By Robert S. Boyer
In contrast to so much texts on good judgment and arithmetic, this ebook is ready easy methods to turn out theorems instead of facts of particular effects. We provide our solutions to such questions as: - whilst may still induction be used? - How does one invent a suitable induction argument? - while may still a definition be extended?
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Additional info for A Computational Logic
Xn) be any m e m b e r of the domain of FO such that ( FO XI . . Xn ) ^ body[F0 ' ] . Let G be a partially correct function with (XI, . . , Xn) in its domain such that ( G XI . . Xn) = (FO XI . . Xn) . Let G ' be the extension of G. By applying the lemma to FO, FO ' , G, G ' , and (XI, . . , Xn) we infer that body[F0 ' ] = body[G ' ] . But ( FO XI . . Xn) = (GX1 . . Xn) = b o d y [ G ' ] = b o d y [ F 0 ' ] . D. Before proving that the domain of FO is Dn we adopt the notational convention that body ' is an abbreviation for the result of substituting Yi for Xi in body[F0 * ] .
Vz>) = (R (MU1. Uz) (M V I . . Vz) ) . Note that RM is well founded. If p is not a theorem there must exist a z-tuple (XI, . . , Xz) such that ( P XI . . Xz ) = F . Let (XI, . . , Xz) be an RM-minimal such z-tuple. E. SHELLS / 35 We now consider the cases on which, if any, of the qi are true on the chosen z-tuple. Case XI . case Xz ) 1. , suppose ( QI XI . . Xz) = F, ( Q2 . Xz ) = F, . . , and ( Qk XI . . Xz ) = F . Then by the base ( P X 1 . . Xz) ^ F , contradicting the assumption that ( P XI .
We restrict the new variables by appealing to the type restriction lemma noted when FLATTEN was introduced. 1. Let us appeal to the induction principle. The recursive terms in the conjecture suggest three inductions. They merge into two likely candidate inductions. However, only one is unflawed. We will induct according to the following scheme: (AND (IMPLIES (NOT (LISTP A)) (p A Y ANS)) (IMPLIES (AND (LISTP A) (p (CDR A) Y ANS)) (p A Y ANS))). LESSP establishes (COUNT A) decreases according to the relation LESSP in the induction step The above induction scheme generates conjectures: that the measure well-founded of t-he scheme.