TSTP Solution File: KLE144+2 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE144+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 : n009.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 05:36:02 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
% 0.08/0.13 % Problem : KLE144+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 : n009.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 11:03:50 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.19/0.48 Command-line arguments: --no-flatten-goal
% 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, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 0.19/0.49 fof(distributivity2, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 0.19/0.49 fof(goals, conjecture, ![X0]: (leq(strong_iteration(star(X0)), strong_iteration(one)) & leq(strong_iteration(one), strong_iteration(star(X0))))).
% 0.19/0.49 fof(idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 0.19/0.49 fof(infty_coinduction, axiom, ![A2, B2, C2]: (leq(C2, addition(multiplication(A2, C2), B2)) => leq(C2, multiplication(strong_iteration(A2), B2)))).
% 0.19/0.49 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 0.19/0.49 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 0.19/0.49 fof(order, axiom, ![B2, A2_2]: (leq(A2_2, B2) <=> addition(A2_2, B2)=B2)).
% 0.19/0.49 fof(star_unfold1, axiom, ![A3]: addition(one, multiplication(A3, star(A3)))=star(A3)).
% 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 (idempotence): addition(X, X) = X.
% 0.19/0.49 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.19/0.49 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.19/0.49 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.19/0.49 Axiom 5 (order): fresh5(X, X, Y, Z) = true.
% 0.19/0.49 Axiom 6 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.19/0.49 Axiom 7 (infty_coinduction): fresh4(X, X, Y, Z, W) = true.
% 0.19/0.49 Axiom 8 (star_unfold1): addition(one, multiplication(X, star(X))) = star(X).
% 0.19/0.49 Axiom 9 (order): fresh5(addition(X, Y), Y, X, Y) = leq(X, Y).
% 0.19/0.49 Axiom 10 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.19/0.49 Axiom 11 (infty_coinduction): fresh4(leq(X, addition(multiplication(Y, X), Z)), true, Y, Z, X) = leq(X, multiplication(strong_iteration(Y), Z)).
% 0.19/0.49
% 0.19/0.49 Lemma 12: leq(X, multiplication(strong_iteration(addition(Y, one)), Z)) = true.
% 0.19/0.49 Proof:
% 0.19/0.49 leq(X, multiplication(strong_iteration(addition(Y, one)), Z))
% 0.19/0.49 = { by axiom 11 (infty_coinduction) R->L }
% 0.19/0.49 fresh4(leq(X, addition(multiplication(addition(Y, one), X), Z)), true, addition(Y, one), Z, X)
% 0.19/0.49 = { by axiom 2 (additive_commutativity) R->L }
% 0.19/0.49 fresh4(leq(X, addition(multiplication(addition(one, Y), X), Z)), true, addition(Y, one), Z, X)
% 0.19/0.49 = { by axiom 10 (distributivity2) }
% 0.19/0.49 fresh4(leq(X, addition(addition(multiplication(one, X), multiplication(Y, X)), Z)), true, addition(Y, one), Z, X)
% 0.19/0.49 = { by axiom 4 (multiplicative_left_identity) }
% 0.19/0.49 fresh4(leq(X, addition(addition(X, multiplication(Y, X)), Z)), true, addition(Y, one), Z, X)
% 0.19/0.49 = { by axiom 6 (additive_associativity) R->L }
% 0.19/0.49 fresh4(leq(X, addition(X, addition(multiplication(Y, X), Z))), true, addition(Y, one), Z, X)
% 0.19/0.49 = { by axiom 9 (order) R->L }
% 0.19/0.49 fresh4(fresh5(addition(X, addition(X, addition(multiplication(Y, X), Z))), addition(X, addition(multiplication(Y, X), Z)), X, addition(X, addition(multiplication(Y, X), Z))), true, addition(Y, one), Z, X)
% 0.19/0.49 = { by axiom 6 (additive_associativity) }
% 0.19/0.49 fresh4(fresh5(addition(addition(X, X), addition(multiplication(Y, X), Z)), addition(X, addition(multiplication(Y, X), Z)), X, addition(X, addition(multiplication(Y, X), Z))), true, addition(Y, one), Z, X)
% 0.19/0.49 = { by axiom 1 (idempotence) }
% 0.19/0.50 fresh4(fresh5(addition(X, addition(multiplication(Y, X), Z)), addition(X, addition(multiplication(Y, X), Z)), X, addition(X, addition(multiplication(Y, X), Z))), true, addition(Y, one), Z, X)
% 0.19/0.50 = { by axiom 5 (order) }
% 0.19/0.50 fresh4(true, true, addition(Y, one), Z, X)
% 0.19/0.50 = { by axiom 7 (infty_coinduction) }
% 0.19/0.50 true
% 0.19/0.50
% 0.19/0.50 Goal 1 (goals): tuple(leq(strong_iteration(one), strong_iteration(star(x0))), leq(strong_iteration(star(x0_2)), strong_iteration(one))) = tuple(true, true).
% 0.19/0.50 Proof:
% 0.19/0.50 tuple(leq(strong_iteration(one), strong_iteration(star(x0))), leq(strong_iteration(star(x0_2)), strong_iteration(one)))
% 0.19/0.50 = { by axiom 3 (multiplicative_right_identity) R->L }
% 0.19/0.50 tuple(leq(strong_iteration(one), strong_iteration(star(x0))), leq(strong_iteration(star(x0_2)), multiplication(strong_iteration(one), one)))
% 0.19/0.50 = { by axiom 1 (idempotence) R->L }
% 0.19/0.50 tuple(leq(strong_iteration(one), strong_iteration(star(x0))), leq(strong_iteration(star(x0_2)), multiplication(strong_iteration(addition(one, one)), one)))
% 0.19/0.50 = { by lemma 12 }
% 0.19/0.50 tuple(leq(strong_iteration(one), strong_iteration(star(x0))), true)
% 0.19/0.50 = { by axiom 8 (star_unfold1) R->L }
% 0.19/0.50 tuple(leq(strong_iteration(one), strong_iteration(addition(one, multiplication(x0, star(x0))))), true)
% 0.19/0.50 = { by axiom 2 (additive_commutativity) R->L }
% 0.19/0.50 tuple(leq(strong_iteration(one), strong_iteration(addition(multiplication(x0, star(x0)), one))), true)
% 0.19/0.50 = { by axiom 3 (multiplicative_right_identity) R->L }
% 0.19/0.50 tuple(leq(strong_iteration(one), multiplication(strong_iteration(addition(multiplication(x0, star(x0)), one)), one)), true)
% 0.19/0.50 = { by lemma 12 }
% 0.19/0.50 tuple(true, true)
% 0.19/0.50 % SZS output end Proof
% 0.19/0.50
% 0.19/0.50 RESULT: Theorem (the conjecture is true).
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