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 ('a, 'b) t = ('a, 'b) Typerep_kernel.Std.Type_equal.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

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
val lift : ('a, 'b) t -> ('a X.t, 'b X.t) t
module Lift2 : 
functor (X : T.T2) -> sig .. end
val lift : ('a1, 'b1) t -> ('a2, 'b2) t -> (('a1, 'a2) X.t, ('b1, 'b2) X.t) t
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. The typical usage is:
      type 'a t
      include Injective with type 'a t := 'a t

For example, 'a list is an injective type, because whenever 'a list = 'b list, we know that 'a = 'b. On the other hand, if we define:

      type 'a t = unit

then clearly t isn't injective, because, e.g. int t = bool t, but int <> bool.

If module M : Injective, then M.strip provides a way to get a proof that two types are equal from a proof that both types transformed by M.t are equal.

OCaml has no built-in language feature to state that a type is injective, which is why we have module type Injective. However, OCaml can infer that a type is injective, and we can use this to match Injective. A typical implementation will look like this:

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

This will not type check for all type constructors (certainly not for non-injective ones!), but it's always safe to try the above implementation if you are unsure. If OCaml accepts this definition, then the type is injective. On the other hand, if OCaml doesn't, then type type may or may not be injective. For example, if the definition of the type depends on abstract types that match Injective, OCaml will not automatically use their injectivity, and one will have to write a more complicated definition of strip that causes OCaml to use that fact. For example:

      module F (M : Type_equal.Injective) : Type_equal.Injective = struct
        type 'a t = 'a M.t * int

        let strip (type a) (type b)
            (e : (a t, b t) Type_equal.t) : (a, b) Type_equal.t =
          let e1, _ = Type_equal.detuple2 e in
          M.strip e1

If in the definition of F we had written the simpler implementation of strip that didn't use M.strip, then OCaml would have reported a type error.

type 'a t
val strip : ('a t, 'b t) equal -> ('a, 'b) equal
module type Injective2 = sig .. end
Injective2 is for a binary type that is injective in both type arguments.
type ('a1, 'a2) t
val strip : (('a1, 'a2) t, ('b1, 'b2) t) equal -> ('a1, 'b1) equal * ('a2, 'b2) equal
module Composition_preserves_injectivity : 
functor (M1 : Injective) ->
functor (M2 : Injective) -> Injective with type 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. Unlike values of type Type_equal.t, values of type Id.t do have semantic content and must have a nontrivial runtime representation.
type 'a t
module Uid : Unique_id_intf.Id
Every Id.t contains a unique id that is distinct from the Uid.t in any other Id.t.
val uid : 'a t -> Uid.t
create ~name defines a new type identity. Two calls to create will result in two distinct identifiers, even for the same arguments with the same type. If the type 'a doesn't support sexp conversion, then a good practice is to have the converter be <:sexp_of< _ >>, (or sexp_of_opaque, if not using pa_sexp).
val create : name:string -> ('a -> Sexplib.Sexp.t) -> 'a t
val hash : 'a t -> int
val name : 'a t -> string
val to_sexp : 'a t -> 'a -> Sexplib.Sexp.t
val same : 'a t -> 'b t -> bool
same_witness t1 t2 and same_witness_exn t1 t2 return a type equality proof iff the two identifiers are physically equal. This is a useful way to achieve a sort of dynamic typing.

The following two idioms are semantically equivalent; however, using same and same_witness_exn instead of matching on same_witness has better performance because same_witness would allocate an intermediate Or_error.t.

        match same_witness ta tb with
        | None -> ...
        | Some e -> ...
        if not (same ta tb)
        then ...
          let e = same_witness_exn ta tb in
val same_witness : 'a t -> 'b t -> ('a, 'b) equal Or_error.t
val same_witness_exn : 'a t -> 'b t -> ('a, 'b) equal
val sexp_of_t : ('a -> Sexplib.Sexp.t) -> 'a t -> Sexplib.Sexp.t