{-# LANGUAGE AllowAmbiguousTypes #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}

-- |
-- Module      : Data.Type.Universe.Subset
-- Copyright   : (c) Justin Le 2018
-- License     : BSD3
--
-- Maintainer  : justin@jle.im
-- Stability   : experimental
-- Portability : non-portable
--
-- Represent a decidable subset of a type-level collection.
module Data.Type.Universe.Subset (
  -- * Subset
  Subset,
  WitSubset (..),
  makeSubset,

  -- ** Subset manipulation
  intersection,
  union,
  symDiff,
  mergeSubset,
  imergeSubset,
  mapSubset,
  imapSubset,

  -- ** Subset extraction
  subsetToList,

  -- ** Subset tests
  subsetToAny,
  subsetToAll,
  subsetToNone,

  -- ** Subset construction
  emptySubset,
  fullSubset,
) where

import Control.Applicative
import Data.Kind
import Data.Monoid (Alt (..))
import Data.Singletons
import Data.Singletons.Decide
import Data.Type.Functor.Product
import Data.Type.Predicate
import Data.Type.Predicate.Logic
import Data.Type.Predicate.Quantification
import Data.Type.Universe

-- | A @'WitSubset' f p @@ as@ describes a /decidable/ subset of type-level
-- collection @as@.
newtype WitSubset f p (as :: f k) = WitSubset
  { forall k (f :: * -> *) (p :: k ~> *) (as :: f k).
WitSubset f p as
-> forall (a :: k). Elem f as a -> Decision (p @@ a)
runWitSubset :: forall a. Elem f as a -> Decision (p @@ a)
  }

-- | A @'Subset' f p@ is a predicate that some decidable subset of an input
-- @as@ is true.
data Subset f :: (k ~> Type) -> (f k ~> Type)

type instance Apply (Subset f p) as = WitSubset f p as

instance (Universe f, Decidable p) => Decidable (Subset f p)
instance (Universe f, Decidable p) => Provable (Subset f p) where
  prove :: Prove (Subset f p)
prove = forall (f :: * -> *) k (p :: k ~> *) (as :: f k).
Universe f =>
(forall (a :: k). Elem f as a -> Sing a -> Decision (p @@ a))
-> Sing as -> Subset f p @@ as
makeSubset @f @_ @p (\Elem f a a
_ -> forall {k1} (p :: k1 ~> *). Decidable p => Decide p
forall (p :: k ~> *). Decidable p => Decide p
decide @p)

-- | Create a 'Subset' from a predicate.
makeSubset ::
  forall f k p (as :: f k).
  Universe f =>
  (forall a. Elem f as a -> Sing a -> Decision (p @@ a)) ->
  Sing as ->
  Subset f p @@ as
makeSubset :: forall (f :: * -> *) k (p :: k ~> *) (as :: f k).
Universe f =>
(forall (a :: k). Elem f as a -> Sing a -> Decision (p @@ a))
-> Sing as -> Subset f p @@ as
makeSubset forall (a :: k). Elem f as a -> Sing a -> Decision (p @@ a)
f Sing as
xs = (forall (a :: k). Elem f as a -> Decision (p @@ a))
-> WitSubset f p as
forall k (f :: * -> *) (p :: k ~> *) (as :: f k).
(forall (a :: k). Elem f as a -> Decision (p @@ a))
-> WitSubset f p as
WitSubset ((forall (a :: k). Elem f as a -> Decision (p @@ a))
 -> WitSubset f p as)
-> (forall (a :: k). Elem f as a -> Decision (p @@ a))
-> WitSubset f p as
forall a b. (a -> b) -> a -> b
$ \Elem f as a
i -> Elem f as a -> Sing a -> Decision (p @@ a)
forall (a :: k). Elem f as a -> Sing a -> Decision (p @@ a)
f Elem f as a
i (Elem f as a -> Sing as -> Sing a
forall {k} (f :: * -> *) (as :: f k) (a :: k).
FProd f =>
Elem f as a -> Sing as -> Sing a
indexSing Elem f as a
i Sing as
xs)

-- | Turn a 'Subset' into a list (or any 'Alternative') of satisfied
-- predicates.
--
-- List is meant to include no duplicates.
subsetToList ::
  forall f p t.
  (Universe f, Alternative t) =>
  (Subset f p --># Any f p) t
subsetToList :: forall {k} (f :: * -> *) (p :: k ~> *) (t :: * -> *).
(Universe f, Alternative t) =>
(-->#) (Subset f p) (Any f p) t
subsetToList Sing a
xs Subset f p @@ a
s = Alt t (Any f p @@ a) -> t (Any f p @@ a)
forall {k} (f :: k -> *) (a :: k). Alt f a -> f a
getAlt (Alt t (Any f p @@ a) -> t (Any f p @@ a))
-> Alt t (Any f p @@ a) -> t (Any f p @@ a)
forall a b. (a -> b) -> a -> b
$ ((forall (a :: k). Elem f a a -> Sing a -> Alt t (Any f p @@ a))
-> Sing a -> Alt t (Any f p @@ a)
forall (f :: * -> *) k (as :: f k) m.
(FProd f, Monoid m) =>
(forall (a :: k). Elem f as a -> Sing a -> m) -> Sing as -> m
`ifoldMapSing` Sing a
xs) ((forall (a :: k). Elem f a a -> Sing a -> Alt t (Any f p @@ a))
 -> Alt t (Any f p @@ a))
-> (forall (a :: k). Elem f a a -> Sing a -> Alt t (Any f p @@ a))
-> Alt t (Any f p @@ a)
forall a b. (a -> b) -> a -> b
$ \Elem f a a
i Sing a
_ -> t (Any f p @@ a) -> Alt t (Any f p @@ a)
forall {k} (f :: k -> *) (a :: k). f a -> Alt f a
Alt (t (Any f p @@ a) -> Alt t (Any f p @@ a))
-> t (Any f p @@ a) -> Alt t (Any f p @@ a)
forall a b. (a -> b) -> a -> b
$ case WitSubset f p a -> forall (a :: k). Elem f a a -> Decision (p @@ a)
forall k (f :: * -> *) (p :: k ~> *) (as :: f k).
WitSubset f p as
-> forall (a :: k). Elem f as a -> Decision (p @@ a)
runWitSubset Subset f p @@ a
WitSubset f p a
s Elem f a a
i of
  Proved p @@ a
p -> (Any f p @@ a) -> t (Any f p @@ a)
forall a. a -> t a
forall (f :: * -> *) a. Applicative f => a -> f a
pure ((Any f p @@ a) -> t (Any f p @@ a))
-> (Any f p @@ a) -> t (Any f p @@ a)
forall a b. (a -> b) -> a -> b
$ Elem f a a -> (p @@ a) -> WitAny f p a
forall {k} (f :: * -> *) (b :: f k) (a1 :: k) (a :: k ~> *).
Elem f b a1 -> (a @@ a1) -> WitAny f a b
WitAny Elem f a a
i p @@ a
p
  Disproved Refuted (p @@ a)
_ -> t (Any f p @@ a)
t (WitAny f p a)
forall a. t a
forall (f :: * -> *) a. Alternative f => f a
empty

-- | Restrict a 'Subset' to a single (arbitrary) member, or fail if none
-- exists.
subsetToAny ::
  forall f p.
  Universe f =>
  Subset f p -?> Any f p
subsetToAny :: forall {k} (f :: * -> *) (p :: k ~> *).
Universe f =>
Subset f p -?> Any f p
subsetToAny Sing a
xs Subset f p @@ a
s = (forall (a :: k). Elem f a a -> Sing a -> Decision (p @@ a))
-> Sing a -> Decision (Apply (Any f p) a)
forall k (p :: k ~> *) (as :: f k).
(forall (a :: k). Elem f as a -> Sing a -> Decision (p @@ a))
-> Sing as -> Decision (Any f p @@ as)
forall (f :: * -> *) k (p :: k ~> *) (as :: f k).
Universe f =>
(forall (a :: k). Elem f as a -> Sing a -> Decision (p @@ a))
-> Sing as -> Decision (Any f p @@ as)
idecideAny (\Elem f a a
i Sing a
_ -> WitSubset f p a -> forall (a :: k). Elem f a a -> Decision (p @@ a)
forall k (f :: * -> *) (p :: k ~> *) (as :: f k).
WitSubset f p as
-> forall (a :: k). Elem f as a -> Decision (p @@ a)
runWitSubset Subset f p @@ a
WitSubset f p a
s Elem f a a
i) Sing a
xs

-- | Construct an empty subset.
emptySubset :: forall f as. (Universe f, SingI as) => Subset f Impossible @@ as
emptySubset :: forall {k} (f :: * -> *) (as :: f k).
(Universe f, SingI as) =>
Subset f Impossible @@ as
emptySubset = forall {k1} (p :: k1 ~> *). Provable p => Prove p
forall (p :: f k ~> *). Provable p => Prove p
prove @(Subset f Impossible) Sing as
forall {k} (a :: k). SingI a => Sing a
sing

-- | Construct a full subset
fullSubset :: forall f as. (Universe f, SingI as) => Subset f Evident @@ as
fullSubset :: forall {k} (f :: * -> *) (as :: f k).
(Universe f, SingI as) =>
Subset f Evident @@ as
fullSubset = forall {k1} (p :: k1 ~> *). Provable p => Prove p
forall (p :: f k ~> *). Provable p => Prove p
prove @(Subset f Evident) Sing as
forall {k} (a :: k). SingI a => Sing a
sing

-- | Test if a subset is empty.
subsetToNone :: forall f p. Universe f => Subset f p -?> None f p
subsetToNone :: forall {k} (f :: * -> *) (p :: k ~> *).
Universe f =>
Subset f p -?> None f p
subsetToNone Sing a
xs Subset f p @@ a
s = (forall (a :: k). Elem f a a -> Sing a -> Decision (p @@ a))
-> Sing a -> Decision (Apply (Not (Any f p)) a)
forall (f :: * -> *) k (p :: k ~> *) (as :: f k).
Universe f =>
(forall (a :: k). Elem f as a -> Sing a -> Decision (p @@ a))
-> Sing as -> Decision (None f p @@ as)
idecideNone (\Elem f a a
i Sing a
_ -> WitSubset f p a -> forall (a :: k). Elem f a a -> Decision (p @@ a)
forall k (f :: * -> *) (p :: k ~> *) (as :: f k).
WitSubset f p as
-> forall (a :: k). Elem f as a -> Decision (p @@ a)
runWitSubset Subset f p @@ a
WitSubset f p a
s Elem f a a
i) Sing a
xs

-- | Combine two subsets based on a decision function
imergeSubset ::
  forall f k p q r (as :: f k).
  () =>
  (forall a. Elem f as a -> Decision (p @@ a) -> Decision (q @@ a) -> Decision (r @@ a)) ->
  Subset f p @@ as ->
  Subset f q @@ as ->
  Subset f r @@ as
imergeSubset :: forall (f :: * -> *) k (p :: k ~> *) (q :: k ~> *) (r :: k ~> *)
       (as :: f k).
(forall (a :: k).
 Elem f as a
 -> Decision (p @@ a) -> Decision (q @@ a) -> Decision (r @@ a))
-> (Subset f p @@ as) -> (Subset f q @@ as) -> Subset f r @@ as
imergeSubset forall (a :: k).
Elem f as a
-> Decision (p @@ a) -> Decision (q @@ a) -> Decision (r @@ a)
f Subset f p @@ as
ps Subset f q @@ as
qs = (forall (a :: k). Elem f as a -> Decision (r @@ a))
-> WitSubset f r as
forall k (f :: * -> *) (p :: k ~> *) (as :: f k).
(forall (a :: k). Elem f as a -> Decision (p @@ a))
-> WitSubset f p as
WitSubset ((forall (a :: k). Elem f as a -> Decision (r @@ a))
 -> WitSubset f r as)
-> (forall (a :: k). Elem f as a -> Decision (r @@ a))
-> WitSubset f r as
forall a b. (a -> b) -> a -> b
$ \Elem f as a
i ->
  Elem f as a
-> Decision (p @@ a) -> Decision (q @@ a) -> Decision (r @@ a)
forall (a :: k).
Elem f as a
-> Decision (p @@ a) -> Decision (q @@ a) -> Decision (r @@ a)
f Elem f as a
i (WitSubset f p as
-> forall (a :: k). Elem f as a -> Decision (p @@ a)
forall k (f :: * -> *) (p :: k ~> *) (as :: f k).
WitSubset f p as
-> forall (a :: k). Elem f as a -> Decision (p @@ a)
runWitSubset Subset f p @@ as
WitSubset f p as
ps Elem f as a
i) (WitSubset f q as
-> forall (a :: k). Elem f as a -> Decision (q @@ a)
forall k (f :: * -> *) (p :: k ~> *) (as :: f k).
WitSubset f p as
-> forall (a :: k). Elem f as a -> Decision (p @@ a)
runWitSubset Subset f q @@ as
WitSubset f q as
qs Elem f as a
i)

-- | Combine two subsets based on a decision function
mergeSubset ::
  forall f k p q r (as :: f k).
  () =>
  (forall a. Decision (p @@ a) -> Decision (q @@ a) -> Decision (r @@ a)) ->
  Subset f p @@ as ->
  Subset f q @@ as ->
  Subset f r @@ as
mergeSubset :: forall (f :: * -> *) k (p :: k ~> *) (q :: k ~> *) (r :: k ~> *)
       (as :: f k).
(forall (a :: k).
 Decision (p @@ a) -> Decision (q @@ a) -> Decision (r @@ a))
-> (Subset f p @@ as) -> (Subset f q @@ as) -> Subset f r @@ as
mergeSubset forall (a :: k).
Decision (p @@ a) -> Decision (q @@ a) -> Decision (r @@ a)
f = (forall (a :: k).
 Elem f as a
 -> Decision (p @@ a) -> Decision (q @@ a) -> Decision (r @@ a))
-> Apply (Subset f p) as
-> Apply (Subset f q) as
-> Apply (Subset f r) as
forall (f :: * -> *) k (p :: k ~> *) (q :: k ~> *) (r :: k ~> *)
       (as :: f k).
(forall (a :: k).
 Elem f as a
 -> Decision (p @@ a) -> Decision (q @@ a) -> Decision (r @@ a))
-> (Subset f p @@ as) -> (Subset f q @@ as) -> Subset f r @@ as
imergeSubset (\(Elem f as a
_ :: Elem f as a) Decision (p @@ a)
p -> forall (a :: k).
Decision (p @@ a) -> Decision (q @@ a) -> Decision (r @@ a)
f @a Decision (p @@ a)
p)

-- | Subset intersection
intersection ::
  forall f p q.
  () =>
  ((Subset f p &&& Subset f q) --> Subset f (p &&& q))
intersection :: forall {k} (f :: * -> *) (p :: k ~> *) (q :: k ~> *) (a :: f k).
Sing a
-> ((Subset f p &&& Subset f q) @@ a) -> Subset f (p &&& q) @@ a
intersection Sing a
_ = (WitSubset f p a
 -> WitSubset f q a -> Apply (Subset f (p &&& q)) a)
-> (WitSubset f p a, WitSubset f q a)
-> Apply (Subset f (p &&& q)) a
forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry ((WitSubset f p a
  -> WitSubset f q a -> Apply (Subset f (p &&& q)) a)
 -> (WitSubset f p a, WitSubset f q a)
 -> Apply (Subset f (p &&& q)) a)
-> (WitSubset f p a
    -> WitSubset f q a -> Apply (Subset f (p &&& q)) a)
-> (WitSubset f p a, WitSubset f q a)
-> Apply (Subset f (p &&& q)) a
forall a b. (a -> b) -> a -> b
$ (forall (a :: k).
 Elem f a a
 -> Decision (p @@ a)
 -> Decision (q @@ a)
 -> Decision ((p &&& q) @@ a))
-> (Subset f p @@ a)
-> (Subset f q @@ a)
-> Apply (Subset f (p &&& q)) a
forall (f :: * -> *) k (p :: k ~> *) (q :: k ~> *) (r :: k ~> *)
       (as :: f k).
(forall (a :: k).
 Elem f as a
 -> Decision (p @@ a) -> Decision (q @@ a) -> Decision (r @@ a))
-> (Subset f p @@ as) -> (Subset f q @@ as) -> Subset f r @@ as
imergeSubset ((forall (a :: k).
  Elem f a a
  -> Decision (p @@ a)
  -> Decision (q @@ a)
  -> Decision ((p &&& q) @@ a))
 -> (Subset f p @@ a)
 -> (Subset f q @@ a)
 -> Apply (Subset f (p &&& q)) a)
-> (forall (a :: k).
    Elem f a a
    -> Decision (p @@ a)
    -> Decision (q @@ a)
    -> Decision ((p &&& q) @@ a))
-> (Subset f p @@ a)
-> (Subset f q @@ a)
-> Apply (Subset f (p &&& q)) a
forall a b. (a -> b) -> a -> b
$ \(Elem f a a
_ :: Elem f as a) -> forall {k1} (p :: k1 ~> *) (q :: k1 ~> *) (a :: k1).
Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p &&& q) @@ a)
forall (p :: k ~> *) (q :: k ~> *) (a :: k).
Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p &&& q) @@ a)
decideAnd @p @q @a

-- | Subset union (left-biased)
union ::
  forall f p q.
  () =>
  ((Subset f p &&& Subset f q) --> Subset f (p ||| q))
union :: forall {k} (f :: * -> *) (p :: k ~> *) (q :: k ~> *) (a :: f k).
Sing a
-> ((Subset f p &&& Subset f q) @@ a) -> Subset f (p ||| q) @@ a
union Sing a
_ = (WitSubset f p a
 -> WitSubset f q a -> Apply (Subset f (p ||| q)) a)
-> (WitSubset f p a, WitSubset f q a)
-> Apply (Subset f (p ||| q)) a
forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry ((WitSubset f p a
  -> WitSubset f q a -> Apply (Subset f (p ||| q)) a)
 -> (WitSubset f p a, WitSubset f q a)
 -> Apply (Subset f (p ||| q)) a)
-> (WitSubset f p a
    -> WitSubset f q a -> Apply (Subset f (p ||| q)) a)
-> (WitSubset f p a, WitSubset f q a)
-> Apply (Subset f (p ||| q)) a
forall a b. (a -> b) -> a -> b
$ (forall (a :: k).
 Elem f a a
 -> Decision (p @@ a)
 -> Decision (q @@ a)
 -> Decision ((p ||| q) @@ a))
-> (Subset f p @@ a)
-> (Subset f q @@ a)
-> Apply (Subset f (p ||| q)) a
forall (f :: * -> *) k (p :: k ~> *) (q :: k ~> *) (r :: k ~> *)
       (as :: f k).
(forall (a :: k).
 Elem f as a
 -> Decision (p @@ a) -> Decision (q @@ a) -> Decision (r @@ a))
-> (Subset f p @@ as) -> (Subset f q @@ as) -> Subset f r @@ as
imergeSubset ((forall (a :: k).
  Elem f a a
  -> Decision (p @@ a)
  -> Decision (q @@ a)
  -> Decision ((p ||| q) @@ a))
 -> (Subset f p @@ a)
 -> (Subset f q @@ a)
 -> Apply (Subset f (p ||| q)) a)
-> (forall (a :: k).
    Elem f a a
    -> Decision (p @@ a)
    -> Decision (q @@ a)
    -> Decision ((p ||| q) @@ a))
-> (Subset f p @@ a)
-> (Subset f q @@ a)
-> Apply (Subset f (p ||| q)) a
forall a b. (a -> b) -> a -> b
$ \(Elem f a a
_ :: Elem f as a) -> forall {k1} (p :: k1 ~> *) (q :: k1 ~> *) (a :: k1).
Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p ||| q) @@ a)
forall (p :: k ~> *) (q :: k ~> *) (a :: k).
Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p ||| q) @@ a)
decideOr @p @q @a

-- | Symmetric subset difference
symDiff ::
  forall f p q.
  () =>
  ((Subset f p &&& Subset f q) --> Subset f (p ^^^ q))
symDiff :: forall {k} (f :: * -> *) (p :: k ~> *) (q :: k ~> *) (a :: f k).
Sing a
-> ((Subset f p &&& Subset f q) @@ a) -> Subset f (p ^^^ q) @@ a
symDiff Sing a
_ = (WitSubset f p a
 -> WitSubset f q a -> Apply (Subset f (p ^^^ q)) a)
-> (WitSubset f p a, WitSubset f q a)
-> Apply (Subset f (p ^^^ q)) a
forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry ((WitSubset f p a
  -> WitSubset f q a -> Apply (Subset f (p ^^^ q)) a)
 -> (WitSubset f p a, WitSubset f q a)
 -> Apply (Subset f (p ^^^ q)) a)
-> (WitSubset f p a
    -> WitSubset f q a -> Apply (Subset f (p ^^^ q)) a)
-> (WitSubset f p a, WitSubset f q a)
-> Apply (Subset f (p ^^^ q)) a
forall a b. (a -> b) -> a -> b
$ (forall (a :: k).
 Elem f a a
 -> Decision (p @@ a)
 -> Decision (q @@ a)
 -> Decision ((p ^^^ q) @@ a))
-> (Subset f p @@ a)
-> (Subset f q @@ a)
-> Apply (Subset f (p ^^^ q)) a
forall (f :: * -> *) k (p :: k ~> *) (q :: k ~> *) (r :: k ~> *)
       (as :: f k).
(forall (a :: k).
 Elem f as a
 -> Decision (p @@ a) -> Decision (q @@ a) -> Decision (r @@ a))
-> (Subset f p @@ as) -> (Subset f q @@ as) -> Subset f r @@ as
imergeSubset ((forall (a :: k).
  Elem f a a
  -> Decision (p @@ a)
  -> Decision (q @@ a)
  -> Decision ((p ^^^ q) @@ a))
 -> (Subset f p @@ a)
 -> (Subset f q @@ a)
 -> Apply (Subset f (p ^^^ q)) a)
-> (forall (a :: k).
    Elem f a a
    -> Decision (p @@ a)
    -> Decision (q @@ a)
    -> Decision ((p ^^^ q) @@ a))
-> (Subset f p @@ a)
-> (Subset f q @@ a)
-> Apply (Subset f (p ^^^ q)) a
forall a b. (a -> b) -> a -> b
$ \(Elem f a a
_ :: Elem f as a) -> forall {k1} (p :: k1 ~> *) (q :: k1 ~> *) (a :: k1).
Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p ^^^ q) @@ a)
forall (p :: k ~> *) (q :: k ~> *) (a :: k).
Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p ^^^ q) @@ a)
decideXor @p @q @a

-- | Test if a subset is equal to the entire original collection
subsetToAll ::
  forall f p.
  Universe f =>
  Subset f p -?> All f p
subsetToAll :: forall {k} (f :: * -> *) (p :: k ~> *).
Universe f =>
Subset f p -?> All f p
subsetToAll Sing a
xs Subset f p @@ a
s = (forall (a :: k). Elem f a a -> Sing a -> Decision (p @@ a))
-> Sing a -> Decision (Apply (All f p) a)
forall k (p :: k ~> *) (as :: f k).
(forall (a :: k). Elem f as a -> Sing a -> Decision (p @@ a))
-> Sing as -> Decision (All f p @@ as)
forall (f :: * -> *) k (p :: k ~> *) (as :: f k).
Universe f =>
(forall (a :: k). Elem f as a -> Sing a -> Decision (p @@ a))
-> Sing as -> Decision (All f p @@ as)
idecideAll (\Elem f a a
i Sing a
_ -> WitSubset f p a -> forall (a :: k). Elem f a a -> Decision (p @@ a)
forall k (f :: * -> *) (p :: k ~> *) (as :: f k).
WitSubset f p as
-> forall (a :: k). Elem f as a -> Decision (p @@ a)
runWitSubset Subset f p @@ a
WitSubset f p a
s Elem f a a
i) Sing a
xs

-- | 'mapSubset', but providing an 'Elem'.
imapSubset ::
  (forall a. Elem f as a -> p @@ a -> q @@ a) ->
  (forall a. Elem f as a -> q @@ a -> p @@ a) ->
  Subset f p @@ as ->
  Subset f q @@ as
imapSubset :: forall {k} (f :: * -> *) (as :: f k) (p :: k ~> *) (q :: k ~> *).
(forall (a :: k). Elem f as a -> (p @@ a) -> q @@ a)
-> (forall (a :: k). Elem f as a -> (q @@ a) -> p @@ a)
-> (Subset f p @@ as)
-> Subset f q @@ as
imapSubset forall (a :: k). Elem f as a -> (p @@ a) -> q @@ a
f forall (a :: k). Elem f as a -> (q @@ a) -> p @@ a
g Subset f p @@ as
s = (forall (a :: k). Elem f as a -> Decision (q @@ a))
-> WitSubset f q as
forall k (f :: * -> *) (p :: k ~> *) (as :: f k).
(forall (a :: k). Elem f as a -> Decision (p @@ a))
-> WitSubset f p as
WitSubset ((forall (a :: k). Elem f as a -> Decision (q @@ a))
 -> WitSubset f q as)
-> (forall (a :: k). Elem f as a -> Decision (q @@ a))
-> WitSubset f q as
forall a b. (a -> b) -> a -> b
$ \Elem f as a
i ->
  (Apply p a -> q @@ a)
-> ((q @@ a) -> Apply p a)
-> Decision (Apply p a)
-> Decision (q @@ a)
forall a b. (a -> b) -> (b -> a) -> Decision a -> Decision b
mapDecision (Elem f as a -> Apply p a -> q @@ a
forall (a :: k). Elem f as a -> (p @@ a) -> q @@ a
f Elem f as a
i) (Elem f as a -> (q @@ a) -> Apply p a
forall (a :: k). Elem f as a -> (q @@ a) -> p @@ a
g Elem f as a
i) (WitSubset f p as
-> forall (a :: k). Elem f as a -> Decision (p @@ a)
forall k (f :: * -> *) (p :: k ~> *) (as :: f k).
WitSubset f p as
-> forall (a :: k). Elem f as a -> Decision (p @@ a)
runWitSubset Subset f p @@ as
WitSubset f p as
s Elem f as a
i)

-- | Map a bidirectional implication over a subset described by that
-- implication.
--
-- Implication needs to be bidirectional, or otherwise we can't produce
-- a /decidable/ subset as a result.
mapSubset ::
  Universe f =>
  (p --> q) ->
  (q --> p) ->
  (Subset f p --> Subset f q)
mapSubset :: forall {k} (f :: * -> *) (p :: k ~> *) (q :: k ~> *).
Universe f =>
(p --> q) -> (q --> p) -> Subset f p --> Subset f q
mapSubset p --> q
f q --> p
g Sing a
xs =
  Sing a
-> (SingI a => (Subset f p @@ a) -> Subset f q @@ a)
-> (Subset f p @@ a)
-> Subset f q @@ a
forall {k} (n :: k) r. Sing n -> (SingI n => r) -> r
withSingI Sing a
xs ((SingI a => (Subset f p @@ a) -> Subset f q @@ a)
 -> (Subset f p @@ a) -> Subset f q @@ a)
-> (SingI a => (Subset f p @@ a) -> Subset f q @@ a)
-> (Subset f p @@ a)
-> Subset f q @@ a
forall a b. (a -> b) -> a -> b
$
    (forall (a :: k). Elem f a a -> (p @@ a) -> q @@ a)
-> (forall (a :: k). Elem f a a -> (q @@ a) -> p @@ a)
-> (Subset f p @@ a)
-> Subset f q @@ a
forall {k} (f :: * -> *) (as :: f k) (p :: k ~> *) (q :: k ~> *).
(forall (a :: k). Elem f as a -> (p @@ a) -> q @@ a)
-> (forall (a :: k). Elem f as a -> (q @@ a) -> p @@ a)
-> (Subset f p @@ as)
-> Subset f q @@ as
imapSubset
      (\Elem f a a
i -> Sing a -> (p @@ a) -> q @@ a
p --> q
f (Elem f a a -> Sing a -> Sing a
forall {k} (f :: * -> *) (as :: f k) (a :: k).
FProd f =>
Elem f as a -> Sing as -> Sing a
indexSing Elem f a a
i Sing a
xs))
      (\Elem f a a
i -> Sing a -> (q @@ a) -> p @@ a
q --> p
g (Elem f a a -> Sing a -> Sing a
forall {k} (f :: * -> *) (as :: f k) (a :: k).
FProd f =>
Elem f as a -> Sing as -> Sing a
indexSing Elem f a a
i Sing a
xs))