The type of keys.

memoize is a memoization combinator for the type key. The function call memoize f produces a function f' that behaves extensionally like f, but is memoized.

The type of memoization tables.

visibly_memoize is a memoization combinator that exposes the memoization table. The function call visibly_memoize f returns a pair of a memoized function f' and a memoization table.

fix is a recursive memoization combinator.

Cycle is raised by defensive_fix when a dependency cycle is detected.

defensive_fix works like fix, except it detects circular dependencies, which can arise if the second-order function supplied by the user does not follow a well-founded recursion pattern. When the user invokes f x, where f is the function returned by defensive_fix, if a cyclic dependency is detected, then Cycle (zs, z) is raised, where the list zs begins with z and continues with a series of intermediate keys, leading back to z. Note that undetected divergence remains possible; this corresponds to an infinite dependency chain, without a cycle.

curried can be used to obtain a curried version of fix or defensive_fix in a concrete instance where the type key is a product type.