%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE129+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n014.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 0.30s 0.74s
% Output : Proof 0.30s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : KLE129+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.07 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.19/0.45 % Computer : n014.cluster.edu
% 0.19/0.45 % Model : x86_64 x86_64
% 0.19/0.45 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.19/0.45 % Memory : 8046.5625MB
% 0.19/0.45 % OS : Linux 6.8.0-71-generic
% 0.19/0.45 % CPULimit : 300
% 0.19/0.45 % WCLimit : 300
% 0.19/0.45 % DateTime : Sun Sep 27 13:11:31 UTC 2026
% 0.19/0.45 % CPUTime :
% 0.19/0.46 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.30/0.74 Command-line arguments: --flatten-regeneralise
% 0.30/0.74
% 0.30/0.74 % SZS status Theorem
% 0.30/0.74
% 0.30/0.75 % SZS output start Proof
% 0.30/0.75 Axiom 1 (complement): c(X) = antidomain(domain(X)).
% 0.30/0.75 Axiom 2 (domain4): domain(X) = antidomain(antidomain(X)).
% 0.30/0.75 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.30/0.75 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.30/0.75 Axiom 5 (additive_idempotence): addition(X, X) = X.
% 0.30/0.75 Axiom 6 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.30/0.75 Axiom 7 (additive_identity): addition(X, zero) = X.
% 0.30/0.75 Axiom 8 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 0.30/0.75 Axiom 9 (divergence1): forward_diamond(X, divergence(X)) = divergence(X).
% 0.30/0.75 Axiom 10 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 0.30/0.75 Axiom 11 (domain1): multiplication(antidomain(X), X) = zero.
% 0.30/0.75 Axiom 12 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 0.30/0.75 Axiom 13 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))).
% 0.30/0.75 Axiom 14 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 0.30/0.75 Axiom 15 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.30/0.75 Axiom 16 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.30/0.75 Axiom 17 (goals): ifeq3(addition(domain(X), forward_diamond(x0, domain(X))), forward_diamond(x0, domain(X)), domain(X), zero) = zero.
% 0.30/0.75
% 0.30/0.75 Lemma 18: addition(domain(X), antidomain(X)) = one.
% 0.30/0.75 Proof:
% 0.30/0.75 addition(domain(X), antidomain(X))
% 0.30/0.75 = { by axiom 2 (domain4) }
% 0.30/0.75 addition(antidomain(antidomain(X)), antidomain(X))
% 0.30/0.75 = { by axiom 14 (domain3) }
% 0.30/0.75 one
% 0.30/0.75
% 0.30/0.75 Lemma 19: domain_difference(antidomain(X), X) = antidomain(X).
% 0.30/0.75 Proof:
% 0.30/0.75 domain_difference(antidomain(X), X)
% 0.30/0.75 = { by axiom 10 (domain_difference) }
% 0.30/0.75 multiplication(domain(antidomain(X)), antidomain(X))
% 0.30/0.75 = { by axiom 7 (additive_identity) R->L }
% 0.30/0.75 addition(multiplication(domain(antidomain(X)), antidomain(X)), zero)
% 0.30/0.75 = { by axiom 11 (domain1) R->L }
% 0.30/0.75 addition(multiplication(domain(antidomain(X)), antidomain(X)), multiplication(antidomain(antidomain(X)), antidomain(X)))
% 0.30/0.75 = { by axiom 16 (left_distributivity) R->L }
% 0.30/0.75 multiplication(addition(domain(antidomain(X)), antidomain(antidomain(X))), antidomain(X))
% 0.30/0.75 = { by lemma 18 }
% 0.30/0.75 multiplication(one, antidomain(X))
% 0.30/0.75 = { by axiom 4 (multiplicative_left_identity) }
% 0.30/0.75 antidomain(X)
% 0.30/0.75
% 0.30/0.75 Lemma 20: domain(antidomain(X)) = c(X).
% 0.30/0.75 Proof:
% 0.30/0.75 domain(antidomain(X))
% 0.30/0.75 = { by axiom 2 (domain4) }
% 0.30/0.75 antidomain(antidomain(antidomain(X)))
% 0.30/0.75 = { by axiom 2 (domain4) R->L }
% 0.30/0.75 antidomain(domain(X))
% 0.30/0.75 = { by axiom 1 (complement) R->L }
% 0.30/0.75 c(X)
% 0.30/0.75
% 0.30/0.75 Lemma 21: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 0.30/0.75 Proof:
% 0.30/0.75 multiplication(antidomain(X), addition(X, Y))
% 0.30/0.75 = { by axiom 6 (additive_commutativity) R->L }
% 0.30/0.75 multiplication(antidomain(X), addition(Y, X))
% 0.30/0.75 = { by axiom 15 (right_distributivity) }
% 0.30/0.75 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 0.30/0.75 = { by axiom 11 (domain1) }
% 0.30/0.75 addition(multiplication(antidomain(X), Y), zero)
% 0.30/0.75 = { by axiom 7 (additive_identity) }
% 0.30/0.75 multiplication(antidomain(X), Y)
% 0.30/0.75
% 0.30/0.75 Lemma 22: antidomain(X) = c(X).
% 0.30/0.75 Proof:
% 0.30/0.75 antidomain(X)
% 0.30/0.75 = { by lemma 19 R->L }
% 0.30/0.75 domain_difference(antidomain(X), X)
% 0.30/0.75 = { by axiom 10 (domain_difference) }
% 0.30/0.75 multiplication(domain(antidomain(X)), antidomain(X))
% 0.30/0.75 = { by lemma 20 }
% 0.30/0.75 multiplication(c(X), antidomain(X))
% 0.30/0.75 = { by axiom 1 (complement) }
% 0.30/0.75 multiplication(antidomain(domain(X)), antidomain(X))
% 0.30/0.75 = { by lemma 21 R->L }
% 0.30/0.75 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 0.30/0.75 = { by lemma 18 }
% 0.30/0.75 multiplication(antidomain(domain(X)), one)
% 0.30/0.75 = { by axiom 3 (multiplicative_right_identity) }
% 0.30/0.75 antidomain(domain(X))
% 0.30/0.75 = { by axiom 1 (complement) R->L }
% 0.30/0.75 c(X)
% 0.30/0.75
% 0.30/0.75 Lemma 23: domain(c(X)) = c(domain(X)).
% 0.30/0.75 Proof:
% 0.30/0.75 domain(c(X))
% 0.30/0.75 = { by axiom 1 (complement) }
% 0.30/0.75 domain(antidomain(domain(X)))
% 0.30/0.75 = { by lemma 20 }
% 0.30/0.75 c(domain(X))
% 0.30/0.75
% 0.30/0.75 Lemma 24: c(c(X)) = domain(domain(domain(X))).
% 0.30/0.75 Proof:
% 0.30/0.75 c(c(X))
% 0.30/0.75 = { by lemma 20 R->L }
% 0.30/0.75 domain(antidomain(c(X)))
% 0.30/0.75 = { by axiom 1 (complement) }
% 0.30/0.75 domain(antidomain(antidomain(domain(X))))
% 0.30/0.75 = { by axiom 2 (domain4) R->L }
% 0.30/0.75 domain(domain(domain(X)))
% 0.30/0.75
% 0.30/0.75 Lemma 25: domain(domain(domain(X))) = domain(X).
% 0.30/0.75 Proof:
% 0.30/0.75 domain(domain(domain(X)))
% 0.30/0.75 = { by lemma 24 R->L }
% 0.30/0.75 c(c(X))
% 0.30/0.75 = { by lemma 22 R->L }
% 0.30/0.75 c(antidomain(X))
% 0.30/0.75 = { by lemma 22 R->L }
% 0.30/0.75 antidomain(antidomain(X))
% 0.30/0.75 = { by axiom 2 (domain4) R->L }
% 0.30/0.75 domain(X)
% 0.30/0.75
% 0.30/0.75 Lemma 26: forward_box(one, X) = domain(domain(domain(domain(domain(X))))).
% 0.30/0.75 Proof:
% 0.30/0.75 forward_box(one, X)
% 0.30/0.75 = { by axiom 13 (forward_box) }
% 0.30/0.75 c(forward_diamond(one, c(X)))
% 0.30/0.75 = { by axiom 8 (forward_diamond) }
% 0.30/0.75 c(domain(multiplication(one, domain(c(X)))))
% 0.30/0.75 = { by axiom 4 (multiplicative_left_identity) }
% 0.30/0.75 c(domain(domain(c(X))))
% 0.30/0.75 = { by lemma 23 }
% 0.30/0.75 c(domain(c(domain(X))))
% 0.30/0.75 = { by lemma 23 }
% 0.30/0.75 c(c(domain(domain(X))))
% 0.30/0.75 = { by lemma 24 }
% 0.30/0.75 domain(domain(domain(domain(domain(X)))))
% 0.30/0.75
% 0.30/0.75 Lemma 27: domain(divergence(X)) = divergence(X).
% 0.30/0.75 Proof:
% 0.30/0.75 domain(divergence(X))
% 0.30/0.75 = { by axiom 9 (divergence1) R->L }
% 0.30/0.75 domain(forward_diamond(X, divergence(X)))
% 0.30/0.75 = { by axiom 8 (forward_diamond) }
% 0.30/0.75 domain(domain(multiplication(X, domain(divergence(X)))))
% 0.30/0.75 = { by lemma 25 R->L }
% 0.30/0.75 domain(domain(domain(domain(multiplication(X, domain(divergence(X)))))))
% 0.30/0.75 = { by lemma 25 R->L }
% 0.30/0.75 domain(domain(domain(domain(domain(domain(multiplication(X, domain(divergence(X)))))))))
% 0.30/0.75 = { by lemma 26 R->L }
% 0.30/0.75 forward_box(one, domain(multiplication(X, domain(divergence(X)))))
% 0.30/0.75 = { by axiom 13 (forward_box) }
% 0.30/0.75 c(forward_diamond(one, c(domain(multiplication(X, domain(divergence(X)))))))
% 0.30/0.75 = { by axiom 1 (complement) }
% 0.30/0.75 c(forward_diamond(one, antidomain(domain(domain(multiplication(X, domain(divergence(X))))))))
% 0.30/0.75 = { by axiom 3 (multiplicative_right_identity) R->L }
% 0.30/0.75 c(forward_diamond(one, multiplication(antidomain(domain(domain(multiplication(X, domain(divergence(X)))))), one)))
% 0.30/0.75 = { by lemma 18 R->L }
% 0.30/0.75 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)))))))))
% 0.30/0.75 = { by axiom 1 (complement) R->L }
% 0.30/0.75 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))))))))
% 0.30/0.75 = { by lemma 21 }
% 0.30/0.76 c(forward_diamond(one, multiplication(antidomain(domain(domain(multiplication(X, domain(divergence(X)))))), c(multiplication(X, domain(divergence(X)))))))
% 0.30/0.76 = { by axiom 1 (complement) R->L }
% 0.30/0.76 c(forward_diamond(one, multiplication(c(domain(multiplication(X, domain(divergence(X))))), c(multiplication(X, domain(divergence(X)))))))
% 0.30/0.76 = { by lemma 20 R->L }
% 0.30/0.76 c(forward_diamond(one, multiplication(domain(antidomain(domain(multiplication(X, domain(divergence(X)))))), c(multiplication(X, domain(divergence(X)))))))
% 0.30/0.76 = { by axiom 1 (complement) }
% 0.30/0.76 c(forward_diamond(one, multiplication(domain(antidomain(domain(multiplication(X, domain(divergence(X)))))), antidomain(domain(multiplication(X, domain(divergence(X))))))))
% 0.30/0.76 = { by axiom 10 (domain_difference) R->L }
% 0.30/0.76 c(forward_diamond(one, domain_difference(antidomain(domain(multiplication(X, domain(divergence(X))))), domain(multiplication(X, domain(divergence(X)))))))
% 0.30/0.76 = { by lemma 19 }
% 0.30/0.76 c(forward_diamond(one, antidomain(domain(multiplication(X, domain(divergence(X)))))))
% 0.30/0.76 = { by axiom 1 (complement) R->L }
% 0.30/0.76 c(forward_diamond(one, c(multiplication(X, domain(divergence(X))))))
% 0.30/0.76 = { by axiom 13 (forward_box) R->L }
% 0.30/0.76 forward_box(one, multiplication(X, domain(divergence(X))))
% 0.30/0.76 = { by lemma 26 }
% 0.30/0.76 domain(domain(domain(domain(domain(multiplication(X, domain(divergence(X))))))))
% 0.30/0.76 = { by lemma 25 }
% 0.30/0.76 domain(domain(domain(multiplication(X, domain(divergence(X))))))
% 0.30/0.76 = { by lemma 25 }
% 0.30/0.76 domain(multiplication(X, domain(divergence(X))))
% 0.30/0.76 = { by axiom 8 (forward_diamond) R->L }
% 0.30/0.76 forward_diamond(X, divergence(X))
% 0.30/0.76 = { by axiom 9 (divergence1) }
% 0.30/0.76 divergence(X)
% 0.30/0.76
% 0.30/0.76 Goal 1 (goals_1): divergence(x0) = zero.
% 0.30/0.76 Proof:
% 0.30/0.76 divergence(x0)
% 0.30/0.76 = { by axiom 12 (ifeq_axiom) R->L }
% 0.30/0.76 ifeq3(divergence(x0), divergence(x0), divergence(x0), zero)
% 0.30/0.76 = { by axiom 5 (additive_idempotence) R->L }
% 0.30/0.76 ifeq3(addition(divergence(x0), divergence(x0)), divergence(x0), divergence(x0), zero)
% 0.30/0.76 = { by axiom 9 (divergence1) R->L }
% 0.30/0.76 ifeq3(addition(divergence(x0), divergence(x0)), forward_diamond(x0, divergence(x0)), divergence(x0), zero)
% 0.30/0.76 = { by axiom 9 (divergence1) R->L }
% 0.30/0.76 ifeq3(addition(divergence(x0), forward_diamond(x0, divergence(x0))), forward_diamond(x0, divergence(x0)), divergence(x0), zero)
% 0.30/0.76 = { by lemma 27 R->L }
% 0.30/0.76 ifeq3(addition(divergence(x0), forward_diamond(x0, divergence(x0))), forward_diamond(x0, divergence(x0)), domain(divergence(x0)), zero)
% 0.30/0.76 = { by lemma 27 R->L }
% 0.30/0.76 ifeq3(addition(divergence(x0), forward_diamond(x0, divergence(x0))), forward_diamond(x0, domain(divergence(x0))), domain(divergence(x0)), zero)
% 0.30/0.76 = { by lemma 27 R->L }
% 0.30/0.76 ifeq3(addition(divergence(x0), forward_diamond(x0, domain(divergence(x0)))), forward_diamond(x0, domain(divergence(x0))), domain(divergence(x0)), zero)
% 0.30/0.76 = { by lemma 27 R->L }
% 0.30/0.76 ifeq3(addition(domain(divergence(x0)), forward_diamond(x0, domain(divergence(x0)))), forward_diamond(x0, domain(divergence(x0))), domain(divergence(x0)), zero)
% 0.30/0.76 = { by axiom 17 (goals) }
% 0.30/0.76 zero
% 0.30/0.76 % SZS output end Proof
% 0.30/0.76
% 0.30/0.76 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------