%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE131+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n005.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 0.58s 0.55s
% Output : Proof 0.58s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE131+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.37 % Computer : n005.cluster.edu
% 0.09/0.37 % Model : x86_64 x86_64
% 0.09/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.37 % Memory : 8046.5625MB
% 0.09/0.37 % OS : Linux 6.8.0-71-generic
% 0.09/0.37 % CPULimit : 300
% 0.09/0.37 % WCLimit : 300
% 0.09/0.37 % DateTime : Sun Sep 27 13:11:16 UTC 2026
% 0.15/0.37 % CPUTime :
% 0.15/0.37 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.58/0.55 Command-line arguments: --flatten --complete-subsets
% 0.58/0.55
% 0.58/0.55 % SZS status Theorem
% 0.58/0.55
% 0.58/0.57 % SZS output start Proof
% 0.58/0.57 Axiom 1 (complement): c(X) = antidomain(domain(X)).
% 0.58/0.57 Axiom 2 (domain4): domain(X) = antidomain(antidomain(X)).
% 0.58/0.57 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.58/0.57 Axiom 4 (right_annihilation): multiplication(X, zero) = zero.
% 0.58/0.57 Axiom 5 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.58/0.57 Axiom 6 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.58/0.57 Axiom 7 (additive_identity): addition(X, zero) = X.
% 0.58/0.57 Axiom 8 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 0.58/0.57 Axiom 9 (divergence1): forward_diamond(X, divergence(X)) = divergence(X).
% 0.58/0.57 Axiom 10 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 0.58/0.57 Axiom 11 (domain1): multiplication(antidomain(X), X) = zero.
% 0.58/0.57 Axiom 12 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))).
% 0.58/0.57 Axiom 13 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 0.58/0.57 Axiom 14 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.58/0.57 Axiom 15 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.58/0.57 Axiom 16 (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)))).
% 0.58/0.57
% 0.58/0.57 Lemma 17: addition(domain(X), antidomain(X)) = one.
% 0.58/0.57 Proof:
% 0.58/0.57 addition(domain(X), antidomain(X))
% 0.58/0.57 = { by axiom 2 (domain4) }
% 0.58/0.57 addition(antidomain(antidomain(X)), antidomain(X))
% 0.58/0.57 = { by axiom 13 (domain3) }
% 0.58/0.57 one
% 0.58/0.57
% 0.58/0.57 Lemma 18: domain_difference(antidomain(X), X) = antidomain(X).
% 0.58/0.57 Proof:
% 0.58/0.57 domain_difference(antidomain(X), X)
% 0.58/0.57 = { by axiom 10 (domain_difference) }
% 0.58/0.57 multiplication(domain(antidomain(X)), antidomain(X))
% 0.58/0.57 = { by axiom 7 (additive_identity) R->L }
% 0.58/0.57 addition(multiplication(domain(antidomain(X)), antidomain(X)), zero)
% 0.58/0.57 = { by axiom 11 (domain1) R->L }
% 0.58/0.57 addition(multiplication(domain(antidomain(X)), antidomain(X)), multiplication(antidomain(antidomain(X)), antidomain(X)))
% 0.58/0.57 = { by axiom 15 (left_distributivity) R->L }
% 0.58/0.57 multiplication(addition(domain(antidomain(X)), antidomain(antidomain(X))), antidomain(X))
% 0.58/0.57 = { by lemma 17 }
% 0.58/0.57 multiplication(one, antidomain(X))
% 0.58/0.57 = { by axiom 5 (multiplicative_left_identity) }
% 0.58/0.57 antidomain(X)
% 0.58/0.57
% 0.58/0.57 Lemma 19: domain(antidomain(X)) = c(X).
% 0.58/0.57 Proof:
% 0.58/0.57 domain(antidomain(X))
% 0.58/0.57 = { by axiom 2 (domain4) }
% 0.58/0.57 antidomain(antidomain(antidomain(X)))
% 0.58/0.57 = { by axiom 2 (domain4) R->L }
% 0.58/0.57 antidomain(domain(X))
% 0.58/0.57 = { by axiom 1 (complement) R->L }
% 0.58/0.57 c(X)
% 0.58/0.57
% 0.58/0.57 Lemma 20: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 0.58/0.57 Proof:
% 0.58/0.57 multiplication(antidomain(X), addition(X, Y))
% 0.58/0.57 = { by axiom 6 (additive_commutativity) R->L }
% 0.58/0.57 multiplication(antidomain(X), addition(Y, X))
% 0.58/0.57 = { by axiom 14 (right_distributivity) }
% 0.58/0.57 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 0.58/0.57 = { by axiom 11 (domain1) }
% 0.58/0.57 addition(multiplication(antidomain(X), Y), zero)
% 0.58/0.57 = { by axiom 7 (additive_identity) }
% 0.58/0.57 multiplication(antidomain(X), Y)
% 0.58/0.57
% 0.58/0.57 Lemma 21: antidomain(X) = c(X).
% 0.58/0.57 Proof:
% 0.58/0.57 antidomain(X)
% 0.58/0.57 = { by lemma 18 R->L }
% 0.58/0.57 domain_difference(antidomain(X), X)
% 0.58/0.57 = { by axiom 10 (domain_difference) }
% 0.58/0.57 multiplication(domain(antidomain(X)), antidomain(X))
% 0.58/0.57 = { by lemma 19 }
% 0.58/0.57 multiplication(c(X), antidomain(X))
% 0.58/0.57 = { by axiom 1 (complement) }
% 0.58/0.57 multiplication(antidomain(domain(X)), antidomain(X))
% 0.58/0.57 = { by lemma 20 R->L }
% 0.58/0.57 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 0.58/0.57 = { by lemma 17 }
% 0.58/0.57 multiplication(antidomain(domain(X)), one)
% 0.58/0.57 = { by axiom 3 (multiplicative_right_identity) }
% 0.58/0.57 antidomain(domain(X))
% 0.58/0.57 = { by axiom 1 (complement) R->L }
% 0.58/0.57 c(X)
% 0.58/0.57
% 0.58/0.57 Lemma 22: antidomain(one) = zero.
% 0.58/0.57 Proof:
% 0.58/0.57 antidomain(one)
% 0.58/0.57 = { by axiom 3 (multiplicative_right_identity) R->L }
% 0.58/0.57 multiplication(antidomain(one), one)
% 0.58/0.57 = { by axiom 11 (domain1) }
% 0.58/0.57 zero
% 0.58/0.57
% 0.58/0.57 Lemma 23: domain(zero) = zero.
% 0.58/0.57 Proof:
% 0.58/0.57 domain(zero)
% 0.58/0.57 = { by axiom 2 (domain4) }
% 0.58/0.57 antidomain(antidomain(zero))
% 0.58/0.57 = { by lemma 22 R->L }
% 0.58/0.57 antidomain(antidomain(antidomain(one)))
% 0.58/0.57 = { by axiom 2 (domain4) R->L }
% 0.58/0.57 antidomain(domain(one))
% 0.58/0.57 = { by axiom 7 (additive_identity) R->L }
% 0.58/0.57 antidomain(addition(domain(one), zero))
% 0.58/0.57 = { by lemma 22 R->L }
% 0.58/0.57 antidomain(addition(domain(one), antidomain(one)))
% 0.58/0.57 = { by lemma 17 }
% 0.58/0.57 antidomain(one)
% 0.58/0.57 = { by lemma 22 }
% 0.58/0.57 zero
% 0.58/0.57
% 0.58/0.57 Lemma 24: domain(c(X)) = c(domain(X)).
% 0.58/0.57 Proof:
% 0.58/0.57 domain(c(X))
% 0.58/0.57 = { by axiom 1 (complement) }
% 0.58/0.57 domain(antidomain(domain(X)))
% 0.58/0.57 = { by lemma 19 }
% 0.58/0.57 c(domain(X))
% 0.58/0.57
% 0.58/0.57 Lemma 25: c(c(X)) = domain(domain(domain(X))).
% 0.58/0.57 Proof:
% 0.58/0.57 c(c(X))
% 0.58/0.57 = { by lemma 19 R->L }
% 0.58/0.57 domain(antidomain(c(X)))
% 0.58/0.57 = { by axiom 1 (complement) }
% 0.58/0.57 domain(antidomain(antidomain(domain(X))))
% 0.58/0.57 = { by axiom 2 (domain4) R->L }
% 0.58/0.57 domain(domain(domain(X)))
% 0.58/0.57
% 0.58/0.57 Lemma 26: domain(domain(domain(X))) = domain(X).
% 0.58/0.57 Proof:
% 0.58/0.57 domain(domain(domain(X)))
% 0.58/0.57 = { by lemma 25 R->L }
% 0.58/0.57 c(c(X))
% 0.58/0.57 = { by lemma 21 R->L }
% 0.58/0.57 c(antidomain(X))
% 0.58/0.57 = { by lemma 21 R->L }
% 0.58/0.57 antidomain(antidomain(X))
% 0.58/0.57 = { by axiom 2 (domain4) R->L }
% 0.58/0.57 domain(X)
% 0.58/0.57
% 0.58/0.57 Lemma 27: forward_box(one, X) = domain(domain(domain(domain(domain(X))))).
% 0.58/0.57 Proof:
% 0.58/0.57 forward_box(one, X)
% 0.58/0.57 = { by axiom 12 (forward_box) }
% 0.58/0.57 c(forward_diamond(one, c(X)))
% 0.58/0.57 = { by axiom 8 (forward_diamond) }
% 0.58/0.57 c(domain(multiplication(one, domain(c(X)))))
% 0.58/0.57 = { by axiom 5 (multiplicative_left_identity) }
% 0.58/0.57 c(domain(domain(c(X))))
% 0.58/0.57 = { by lemma 24 }
% 0.58/0.57 c(domain(c(domain(X))))
% 0.58/0.57 = { by lemma 24 }
% 0.58/0.57 c(c(domain(domain(X))))
% 0.58/0.57 = { by lemma 25 }
% 0.58/0.57 domain(domain(domain(domain(domain(X)))))
% 0.58/0.57
% 0.58/0.57 Lemma 28: domain(divergence(X)) = divergence(X).
% 0.58/0.57 Proof:
% 0.58/0.57 domain(divergence(X))
% 0.58/0.57 = { by axiom 9 (divergence1) R->L }
% 0.58/0.57 domain(forward_diamond(X, divergence(X)))
% 0.58/0.57 = { by axiom 8 (forward_diamond) }
% 0.58/0.57 domain(domain(multiplication(X, domain(divergence(X)))))
% 0.58/0.57 = { by lemma 26 R->L }
% 0.58/0.57 domain(domain(domain(domain(multiplication(X, domain(divergence(X)))))))
% 0.58/0.57 = { by lemma 26 R->L }
% 0.58/0.57 domain(domain(domain(domain(domain(domain(multiplication(X, domain(divergence(X)))))))))
% 0.58/0.57 = { by lemma 27 R->L }
% 0.58/0.57 forward_box(one, domain(multiplication(X, domain(divergence(X)))))
% 0.58/0.58 = { by axiom 12 (forward_box) }
% 0.58/0.58 c(forward_diamond(one, c(domain(multiplication(X, domain(divergence(X)))))))
% 0.58/0.58 = { by axiom 1 (complement) }
% 0.58/0.58 c(forward_diamond(one, antidomain(domain(domain(multiplication(X, domain(divergence(X))))))))
% 0.58/0.58 = { by axiom 3 (multiplicative_right_identity) R->L }
% 0.58/0.58 c(forward_diamond(one, multiplication(antidomain(domain(domain(multiplication(X, domain(divergence(X)))))), one)))
% 0.58/0.58 = { by lemma 17 R->L }
% 0.58/0.58 c(forward_diamond(one, multiplication(antidomain(domain(domain(multiplication(X, domain(divergence(X)))))), addition(domain(domain(multiplication(X, domain(divergence(X))))), antidomain(domain(multiplication(X, domain(divergence(X)))))))))
% 0.58/0.58 = { by axiom 1 (complement) R->L }
% 0.58/0.58 c(forward_diamond(one, multiplication(antidomain(domain(domain(multiplication(X, domain(divergence(X)))))), addition(domain(domain(multiplication(X, domain(divergence(X))))), c(multiplication(X, domain(divergence(X))))))))
% 0.58/0.58 = { by lemma 20 }
% 0.58/0.58 c(forward_diamond(one, multiplication(antidomain(domain(domain(multiplication(X, domain(divergence(X)))))), c(multiplication(X, domain(divergence(X)))))))
% 0.58/0.58 = { by axiom 1 (complement) R->L }
% 0.58/0.58 c(forward_diamond(one, multiplication(c(domain(multiplication(X, domain(divergence(X))))), c(multiplication(X, domain(divergence(X)))))))
% 0.58/0.58 = { by lemma 19 R->L }
% 0.58/0.58 c(forward_diamond(one, multiplication(domain(antidomain(domain(multiplication(X, domain(divergence(X)))))), c(multiplication(X, domain(divergence(X)))))))
% 0.58/0.58 = { by axiom 1 (complement) }
% 0.58/0.58 c(forward_diamond(one, multiplication(domain(antidomain(domain(multiplication(X, domain(divergence(X)))))), antidomain(domain(multiplication(X, domain(divergence(X))))))))
% 0.58/0.58 = { by axiom 10 (domain_difference) R->L }
% 0.58/0.58 c(forward_diamond(one, domain_difference(antidomain(domain(multiplication(X, domain(divergence(X))))), domain(multiplication(X, domain(divergence(X)))))))
% 0.58/0.58 = { by lemma 18 }
% 0.58/0.58 c(forward_diamond(one, antidomain(domain(multiplication(X, domain(divergence(X)))))))
% 0.58/0.58 = { by axiom 1 (complement) R->L }
% 0.58/0.58 c(forward_diamond(one, c(multiplication(X, domain(divergence(X))))))
% 0.58/0.58 = { by axiom 12 (forward_box) R->L }
% 0.58/0.58 forward_box(one, multiplication(X, domain(divergence(X))))
% 0.58/0.58 = { by lemma 27 }
% 0.58/0.58 domain(domain(domain(domain(domain(multiplication(X, domain(divergence(X))))))))
% 0.58/0.58 = { by lemma 26 }
% 0.58/0.58 domain(domain(domain(multiplication(X, domain(divergence(X))))))
% 0.58/0.58 = { by lemma 26 }
% 0.58/0.58 domain(multiplication(X, domain(divergence(X))))
% 0.58/0.58 = { by axiom 8 (forward_diamond) R->L }
% 0.58/0.58 forward_diamond(X, divergence(X))
% 0.58/0.58 = { by axiom 9 (divergence1) }
% 0.58/0.58 divergence(X)
% 0.58/0.58
% 0.58/0.58 Lemma 29: domain_difference(X, X) = zero.
% 0.58/0.58 Proof:
% 0.58/0.58 domain_difference(X, X)
% 0.58/0.58 = { by axiom 10 (domain_difference) }
% 0.58/0.58 multiplication(domain(X), antidomain(X))
% 0.58/0.58 = { by axiom 2 (domain4) }
% 0.58/0.58 multiplication(antidomain(antidomain(X)), antidomain(X))
% 0.58/0.58 = { by axiom 11 (domain1) }
% 0.58/0.58 zero
% 0.58/0.58
% 0.58/0.58 Lemma 30: forward_diamond(X, zero) = zero.
% 0.58/0.58 Proof:
% 0.58/0.58 forward_diamond(X, zero)
% 0.58/0.58 = { by axiom 8 (forward_diamond) }
% 0.58/0.58 domain(multiplication(X, domain(zero)))
% 0.58/0.58 = { by lemma 23 }
% 0.58/0.58 domain(multiplication(X, zero))
% 0.58/0.58 = { by axiom 4 (right_annihilation) }
% 0.58/0.58 domain(zero)
% 0.58/0.58 = { by lemma 23 }
% 0.58/0.58 zero
% 0.58/0.58
% 0.58/0.58 Goal 1 (goals_1): divergence(x0) = zero.
% 0.58/0.58 Proof:
% 0.58/0.58 divergence(x0)
% 0.58/0.58 = { by axiom 7 (additive_identity) R->L }
% 0.58/0.58 addition(divergence(x0), zero)
% 0.58/0.58 = { by lemma 30 R->L }
% 0.58/0.58 addition(divergence(x0), forward_diamond(star(x0), zero))
% 0.58/0.58 = { by lemma 29 R->L }
% 0.58/0.58 addition(divergence(x0), forward_diamond(star(x0), domain_difference(divergence(x0), divergence(x0))))
% 0.58/0.58 = { by axiom 9 (divergence1) R->L }
% 0.58/0.58 addition(divergence(x0), forward_diamond(star(x0), domain_difference(divergence(x0), forward_diamond(x0, divergence(x0)))))
% 0.58/0.58 = { by lemma 28 R->L }
% 0.58/0.58 addition(divergence(x0), forward_diamond(star(x0), domain_difference(divergence(x0), forward_diamond(x0, domain(divergence(x0))))))
% 0.58/0.58 = { by lemma 28 R->L }
% 0.58/0.58 addition(divergence(x0), forward_diamond(star(x0), domain_difference(domain(divergence(x0)), forward_diamond(x0, domain(divergence(x0))))))
% 0.58/0.58 = { by lemma 28 R->L }
% 0.58/0.58 addition(domain(divergence(x0)), forward_diamond(star(x0), domain_difference(domain(divergence(x0)), forward_diamond(x0, domain(divergence(x0))))))
% 0.58/0.58 = { by axiom 16 (goals) }
% 0.58/0.58 forward_diamond(star(x0), domain_difference(domain(divergence(x0)), forward_diamond(x0, domain(divergence(x0)))))
% 0.58/0.58 = { by lemma 28 }
% 0.58/0.58 forward_diamond(star(x0), domain_difference(domain(divergence(x0)), forward_diamond(x0, divergence(x0))))
% 0.58/0.58 = { by lemma 28 }
% 0.58/0.58 forward_diamond(star(x0), domain_difference(divergence(x0), forward_diamond(x0, divergence(x0))))
% 0.58/0.58 = { by axiom 9 (divergence1) }
% 0.58/0.58 forward_diamond(star(x0), domain_difference(divergence(x0), divergence(x0)))
% 0.58/0.58 = { by lemma 29 }
% 0.58/0.58 forward_diamond(star(x0), zero)
% 0.58/0.58 = { by lemma 30 }
% 0.58/0.58 zero
% 0.58/0.58 % SZS output end Proof
% 0.58/0.58
% 0.58/0.58 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------