(** 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 runtime.
*)

open! Import
open T

type ('a, 'b) t = T : ('a, 'a) t [@@deriving_inline sexp_of]

val sexp_of_t
  :  ('a -> Sexplib0.Sexp.t)
  -> ('b -> Sexplib0.Sexp.t)
  -> ('a, 'b) t
  -> Sexplib0.Sexp.t

[@@@end]

(** just an alias, needed when [t] gets shadowed below *)
type ('a, 'b) equal = ('a, 'b) t

(** [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

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

(** 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 : T1) : sig
  val lift : ('a, 'b) t -> ('a X.t, 'b X.t) t
end

module Lift2 (X : T2) : sig
  val lift : ('a1, 'b1) t -> ('a2, 'b2) t -> (('a1, 'a2) X.t, ('b1, 'b2) X.t) t
end

module Lift3 (X : T3) : sig
  val lift
    :  ('a1, 'b1) t
    -> ('a2, 'b2) t
    -> ('a3, 'b3) t
    -> (('a1, 'a2, 'a3) X.t, ('b1, 'b2, 'b3) X.t) t
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

(** [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 =
        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 the 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
        ;;
      end
    ]}

    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.
*)
module type Injective = sig
  type 'a t

  val strip : ('a t, 'b t) equal -> ('a, 'b) equal
end

(** [Injective2] is for a binary type that is injective in both type arguments. *)
module type Injective2 = sig
  type ('a1, 'a2) t

  val strip : (('a1, 'a2) t, ('b1, 'b2) t) equal -> ('a1, 'b1) equal * ('a2, 'b2) equal
end

(** [Composition_preserves_injectivity] is a functor that proves that composition of
    injective types is injective. *)
module Composition_preserves_injectivity (M1 : Injective) (M2 : Injective) :
  Injective with type 'a t = 'a M1.t M2.t

(** [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. *)
module Id : sig
  type 'a t [@@deriving_inline sexp_of]

  val sexp_of_t : ('a -> Sexplib0.Sexp.t) -> 'a t -> Sexplib0.Sexp.t

  [@@@end]

  (** Every [Id.t] contains a unique id that is distinct from the [Uid.t] in any other
      [Id.t]. *)
  module Uid : Identifiable.S

  val uid : _ 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 ppx_sexp_conv). *)
  val create : name:string -> ('a -> Sexp.t) -> 'a t

  (** Accessors *)

  val hash : _ t -> int
  val name : _ t -> string
  val to_sexp : 'a t -> 'a -> Sexp.t
  val hash_fold_t : Hash.state -> _ t -> Hash.state

  (** [same_witness t1 t2] and [same_witness_exn t1 t2] return a type equality proof iff
      the two identifiers are the same (i.e., physically equal, resulting from the same
      call to [create]).  This is a useful way to achieve a sort of dynamic typing.
      [same_witness] does not allocate a [Some] every time it is called.

      [same t1 t2 = is_some (same_witness t1 t2)].
  *)

  val same : _ t -> _ t -> bool
  val same_witness : 'a t -> 'b t -> ('a, 'b) equal option
  val same_witness_exn : 'a t -> 'b t -> ('a, 'b) equal
end
