%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE133+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 11:39:52 AM UTC 2026
% Result : Theorem 0.22s 0.59s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE133+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.35 % Computer : n019.cluster.edu
% 0.09/0.35 % Model : x86_64 x86_64
% 0.09/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.35 % Memory : 8046.5625MB
% 0.09/0.35 % OS : Linux 6.8.0-71-generic
% 0.09/0.35 % CPULimit : 300
% 0.09/0.35 % WCLimit : 300
% 0.09/0.35 % DateTime : Sun Sep 27 13:12:33 UTC 2026
% 0.09/0.36 % CPUTime :
% 0.09/0.36 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.22/0.59 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.22/0.59
% 0.22/0.59 % SZS status Theorem
% 0.22/0.59
% 0.22/0.59 % SZS output start Proof
% 0.22/0.59 Axiom 1 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.22/0.59 Axiom 2 (additive_identity): addition(X, zero) = X.
% 0.22/0.59 Axiom 3 (right_annihilation): multiplication(X, zero) = zero.
% 0.22/0.59 Axiom 4 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.22/0.59 Axiom 5 (left_annihilation): multiplication(zero, X) = zero.
% 0.22/0.59 Axiom 6 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.22/0.59 Axiom 7 (domain4): domain(X) = antidomain(antidomain(X)).
% 0.22/0.59 Axiom 8 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 0.22/0.60 Axiom 9 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 0.22/0.60 Axiom 10 (domain1): multiplication(antidomain(X), X) = zero.
% 0.22/0.60 Axiom 11 (goals_1): forward_diamond(x0, forward_diamond(x0, domain(X))) = forward_diamond(x0, domain(X)).
% 0.22/0.60 Axiom 12 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 0.22/0.60 Axiom 13 (goals): addition(domain(X), forward_diamond(star(x0), domain_difference(domain(X), forward_diamond(x0, domain(X))))) = forward_diamond(star(x0), domain_difference(domain(X), forward_diamond(x0, domain(X)))).
% 0.22/0.60
% 0.22/0.60 Lemma 14: antidomain(one) = zero.
% 0.22/0.60 Proof:
% 0.22/0.60 antidomain(one)
% 0.22/0.60 = { by axiom 4 (multiplicative_right_identity) R->L }
% 0.22/0.60 multiplication(antidomain(one), one)
% 0.22/0.60 = { by axiom 10 (domain1) }
% 0.22/0.60 zero
% 0.22/0.60
% 0.22/0.60 Lemma 15: addition(domain(X), antidomain(X)) = one.
% 0.22/0.60 Proof:
% 0.22/0.60 addition(domain(X), antidomain(X))
% 0.22/0.60 = { by axiom 7 (domain4) }
% 0.22/0.60 addition(antidomain(antidomain(X)), antidomain(X))
% 0.22/0.60 = { by axiom 12 (domain3) }
% 0.22/0.60 one
% 0.22/0.60
% 0.22/0.60 Lemma 16: domain(one) = one.
% 0.22/0.60 Proof:
% 0.22/0.60 domain(one)
% 0.22/0.60 = { by axiom 2 (additive_identity) R->L }
% 0.22/0.60 addition(domain(one), zero)
% 0.22/0.60 = { by lemma 14 R->L }
% 0.22/0.60 addition(domain(one), antidomain(one))
% 0.22/0.60 = { by lemma 15 }
% 0.22/0.60 one
% 0.22/0.60
% 0.22/0.60 Lemma 17: domain(zero) = zero.
% 0.22/0.60 Proof:
% 0.22/0.60 domain(zero)
% 0.22/0.60 = { by axiom 7 (domain4) }
% 0.22/0.60 antidomain(antidomain(zero))
% 0.22/0.60 = { by lemma 14 R->L }
% 0.22/0.60 antidomain(antidomain(antidomain(one)))
% 0.22/0.60 = { by axiom 7 (domain4) R->L }
% 0.22/0.60 antidomain(domain(one))
% 0.22/0.60 = { by lemma 16 }
% 0.22/0.60 antidomain(one)
% 0.22/0.60 = { by lemma 14 }
% 0.22/0.60 zero
% 0.22/0.60
% 0.22/0.60 Lemma 18: domain_difference(X, X) = zero.
% 0.22/0.60 Proof:
% 0.22/0.60 domain_difference(X, X)
% 0.22/0.60 = { by axiom 9 (domain_difference) }
% 0.22/0.60 multiplication(domain(X), antidomain(X))
% 0.22/0.60 = { by axiom 7 (domain4) }
% 0.22/0.60 multiplication(antidomain(antidomain(X)), antidomain(X))
% 0.22/0.60 = { by axiom 10 (domain1) }
% 0.22/0.60 zero
% 0.22/0.60
% 0.22/0.60 Lemma 19: forward_diamond(X, zero) = zero.
% 0.22/0.60 Proof:
% 0.22/0.60 forward_diamond(X, zero)
% 0.22/0.60 = { by axiom 8 (forward_diamond) }
% 0.22/0.60 domain(multiplication(X, domain(zero)))
% 0.22/0.60 = { by lemma 17 }
% 0.22/0.60 domain(multiplication(X, zero))
% 0.22/0.60 = { by axiom 3 (right_annihilation) }
% 0.22/0.60 domain(zero)
% 0.22/0.60 = { by lemma 17 }
% 0.22/0.60 zero
% 0.22/0.60
% 0.22/0.60 Lemma 20: forward_diamond(X, one) = domain(X).
% 0.22/0.60 Proof:
% 0.22/0.60 forward_diamond(X, one)
% 0.22/0.60 = { by axiom 8 (forward_diamond) }
% 0.22/0.60 domain(multiplication(X, domain(one)))
% 0.22/0.60 = { by lemma 16 }
% 0.22/0.60 domain(multiplication(X, one))
% 0.22/0.60 = { by axiom 4 (multiplicative_right_identity) }
% 0.22/0.60 domain(X)
% 0.22/0.60
% 0.22/0.60 Lemma 21: forward_diamond(x0, domain(x0)) = domain(x0).
% 0.22/0.60 Proof:
% 0.22/0.60 forward_diamond(x0, domain(x0))
% 0.22/0.60 = { by lemma 20 R->L }
% 0.22/0.60 forward_diamond(x0, forward_diamond(x0, one))
% 0.22/0.60 = { by lemma 16 R->L }
% 0.22/0.60 forward_diamond(x0, forward_diamond(x0, domain(one)))
% 0.22/0.60 = { by axiom 11 (goals_1) }
% 0.22/0.60 forward_diamond(x0, domain(one))
% 0.22/0.60 = { by lemma 16 }
% 0.22/0.60 forward_diamond(x0, one)
% 0.22/0.60 = { by lemma 20 }
% 0.22/0.60 domain(x0)
% 0.22/0.60
% 0.22/0.60 Goal 1 (goals_2): addition(forward_diamond(x0, domain(x3)), forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3))))) = forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))).
% 0.22/0.60 Proof:
% 0.22/0.60 addition(forward_diamond(x0, domain(x3)), forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))))
% 0.22/0.60 = { by axiom 1 (additive_commutativity) }
% 0.22/0.60 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), forward_diamond(x0, domain(x3)))
% 0.22/0.60 = { by axiom 6 (multiplicative_left_identity) R->L }
% 0.22/0.60 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), forward_diamond(multiplication(one, x0), domain(x3)))
% 0.22/0.60 = { by lemma 15 R->L }
% 0.22/0.60 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), forward_diamond(multiplication(addition(domain(x0), antidomain(x0)), x0), domain(x3)))
% 0.22/0.60 = { by axiom 2 (additive_identity) R->L }
% 0.22/0.61 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), forward_diamond(multiplication(addition(addition(domain(x0), zero), antidomain(x0)), x0), domain(x3)))
% 0.22/0.61 = { by lemma 19 R->L }
% 0.22/0.61 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), forward_diamond(multiplication(addition(addition(domain(x0), forward_diamond(star(x0), zero)), antidomain(x0)), x0), domain(x3)))
% 0.22/0.61 = { by lemma 18 R->L }
% 0.22/0.61 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), forward_diamond(multiplication(addition(addition(domain(x0), forward_diamond(star(x0), domain_difference(domain(x0), domain(x0)))), antidomain(x0)), x0), domain(x3)))
% 0.22/0.61 = { by lemma 21 R->L }
% 0.22/0.61 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), forward_diamond(multiplication(addition(addition(domain(x0), forward_diamond(star(x0), domain_difference(domain(x0), forward_diamond(x0, domain(x0))))), antidomain(x0)), x0), domain(x3)))
% 0.22/0.61 = { by axiom 13 (goals) }
% 0.22/0.61 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), forward_diamond(multiplication(addition(forward_diamond(star(x0), domain_difference(domain(x0), forward_diamond(x0, domain(x0)))), antidomain(x0)), x0), domain(x3)))
% 0.22/0.61 = { by lemma 21 }
% 0.22/0.61 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), forward_diamond(multiplication(addition(forward_diamond(star(x0), domain_difference(domain(x0), domain(x0))), antidomain(x0)), x0), domain(x3)))
% 0.22/0.61 = { by lemma 18 }
% 0.22/0.61 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), forward_diamond(multiplication(addition(forward_diamond(star(x0), zero), antidomain(x0)), x0), domain(x3)))
% 0.22/0.61 = { by lemma 19 }
% 0.22/0.61 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), forward_diamond(multiplication(addition(zero, antidomain(x0)), x0), domain(x3)))
% 0.22/0.61 = { by axiom 1 (additive_commutativity) R->L }
% 0.22/0.61 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), forward_diamond(multiplication(addition(antidomain(x0), zero), x0), domain(x3)))
% 0.22/0.61 = { by axiom 2 (additive_identity) }
% 0.22/0.61 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), forward_diamond(multiplication(antidomain(x0), x0), domain(x3)))
% 0.22/0.61 = { by axiom 10 (domain1) }
% 0.22/0.61 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), forward_diamond(zero, domain(x3)))
% 0.22/0.61 = { by axiom 8 (forward_diamond) }
% 0.22/0.61 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), domain(multiplication(zero, domain(domain(x3)))))
% 0.22/0.61 = { by axiom 5 (left_annihilation) }
% 0.22/0.61 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), domain(zero))
% 0.22/0.61 = { by lemma 17 }
% 0.22/0.61 addition(forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3)))), zero)
% 0.22/0.61 = { by axiom 2 (additive_identity) }
% 0.22/0.61 forward_diamond(star(x0), domain_difference(domain(x3), forward_diamond(x0, domain(x3))))
% 0.22/0.61 % SZS output end Proof
% 0.22/0.61
% 0.22/0.61 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------