Higher-order imperative enumeration of binary trees in COQ

Inductive bintree : Set := | Leaf: bintree | Node: bintree -> bintree -> bintree. In the following, I also use some definitions of the Coq library. Type nat is type of Peano numbers generated from O and S. Type Z is type of infinite binary integers (more efficient than nat to perform large concrete computations). At last, list is the polymorphic type of lists. The Coq definition of enumBT is given below. Here, K is the “specification type” of the state DSM (see [Bou06]). It is parametrized both by St the type of the global state and by unit the type of the result. It uses an infix operator -; to denote sequences: “p1 -; p2” is a notation for “bind p1 (fun :unit => p2)”. The main advantage of this CPS-like implementation is to call f as soon as a tree is computed, before to compute the next tree. Moreover, whereas the number of binary trees is exponential in function of 2n (e.g. the number of nodes of a balanced binary tree of height n), this function requires only a memory linear in function of 2n. Fixpoint enumBT (St:Type) (n:nat) (f:bintree -> K St unit) {struct n} : K St unit := match n with | O => f Leaf | (S p) => enumBT p (fun l => enumBT p (fun r => f (Node l r)) -; enumlt p (fun r => f (Node l r) -; f (Node r l))) end with enumlt (St:Type) (n:nat) (f:bintree -> K St unit) {struct n} : K St unit := match n with | O => skip | (S p) => (enumBT p f) -; (enumlt p f) end.