TSTP Solution File: KLE078+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE078+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 : n021.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 05:35:48 EDT 2023
% Result : Theorem 0.18s 0.53s
% Output : Proof 0.18s
% 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 : KLE078+1 : TPTP v8.1.2. Released v4.0.0.
% 0.11/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.11/0.33 % Computer : n021.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Tue Aug 29 11:32:44 EDT 2023
% 0.11/0.33 % CPUTime :
% 0.18/0.53 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.18/0.53
% 0.18/0.53 % SZS status Theorem
% 0.18/0.53
% 0.18/0.54 % SZS output start Proof
% 0.18/0.54 Take the following subset of the input axioms:
% 0.18/0.55 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 0.18/0.55 fof(additive_commutativity, axiom, ![A2, B2]: addition(A2, B2)=addition(B2, A2)).
% 0.18/0.55 fof(additive_idempotence, axiom, ![A2]: addition(A2, A2)=A2).
% 0.18/0.55 fof(additive_identity, axiom, ![A2]: addition(A2, zero)=A2).
% 0.18/0.55 fof(domain1, axiom, ![X0]: addition(X0, multiplication(domain(X0), X0))=multiplication(domain(X0), X0)).
% 0.18/0.55 fof(domain2, axiom, ![X1, X0_2]: domain(multiplication(X0_2, X1))=domain(multiplication(X0_2, domain(X1)))).
% 0.18/0.55 fof(domain3, axiom, ![X0_2]: addition(domain(X0_2), one)=one).
% 0.18/0.55 fof(domain4, axiom, domain(zero)=zero).
% 0.18/0.55 fof(goals, conjecture, ![X0_2]: (![X1_2]: (addition(domain(X1_2), antidomain(X1_2))=one & multiplication(domain(X1_2), antidomain(X1_2))=zero) => domain(antidomain(X0_2))=antidomain(X0_2))).
% 0.18/0.55 fof(left_annihilation, axiom, ![A2]: multiplication(zero, A2)=zero).
% 0.18/0.55 fof(left_distributivity, axiom, ![A2, B2, C2]: multiplication(addition(A2, B2), C2)=addition(multiplication(A2, C2), multiplication(B2, C2))).
% 0.18/0.55 fof(multiplicative_left_identity, axiom, ![A2]: multiplication(one, A2)=A2).
% 0.18/0.55 fof(multiplicative_right_identity, axiom, ![A2]: multiplication(A2, one)=A2).
% 0.18/0.55 fof(right_distributivity, axiom, ![A2, B2, C2]: multiplication(A2, addition(B2, C2))=addition(multiplication(A2, B2), multiplication(A2, C2))).
% 0.18/0.55
% 0.18/0.55 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.18/0.55 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.18/0.55 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.18/0.55 fresh(y, y, x1...xn) = u
% 0.18/0.55 C => fresh(s, t, x1...xn) = v
% 0.18/0.55 where fresh is a fresh function symbol and x1..xn are the free
% 0.18/0.55 variables of u and v.
% 0.18/0.55 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.18/0.55 input problem has no model of domain size 1).
% 0.18/0.55
% 0.18/0.55 The encoding turns the above axioms into the following unit equations and goals:
% 0.18/0.55
% 0.18/0.55 Axiom 1 (domain4): domain(zero) = zero.
% 0.18/0.55 Axiom 2 (additive_idempotence): addition(X, X) = X.
% 0.18/0.55 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.18/0.55 Axiom 4 (additive_identity): addition(X, zero) = X.
% 0.18/0.55 Axiom 5 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.18/0.55 Axiom 6 (left_annihilation): multiplication(zero, X) = zero.
% 0.18/0.55 Axiom 7 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.18/0.55 Axiom 8 (domain3): addition(domain(X), one) = one.
% 0.18/0.55 Axiom 9 (goals): addition(domain(X), antidomain(X)) = one.
% 0.18/0.55 Axiom 10 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.18/0.55 Axiom 11 (goals_1): multiplication(domain(X), antidomain(X)) = zero.
% 0.18/0.55 Axiom 12 (domain2): domain(multiplication(X, Y)) = domain(multiplication(X, domain(Y))).
% 0.18/0.55 Axiom 13 (domain1): addition(X, multiplication(domain(X), X)) = multiplication(domain(X), X).
% 0.18/0.55 Axiom 14 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.18/0.55 Axiom 15 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.18/0.55
% 0.18/0.55 Lemma 16: addition(one, domain(X)) = one.
% 0.18/0.55 Proof:
% 0.18/0.55 addition(one, domain(X))
% 0.18/0.55 = { by axiom 3 (additive_commutativity) R->L }
% 0.18/0.55 addition(domain(X), one)
% 0.18/0.55 = { by axiom 8 (domain3) }
% 0.18/0.55 one
% 0.18/0.55
% 0.18/0.55 Lemma 17: addition(antidomain(X), domain(X)) = one.
% 0.18/0.55 Proof:
% 0.18/0.55 addition(antidomain(X), domain(X))
% 0.18/0.55 = { by axiom 3 (additive_commutativity) R->L }
% 0.18/0.55 addition(domain(X), antidomain(X))
% 0.18/0.55 = { by axiom 9 (goals) }
% 0.18/0.55 one
% 0.18/0.55
% 0.18/0.55 Lemma 18: multiplication(X, addition(one, Y)) = addition(X, multiplication(X, Y)).
% 0.18/0.55 Proof:
% 0.18/0.55 multiplication(X, addition(one, Y))
% 0.18/0.55 = { by axiom 14 (right_distributivity) }
% 0.18/0.55 addition(multiplication(X, one), multiplication(X, Y))
% 0.18/0.55 = { by axiom 5 (multiplicative_right_identity) }
% 0.18/0.55 addition(X, multiplication(X, Y))
% 0.18/0.55
% 0.18/0.55 Lemma 19: domain(multiplication(domain(X), domain(antidomain(X)))) = zero.
% 0.18/0.55 Proof:
% 0.18/0.55 domain(multiplication(domain(X), domain(antidomain(X))))
% 0.18/0.55 = { by axiom 12 (domain2) R->L }
% 0.18/0.55 domain(multiplication(domain(X), antidomain(X)))
% 0.18/0.55 = { by axiom 11 (goals_1) }
% 0.18/0.55 domain(zero)
% 0.18/0.55 = { by axiom 1 (domain4) }
% 0.18/0.55 zero
% 0.18/0.55
% 0.18/0.55 Goal 1 (goals_2): domain(antidomain(x0)) = antidomain(x0).
% 0.18/0.55 Proof:
% 0.18/0.55 domain(antidomain(x0))
% 0.18/0.55 = { by axiom 7 (multiplicative_left_identity) R->L }
% 0.18/0.55 multiplication(one, domain(antidomain(x0)))
% 0.18/0.55 = { by lemma 17 R->L }
% 0.18/0.55 multiplication(addition(antidomain(x0), domain(x0)), domain(antidomain(x0)))
% 0.18/0.55 = { by axiom 15 (left_distributivity) }
% 0.18/0.55 addition(multiplication(antidomain(x0), domain(antidomain(x0))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.55 = { by axiom 5 (multiplicative_right_identity) R->L }
% 0.18/0.55 addition(multiplication(antidomain(x0), multiplication(domain(antidomain(x0)), one)), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.55 = { by lemma 17 R->L }
% 0.18/0.55 addition(multiplication(antidomain(x0), multiplication(domain(antidomain(x0)), addition(antidomain(x0), domain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.55 = { by axiom 2 (additive_idempotence) R->L }
% 0.18/0.55 addition(multiplication(antidomain(x0), multiplication(domain(antidomain(x0)), addition(addition(antidomain(x0), antidomain(x0)), domain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.55 = { by axiom 10 (additive_associativity) R->L }
% 0.18/0.55 addition(multiplication(antidomain(x0), multiplication(domain(antidomain(x0)), addition(antidomain(x0), addition(antidomain(x0), domain(x0))))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.55 = { by lemma 17 }
% 0.18/0.55 addition(multiplication(antidomain(x0), multiplication(domain(antidomain(x0)), addition(antidomain(x0), one))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.55 = { by axiom 3 (additive_commutativity) }
% 0.18/0.55 addition(multiplication(antidomain(x0), multiplication(domain(antidomain(x0)), addition(one, antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.55 = { by lemma 18 }
% 0.18/0.55 addition(multiplication(antidomain(x0), addition(domain(antidomain(x0)), multiplication(domain(antidomain(x0)), antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.55 = { by axiom 13 (domain1) R->L }
% 0.18/0.55 addition(multiplication(antidomain(x0), addition(domain(antidomain(x0)), addition(antidomain(x0), multiplication(domain(antidomain(x0)), antidomain(x0))))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.55 = { by axiom 7 (multiplicative_left_identity) R->L }
% 0.18/0.55 addition(multiplication(antidomain(x0), addition(domain(antidomain(x0)), addition(multiplication(one, antidomain(x0)), multiplication(domain(antidomain(x0)), antidomain(x0))))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by axiom 15 (left_distributivity) R->L }
% 0.18/0.56 addition(multiplication(antidomain(x0), addition(domain(antidomain(x0)), multiplication(addition(one, domain(antidomain(x0))), antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by lemma 16 }
% 0.18/0.56 addition(multiplication(antidomain(x0), addition(domain(antidomain(x0)), multiplication(one, antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by axiom 7 (multiplicative_left_identity) }
% 0.18/0.56 addition(multiplication(antidomain(x0), addition(domain(antidomain(x0)), antidomain(x0))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by axiom 3 (additive_commutativity) R->L }
% 0.18/0.56 addition(multiplication(antidomain(x0), addition(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by axiom 14 (right_distributivity) }
% 0.18/0.56 addition(addition(multiplication(antidomain(x0), antidomain(x0)), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by axiom 4 (additive_identity) R->L }
% 0.18/0.56 addition(addition(addition(multiplication(antidomain(x0), antidomain(x0)), zero), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by axiom 3 (additive_commutativity) }
% 0.18/0.56 addition(addition(addition(zero, multiplication(antidomain(x0), antidomain(x0))), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by axiom 11 (goals_1) R->L }
% 0.18/0.56 addition(addition(addition(multiplication(domain(x0), antidomain(x0)), multiplication(antidomain(x0), antidomain(x0))), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by axiom 15 (left_distributivity) R->L }
% 0.18/0.56 addition(addition(multiplication(addition(domain(x0), antidomain(x0)), antidomain(x0)), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by axiom 3 (additive_commutativity) }
% 0.18/0.56 addition(addition(multiplication(addition(antidomain(x0), domain(x0)), antidomain(x0)), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by lemma 17 }
% 0.18/0.56 addition(addition(multiplication(one, antidomain(x0)), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by axiom 7 (multiplicative_left_identity) }
% 0.18/0.56 addition(addition(antidomain(x0), multiplication(antidomain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by lemma 18 R->L }
% 0.18/0.56 addition(multiplication(antidomain(x0), addition(one, domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by lemma 16 }
% 0.18/0.56 addition(multiplication(antidomain(x0), one), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by axiom 5 (multiplicative_right_identity) }
% 0.18/0.56 addition(antidomain(x0), multiplication(domain(x0), domain(antidomain(x0))))
% 0.18/0.56 = { by axiom 4 (additive_identity) R->L }
% 0.18/0.56 addition(antidomain(x0), addition(multiplication(domain(x0), domain(antidomain(x0))), zero))
% 0.18/0.56 = { by axiom 6 (left_annihilation) R->L }
% 0.18/0.56 addition(antidomain(x0), addition(multiplication(domain(x0), domain(antidomain(x0))), multiplication(zero, multiplication(domain(x0), domain(antidomain(x0))))))
% 0.18/0.56 = { by lemma 19 R->L }
% 0.18/0.56 addition(antidomain(x0), addition(multiplication(domain(x0), domain(antidomain(x0))), multiplication(domain(multiplication(domain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0))))))
% 0.18/0.56 = { by axiom 13 (domain1) }
% 0.18/0.56 addition(antidomain(x0), multiplication(domain(multiplication(domain(x0), domain(antidomain(x0)))), multiplication(domain(x0), domain(antidomain(x0)))))
% 0.18/0.56 = { by lemma 19 }
% 0.18/0.56 addition(antidomain(x0), multiplication(zero, multiplication(domain(x0), domain(antidomain(x0)))))
% 0.18/0.56 = { by axiom 6 (left_annihilation) }
% 0.18/0.56 addition(antidomain(x0), zero)
% 0.18/0.56 = { by axiom 4 (additive_identity) }
% 0.18/0.56 antidomain(x0)
% 0.18/0.56 % SZS output end Proof
% 0.18/0.56
% 0.18/0.56 RESULT: Theorem (the conjecture is true).
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