TSTP Solution File: KLE148+2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE148+2 : 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 05:36:04 EDT 2023
% Result : Theorem 0.21s 0.50s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.13 % Problem : KLE148+2 : TPTP v8.1.2. Released v4.0.0.
% 0.08/0.14 % 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 : n021.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 12:05:59 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.21/0.50 Command-line arguments: --ground-connectedness --complete-subsets
% 0.21/0.50
% 0.21/0.50 % SZS status Theorem
% 0.21/0.50
% 0.21/0.51 % SZS output start Proof
% 0.21/0.51 Take the following subset of the input axioms:
% 0.21/0.51 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 0.21/0.51 fof(additive_commutativity, axiom, ![B2, A3]: addition(A3, B2)=addition(B2, A3)).
% 0.21/0.51 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 0.21/0.51 fof(distributivity1, axiom, ![B2, A3, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 0.21/0.51 fof(goals, conjecture, ![X0, X1]: ((multiplication(X0, X1)=zero => leq(multiplication(X0, strong_iteration(X1)), X0)) & leq(X0, multiplication(X0, strong_iteration(X1))))).
% 0.21/0.51 fof(idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 0.21/0.51 fof(infty_unfold1, axiom, ![A3]: strong_iteration(A3)=addition(multiplication(A3, strong_iteration(A3)), one)).
% 0.21/0.51 fof(left_annihilation, axiom, ![A3]: multiplication(zero, A3)=zero).
% 0.21/0.51 fof(multiplicative_associativity, axiom, ![B2, A3, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 0.21/0.51 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 0.21/0.51 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 0.21/0.51
% 0.21/0.51 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.51 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.51 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.51 fresh(y, y, x1...xn) = u
% 0.21/0.51 C => fresh(s, t, x1...xn) = v
% 0.21/0.51 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.51 variables of u and v.
% 0.21/0.51 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.51 input problem has no model of domain size 1).
% 0.21/0.51
% 0.21/0.51 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.51
% 0.21/0.51 Axiom 1 (goals_1): fresh5(X, X) = zero.
% 0.21/0.51 Axiom 2 (idempotence): addition(X, X) = X.
% 0.21/0.51 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.21/0.51 Axiom 4 (additive_identity): addition(X, zero) = X.
% 0.21/0.51 Axiom 5 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.21/0.51 Axiom 6 (left_annihilation): multiplication(zero, X) = zero.
% 0.21/0.51 Axiom 7 (order): fresh4(X, X, Y, Z) = true.
% 0.21/0.51 Axiom 8 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.21/0.51 Axiom 9 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 0.21/0.51 Axiom 10 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
% 0.21/0.51 Axiom 11 (order): fresh4(addition(X, Y), Y, X, Y) = leq(X, Y).
% 0.21/0.51 Axiom 12 (distributivity1): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.21/0.51 Axiom 13 (goals_1): fresh5(leq(x0, multiplication(x0, strong_iteration(x1))), true) = multiplication(x0_2, x1_2).
% 0.21/0.51
% 0.21/0.51 Lemma 14: addition(X, addition(X, Y)) = addition(X, Y).
% 0.21/0.51 Proof:
% 0.21/0.51 addition(X, addition(X, Y))
% 0.21/0.51 = { by axiom 8 (additive_associativity) }
% 0.21/0.51 addition(addition(X, X), Y)
% 0.21/0.51 = { by axiom 2 (idempotence) }
% 0.21/0.51 addition(X, Y)
% 0.21/0.51
% 0.21/0.51 Lemma 15: addition(one, multiplication(X, strong_iteration(X))) = strong_iteration(X).
% 0.21/0.51 Proof:
% 0.21/0.51 addition(one, multiplication(X, strong_iteration(X)))
% 0.21/0.51 = { by axiom 3 (additive_commutativity) R->L }
% 0.21/0.51 addition(multiplication(X, strong_iteration(X)), one)
% 0.21/0.51 = { by axiom 10 (infty_unfold1) R->L }
% 0.21/0.51 strong_iteration(X)
% 0.21/0.51
% 0.21/0.51 Lemma 16: leq(X, multiplication(X, strong_iteration(Y))) = true.
% 0.21/0.51 Proof:
% 0.21/0.51 leq(X, multiplication(X, strong_iteration(Y)))
% 0.21/0.51 = { by lemma 15 R->L }
% 0.21/0.51 leq(X, multiplication(X, addition(one, multiplication(Y, strong_iteration(Y)))))
% 0.21/0.51 = { by lemma 14 R->L }
% 0.21/0.51 leq(X, multiplication(X, addition(one, addition(one, multiplication(Y, strong_iteration(Y))))))
% 0.21/0.51 = { by lemma 15 }
% 0.21/0.51 leq(X, multiplication(X, addition(one, strong_iteration(Y))))
% 0.21/0.51 = { by axiom 12 (distributivity1) }
% 0.21/0.51 leq(X, addition(multiplication(X, one), multiplication(X, strong_iteration(Y))))
% 0.21/0.51 = { by axiom 5 (multiplicative_right_identity) }
% 0.21/0.51 leq(X, addition(X, multiplication(X, strong_iteration(Y))))
% 0.21/0.51 = { by axiom 11 (order) R->L }
% 0.21/0.51 fresh4(addition(X, addition(X, multiplication(X, strong_iteration(Y)))), addition(X, multiplication(X, strong_iteration(Y))), X, addition(X, multiplication(X, strong_iteration(Y))))
% 0.21/0.51 = { by lemma 14 }
% 0.21/0.51 fresh4(addition(X, multiplication(X, strong_iteration(Y))), addition(X, multiplication(X, strong_iteration(Y))), X, addition(X, multiplication(X, strong_iteration(Y))))
% 0.21/0.51 = { by axiom 7 (order) }
% 0.21/0.51 true
% 0.21/0.51
% 0.21/0.51 Goal 1 (goals): tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), x0_2), leq(x0, multiplication(x0, strong_iteration(x1)))) = tuple(true, true).
% 0.21/0.51 Proof:
% 0.21/0.51 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), x0_2), leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.51 = { by axiom 5 (multiplicative_right_identity) R->L }
% 0.21/0.51 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, one)), leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.51 = { by axiom 4 (additive_identity) R->L }
% 0.21/0.51 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(multiplication(x0_2, one), zero)), leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.51 = { by axiom 3 (additive_commutativity) }
% 0.21/0.51 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(zero, multiplication(x0_2, one))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.51 = { by axiom 6 (left_annihilation) R->L }
% 0.21/0.51 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(multiplication(zero, strong_iteration(x1_2)), multiplication(x0_2, one))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.51 = { by axiom 1 (goals_1) R->L }
% 0.21/0.51 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(multiplication(fresh5(leq(x0, multiplication(x0, strong_iteration(x1))), leq(x0, multiplication(x0, strong_iteration(x1)))), strong_iteration(x1_2)), multiplication(x0_2, one))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.51 = { by lemma 16 }
% 0.21/0.51 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(multiplication(fresh5(leq(x0, multiplication(x0, strong_iteration(x1))), true), strong_iteration(x1_2)), multiplication(x0_2, one))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.51 = { by axiom 13 (goals_1) }
% 0.21/0.51 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(multiplication(multiplication(x0_2, x1_2), strong_iteration(x1_2)), multiplication(x0_2, one))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.52 = { by axiom 9 (multiplicative_associativity) R->L }
% 0.21/0.52 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(multiplication(x0_2, multiplication(x1_2, strong_iteration(x1_2))), multiplication(x0_2, one))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.52 = { by axiom 12 (distributivity1) R->L }
% 0.21/0.52 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, addition(multiplication(x1_2, strong_iteration(x1_2)), one))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.52 = { by axiom 3 (additive_commutativity) }
% 0.21/0.52 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, addition(one, multiplication(x1_2, strong_iteration(x1_2))))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.52 = { by lemma 15 }
% 0.21/0.52 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, strong_iteration(x1_2))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.52 = { by axiom 11 (order) R->L }
% 0.21/0.52 tuple(fresh4(addition(multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, strong_iteration(x1_2))), multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, strong_iteration(x1_2))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.52 = { by axiom 2 (idempotence) }
% 0.21/0.52 tuple(fresh4(multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, strong_iteration(x1_2))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.52 = { by axiom 7 (order) }
% 0.21/0.52 tuple(true, leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.52 = { by lemma 16 R->L }
% 0.21/0.52 tuple(leq(X, multiplication(X, strong_iteration(Y))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 0.21/0.52 = { by lemma 16 }
% 0.21/0.52 tuple(leq(X, multiplication(X, strong_iteration(Y))), true)
% 0.21/0.52 = { by lemma 16 }
% 0.21/0.52 tuple(true, true)
% 0.21/0.52 % SZS output end Proof
% 0.21/0.52
% 0.21/0.52 RESULT: Theorem (the conjecture is true).
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