TSTP Solution File: KLE167+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE167+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 : n017.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 05:36:09 EDT 2023
% Result : Theorem 0.19s 0.46s
% 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
% 0.11/0.12 % Problem : KLE167+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
% 0.12/0.33 % Computer : n017.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Tue Aug 29 11:45:28 EDT 2023
% 0.12/0.33 % CPUTime :
% 0.19/0.46 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 0.19/0.46
% 0.19/0.46 % SZS status Theorem
% 0.19/0.46
% 0.19/0.47 % SZS output start Proof
% 0.19/0.47 Take the following subset of the input axioms:
% 0.19/0.47 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 0.19/0.47 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 0.19/0.47 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 0.19/0.47 fof(distributivity1, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 0.19/0.47 fof(goals, conjecture, ![X0, X1]: ((multiplication(X0, X1)=zero => leq(multiplication(X0, star(X1)), X0)) & leq(X0, multiplication(X0, star(X1))))).
% 0.19/0.47 fof(idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 0.19/0.47 fof(left_annihilation, axiom, ![A3]: multiplication(zero, A3)=zero).
% 0.19/0.47 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 0.19/0.47 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 0.19/0.47 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 0.19/0.47 fof(star_unfold1, axiom, ![A3]: addition(one, multiplication(A3, star(A3)))=star(A3)).
% 0.19/0.47
% 0.19/0.47 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.47 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.47 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.47 fresh(y, y, x1...xn) = u
% 0.19/0.47 C => fresh(s, t, x1...xn) = v
% 0.19/0.47 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.47 variables of u and v.
% 0.19/0.47 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.47 input problem has no model of domain size 1).
% 0.19/0.47
% 0.19/0.47 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.47
% 0.19/0.47 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.19/0.47 Axiom 2 (left_annihilation): multiplication(zero, X) = zero.
% 0.19/0.47 Axiom 3 (idempotence): addition(X, X) = X.
% 0.19/0.47 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.19/0.47 Axiom 5 (additive_identity): addition(X, zero) = X.
% 0.19/0.47 Axiom 6 (goals_1): fresh5(X, X) = zero.
% 0.19/0.47 Axiom 7 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 0.19/0.47 Axiom 8 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.19/0.47 Axiom 9 (star_unfold1): addition(one, multiplication(X, star(X))) = star(X).
% 0.19/0.47 Axiom 10 (distributivity1): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.19/0.47 Axiom 11 (order): fresh4(X, X, Y, Z) = true.
% 0.19/0.47 Axiom 12 (goals_1): fresh5(leq(x0, multiplication(x0, star(x1))), true) = multiplication(x0_2, x1_2).
% 0.19/0.47 Axiom 13 (order): fresh4(addition(X, Y), Y, X, Y) = leq(X, Y).
% 0.19/0.47
% 0.19/0.47 Lemma 14: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
% 0.19/0.47 Proof:
% 0.19/0.47 addition(X, multiplication(X, Y))
% 0.19/0.47 = { by axiom 1 (multiplicative_right_identity) R->L }
% 0.19/0.47 addition(multiplication(X, one), multiplication(X, Y))
% 0.19/0.47 = { by axiom 10 (distributivity1) R->L }
% 0.19/0.47 multiplication(X, addition(one, Y))
% 0.19/0.47 = { by axiom 4 (additive_commutativity) }
% 0.19/0.47 multiplication(X, addition(Y, one))
% 0.19/0.47
% 0.19/0.47 Lemma 15: leq(X, multiplication(X, star(Y))) = true.
% 0.19/0.47 Proof:
% 0.19/0.47 leq(X, multiplication(X, star(Y)))
% 0.19/0.47 = { by axiom 13 (order) R->L }
% 0.19/0.47 fresh4(addition(X, multiplication(X, star(Y))), multiplication(X, star(Y)), X, multiplication(X, star(Y)))
% 0.19/0.47 = { by lemma 14 }
% 0.19/0.47 fresh4(multiplication(X, addition(star(Y), one)), multiplication(X, star(Y)), X, multiplication(X, star(Y)))
% 0.19/0.47 = { by axiom 4 (additive_commutativity) }
% 0.19/0.47 fresh4(multiplication(X, addition(one, star(Y))), multiplication(X, star(Y)), X, multiplication(X, star(Y)))
% 0.19/0.47 = { by axiom 9 (star_unfold1) R->L }
% 0.19/0.47 fresh4(multiplication(X, addition(one, addition(one, multiplication(Y, star(Y))))), multiplication(X, star(Y)), X, multiplication(X, star(Y)))
% 0.19/0.47 = { by axiom 8 (additive_associativity) }
% 0.19/0.47 fresh4(multiplication(X, addition(addition(one, one), multiplication(Y, star(Y)))), multiplication(X, star(Y)), X, multiplication(X, star(Y)))
% 0.19/0.47 = { by axiom 3 (idempotence) }
% 0.19/0.47 fresh4(multiplication(X, addition(one, multiplication(Y, star(Y)))), multiplication(X, star(Y)), X, multiplication(X, star(Y)))
% 0.19/0.47 = { by axiom 9 (star_unfold1) }
% 0.19/0.47 fresh4(multiplication(X, star(Y)), multiplication(X, star(Y)), X, multiplication(X, star(Y)))
% 0.19/0.47 = { by axiom 11 (order) }
% 0.19/0.47 true
% 0.19/0.47
% 0.19/0.47 Goal 1 (goals): tuple(leq(multiplication(x0_2, star(x1_2)), x0_2), leq(x0, multiplication(x0, star(x1)))) = tuple(true, true).
% 0.19/0.47 Proof:
% 0.19/0.47 tuple(leq(multiplication(x0_2, star(x1_2)), x0_2), leq(x0, multiplication(x0, star(x1))))
% 0.19/0.47 = { by axiom 9 (star_unfold1) R->L }
% 0.19/0.47 tuple(leq(multiplication(x0_2, addition(one, multiplication(x1_2, star(x1_2)))), x0_2), leq(x0, multiplication(x0, star(x1))))
% 0.19/0.47 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.47 tuple(leq(multiplication(x0_2, addition(multiplication(x1_2, star(x1_2)), one)), x0_2), leq(x0, multiplication(x0, star(x1))))
% 0.19/0.47 = { by lemma 14 R->L }
% 0.19/0.47 tuple(leq(addition(x0_2, multiplication(x0_2, multiplication(x1_2, star(x1_2)))), x0_2), leq(x0, multiplication(x0, star(x1))))
% 0.19/0.47 = { by axiom 7 (multiplicative_associativity) }
% 0.19/0.47 tuple(leq(addition(x0_2, multiplication(multiplication(x0_2, x1_2), star(x1_2))), x0_2), leq(x0, multiplication(x0, star(x1))))
% 0.19/0.47 = { by axiom 12 (goals_1) R->L }
% 0.19/0.47 tuple(leq(addition(x0_2, multiplication(fresh5(leq(x0, multiplication(x0, star(x1))), true), star(x1_2))), x0_2), leq(x0, multiplication(x0, star(x1))))
% 0.19/0.47 = { by lemma 15 }
% 0.19/0.47 tuple(leq(addition(x0_2, multiplication(fresh5(true, true), star(x1_2))), x0_2), leq(x0, multiplication(x0, star(x1))))
% 0.19/0.47 = { by axiom 6 (goals_1) }
% 0.19/0.47 tuple(leq(addition(x0_2, multiplication(zero, star(x1_2))), x0_2), leq(x0, multiplication(x0, star(x1))))
% 0.19/0.47 = { by axiom 2 (left_annihilation) }
% 0.19/0.47 tuple(leq(addition(x0_2, zero), x0_2), leq(x0, multiplication(x0, star(x1))))
% 0.19/0.47 = { by axiom 5 (additive_identity) }
% 0.19/0.47 tuple(leq(x0_2, x0_2), leq(x0, multiplication(x0, star(x1))))
% 0.19/0.47 = { by axiom 13 (order) R->L }
% 0.19/0.47 tuple(fresh4(addition(x0_2, x0_2), x0_2, x0_2, x0_2), leq(x0, multiplication(x0, star(x1))))
% 0.19/0.47 = { by axiom 3 (idempotence) }
% 0.19/0.47 tuple(fresh4(x0_2, x0_2, x0_2, x0_2), leq(x0, multiplication(x0, star(x1))))
% 0.19/0.47 = { by axiom 11 (order) }
% 0.19/0.47 tuple(true, leq(x0, multiplication(x0, star(x1))))
% 0.19/0.47 = { by lemma 15 }
% 0.19/0.47 tuple(true, true)
% 0.19/0.47 % SZS output end Proof
% 0.19/0.47
% 0.19/0.47 RESULT: Theorem (the conjecture is true).
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