/1.00 addition(multiplication(antidomain(antidomain(domain(multiplication(x0, domain(x1))))), addition(one, antidomain(multiplication(x0, domain(x1))))), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by axiom 3 (additive_commutativity) }
% 3.48/1.00 addition(multiplication(antidomain(antidomain(domain(multiplication(x0, domain(x1))))), addition(antidomain(multiplication(x0, domain(x1))), one)), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by lemma 23 R->L }
% 3.48/1.00 addition(multiplication(antidomain(antidomain(domain(multiplication(x0, domain(x1))))), addition(antidomain(domain(multiplication(x0, domain(x1)))), one)), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by axiom 3 (additive_commutativity) }
% 3.48/1.00 addition(multiplication(antidomain(antidomain(domain(multiplication(x0, domain(x1))))), addition(one, antidomain(domain(multiplication(x0, domain(x1)))))), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by lemma 25 }
% 3.48/1.00 addition(multiplication(antidomain(antidomain(domain(multiplication(x0, domain(x1))))), one), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by axiom 5 (multiplicative_right_identity) }
% 3.48/1.00 addition(antidomain(antidomain(domain(multiplication(x0, domain(x1))))), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by axiom 1 (domain4) R->L }
% 3.48/1.00 addition(domain(domain(multiplication(x0, domain(x1)))), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by lemma 24 }
% 3.48/1.00 addition(domain(multiplication(x0, domain(x1))), multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by lemma 23 R->L }
% 3.48/1.00 addition(domain(multiplication(x0, domain(x1))), multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1))))
% 3.48/1.00 = { by axiom 6 (multiplicative_left_identity) R->L }
% 3.48/1.00 addition(domain(multiplication(x0, domain(x1))), multiplication(one, multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))))
% 3.48/1.00 = { by lemma 25 R->L }
% 3.48/1.00 addition(domain(multiplication(x0, domain(x1))), multiplication(addition(one, antidomain(multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))), multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))))
% 3.48/1.00 = { by lemma 20 R->L }
% 3.48/1.00 addition(domain(multiplication(x0, domain(x1))), multiplication(addition(addition(domain(one), antidomain(one)), antidomain(multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))), multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))))
% 3.48/1.00 = { by lemma 19 }
% 3.48/1.00 addition(domain(multiplication(x0, domain(x1))), multiplication(addition(addition(domain(one), zero), antidomain(multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))), multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))))
% 3.48/1.00 = { by axiom 4 (additive_identity) }
% 3.48/1.00 addition(domain(multiplication(x0, domain(x1))), multiplication(addition(domain(one), antidomain(multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))), multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))))
% 3.48/1.01 = { by axiom 1 (domain4) }
% 3.48/1.01 addition(domain(multiplication(x0, domain(x1))), multiplication(addition(antidomain(antidomain(one)), antidomain(multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))), multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))))
% 3.48/1.01 = { by lemma 19 }
% 3.48/1.01 addition(domain(multiplication(x0, domain(x1))), multiplication(addition(antidomain(zero), antidomain(multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))), multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))))
% 3.48/1.01 = { by axiom 7 (left_annihilation) R->L }
% 3.48/1.01 addition(domain(multiplication(x0, domain(x1))), multiplication(addition(antidomain(multiplication(zero, x1)), antidomain(multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))), multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))))
% 3.48/1.01 = { by axiom 8 (domain1) R->L }
% 3.48/1.01 addition(domain(multiplication(x0, domain(x1))), multiplication(addition(antidomain(multiplication(multiplication(antidomain(multiplication(x0, domain(x1))), multiplication(x0, domain(x1))), x1)), antidomain(multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))), multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))))
% 3.48/1.01 = { by axiom 12 (multiplicative_associativity) R->L }
% 3.48/1.01 addition(domain(multiplication(x0, domain(x1))), multiplication(addition(antidomain(multiplication(antidomain(multiplication(x0, domain(x1))), multiplication(multiplication(x0, domain(x1)), x1))), antidomain(multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))), multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))))
% 3.48/1.01 = { by axiom 12 (multiplicative_associativity) R->L }
% 3.48/1.01 addition(domain(multiplication(x0, domain(x1))), multiplication(addition(antidomain(multiplication(antidomain(multiplication(x0, domain(x1))), multiplication(x0, multiplication(domain(x1), x1)))), antidomain(multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))), multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))))
% 3.48/1.01 = { by lemma 22 }
% 3.48/1.01 addition(domain(multiplication(x0, domain(x1))), multiplication(addition(antidomain(multiplication(antidomain(multiplication(x0, domain(x1))), multiplication(x0, x1))), antidomain(multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1))))), multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))))
% 3.48/1.01 = { by lemma 26 }
% 3.48/1.01 addition(domain(multiplication(x0, domain(x1))), multiplication(antidomain(multiplication(antidomain(multiplication(x0, domain(x1))), domain(multiplication(x0, x1)))), multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))))
% 3.48/1.01 = { by lemma 23 R->L }
% 3.48/1.01 addition(domain(multiplication(x0, domain(x1))), multiplication(antidomain(multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))), multiplication(antidomain(domain(multiplication(x0, domain(x1)))), domain(multiplication(x0, x1)))))
% 3.48/1.01 = { by axiom 8 (domain1) }
% 3.48/1.01 addition(domain(multiplication(x0, domain(x1))), zero)
% 3.48/1.01 = { by axiom 4 (additive_identity) }
% 3.48/1.01 domain(multiplication(x0, domain(x1)))
% 3.48/1.01 % SZS output end Proof
% 3.48/1.01
% 3.48/1.01 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------