TSTP Solution File: KLE081+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE081+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 : n005.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 0.19s 0.48s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : KLE081+1 : TPTP v8.1.2. Released v4.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n005.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Tue Aug 29 12:33:52 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.48 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.19/0.48
% 0.19/0.48 % SZS status Theorem
% 0.19/0.48
% 0.19/0.49 % SZS output start Proof
% 0.19/0.49 Take the following subset of the input axioms:
% 0.19/0.49 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 0.19/0.49 fof(additive_commutativity, axiom, ![A2, B2]: addition(A2, B2)=addition(B2, A2)).
% 0.19/0.49 fof(additive_idempotence, axiom, ![A2]: addition(A2, A2)=A2).
% 0.19/0.49 fof(additive_identity, axiom, ![A2]: addition(A2, zero)=A2).
% 0.19/0.49 fof(domain1, axiom, ![X0]: addition(X0, multiplication(domain(X0), X0))=multiplication(domain(X0), X0)).
% 0.19/0.49 fof(domain3, axiom, ![X0_2]: addition(domain(X0_2), one)=one).
% 0.19/0.49 fof(domain5, axiom, ![X1, X0_2]: domain(addition(X0_2, X1))=addition(domain(X0_2), domain(X1))).
% 0.19/0.49 fof(goals, conjecture, ![X0_2]: (![X1_2]: (addition(domain(X1_2), antidomain(X1_2))=one & multiplication(domain(X1_2), antidomain(X1_2))=zero) => multiplication(antidomain(X0_2), X0_2)=zero)).
% 0.19/0.49 fof(left_annihilation, axiom, ![A2]: multiplication(zero, A2)=zero).
% 0.19/0.49 fof(left_distributivity, axiom, ![A2, B2, C2]: multiplication(addition(A2, B2), C2)=addition(multiplication(A2, C2), multiplication(B2, C2))).
% 0.19/0.49 fof(multiplicative_associativity, axiom, ![A2, B2, C2]: multiplication(A2, multiplication(B2, C2))=multiplication(multiplication(A2, B2), C2)).
% 0.19/0.49 fof(multiplicative_left_identity, axiom, ![A2]: multiplication(one, A2)=A2).
% 0.19/0.49
% 0.19/0.49 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.49 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.49 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.49 fresh(y, y, x1...xn) = u
% 0.19/0.49 C => fresh(s, t, x1...xn) = v
% 0.19/0.49 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.49 variables of u and v.
% 0.19/0.49 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.49 input problem has no model of domain size 1).
% 0.19/0.49
% 0.19/0.49 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.49
% 0.19/0.49 Axiom 1 (left_annihilation): multiplication(zero, X) = zero.
% 0.19/0.49 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.19/0.49 Axiom 3 (additive_idempotence): addition(X, X) = X.
% 0.19/0.49 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.19/0.49 Axiom 5 (additive_identity): addition(X, zero) = X.
% 0.19/0.49 Axiom 6 (domain3): addition(domain(X), one) = one.
% 0.19/0.49 Axiom 7 (goals_1): multiplication(domain(X), antidomain(X)) = zero.
% 0.19/0.49 Axiom 8 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 0.19/0.49 Axiom 9 (domain5): domain(addition(X, Y)) = addition(domain(X), domain(Y)).
% 0.19/0.49 Axiom 10 (goals): addition(domain(X), antidomain(X)) = one.
% 0.19/0.49 Axiom 11 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.19/0.49 Axiom 12 (domain1): addition(X, multiplication(domain(X), X)) = multiplication(domain(X), X).
% 0.19/0.49 Axiom 13 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.19/0.49
% 0.19/0.49 Lemma 14: addition(antidomain(X), domain(X)) = one.
% 0.19/0.49 Proof:
% 0.19/0.49 addition(antidomain(X), domain(X))
% 0.19/0.49 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.49 addition(domain(X), antidomain(X))
% 0.19/0.49 = { by axiom 10 (goals) }
% 0.19/0.49 one
% 0.19/0.49
% 0.19/0.49 Goal 1 (goals_2): multiplication(antidomain(x0), x0) = zero.
% 0.19/0.49 Proof:
% 0.19/0.49 multiplication(antidomain(x0), x0)
% 0.19/0.49 = { by axiom 2 (multiplicative_left_identity) R->L }
% 0.19/0.49 multiplication(one, multiplication(antidomain(x0), x0))
% 0.19/0.49 = { by axiom 6 (domain3) R->L }
% 0.19/0.49 multiplication(addition(domain(multiplication(antidomain(x0), x0)), one), multiplication(antidomain(x0), x0))
% 0.19/0.49 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.49 multiplication(addition(one, domain(multiplication(antidomain(x0), x0))), multiplication(antidomain(x0), x0))
% 0.19/0.49 = { by axiom 13 (left_distributivity) }
% 0.19/0.49 addition(multiplication(one, multiplication(antidomain(x0), x0)), multiplication(domain(multiplication(antidomain(x0), x0)), multiplication(antidomain(x0), x0)))
% 0.19/0.49 = { by axiom 2 (multiplicative_left_identity) }
% 0.19/0.49 addition(multiplication(antidomain(x0), x0), multiplication(domain(multiplication(antidomain(x0), x0)), multiplication(antidomain(x0), x0)))
% 0.19/0.49 = { by axiom 12 (domain1) }
% 0.19/0.49 multiplication(domain(multiplication(antidomain(x0), x0)), multiplication(antidomain(x0), x0))
% 0.19/0.49 = { by axiom 8 (multiplicative_associativity) }
% 0.19/0.49 multiplication(multiplication(domain(multiplication(antidomain(x0), x0)), antidomain(x0)), x0)
% 0.19/0.49 = { by axiom 5 (additive_identity) R->L }
% 0.19/0.49 multiplication(addition(multiplication(domain(multiplication(antidomain(x0), x0)), antidomain(x0)), zero), x0)
% 0.19/0.49 = { by axiom 7 (goals_1) R->L }
% 0.19/0.49 multiplication(addition(multiplication(domain(multiplication(antidomain(x0), x0)), antidomain(x0)), multiplication(domain(x0), antidomain(x0))), x0)
% 0.19/0.49 = { by axiom 13 (left_distributivity) R->L }
% 0.19/0.49 multiplication(multiplication(addition(domain(multiplication(antidomain(x0), x0)), domain(x0)), antidomain(x0)), x0)
% 0.19/0.49 = { by axiom 9 (domain5) R->L }
% 0.19/0.49 multiplication(multiplication(domain(addition(multiplication(antidomain(x0), x0), x0)), antidomain(x0)), x0)
% 0.19/0.49 = { by axiom 2 (multiplicative_left_identity) R->L }
% 0.19/0.49 multiplication(multiplication(domain(addition(multiplication(antidomain(x0), x0), multiplication(one, x0))), antidomain(x0)), x0)
% 0.19/0.49 = { by axiom 13 (left_distributivity) R->L }
% 0.19/0.49 multiplication(multiplication(domain(multiplication(addition(antidomain(x0), one), x0)), antidomain(x0)), x0)
% 0.19/0.49 = { by lemma 14 R->L }
% 0.19/0.49 multiplication(multiplication(domain(multiplication(addition(antidomain(x0), addition(antidomain(x0), domain(x0))), x0)), antidomain(x0)), x0)
% 0.19/0.49 = { by axiom 11 (additive_associativity) }
% 0.19/0.49 multiplication(multiplication(domain(multiplication(addition(addition(antidomain(x0), antidomain(x0)), domain(x0)), x0)), antidomain(x0)), x0)
% 0.19/0.49 = { by axiom 3 (additive_idempotence) }
% 0.19/0.49 multiplication(multiplication(domain(multiplication(addition(antidomain(x0), domain(x0)), x0)), antidomain(x0)), x0)
% 0.19/0.49 = { by lemma 14 }
% 0.19/0.49 multiplication(multiplication(domain(multiplication(one, x0)), antidomain(x0)), x0)
% 0.19/0.49 = { by axiom 2 (multiplicative_left_identity) }
% 0.19/0.49 multiplication(multiplication(domain(x0), antidomain(x0)), x0)
% 0.19/0.49 = { by axiom 7 (goals_1) }
% 0.19/0.49 multiplication(zero, x0)
% 0.19/0.49 = { by axiom 1 (left_annihilation) }
% 0.19/0.49 zero
% 0.19/0.49 % SZS output end Proof
% 0.19/0.49
% 0.19/0.49 RESULT: Theorem (the conjecture is true).
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