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Joined 2026-09-21 Message
Incoming links: Mutual recursion
Add
Fixes [Ident] [Expr] to the abstract syntax tree. For the stipulated nonempty, equally long lists of distinct names and lambda abstractions, define all programming closures against one common extended semantic environment:eval (Fixes xs es) rho = result (head vs)
where
rho' = extend rho (zip xs vs)
vs = map (\e -> close e rho') esExpanding the auxiliary function, if ,Every body can therefore call every member of the family, including itself. Constructing the family performs no body computations in the arbitrary computational monad. As in part (c), host lazy evaluation ties the recursive semantic environment; a monadic fixed-point operator is unnecessary.
For example, with two bodies implementing evenness and oddness, respectively, each programming closure sees both names in . Returning the first programming closure still retains access to the second. Building each programming closure in an independently extended old semantic environment would lose this mutual recursion. For , the definition agrees with
Fix.The constructors For example,
ZigType and ZagType record which of the two alternating positions contains the last element. The tail types swap the two type parameters, so the return type of lastZag swaps them back:lastZig :: Zig a b -> ZigOrZagType a b
lastZig Nil = error "lastZig: empty input"
lastZig (Cins x Nal) = ZigType x
lastZig (Cins _ zs) = lastZag zs
lastZag :: Zag a b -> ZigOrZagType b a
lastZag Nal = error "lastZag: empty input"
lastZag (Cans x Nil) = ZagType x
lastZag (Cans _ zs) = lastZig zslastZig passes a tail of type Zag b a to lastZag, whose result then has type ZigOrZagType a b, exactly as required. Conversely, lastZag passes a Zig b a tail to lastZig. A singleton returns its element in the appropriate constructor; a longer input discards one element and uses mutual recursion on the shorter tail. This proves correctness by structural induction on the finite alternating sequence. The three source examples give ZigType 2, ZagType "D", and ZigType True.Map each current first-position value with the first function, then swap the functions when descending to the other algebraic data type:All four type variables are independent: mapping may change both element types. There is no
mapZig :: (a -> c) -> (b -> d) -> Zig a b -> Zig c d
mapZig _ _ Nil = Nil
mapZig f g (Cins x zs) = Cins (f x) (mapZag g f zs)
mapZag :: (a -> c) -> (b -> d) -> Zag a b -> Zag c d
mapZag _ _ Nal = Nal
mapZag f g (Cans x zs) = Cans (f x) (mapZig g f zs)Enum or numeric constraint on the mapping functions themselves; constraints arise only from particular arguments such as succ or (+2). The mutual recursion preserves every constructor and applies the intended function at each alternating position. The results are Cins 3 (Cans 'B' (Cins 4 Nal)) and Cans "D" (Cins True (Cans "E" Nil)), respectively.