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
b are equal.
One can think of such a value as a proof of type equality. The
has operations for constructing and manipulating such proofs. For example, the
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
b are equal,
there are two ways to use
t to safely convert a value of type
a to a value of type
Type_equal.conv or pattern matching on
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
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.
trans construct proofs that type equality is reflexive,
symmetric, and transitive.
conv t x uses the type equality
t : (a, b) t as evidence to safely cast
a to type
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
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
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
b, then for any 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
Lift* functors do this.
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
'a list is an injective type, because whenever
'a list = 'b list, we
'b. On the other hand, if we define:
type 'a t = unit
t isn't injective, because, e.g.
int t = bool t, but
int <> bool.
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
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
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 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
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
M.strip, then OCaml would have reported a type error.
Id provides identifiers for types, and the ability to test (via
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.
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
'a doesn't support sexp conversion, then a good practice is to have the
<:sexp_of< _ >>, (or
sexp_of_opaque, if not using pa_sexp).
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
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).