TSTP Solution File: KLE099+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE099+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 : n024.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 05:35:53 EDT 2023
% Result : Theorem 0.20s 0.65s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : KLE099+1 : TPTP v8.1.2. Released v4.0.0.
% 0.11/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n024.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Tue Aug 29 12:24:38 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.65 Command-line arguments: --flatten
% 0.20/0.65
% 0.20/0.65 % SZS status Theorem
% 0.20/0.65
% 0.20/0.65 % SZS output start Proof
% 0.20/0.65 Take the following subset of the input axioms:
% 0.20/0.66 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
% 0.20/0.66 fof(additive_identity, axiom, ![A2]: addition(A2, zero)=A2).
% 0.20/0.66 fof(complement, axiom, ![X0]: c(X0)=antidomain(domain(X0))).
% 0.20/0.66 fof(domain1, axiom, ![X0_2]: multiplication(antidomain(X0_2), X0_2)=zero).
% 0.20/0.66 fof(domain3, axiom, ![X0_2]: addition(antidomain(antidomain(X0_2)), antidomain(X0_2))=one).
% 0.20/0.66 fof(domain4, axiom, ![X0_2]: domain(X0_2)=antidomain(antidomain(X0_2))).
% 0.20/0.66 fof(domain_difference, axiom, ![X1, X0_2]: domain_difference(X0_2, X1)=multiplication(domain(X0_2), antidomain(X1))).
% 0.20/0.66 fof(forward_diamond, axiom, ![X0_2, X1_2]: forward_diamond(X0_2, X1_2)=domain(multiplication(X0_2, domain(X1_2)))).
% 0.20/0.66 fof(goals, conjecture, ![X2, X0_2, X1_2]: (addition(forward_diamond(X0_2, domain(X1_2)), domain(X2))=domain(X2) => multiplication(antidomain(X2), multiplication(X0_2, domain(X1_2)))=zero)).
% 0.20/0.66 fof(left_annihilation, axiom, ![A2]: multiplication(zero, A2)=zero).
% 0.20/0.66 fof(left_distributivity, axiom, ![C, A2, B2]: multiplication(addition(A2, B2), C)=addition(multiplication(A2, C), multiplication(B2, C))).
% 0.20/0.66 fof(multiplicative_associativity, axiom, ![A2, B2, C2]: multiplication(A2, multiplication(B2, C2))=multiplication(multiplication(A2, B2), C2)).
% 0.20/0.66 fof(multiplicative_left_identity, axiom, ![A2]: multiplication(one, A2)=A2).
% 0.20/0.66 fof(multiplicative_right_identity, axiom, ![A2]: multiplication(A2, one)=A2).
% 0.20/0.66 fof(right_distributivity, axiom, ![A2, B2, C2]: multiplication(A2, addition(B2, C2))=addition(multiplication(A2, B2), multiplication(A2, C2))).
% 0.20/0.66
% 0.20/0.66 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.66 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.66 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.66 fresh(y, y, x1...xn) = u
% 0.20/0.66 C => fresh(s, t, x1...xn) = v
% 0.20/0.66 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.66 variables of u and v.
% 0.20/0.66 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.66 input problem has no model of domain size 1).
% 0.20/0.66
% 0.20/0.66 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.66
% 0.20/0.66 Axiom 1 (complement): c(X) = antidomain(domain(X)).
% 0.20/0.66 Axiom 2 (domain4): domain(X) = antidomain(antidomain(X)).
% 0.20/0.66 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.20/0.66 Axiom 4 (additive_identity): addition(X, zero) = X.
% 0.20/0.66 Axiom 5 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.20/0.66 Axiom 6 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.20/0.66 Axiom 7 (left_annihilation): multiplication(zero, X) = zero.
% 0.20/0.66 Axiom 8 (domain1): multiplication(antidomain(X), X) = zero.
% 0.20/0.66 Axiom 9 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 0.20/0.66 Axiom 10 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 0.20/0.66 Axiom 11 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 0.20/0.66 Axiom 12 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 0.20/0.66 Axiom 13 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.20/0.66 Axiom 14 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.20/0.66 Axiom 15 (goals): addition(forward_diamond(x0, domain(x1)), domain(x2)) = domain(x2).
% 0.20/0.66
% 0.20/0.66 Lemma 16: addition(domain(X), antidomain(X)) = one.
% 0.20/0.66 Proof:
% 0.20/0.66 addition(domain(X), antidomain(X))
% 0.20/0.66 = { by axiom 2 (domain4) }
% 0.20/0.66 addition(antidomain(antidomain(X)), antidomain(X))
% 0.20/0.66 = { by axiom 12 (domain3) }
% 0.20/0.66 one
% 0.20/0.66
% 0.20/0.66 Lemma 17: multiplication(domain(X), X) = X.
% 0.20/0.66 Proof:
% 0.20/0.66 multiplication(domain(X), X)
% 0.20/0.66 = { by axiom 4 (additive_identity) R->L }
% 0.20/0.66 addition(multiplication(domain(X), X), zero)
% 0.20/0.66 = { by axiom 8 (domain1) R->L }
% 0.20/0.66 addition(multiplication(domain(X), X), multiplication(antidomain(X), X))
% 0.20/0.66 = { by axiom 14 (left_distributivity) R->L }
% 0.20/0.66 multiplication(addition(domain(X), antidomain(X)), X)
% 0.20/0.66 = { by lemma 16 }
% 0.20/0.66 multiplication(one, X)
% 0.20/0.66 = { by axiom 6 (multiplicative_left_identity) }
% 0.20/0.66 X
% 0.20/0.66
% 0.20/0.66 Lemma 18: domain_difference(antidomain(X), X) = antidomain(X).
% 0.20/0.66 Proof:
% 0.20/0.66 domain_difference(antidomain(X), X)
% 0.20/0.66 = { by axiom 10 (domain_difference) }
% 0.20/0.66 multiplication(domain(antidomain(X)), antidomain(X))
% 0.20/0.66 = { by lemma 17 }
% 0.20/0.66 antidomain(X)
% 0.20/0.66
% 0.20/0.66 Lemma 19: domain(antidomain(X)) = c(X).
% 0.20/0.66 Proof:
% 0.20/0.66 domain(antidomain(X))
% 0.20/0.66 = { by axiom 2 (domain4) }
% 0.20/0.66 antidomain(antidomain(antidomain(X)))
% 0.20/0.66 = { by axiom 2 (domain4) R->L }
% 0.20/0.66 antidomain(domain(X))
% 0.20/0.66 = { by axiom 1 (complement) R->L }
% 0.20/0.66 c(X)
% 0.20/0.66
% 0.20/0.66 Lemma 20: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 0.20/0.66 Proof:
% 0.20/0.66 multiplication(antidomain(X), addition(X, Y))
% 0.20/0.66 = { by axiom 3 (additive_commutativity) R->L }
% 0.20/0.66 multiplication(antidomain(X), addition(Y, X))
% 0.20/0.66 = { by axiom 13 (right_distributivity) }
% 0.20/0.66 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 0.20/0.66 = { by axiom 8 (domain1) }
% 0.20/0.66 addition(multiplication(antidomain(X), Y), zero)
% 0.20/0.66 = { by axiom 4 (additive_identity) }
% 0.20/0.66 multiplication(antidomain(X), Y)
% 0.20/0.66
% 0.20/0.66 Lemma 21: antidomain(X) = c(X).
% 0.20/0.66 Proof:
% 0.20/0.66 antidomain(X)
% 0.20/0.66 = { by lemma 18 R->L }
% 0.20/0.66 domain_difference(antidomain(X), X)
% 0.20/0.66 = { by axiom 10 (domain_difference) }
% 0.20/0.66 multiplication(domain(antidomain(X)), antidomain(X))
% 0.20/0.66 = { by lemma 19 }
% 0.20/0.66 multiplication(c(X), antidomain(X))
% 0.20/0.66 = { by axiom 1 (complement) }
% 0.20/0.66 multiplication(antidomain(domain(X)), antidomain(X))
% 0.20/0.66 = { by lemma 20 R->L }
% 0.20/0.66 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 0.20/0.66 = { by lemma 16 }
% 0.20/0.66 multiplication(antidomain(domain(X)), one)
% 0.20/0.66 = { by axiom 5 (multiplicative_right_identity) }
% 0.20/0.66 antidomain(domain(X))
% 0.20/0.66 = { by axiom 1 (complement) R->L }
% 0.20/0.66 c(X)
% 0.20/0.66
% 0.20/0.66 Lemma 22: antidomain(c(X)) = c(antidomain(X)).
% 0.20/0.66 Proof:
% 0.20/0.66 antidomain(c(X))
% 0.20/0.66 = { by lemma 19 R->L }
% 0.20/0.66 antidomain(domain(antidomain(X)))
% 0.20/0.66 = { by axiom 1 (complement) R->L }
% 0.20/0.66 c(antidomain(X))
% 0.20/0.66
% 0.20/0.66 Lemma 23: c(c(X)) = domain(X).
% 0.20/0.66 Proof:
% 0.20/0.66 c(c(X))
% 0.20/0.66 = { by lemma 21 R->L }
% 0.20/0.66 c(antidomain(X))
% 0.20/0.66 = { by lemma 22 R->L }
% 0.20/0.66 antidomain(c(X))
% 0.20/0.66 = { by axiom 5 (multiplicative_right_identity) R->L }
% 0.20/0.66 multiplication(antidomain(c(X)), one)
% 0.20/0.66 = { by lemma 16 R->L }
% 0.20/0.66 multiplication(antidomain(c(X)), addition(domain(antidomain(X)), antidomain(antidomain(X))))
% 0.20/0.66 = { by lemma 19 }
% 0.20/0.66 multiplication(antidomain(c(X)), addition(c(X), antidomain(antidomain(X))))
% 0.20/0.66 = { by axiom 2 (domain4) R->L }
% 0.20/0.66 multiplication(antidomain(c(X)), addition(c(X), domain(X)))
% 0.20/0.66 = { by lemma 20 }
% 0.20/0.66 multiplication(antidomain(c(X)), domain(X))
% 0.20/0.66 = { by lemma 22 }
% 0.20/0.66 multiplication(c(antidomain(X)), domain(X))
% 0.20/0.66 = { by lemma 19 R->L }
% 0.20/0.66 multiplication(domain(antidomain(antidomain(X))), domain(X))
% 0.20/0.66 = { by axiom 2 (domain4) }
% 0.20/0.66 multiplication(domain(antidomain(antidomain(X))), antidomain(antidomain(X)))
% 0.20/0.66 = { by axiom 10 (domain_difference) R->L }
% 0.20/0.66 domain_difference(antidomain(antidomain(X)), antidomain(X))
% 0.20/0.66 = { by lemma 18 }
% 0.20/0.66 antidomain(antidomain(X))
% 0.20/0.66 = { by axiom 2 (domain4) R->L }
% 0.20/0.66 domain(X)
% 0.20/0.66
% 0.20/0.66 Goal 1 (goals_1): multiplication(antidomain(x2), multiplication(x0, domain(x1))) = zero.
% 0.20/0.66 Proof:
% 0.20/0.66 multiplication(antidomain(x2), multiplication(x0, domain(x1)))
% 0.20/0.66 = { by lemma 17 R->L }
% 0.20/0.66 multiplication(antidomain(x2), multiplication(domain(multiplication(x0, domain(x1))), multiplication(x0, domain(x1))))
% 0.20/0.66 = { by lemma 23 R->L }
% 0.20/0.66 multiplication(antidomain(x2), multiplication(domain(multiplication(x0, c(c(x1)))), multiplication(x0, domain(x1))))
% 0.20/0.66 = { by axiom 1 (complement) }
% 0.20/0.66 multiplication(antidomain(x2), multiplication(domain(multiplication(x0, c(antidomain(domain(x1))))), multiplication(x0, domain(x1))))
% 0.20/0.66 = { by lemma 21 }
% 0.20/0.66 multiplication(antidomain(x2), multiplication(domain(multiplication(x0, c(c(domain(x1))))), multiplication(x0, domain(x1))))
% 0.20/0.66 = { by lemma 23 }
% 0.20/0.66 multiplication(antidomain(x2), multiplication(domain(multiplication(x0, domain(domain(x1)))), multiplication(x0, domain(x1))))
% 0.20/0.66 = { by axiom 9 (forward_diamond) R->L }
% 0.20/0.66 multiplication(antidomain(x2), multiplication(forward_diamond(x0, domain(x1)), multiplication(x0, domain(x1))))
% 0.20/0.66 = { by axiom 11 (multiplicative_associativity) }
% 0.20/0.66 multiplication(multiplication(antidomain(x2), forward_diamond(x0, domain(x1))), multiplication(x0, domain(x1)))
% 0.20/0.66 = { by lemma 21 }
% 0.20/0.66 multiplication(multiplication(c(x2), forward_diamond(x0, domain(x1))), multiplication(x0, domain(x1)))
% 0.20/0.66 = { by axiom 1 (complement) }
% 0.20/0.66 multiplication(multiplication(antidomain(domain(x2)), forward_diamond(x0, domain(x1))), multiplication(x0, domain(x1)))
% 0.20/0.66 = { by lemma 20 R->L }
% 0.20/0.66 multiplication(multiplication(antidomain(domain(x2)), addition(domain(x2), forward_diamond(x0, domain(x1)))), multiplication(x0, domain(x1)))
% 0.20/0.66 = { by axiom 3 (additive_commutativity) R->L }
% 0.20/0.66 multiplication(multiplication(antidomain(domain(x2)), addition(forward_diamond(x0, domain(x1)), domain(x2))), multiplication(x0, domain(x1)))
% 0.20/0.66 = { by axiom 1 (complement) R->L }
% 0.20/0.66 multiplication(multiplication(c(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))), multiplication(x0, domain(x1)))
% 0.20/0.66 = { by lemma 21 R->L }
% 0.20/0.66 multiplication(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))), multiplication(x0, domain(x1)))
% 0.20/0.66 = { by axiom 15 (goals) }
% 0.20/0.66 multiplication(multiplication(antidomain(x2), domain(x2)), multiplication(x0, domain(x1)))
% 0.20/0.66 = { by lemma 21 }
% 0.20/0.66 multiplication(multiplication(c(x2), domain(x2)), multiplication(x0, domain(x1)))
% 0.20/0.66 = { by axiom 1 (complement) }
% 0.20/0.66 multiplication(multiplication(antidomain(domain(x2)), domain(x2)), multiplication(x0, domain(x1)))
% 0.20/0.66 = { by axiom 8 (domain1) }
% 0.20/0.66 multiplication(zero, multiplication(x0, domain(x1)))
% 0.20/0.66 = { by axiom 7 (left_annihilation) }
% 0.20/0.66 zero
% 0.20/0.66 % SZS output end Proof
% 0.20/0.66
% 0.20/0.66 RESULT: Theorem (the conjecture is true).
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