TSTP Solution File: KLE128+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE128+1 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n018.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 05:35:58 EDT 2023
% Result : Theorem 37.24s 5.15s
% Output : Proof 37.24s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.09/0.13 % Problem : KLE128+1 : TPTP v8.1.2. Released v4.0.0.
% 0.09/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n018.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Tue Aug 29 12:09:33 EDT 2023
% 0.14/0.35 % CPUTime :
% 37.24/5.15 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 37.24/5.15
% 37.24/5.15 % SZS status Theorem
% 37.24/5.15
% 37.24/5.16 % SZS output start Proof
% 37.24/5.16 Take the following subset of the input axioms:
% 37.24/5.16 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 37.24/5.16 fof(additive_commutativity, axiom, ![A2, B2]: addition(A2, B2)=addition(B2, A2)).
% 37.24/5.16 fof(additive_idempotence, axiom, ![A2]: addition(A2, A2)=A2).
% 37.24/5.16 fof(additive_identity, axiom, ![A2]: addition(A2, zero)=A2).
% 37.24/5.16 fof(codomain1, axiom, ![X0]: multiplication(X0, coantidomain(X0))=zero).
% 37.24/5.16 fof(divergence2, axiom, ![X1, X2, X0_2]: (addition(domain(X0_2), addition(forward_diamond(X1, domain(X0_2)), domain(X2)))=addition(forward_diamond(X1, domain(X0_2)), domain(X2)) => addition(domain(X0_2), addition(divergence(X1), forward_diamond(star(X1), domain(X2))))=addition(divergence(X1), forward_diamond(star(X1), domain(X2))))).
% 37.24/5.16 fof(domain1, axiom, ![X0_2]: multiplication(antidomain(X0_2), X0_2)=zero).
% 37.24/5.16 fof(domain2, axiom, ![X1_2, X0_2]: addition(antidomain(multiplication(X0_2, X1_2)), antidomain(multiplication(X0_2, antidomain(antidomain(X1_2)))))=antidomain(multiplication(X0_2, antidomain(antidomain(X1_2))))).
% 37.24/5.16 fof(domain3, axiom, ![X0_2]: addition(antidomain(antidomain(X0_2)), antidomain(X0_2))=one).
% 37.24/5.16 fof(domain4, axiom, ![X0_2]: domain(X0_2)=antidomain(antidomain(X0_2))).
% 37.24/5.16 fof(forward_diamond, axiom, ![X1_2, X0_2]: forward_diamond(X0_2, X1_2)=domain(multiplication(X0_2, domain(X1_2)))).
% 37.24/5.16 fof(goals, conjecture, ![X0_2]: (divergence(X0_2)=zero => ![X1_2]: (addition(domain(X1_2), forward_diamond(X0_2, domain(X1_2)))=forward_diamond(X0_2, domain(X1_2)) => domain(X1_2)=zero))).
% 37.24/5.16 fof(multiplicative_right_identity, axiom, ![A2]: multiplication(A2, one)=A2).
% 37.24/5.16
% 37.24/5.16 Now clausify the problem and encode Horn clauses using encoding 3 of
% 37.24/5.16 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 37.24/5.16 We repeatedly replace C & s=t => u=v by the two clauses:
% 37.24/5.16 fresh(y, y, x1...xn) = u
% 37.24/5.16 C => fresh(s, t, x1...xn) = v
% 37.24/5.16 where fresh is a fresh function symbol and x1..xn are the free
% 37.24/5.16 variables of u and v.
% 37.24/5.16 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 37.24/5.16 input problem has no model of domain size 1).
% 37.24/5.16
% 37.24/5.16 The encoding turns the above axioms into the following unit equations and goals:
% 37.24/5.16
% 37.24/5.16 Axiom 1 (goals_1): divergence(x0) = zero.
% 37.24/5.16 Axiom 2 (additive_idempotence): addition(X, X) = X.
% 37.24/5.16 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 37.24/5.16 Axiom 4 (additive_identity): addition(X, zero) = X.
% 37.24/5.16 Axiom 5 (multiplicative_right_identity): multiplication(X, one) = X.
% 37.24/5.16 Axiom 6 (domain4): domain(X) = antidomain(antidomain(X)).
% 37.24/5.16 Axiom 7 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 37.24/5.16 Axiom 8 (domain1): multiplication(antidomain(X), X) = zero.
% 37.24/5.16 Axiom 9 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 37.24/5.16 Axiom 10 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 37.24/5.16 Axiom 11 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 37.24/5.16 Axiom 12 (goals): addition(domain(x1), forward_diamond(x0, domain(x1))) = forward_diamond(x0, domain(x1)).
% 37.24/5.16 Axiom 13 (divergence2): fresh2(X, X, Y, Z, W) = addition(divergence(Z), forward_diamond(star(Z), domain(W))).
% 37.24/5.16 Axiom 14 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 37.24/5.16 Axiom 15 (divergence2): fresh2(addition(domain(X), addition(forward_diamond(Y, domain(X)), domain(Z))), addition(forward_diamond(Y, domain(X)), domain(Z)), X, Y, Z) = addition(domain(X), addition(divergence(Y), forward_diamond(star(Y), domain(Z)))).
% 37.24/5.16
% 37.24/5.16 Lemma 16: antidomain(one) = zero.
% 37.24/5.16 Proof:
% 37.24/5.16 antidomain(one)
% 37.24/5.16 = { by axiom 5 (multiplicative_right_identity) R->L }
% 37.24/5.16 multiplication(antidomain(one), one)
% 37.24/5.16 = { by axiom 8 (domain1) }
% 37.24/5.16 zero
% 37.24/5.16
% 37.24/5.16 Lemma 17: addition(domain(X), antidomain(X)) = one.
% 37.24/5.16 Proof:
% 37.24/5.16 addition(domain(X), antidomain(X))
% 37.24/5.16 = { by axiom 6 (domain4) }
% 37.24/5.16 addition(antidomain(antidomain(X)), antidomain(X))
% 37.24/5.16 = { by axiom 11 (domain3) }
% 37.24/5.16 one
% 37.24/5.16
% 37.24/5.16 Lemma 18: addition(one, antidomain(X)) = one.
% 37.24/5.16 Proof:
% 37.24/5.16 addition(one, antidomain(X))
% 37.24/5.16 = { by axiom 3 (additive_commutativity) R->L }
% 37.24/5.16 addition(antidomain(X), one)
% 37.24/5.16 = { by lemma 17 R->L }
% 37.24/5.16 addition(antidomain(X), addition(domain(X), antidomain(X)))
% 37.24/5.16 = { by axiom 2 (additive_idempotence) R->L }
% 37.24/5.16 addition(antidomain(X), addition(addition(domain(X), domain(X)), antidomain(X)))
% 37.24/5.16 = { by axiom 9 (additive_associativity) R->L }
% 37.24/5.16 addition(antidomain(X), addition(domain(X), addition(domain(X), antidomain(X))))
% 37.24/5.16 = { by axiom 3 (additive_commutativity) R->L }
% 37.24/5.16 addition(antidomain(X), addition(domain(X), addition(antidomain(X), domain(X))))
% 37.24/5.16 = { by axiom 9 (additive_associativity) }
% 37.24/5.16 addition(addition(antidomain(X), domain(X)), addition(antidomain(X), domain(X)))
% 37.24/5.16 = { by axiom 2 (additive_idempotence) }
% 37.24/5.16 addition(antidomain(X), domain(X))
% 37.24/5.16 = { by axiom 3 (additive_commutativity) }
% 37.24/5.16 addition(domain(X), antidomain(X))
% 37.24/5.16 = { by lemma 17 }
% 37.24/5.16 one
% 37.24/5.16
% 37.24/5.16 Lemma 19: addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))) = antidomain(multiplication(X, domain(Y))).
% 37.24/5.16 Proof:
% 37.24/5.16 addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y))))
% 37.24/5.16 = { by axiom 6 (domain4) }
% 37.24/5.16 addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y)))))
% 37.24/5.16 = { by axiom 14 (domain2) }
% 37.24/5.16 antidomain(multiplication(X, antidomain(antidomain(Y))))
% 37.24/5.16 = { by axiom 6 (domain4) R->L }
% 37.24/5.16 antidomain(multiplication(X, domain(Y)))
% 37.24/5.16
% 37.24/5.16 Lemma 20: forward_diamond(X, domain(coantidomain(X))) = zero.
% 37.24/5.16 Proof:
% 37.24/5.16 forward_diamond(X, domain(coantidomain(X)))
% 37.24/5.16 = { by axiom 10 (forward_diamond) }
% 37.24/5.16 domain(multiplication(X, domain(domain(coantidomain(X)))))
% 37.24/5.16 = { by axiom 6 (domain4) }
% 37.24/5.16 antidomain(antidomain(multiplication(X, domain(domain(coantidomain(X))))))
% 37.24/5.16 = { by lemma 19 R->L }
% 37.24/5.16 antidomain(addition(antidomain(multiplication(X, domain(coantidomain(X)))), antidomain(multiplication(X, domain(domain(coantidomain(X)))))))
% 37.24/5.16 = { by lemma 19 R->L }
% 37.24/5.16 antidomain(addition(addition(antidomain(multiplication(X, coantidomain(X))), antidomain(multiplication(X, domain(coantidomain(X))))), antidomain(multiplication(X, domain(domain(coantidomain(X)))))))
% 37.24/5.16 = { by axiom 7 (codomain1) }
% 37.24/5.16 antidomain(addition(addition(antidomain(zero), antidomain(multiplication(X, domain(coantidomain(X))))), antidomain(multiplication(X, domain(domain(coantidomain(X)))))))
% 37.24/5.16 = { by lemma 16 R->L }
% 37.24/5.16 antidomain(addition(addition(antidomain(antidomain(one)), antidomain(multiplication(X, domain(coantidomain(X))))), antidomain(multiplication(X, domain(domain(coantidomain(X)))))))
% 37.24/5.16 = { by axiom 6 (domain4) R->L }
% 37.24/5.16 antidomain(addition(addition(domain(one), antidomain(multiplication(X, domain(coantidomain(X))))), antidomain(multiplication(X, domain(domain(coantidomain(X)))))))
% 37.24/5.16 = { by axiom 4 (additive_identity) R->L }
% 37.24/5.16 antidomain(addition(addition(addition(domain(one), zero), antidomain(multiplication(X, domain(coantidomain(X))))), antidomain(multiplication(X, domain(domain(coantidomain(X)))))))
% 37.24/5.16 = { by lemma 16 R->L }
% 37.24/5.16 antidomain(addition(addition(addition(domain(one), antidomain(one)), antidomain(multiplication(X, domain(coantidomain(X))))), antidomain(multiplication(X, domain(domain(coantidomain(X)))))))
% 37.24/5.16 = { by lemma 17 }
% 37.24/5.16 antidomain(addition(addition(one, antidomain(multiplication(X, domain(coantidomain(X))))), antidomain(multiplication(X, domain(domain(coantidomain(X)))))))
% 37.24/5.16 = { by lemma 18 }
% 37.24/5.16 antidomain(addition(one, antidomain(multiplication(X, domain(domain(coantidomain(X)))))))
% 37.24/5.16 = { by lemma 18 }
% 37.24/5.16 antidomain(one)
% 37.24/5.16 = { by lemma 16 }
% 37.24/5.16 zero
% 37.24/5.16
% 37.24/5.16 Lemma 21: fresh2(X, X, Y, x0, Z) = forward_diamond(star(x0), domain(Z)).
% 37.24/5.16 Proof:
% 37.24/5.16 fresh2(X, X, Y, x0, Z)
% 37.24/5.16 = { by axiom 13 (divergence2) }
% 37.24/5.16 addition(divergence(x0), forward_diamond(star(x0), domain(Z)))
% 37.24/5.16 = { by axiom 1 (goals_1) }
% 37.24/5.16 addition(zero, forward_diamond(star(x0), domain(Z)))
% 37.24/5.16 = { by axiom 3 (additive_commutativity) R->L }
% 37.24/5.16 addition(forward_diamond(star(x0), domain(Z)), zero)
% 37.24/5.16 = { by axiom 4 (additive_identity) }
% 37.24/5.16 forward_diamond(star(x0), domain(Z))
% 37.24/5.16
% 37.24/5.16 Goal 1 (goals_2): domain(x1) = zero.
% 37.24/5.16 Proof:
% 37.24/5.16 domain(x1)
% 37.24/5.16 = { by axiom 4 (additive_identity) R->L }
% 37.24/5.16 addition(domain(x1), zero)
% 37.24/5.16 = { by lemma 20 R->L }
% 37.24/5.16 addition(domain(x1), forward_diamond(star(x0), domain(coantidomain(star(x0)))))
% 37.24/5.16 = { by lemma 21 R->L }
% 37.24/5.16 addition(domain(x1), fresh2(X, X, Y, x0, coantidomain(star(x0))))
% 37.24/5.16 = { by axiom 13 (divergence2) }
% 37.24/5.16 addition(domain(x1), addition(divergence(x0), forward_diamond(star(x0), domain(coantidomain(star(x0))))))
% 37.24/5.16 = { by axiom 15 (divergence2) R->L }
% 37.24/5.16 fresh2(addition(domain(x1), addition(forward_diamond(x0, domain(x1)), domain(coantidomain(star(x0))))), addition(forward_diamond(x0, domain(x1)), domain(coantidomain(star(x0)))), x1, x0, coantidomain(star(x0)))
% 37.24/5.16 = { by axiom 3 (additive_commutativity) }
% 37.24/5.16 fresh2(addition(domain(x1), addition(forward_diamond(x0, domain(x1)), domain(coantidomain(star(x0))))), addition(domain(coantidomain(star(x0))), forward_diamond(x0, domain(x1))), x1, x0, coantidomain(star(x0)))
% 37.24/5.16 = { by axiom 9 (additive_associativity) }
% 37.24/5.16 fresh2(addition(addition(domain(x1), forward_diamond(x0, domain(x1))), domain(coantidomain(star(x0)))), addition(domain(coantidomain(star(x0))), forward_diamond(x0, domain(x1))), x1, x0, coantidomain(star(x0)))
% 37.24/5.16 = { by axiom 12 (goals) }
% 37.24/5.16 fresh2(addition(forward_diamond(x0, domain(x1)), domain(coantidomain(star(x0)))), addition(domain(coantidomain(star(x0))), forward_diamond(x0, domain(x1))), x1, x0, coantidomain(star(x0)))
% 37.24/5.16 = { by axiom 3 (additive_commutativity) }
% 37.24/5.16 fresh2(addition(domain(coantidomain(star(x0))), forward_diamond(x0, domain(x1))), addition(domain(coantidomain(star(x0))), forward_diamond(x0, domain(x1))), x1, x0, coantidomain(star(x0)))
% 37.24/5.16 = { by lemma 21 }
% 37.24/5.16 forward_diamond(star(x0), domain(coantidomain(star(x0))))
% 37.24/5.16 = { by lemma 20 }
% 37.24/5.17 zero
% 37.24/5.17 % SZS output end Proof
% 37.24/5.18
% 37.24/5.18 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------