二叉树的结构归纳

Question

Consider the following function definitions:

data Tree a = Leaf a | Node a (Tree a) (Tree a)

sumLeaves :: Tree a -> Integer
sumLeaves (Leaf x) = 1 
sumLeaves (Node _ lt rt) = sumLeaves lt + sumLeaves rt 

sumNodes :: Tree a -> Integer
sumNodes (Leaf x) = 0 
sumNodes (Node _ lt rt) = 1 + sumNodes lt + sumNodes rt 

Task: Prove by structural induction that for all finite trees t:: Tree a holds:

sumLeaves t = sumNodes t + 1

I now have:

sumLeaves t = sumNodes t + 1

I.A. t = (Tree 0) = (Leaf 0)

sumLeaves (Leaf 0) = 1 sumNodes (Leaf 0) + 1

0 + 1 = 1

1 = 1

I.V.: We assume that the statement holds for an arbitrary t.

IS:

Now what is the induction step, I really have problems with solving this last step.

Answer 1

Similar to the induction over natural numbers, where the induction step is assuming that some hypothesis holds for all numbers less than n , and then proving the hypothesis for n, structural induction step assumes that hypothesis holds for all structures that are substructures of a given structure .

In your case, correct assumption would be that the statement holds for trees lt and rt that are subtrees of the arbitrary t@(Node x lt rt) , and then proving the statement for t .