List Induction (for finite lists) Recall the list induction principle to prove a

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List Induction (for finite lists) Recall the list induction principle to prove a property P for finite lists of type a. [P([])^forall X:: a. forall xs:: [a].P(xs) rightarrow P(x: xs) rightarrow P(x: xs) rightarrow forall ys:: [a].P(ys) Thus, for a property P of lists, to show that forall ys:: [a].P(ys) it is enough to show two things: P([]) forall x:: a. forall xs:: [a]. P(xs) rightarrow P(x: xs) Here are some definitions^1. head(h: t) = h head[] = last[x] = x last(h: t) = last t last[] = reverse[] = [] reverse(h: t) = (reverse t) + +[h] map f[] = [] map f (h: t) = (f h): map f t (f.g)x = f(g x) Prove the following by list induction on m. forall m: [a]. map(x rightarrow x) m = m forall m: [a], map(f. g) m = ((map f). (map g)) m forall m: [a], head (reverse m) = last m forall m: [a]. last (reverse m) = head m

Explanation / Answer

first step:itis to prove the given statement for the first natural number the second step known as aa

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