TSTP Solution File: KLE146+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE146+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 05:36:03 EDT 2023
% Result : Theorem 0.11s 0.35s
% Output : Proof 0.11s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.09/0.11 % Problem : KLE146+1 : TPTP v8.1.2. Released v4.0.0.
% 0.09/0.11 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.11/0.31 % Computer : n032.cluster.edu
% 0.11/0.31 % Model : x86_64 x86_64
% 0.11/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.31 % Memory : 8042.1875MB
% 0.11/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.31 % CPULimit : 300
% 0.11/0.31 % WCLimit : 300
% 0.11/0.31 % DateTime : Tue Aug 29 12:40:31 EDT 2023
% 0.11/0.31 % CPUTime :
% 0.11/0.35 Command-line arguments: --no-flatten-goal
% 0.11/0.35
% 0.11/0.35 % SZS status Theorem
% 0.11/0.35
% 0.11/0.35 % SZS output start Proof
% 0.11/0.35 Take the following subset of the input axioms:
% 0.11/0.35 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 0.11/0.35 fof(additive_commutativity, axiom, ![B2, A3]: addition(A3, B2)=addition(B2, A3)).
% 0.11/0.35 fof(goals, conjecture, ![X0]: leq(one, strong_iteration(X0))).
% 0.11/0.35 fof(idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 0.11/0.35 fof(infty_unfold1, axiom, ![A3]: strong_iteration(A3)=addition(multiplication(A3, strong_iteration(A3)), one)).
% 0.11/0.35 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 0.11/0.35
% 0.11/0.35 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.11/0.35 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.11/0.35 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.11/0.35 fresh(y, y, x1...xn) = u
% 0.11/0.35 C => fresh(s, t, x1...xn) = v
% 0.11/0.35 where fresh is a fresh function symbol and x1..xn are the free
% 0.11/0.35 variables of u and v.
% 0.11/0.35 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.11/0.35 input problem has no model of domain size 1).
% 0.11/0.35
% 0.11/0.35 The encoding turns the above axioms into the following unit equations and goals:
% 0.11/0.35
% 0.11/0.35 Axiom 1 (idempotence): addition(X, X) = X.
% 0.11/0.35 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.11/0.35 Axiom 3 (order): fresh5(X, X, Y, Z) = true.
% 0.11/0.35 Axiom 4 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.11/0.35 Axiom 5 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
% 0.11/0.35 Axiom 6 (order): fresh5(addition(X, Y), Y, X, Y) = leq(X, Y).
% 0.11/0.35
% 0.11/0.35 Lemma 7: addition(one, multiplication(X, strong_iteration(X))) = strong_iteration(X).
% 0.11/0.35 Proof:
% 0.11/0.35 addition(one, multiplication(X, strong_iteration(X)))
% 0.11/0.35 = { by axiom 2 (additive_commutativity) R->L }
% 0.11/0.35 addition(multiplication(X, strong_iteration(X)), one)
% 0.11/0.35 = { by axiom 5 (infty_unfold1) R->L }
% 0.11/0.35 strong_iteration(X)
% 0.11/0.35
% 0.11/0.35 Goal 1 (goals): leq(one, strong_iteration(x0)) = true.
% 0.11/0.35 Proof:
% 0.11/0.35 leq(one, strong_iteration(x0))
% 0.11/0.35 = { by axiom 6 (order) R->L }
% 0.11/0.35 fresh5(addition(one, strong_iteration(x0)), strong_iteration(x0), one, strong_iteration(x0))
% 0.11/0.35 = { by lemma 7 R->L }
% 0.11/0.35 fresh5(addition(one, addition(one, multiplication(x0, strong_iteration(x0)))), strong_iteration(x0), one, strong_iteration(x0))
% 0.11/0.35 = { by axiom 4 (additive_associativity) }
% 0.11/0.35 fresh5(addition(addition(one, one), multiplication(x0, strong_iteration(x0))), strong_iteration(x0), one, strong_iteration(x0))
% 0.11/0.35 = { by axiom 1 (idempotence) }
% 0.11/0.35 fresh5(addition(one, multiplication(x0, strong_iteration(x0))), strong_iteration(x0), one, strong_iteration(x0))
% 0.11/0.35 = { by lemma 7 }
% 0.11/0.35 fresh5(strong_iteration(x0), strong_iteration(x0), one, strong_iteration(x0))
% 0.11/0.35 = { by axiom 3 (order) }
% 0.11/0.35 true
% 0.11/0.35 % SZS output end Proof
% 0.11/0.35
% 0.11/0.35 RESULT: Theorem (the conjecture is true).
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