In languages such as Agda, Idris, or Haskell with type extensions, there is a = type sort of like the following
data a :~: b where
Refl :: a :~: a
a :~: b means that a and b are the same.
Can such a type be defined in the calculus of constructions or Morte (which is programming language based on the calculus of construction)?
The standard Church-encoding of a :~: b in CoC is:
(a :~: b) =
forall (P :: * -> * -> *).
(forall c :: *. P c c) ->
P a b
Refl being
Refl a :: a :~: a
Refl a =
\ (P :: * -> * -> *)
(h :: forall (c::*). P c c) ->
h a
The above formulates equality between types. For equality between terms, the :~: relation must take an additional argument t :: *, where a b :: t.
((:~:) t a b) =
forall (P :: t -> t -> *).
(forall c :: t. P c c) ->
P a b
Refl t a :: (:~:) t a a
Refl t a =
\ (P :: t -> t -> *)
(h :: forall (c :: t). P c c) ->
h a
来源：https://stackoverflow.com/questions/36181738/refl-thing-in-calculus-of-constructions