TSTP Solution File: KLE142+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE142+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 : n032.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 05:36:01 EDT 2023
% Result : Theorem 0.13s 0.39s
% Output : Proof 0.13s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.02/0.10 % Problem : KLE142+1 : TPTP v8.1.2. Released v4.0.0.
% 0.02/0.11 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.09/0.30 % Computer : n032.cluster.edu
% 0.09/0.30 % Model : x86_64 x86_64
% 0.09/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.30 % Memory : 8042.1875MB
% 0.09/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.09/0.30 % CPULimit : 300
% 0.09/0.30 % WCLimit : 300
% 0.09/0.30 % DateTime : Tue Aug 29 12:28:31 EDT 2023
% 0.09/0.30 % CPUTime :
% 0.13/0.39 Command-line arguments: --no-flatten-goal
% 0.13/0.39
% 0.13/0.39 % SZS status Theorem
% 0.13/0.39
% 0.13/0.39 % SZS output start Proof
% 0.13/0.39 Take the following subset of the input axioms:
% 0.13/0.39 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 0.13/0.39 fof(additive_commutativity, axiom, ![B2, A3]: addition(A3, B2)=addition(B2, A3)).
% 0.13/0.39 fof(distributivity2, axiom, ![B2, A3, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 0.13/0.39 fof(goals, conjecture, ![X0]: strong_iteration(strong_iteration(X0))=strong_iteration(one)).
% 0.13/0.39 fof(idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 0.13/0.39 fof(infty_coinduction, axiom, ![A2, B2, C2]: (leq(C2, addition(multiplication(A2, C2), B2)) => leq(C2, multiplication(strong_iteration(A2), B2)))).
% 0.13/0.39 fof(infty_unfold1, axiom, ![A3]: strong_iteration(A3)=addition(multiplication(A3, strong_iteration(A3)), one)).
% 0.13/0.39 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 0.13/0.39 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 0.13/0.39 fof(order, axiom, ![B2, A2_2]: (leq(A2_2, B2) <=> addition(A2_2, B2)=B2)).
% 0.13/0.39
% 0.13/0.39 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.13/0.39 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.13/0.39 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.13/0.39 fresh(y, y, x1...xn) = u
% 0.13/0.39 C => fresh(s, t, x1...xn) = v
% 0.13/0.39 where fresh is a fresh function symbol and x1..xn are the free
% 0.13/0.39 variables of u and v.
% 0.13/0.39 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.13/0.39 input problem has no model of domain size 1).
% 0.13/0.39
% 0.13/0.39 The encoding turns the above axioms into the following unit equations and goals:
% 0.13/0.39
% 0.13/0.39 Axiom 1 (idempotence): addition(X, X) = X.
% 0.13/0.39 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.13/0.39 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.13/0.39 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.13/0.39 Axiom 5 (order_1): fresh(X, X, Y, Z) = Z.
% 0.13/0.39 Axiom 6 (order): fresh5(X, X, Y, Z) = true.
% 0.13/0.39 Axiom 7 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.13/0.39 Axiom 8 (infty_coinduction): fresh4(X, X, Y, Z, W) = true.
% 0.13/0.39 Axiom 9 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
% 0.13/0.39 Axiom 10 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 0.13/0.39 Axiom 11 (order): fresh5(addition(X, Y), Y, X, Y) = leq(X, Y).
% 0.13/0.39 Axiom 12 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.13/0.39 Axiom 13 (infty_coinduction): fresh4(leq(X, addition(multiplication(Y, X), Z)), true, Y, Z, X) = leq(X, multiplication(strong_iteration(Y), Z)).
% 0.13/0.39
% 0.13/0.39 Lemma 14: addition(one, multiplication(X, strong_iteration(X))) = strong_iteration(X).
% 0.13/0.39 Proof:
% 0.13/0.39 addition(one, multiplication(X, strong_iteration(X)))
% 0.13/0.39 = { by axiom 2 (additive_commutativity) R->L }
% 0.13/0.39 addition(multiplication(X, strong_iteration(X)), one)
% 0.13/0.39 = { by axiom 9 (infty_unfold1) R->L }
% 0.13/0.39 strong_iteration(X)
% 0.13/0.39
% 0.13/0.39 Lemma 15: addition(X, addition(X, Y)) = addition(X, Y).
% 0.13/0.39 Proof:
% 0.13/0.39 addition(X, addition(X, Y))
% 0.13/0.39 = { by axiom 7 (additive_associativity) }
% 0.13/0.39 addition(addition(X, X), Y)
% 0.13/0.39 = { by axiom 1 (idempotence) }
% 0.13/0.39 addition(X, Y)
% 0.13/0.39
% 0.13/0.39 Lemma 16: addition(one, strong_iteration(X)) = strong_iteration(X).
% 0.13/0.39 Proof:
% 0.13/0.39 addition(one, strong_iteration(X))
% 0.13/0.39 = { by lemma 14 R->L }
% 0.13/0.39 addition(one, addition(one, multiplication(X, strong_iteration(X))))
% 0.13/0.39 = { by lemma 15 }
% 0.13/0.39 addition(one, multiplication(X, strong_iteration(X)))
% 0.13/0.39 = { by lemma 14 }
% 0.13/0.39 strong_iteration(X)
% 0.13/0.39
% 0.13/0.39 Lemma 17: leq(X, multiplication(strong_iteration(addition(Y, one)), Z)) = true.
% 0.13/0.39 Proof:
% 0.13/0.39 leq(X, multiplication(strong_iteration(addition(Y, one)), Z))
% 0.13/0.39 = { by axiom 13 (infty_coinduction) R->L }
% 0.13/0.39 fresh4(leq(X, addition(multiplication(addition(Y, one), X), Z)), true, addition(Y, one), Z, X)
% 0.13/0.39 = { by axiom 2 (additive_commutativity) R->L }
% 0.13/0.39 fresh4(leq(X, addition(multiplication(addition(one, Y), X), Z)), true, addition(Y, one), Z, X)
% 0.13/0.39 = { by axiom 12 (distributivity2) }
% 0.13/0.39 fresh4(leq(X, addition(addition(multiplication(one, X), multiplication(Y, X)), Z)), true, addition(Y, one), Z, X)
% 0.13/0.39 = { by axiom 4 (multiplicative_left_identity) }
% 0.13/0.39 fresh4(leq(X, addition(addition(X, multiplication(Y, X)), Z)), true, addition(Y, one), Z, X)
% 0.13/0.39 = { by axiom 7 (additive_associativity) R->L }
% 0.13/0.39 fresh4(leq(X, addition(X, addition(multiplication(Y, X), Z))), true, addition(Y, one), Z, X)
% 0.13/0.39 = { by axiom 11 (order) R->L }
% 0.13/0.39 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)
% 0.13/0.39 = { by lemma 15 }
% 0.13/0.39 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)
% 0.13/0.39 = { by axiom 6 (order) }
% 0.13/0.39 fresh4(true, true, addition(Y, one), Z, X)
% 0.13/0.39 = { by axiom 8 (infty_coinduction) }
% 0.13/0.39 true
% 0.13/0.39
% 0.13/0.39 Goal 1 (goals): strong_iteration(strong_iteration(x0)) = strong_iteration(one).
% 0.13/0.39 Proof:
% 0.13/0.39 strong_iteration(strong_iteration(x0))
% 0.13/0.39 = { by lemma 16 R->L }
% 0.13/0.39 addition(one, strong_iteration(strong_iteration(x0)))
% 0.13/0.39 = { by axiom 5 (order_1) R->L }
% 0.13/0.39 addition(one, fresh(true, true, strong_iteration(one), strong_iteration(strong_iteration(x0))))
% 0.13/0.39 = { by lemma 17 R->L }
% 0.13/0.39 addition(one, fresh(leq(strong_iteration(one), multiplication(strong_iteration(addition(multiplication(x0, strong_iteration(x0)), one)), one)), true, strong_iteration(one), strong_iteration(strong_iteration(x0))))
% 0.13/0.39 = { by axiom 3 (multiplicative_right_identity) }
% 0.13/0.39 addition(one, fresh(leq(strong_iteration(one), strong_iteration(addition(multiplication(x0, strong_iteration(x0)), one))), true, strong_iteration(one), strong_iteration(strong_iteration(x0))))
% 0.13/0.39 = { by axiom 2 (additive_commutativity) }
% 0.13/0.39 addition(one, fresh(leq(strong_iteration(one), strong_iteration(addition(one, multiplication(x0, strong_iteration(x0))))), true, strong_iteration(one), strong_iteration(strong_iteration(x0))))
% 0.13/0.39 = { by lemma 14 }
% 0.13/0.39 addition(one, fresh(leq(strong_iteration(one), strong_iteration(strong_iteration(x0))), true, strong_iteration(one), strong_iteration(strong_iteration(x0))))
% 0.13/0.39 = { by axiom 10 (order_1) }
% 0.13/0.39 addition(one, addition(strong_iteration(one), strong_iteration(strong_iteration(x0))))
% 0.13/0.39 = { by axiom 7 (additive_associativity) }
% 0.13/0.39 addition(addition(one, strong_iteration(one)), strong_iteration(strong_iteration(x0)))
% 0.13/0.40 = { by lemma 16 }
% 0.13/0.40 addition(strong_iteration(one), strong_iteration(strong_iteration(x0)))
% 0.13/0.40 = { by axiom 2 (additive_commutativity) }
% 0.13/0.40 addition(strong_iteration(strong_iteration(x0)), strong_iteration(one))
% 0.13/0.40 = { by axiom 10 (order_1) R->L }
% 0.13/0.40 fresh(leq(strong_iteration(strong_iteration(x0)), strong_iteration(one)), true, strong_iteration(strong_iteration(x0)), strong_iteration(one))
% 0.13/0.40 = { by axiom 3 (multiplicative_right_identity) R->L }
% 0.13/0.40 fresh(leq(strong_iteration(strong_iteration(x0)), multiplication(strong_iteration(one), one)), true, strong_iteration(strong_iteration(x0)), strong_iteration(one))
% 0.13/0.40 = { by axiom 1 (idempotence) R->L }
% 0.13/0.40 fresh(leq(strong_iteration(strong_iteration(x0)), multiplication(strong_iteration(addition(one, one)), one)), true, strong_iteration(strong_iteration(x0)), strong_iteration(one))
% 0.13/0.40 = { by lemma 17 }
% 0.13/0.40 fresh(true, true, strong_iteration(strong_iteration(x0)), strong_iteration(one))
% 0.13/0.40 = { by axiom 5 (order_1) }
% 0.13/0.40 strong_iteration(one)
% 0.13/0.40 % SZS output end Proof
% 0.13/0.40
% 0.13/0.40 RESULT: Theorem (the conjecture is true).
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