mirror of
https://github.com/idris-lang/Idris2.git
synced 2024-12-21 10:41:59 +03:00
97 lines
3.6 KiB
Idris
97 lines
3.6 KiB
Idris
||| This module is inspired by the open union used in the paper
|
|
||| Freer Monads, More Extensible Effects
|
|
||| by Oleg Kiselyov and Hiromi Ishii
|
|
|||
|
|
||| By using an AtIndex proof, we are able to get rid of all of the unsafe
|
|
||| coercions in the original module.
|
|
|
|
module Data.OpenUnion
|
|
|
|
import Data.DPair
|
|
import Data.List.AtIndex
|
|
import Data.List.HasLength
|
|
import Data.Nat
|
|
import Data.Nat.Order.Properties
|
|
import Decidable.Equality
|
|
import Syntax.WithProof
|
|
|
|
%default total
|
|
|
|
||| An open union of families is an index picking a family out together with
|
|
||| a value in the family thus picked.
|
|
public export
|
|
data Union : (elt : a -> Type) -> (ts : List a) -> Type where
|
|
Element : (k : Nat) -> (0 _ : AtIndex t ts k) -> elt t -> Union elt ts
|
|
|
|
||| An empty open union of families is empty
|
|
public export
|
|
Uninhabited (Union elt []) where
|
|
uninhabited (Element _ p _) = void (uninhabited p)
|
|
|
|
||| Injecting a value into an open union, provided we know the index of
|
|
||| the appropriate type family.
|
|
inj' : (k : Nat) -> (0 _ : AtIndex t ts k) -> elt t -> Union elt ts
|
|
inj' = Element
|
|
|
|
||| Projecting out of an open union, provided we know the index of the
|
|
||| appropriate type family. This may obviously fail if the value stored
|
|
||| actually corresponds to another family.
|
|
prj' : (k : Nat) -> (0 _ : AtIndex t ts k) -> Union elt ts -> Maybe (elt t)
|
|
prj' k p (Element k' q t) with (decEq k k')
|
|
prj' k p (Element k q t) | Yes Refl = rewrite atIndexUnique p q in Just t
|
|
prj' k p (Element k' q t) | No neq = Nothing
|
|
|
|
||| Given that equality of type families is not decidable, we have to
|
|
||| rely on the interface `Member` to automatically find the index of a
|
|
||| given family.
|
|
public export
|
|
inj : Member t ts => elt t -> Union elt ts
|
|
inj = let (Element n p) = isMember t ts in inj' n p
|
|
|
|
||| Given that equality of type families is not decidable, we have to
|
|
||| rely on the interface `Member` to automatically find the index of a
|
|
||| given family.
|
|
public export
|
|
prj : Member t ts => Union elt ts -> Maybe (elt t)
|
|
prj = let (Element n p) = isMember t ts in prj' n p
|
|
|
|
||| By doing a bit of arithmetic we can figure out whether the union's value
|
|
||| came from the left or the right list used in the index.
|
|
public export
|
|
split : Subset Nat (HasLength ss) ->
|
|
Union elt (ss ++ ts) -> Either (Union elt ss) (Union elt ts)
|
|
split m (Element n p t) with (@@ lt n (fst m))
|
|
split m (Element n p t) | (True ** lt)
|
|
= Left (Element n (strengthenL m lt p) t)
|
|
split m (Element n p t) | (False ** notlt)
|
|
= let 0 lte : lte (fst m) n === True
|
|
= LteIslte (fst m) n (notltIsGTE n (fst m) notlt)
|
|
in Right (Element (minus n (fst m)) (strengthenR m lte p) t)
|
|
|
|
||| We can inspect an open union over a non-empty list of families to check
|
|
||| whether the value it contains belongs either to the first family or any
|
|
||| other in the tail.
|
|
public export
|
|
decomp : Union elt (t :: ts) -> Either (Union elt ts) (elt t)
|
|
decomp (Element 0 (Z) t) = Right t
|
|
decomp (Element (S n) (S p) t) = Left (Element n p t)
|
|
|
|
||| An open union over a singleton list is just a wrapper
|
|
public export
|
|
decomp0 : Union elt [t] -> elt t
|
|
decomp0 elt = case decomp elt of
|
|
Left t => absurd t
|
|
Right t => t
|
|
|
|
||| Inserting new union members on the right leaves the index unchanged.
|
|
public export
|
|
weakenR : Union elt ts -> Union elt (ts ++ us)
|
|
weakenR (Element n p t) = Element n (weakenR p) t
|
|
|
|
||| Inserting new union members on the left, requires shifting the index by
|
|
||| the number of members introduced. Note that this number is the only
|
|
||| thing we need to keep around at runtime.
|
|
public export
|
|
weakenL : Subset Nat (HasLength ss) -> Union elt ts -> Union elt (ss ++ ts)
|
|
weakenL m (Element n p t) = Element (fst m + n) (weakenL m p) t
|