package base
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type ('a, 'b) t =
| T : ( 'a, 'a ) t
val sexp_of_t : ( 'a -> Sexplib0.Sexp.t ) -> ( 'b -> Sexplib0.Sexp.t ) -> ( 'a, 'b ) t -> Sexplib0.Sexp.t
type ('a, 'b) equal = ( 'a, 'b ) t

just an alias, needed when t gets shadowed below

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

val refl : ( 'a, 'a ) t
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 (X : T.T1) : sig ... end
module Lift2 (X : T.T2) : sig ... end
module Lift3 (X : T.T3) : sig ... end

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

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

module type Injective2 = sig ... end

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

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 runtime 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.