Mapping for containers with two element types.
Bi_mappable implements the Haskell notion of a bifunctor: a container that contains two distinct element types, both of which can be non-monadically, covariantly mapped over.
Common examples include:
- maps and associative lists, where the two types are keys and values;
- result types, where the two types are success and failure.
Signatures
For input and output signatures for this module's functors, see Bi_mappable_types.
Making full bi-mappable type modules
These functors build full implementations of bi-mappability given the basic minimal definitions above.
Make2 implements S2 for an arity-2 bi-mappable container.
Make1_left implements S1_left for an arity-1 bi-mappable container with floating left type.
Make1_right implements S1_right for an arity-1 bi-mappable container with floating right type.
Make0 implements S0 for an arity-0 bi-mappable container.
Fixing types
We can convert arity-2 modules to arity-1 modules, and arity-1 modules to arity-0 modules, by fixing types. The various FixX_Y functors achieve this.
Arity-2
Fix2_left (I) (Left) fixes the left type of I to Left, making it an S1_right.
Fix2_right (S) (Left) fixes the right type of S to Right, making it an S1_left.
Fix2_both (S) (Left) (Right) fixes the types of S to Left and Right, making it an S0.
Arity-1
Fix1_left (S) (Left) fixes the floating left type of S to Left, making it an S0.
Fix1_right (I) (Right) fixes the floating right type of S to Right, making it an S0.
Converting bi-mappable modules to mappable modules
By ignoring values of either the left or the right type, we can derive Mappable modules from bi-mappable ones. Since the various S n signatures contain functions for doing this on an ad-hoc basis, the functors below are mainly for use when one needs actual Mappable instances.
This reflects the 'clowns to the left of me, jokers to the right' (the technical term!) set-up in Haskell; each Map_leftX functor implements a Clown; each Map_rightX functor implements a a Joker.
Since, unlike Haskell, we can't partially apply type constructors in OCaml, there are no arity-2 conversions available, and the arity-1 conversions only work if their direction is the one with a floating type. To rectify this, use Fix2_left and friends.
Arity-1
Mapping over the left type of an arity-1 bi-mappable container with a floating left type.
Mapping over the right type of an arity-1 bi-mappable container with a floating right type.
Arity-0
Mapping over the left type of an arity-0 bi-mappable container.
Mapping over the right type. of an arity-0 bi-mappable container.
Chaining containers
Chaining a mappable on the outside of a bi-mappable
These functors let us compose an inner bi-mappable container with an outer mappable container, producing a bi-map.
For example, we can make associative lists bi-mappable by composing a bi-map over pairs (a * b) with a map over lists.
In Haskell terms, this is a Tannen.
Chain_Bi2_Map1 (Bi) (Map) composes a bi-map Bi on an inner arity-2 container over a map Map over an outer arity-1 container.
Chain_Bi1_left_Map1 (Bi) (Map) composes a bi-map Bi on an inner arity-1 container with floating left type over a map Map over an outer arity-1 container.
Chain_Bi1_right_Map1 (Bi) (Map) composes a bi-map Bi on an inner arity-1 container with floating right type over a map Map over an outer arity-1 container.
Chain_Bi0_Map1 (Bi) (Map) composes a bi-map Bi on an inner arity-0 container over a map Map over an outer arity-1 container.
Chaining mappables on the inside of a bi-mappable
These functors let us compose one or two inner mappable containers with an outer bi-mappable container, producing a bi-mappable.
In Haskell terms, this is a Biff.
Chain_Map1_Bi2 (LMap) (RMap) (Bi) composes an inner arity-1 map LMap on the left, and another such map RMap on the right, of an arity-2 bi-traversal Bi.
Chain_Map1_Bi1_left (LMap) (Bi) composes an inner arity-1 map LMap on the left of an arity-1 bi-map Bi with floating left type.
Chain_Map1_Bi1_right (RMap) (Bi) composes an inner arity-1 map LMap on the right of an arity-1 bi-map Bi with floating right type.