# Module Std_kernel.Type_equal

    module Type_equal: Type_equal

    type ('a, 'b) t = ('a, 'b) Typerep_lib.Std.Type_equal.t =

`|` | `T : `('a0, 'a0) 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.

It is always safe to conclude that if type `a` equals `b`, then for any type `'a t`, type `a t` equals `b t`. The OCaml type checker uses this fact when it can. However, sometimes, e.g. when using `conv`, one needs to explicitly use this fact to construct an appropriate `Type_equal.t`. The `Lift*` functors do this.

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

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

    module Lift3: functor (X : T.T3) -> 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.