`type order =`

1. `| Lt`
2. `| Eq`
3. `| Gt`
   (*

   An algebraic datatype for ordering.

   Traditional OCaml code, under the influence of C comparison functions, has used int-returning comparisons (< 0, 0 or > 0). Using an algebraic datatype instead is actually nicer, both for comparison producers (no arbitrary choice of a positive and negative value) and consumers (nice pattern-matching elimination).

   *)

`type 'a ord = 'a -> 'a -> order`

The type of ordering functions returning an `order` variant.

`type 'a comp = 'a -> 'a -> int`

The legacy int-returning comparisons :

- compare a b < 0 means a < b
- compare a b = 0 means a = b
- compare a b > 0 means a > b

`module type Comp = sig ... end`

We use `compare` as member name instead of `comp`, so that the Comp modules can be used as the legacy OrderedType interface.

`module type Ord = sig ... end`
`val ord0 : int -> order`
`val ord : 'a comp -> 'a ord`

Returns a variant ordering from a legacy comparison

`module Ord (Comp : Comp) : Ord with type t = Comp.t`
`val comp0 : order -> int`
`val comp : 'a ord -> 'a comp`

Returns an legacy comparison from a variant ordering

`module Comp (Ord : Ord) : Comp with type t = Ord.t`
`val poly_comp : 'a comp`
`val poly_ord : 'a ord`
`val poly : 'a ord`

Polymorphic comparison functions, based on the `Pervasives.compare` function from inria's stdlib, have polymorphic types: they claim to be able to compare values of any type. In practice, they work for only some types, may fail on function types and may not terminate on cyclic values.

They work by runtime magic, inspecting the values in an untyped way. While being an useful hack for base types and simple composite types (say `(int * float) list`, they do not play well with functions, type abstractions, and structures that would need a finer notion of equality/comparison. For example, if one represent sets as balanced binary tree, one may want set with equal elements but different balancings to be equal, which would not be the case using the polymorphic equality function.

When possible, you should therefore avoid relying on these polymorphic comparison functions. You should be especially careful if your data structure may later evolve to allow cyclic data structures or functions.

`val rev_ord0 : order -> order`
`val rev_comp0 : int -> int`
`val rev_ord : 'a ord -> 'a ord`
`val rev_comp : 'a comp -> 'a comp`
`val rev : 'a ord -> 'a ord`

Reverse a given ordering. If `Int.ord` sorts integer by increasing order, `rev Int.ord` will sort them by decreasing order.

`module RevOrd (Ord : Ord) : Ord with type t = Ord.t`
`module RevComp (Comp : Comp) : Comp with type t = Comp.t`
`module Rev (Ord : Ord) : Ord with type t = Ord.t`
`type 'a eq = 'a -> 'a -> bool`

The type for equality function.

All ordered types also support equality, as equality can be derived from ordering. However, there are also cases where elements may be compared for equality, but have no natural ordering. It is therefore useful to provide equality as an independent notion.

`val eq_ord0 : order -> bool`
`val eq_comp0 : int -> bool`
`val eq_ord : 'a ord -> 'a eq`
`val eq_comp : 'a comp -> 'a eq`
`val eq : 'a ord -> 'a eq`

Derives an equality function from an ordering function.

`module type Eq = sig ... end`
`module EqOrd (Ord : Ord) : Eq with type t = Ord.t`
`module EqComp (Comp : Comp) : Eq with type t = Comp.t`
`module Eq (Ord : Ord) : Eq with type t = Ord.t`
`type 'a choice = 'a -> 'a -> 'a`

choice functions, see `min` and `max`.

`val min_ord : 'a ord -> 'a choice`
`val max_ord : 'a ord -> 'a choice`
`val min_comp : 'a comp -> 'a choice`
`val max_comp : 'a comp -> 'a choice`
`val min : 'a ord -> 'a choice`

`min ord` will choose the smallest element, according to `ord`. For example, `min Int.ord 1 2` will return `1`.

      (* the minimum element of a list *)
      let list_min ord = List.reduce (min ord)

`val max : 'a ord -> 'a choice`

`max ord` will choose the biggest element according to `ord`.

`val bin_comp : 'a comp -> 'a -> 'a -> 'b comp -> 'b -> 'b -> int`
`val bin_ord : 'a ord -> 'a -> 'a -> 'b ord -> 'b -> 'b -> order`

binary lifting of the comparison function, using lexicographic order: `bin_ord ord1 v1 v1' ord2 v2 v2'` is `ord2 v2 v2'` if `ord1 v1 v1' = Eq`, and `ord1 v1 v1'` otherwise.

`val bin_eq : 'a eq -> 'a -> 'a -> 'b eq -> 'b -> 'b -> bool`
`val map_eq : ('a -> 'b) -> 'b eq -> 'a eq`
`val map_comp : ('a -> 'b) -> 'b comp -> 'a comp`
`val map_ord : ('a -> 'b) -> 'b ord -> 'a ord`

These functions extend an existing equality/comparison/ordering to a new domain through a mapping function. For example, to order sets by their cardinality, use `map_ord Set.cardinal Int.ord`. The input of the mapping function is the type you want to compare, so this is the reverse of `List.map`.

`module Incubator : sig ... end`