Module Type_equal

module Type_equal: sig .. end

For representing type equalities otherwise not known by the type-checker.

The purpose of Type_equal is to represent type equalities that the type checker otherwise would not know, perhaps because the type equality depends on dynamic data, or perhaps because the type system isn't powerful enough.

A value of type (a, b) Type_equal.t represents that types a and b are equal. One can think of such a value as a proof of type equality. The Type_equal module has operations for constructing and manipulating such proofs. For example, the functions refl, sym, and trans express the usual properties of reflexivity, symmetry, and transitivity of equality.

If one has a value t : (a, b) Type_equal.t that proves types a and b are equal, there are two ways to use t to safely convert a value of type a to a value of type b: Type_equal.conv or pattern matching on Type_equal.T:

      let f (type a) (type b) (t : (a, b) Type_equal.t) (a : a) : b =
        Type_equal.conv t a

      let f (type a) (type b) (t : (a, b) Type_equal.t) (a : a) : b =
        let Type_equal.T = t in a
    

At runtime, conversion by either means is just the identity -- nothing is changing about the value. Consistent with this, a value of type Type_equal.t is always just a constructor Type_equal.T; the value has no interesting semantic content. Type_equal gets its power from the ability to, in a type-safe way, prove to the type checker that two types are equal. The Type_equal.t value that is passed is necessary for the type-checker's rules to be correct, but the compiler, could, in principle, not pass around values of type Type_equal.t at run time.

type ('_, '_) t = 

type ('a, 'b) equal = ('a, 'b) t 

just an alias, needed when t gets shadowed below

val refl : ('a, 'a) t

refl, sym, and trans construct proofs that type equality is reflexive, symmetric, and transitive.

val sym : ('a, 'b) t -> ('b, 'a) t

val trans : ('a, 'b) t -> ('b, 'c) t -> ('a, 'c) t

val conv : ('a, 'b) t -> 'a -> 'b

conv t x uses the type equality t : (a, b) t as evidence to safely cast x from type a to type b. conv is semantically just the identity function.

In a program that has t : (a, b) t where one has a value of type a that one wants to treat as a value of type b, it is often sufficient to pattern match on Type_equal.T rather than use conv. However, there are situations where OCaml's type checker will not use the type equality a = b, and one must use conv. For example:

      module F (M1 : sig type t end) (M2 : sig type t end) : sig
        val f : (M1.t, M2.t) equal -> M1.t -> M2.t
      end = struct
        let f equal (m1 : M1.t) = conv equal m1
      end
    

If one wrote the body of F using pattern matching on T:

      let f (T : (M1.t, M2.t) equal) (m1 : M1.t) = (m1 : M2.t)
    

this would give a type error.

module Lift: 
functor (X : T.T1) -> sig .. end

module Lift2: 
functor (X : T.T2) -> sig .. end

val detuple2 : ('a1 * 'a2, 'b1 * 'b2) t ->
       ('a1, 'b1) t * ('a2, 'b2) t

tuple2 and detuple2 convert between equality on a 2-tuple and its components.

val tuple2 : ('a1, 'b1) t ->
       ('a2, 'b2) t -> ('a1 * 'a2, 'b1 * 'b2) t

module type Injective = sig .. end

Injective is an interface that states that a type is injective, where the type is viewed as a function from types to other types.

module type Injective2 = sig .. end

Injective2 is for a binary type that is injective in both type arguments.

module Composition_preserves_injectivity: 
functor (M1 : Injective) -> 
functor (M2 : Injective) -> Injective  with type 'a t = 'a M1.t M2.t

Composition_preserves_injectivity is a functor that proves that composition of injective types is injective.

module Id: sig .. end

Id provides identifiers for types, and the ability to test (via Id.same) at run-time if two identifiers are equal, and if so to get a proof of equality of their types.