mirror of
https://github.com/idris-lang/Idris2.git
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360 lines
13 KiB
Idris
360 lines
13 KiB
Idris
------------------------------------------------------------------------
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-- This module is based on the following papers:
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-- Categories of Containers
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-- Abbott, Altenkirch, Ghani
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-- Derivatives of Containers
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-- Abbott, Altenkirch, Ghani, McBride
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------------------------------------------------------------------------
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module Data.Container
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import Data.Either
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import Decidable.Equality
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%default total
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------------------------------------------------------------------------
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-- Container and their morphisms
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-- * Extension is a functor from Container to Type
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-- Objects of the category of containers
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namespace Container
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public export
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record Container where
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constructor MkContainer
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Shape : Type
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Position : Shape -> Type
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public export
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record Extension (c : Container) (x : Type) where
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constructor MkExtension
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shape : Shape c
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payloads : Position c shape -> x
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||| The image of a container by @Extension@ is a functor
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public export
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Functor (Extension c) where map f (MkExtension s p) = MkExtension s (f . p)
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-- Morphisms of the category of containers
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namespace Morphism
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public export
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record Morphism (c, d : Container) where
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constructor MkMorphism
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shapeMorphism : Shape c -> Shape d
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positionMorphism : {s : Shape c} -> Position d (shapeMorphism s) -> Position c s
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public export
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Extension : Morphism c d -> Extension c x -> Extension d x
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Extension phi (MkExtension s p)
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= MkExtension (shapeMorphism phi s) (p . positionMorphism phi)
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------------------------------------------------------------------------
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-- Combinators to build containers
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namespace Combinators
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-- Constant
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public export
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Const : Type -> Container
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Const k = MkContainer k (const Void)
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export
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toConst : k -> Extension (Const k) x
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toConst v = MkExtension v absurd
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export
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fromConst : Extension (Const k) x -> k
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fromConst (MkExtension v _) = v
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-- Identity
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public export
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Identity : Container
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Identity = MkContainer () (\ () => ())
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export
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toIdentity : x -> Extension Identity x
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toIdentity v = MkExtension () (const v)
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export
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fromIdentity : Extension Identity x -> x
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fromIdentity (MkExtension () chld) = chld ()
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-- Composition
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public export
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Compose : (d, c : Container) -> Container
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Compose d c = MkContainer
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(Extension d (Shape c))
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(\ (MkExtension shp chld) => (p : Position d shp ** Position c (chld p)))
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export
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toCompose : (Extension d . Extension c) x -> Extension (Compose d c) x
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toCompose (MkExtension shp1 chld)
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= MkExtension (MkExtension shp1 (shape . chld)) (\ (p ** q) => payloads (chld p) q)
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export
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fromCompose : Extension (Compose d c) x -> (Extension d . Extension c) x
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fromCompose (MkExtension (MkExtension shp1 shp2) chld)
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= MkExtension shp1 (\ p => MkExtension (shp2 p) (\ q => chld (p ** q)))
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-- Direct sum
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public export
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Sum : (c, d : Container) -> Container
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Sum c d = MkContainer (Either (Shape c) (Shape d)) (either (Position c) (Position d))
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export
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toSum : Either (Extension c x) (Extension d x) -> Extension (Sum c d) x
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toSum (Left (MkExtension shp chld)) = MkExtension (Left shp) chld
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toSum (Right (MkExtension shp chld)) = MkExtension (Right shp) chld
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export
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fromSum : Extension (Sum c d) x -> Either (Extension c x) (Extension d x)
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fromSum (MkExtension (Left shp) chld) = Left (MkExtension shp chld)
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fromSum (MkExtension (Right shp) chld) = Right (MkExtension shp chld)
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-- Pairing
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public export
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Pair : (c, d : Container) -> Container
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Pair c d = MkContainer (Shape c, Shape d) (\ (p, q) => Either (Position c p) (Position d q))
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export
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toPair : (Extension c x, Extension d x) -> Extension (Pair c d) x
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toPair (MkExtension shp1 chld1, MkExtension shp2 chld2)
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= MkExtension (shp1, shp2) (either chld1 chld2)
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export
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fromPair : Extension (Pair c d) x -> (Extension c x, Extension d x)
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fromPair (MkExtension (shp1, shp2) chld)
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= (MkExtension shp1 (chld . Left), MkExtension shp2 (chld . Right))
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-- Branching over a Type
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public export
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Exponential : Type -> Container -> Container
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Exponential k c = MkContainer (k -> Shape c) (\ p => (v : k ** Position c (p v)))
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export
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toExponential : (k -> Extension c x) -> Extension (Exponential k c) x
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toExponential f = MkExtension (shape . f) (\ (v ** p) => payloads (f v) p)
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export
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fromExponential : Extension (Exponential k c) x -> (k -> Extension c x)
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fromExponential (MkExtension shp chld) k = MkExtension (shp k) (\ p => chld (k ** p))
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------------------------------------------------------------------------
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-- Taking various fixpoints of containers
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namespace Initial
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public export
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data W : Container -> Type where
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MkW : Extension c (W c) -> W c
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export
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map : Morphism c d -> W c -> W d
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map f (MkW (MkExtension shp chld)) = MkW $ Extension f (MkExtension shp (\ p => map f (chld p)))
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-- Container.map inlined because of -------------------^
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-- termination checking
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export
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foldr : (Extension c x -> x) -> W c -> x
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foldr alg (MkW (MkExtension shp chld)) = alg (MkExtension shp (\ p => foldr alg (chld p)))
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export
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para : (Extension c (x, W c) -> x) -> W c -> x
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para alg (MkW (MkExtension shp chld)) = alg (MkExtension shp (\ p => let w = chld p in (para alg w, w)))
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namespace Monad
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||| @Free@ is a wrapper around @W@ to make it inference friendly.
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||| Without this wrapper, neither @pure@ nor @bind@ are able to infer their @c@ argument.
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public export
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record Free (c : Container) (x : Type) where
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constructor MkFree
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runFree : W (Sum c (Const x))
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export
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pure : x -> Free c x
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pure x = MkFree $ MkW (toSum (Right (toConst x)))
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export
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(>>=) : Free c x -> (x -> Free c y) -> Free c y
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(>>=) (MkFree mx) k = foldr (alg . fromSum {c} {d = Const x}) mx where
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alg : Either (Extension c (Free c y)) (Extension (Const x) (Free c y)) -> Free c y
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alg = either (MkFree . MkW . toSum {c} {d = Const y} . Left . map (runFree {c}))
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(k . fromConst {k = x})
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export
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join : Free c (Free c x) -> Free c x
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join = (>>= id)
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namespace Final
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public export
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data M : Container -> Type where
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MkM : Extension c (Inf (M c)) -> M c
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export
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unfoldr : (s -> Extension c s) -> s -> M c
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unfoldr next seed =
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let (MkExtension shp chld) = next seed in
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MkM (MkExtension shp (\ p => unfoldr next (chld p)))
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namespace Comonad
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||| @Cofree@ is a wrapper around @M@ to make it inference friendly.
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||| Without this wrapper, neither @extract@ nor @extend@ are able to infer their @c@ argument.
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public export
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record Cofree (c : Container) (x : Type) where
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constructor MkCofree
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runCofree : M (Pair (Const x) c)
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export
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extract : Cofree c x -> x
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extract (MkCofree (MkM m)) = fst (shape m)
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export
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extend : (Cofree c a -> b) -> Cofree c a -> Cofree c b
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extend alg = MkCofree . unfoldr next . runCofree where
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next : M (Pair (Const a) c) -> Extension (Pair (Const b) c) (M (Pair (Const a) c))
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next m@(MkM layer) =
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let (_, (MkExtension shp chld)) = fromPair {c = Const a} layer in
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let b = toConst (alg (MkCofree m)) in
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toPair (b, MkExtension shp (\ p => chld p))
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-- Eta-expanded to force Inf ------^
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export
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duplicate : Cofree c a -> Cofree c (Cofree c a)
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duplicate = extend id
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------------------------------------------------------------------------
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-- Derivative
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namespace Derivative
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public export
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Derivative : Container -> Container
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Derivative c = MkContainer
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(s : Shape c ** Position c s)
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(\ (s ** p) => (p' : Position c s ** Not (p === p')))
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export
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hole : (v : Extension (Derivative c) x) -> Position c (fst (shape v))
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hole (MkExtension (shp ** p) _) = p
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export
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unplug : (v : Extension c x) -> Position c (shape v) -> (Extension (Derivative c) x, x)
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unplug (MkExtension shp chld) p = (MkExtension (shp ** p) (chld . fst), chld p)
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export
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plug : (v : Extension (Derivative c) x) -> DecEq (Position c (fst (shape v))) => x -> Extension c x
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plug (MkExtension (shp ** p) chld) x = MkExtension shp $ \ p' => case decEq p p' of
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Yes eq => x
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No neq => chld (p' ** neq)
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export
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toConst : Extension (Const Void) x -> Extension (Derivative (Const k)) x
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toConst v = absurd (fromConst v)
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export
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fromConst : Extension (Derivative (Const k)) x -> Extension (Const Void) x
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fromConst v = absurd (hole {c = Const _} v)
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export
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toIdentity : Extension (Derivative Identity) x
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toIdentity = MkExtension (() ** ()) (\ (() ** eq) => absurd (eq Refl))
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export
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toSum : Extension (Sum (Derivative c) (Derivative d)) x ->
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Extension (Derivative (Sum c d)) x
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toSum v = case fromSum {c = Derivative c} {d = Derivative d} v of
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Left (MkExtension (shp ** p) chld) => MkExtension (Left shp ** p) chld
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Right (MkExtension (shp ** p) chld) => MkExtension (Right shp ** p) chld
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export
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fromSum : Extension (Derivative (Sum c d)) x ->
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Extension (Sum (Derivative c) (Derivative d)) x
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fromSum (MkExtension (shp ** p) chld) = toSum {c = Derivative c} {d = Derivative d} $ case shp of
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Left shp => Left (MkExtension (shp ** p) chld)
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Right shp => Right (MkExtension (shp ** p) chld)
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export
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toPair : Extension (Sum (Pair (Derivative c) d) (Pair c (Derivative d))) x ->
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Extension (Derivative (Pair c d)) x
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toPair v = case fromSum {c = Pair (Derivative c) d} {d = Pair c (Derivative d)} v of
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Left p => let (MkExtension (shp1 ** p1) chld1, MkExtension shp2 chld2) = fromPair {c = Derivative c} p in
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MkExtension ((shp1, shp2) ** Left p1) $ \ (p' ** neq) => case p' of
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Left p1' => chld1 (p1' ** (\prf => neq $ cong Left prf))
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Right p2' => chld2 p2'
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Right p => let (MkExtension shp1 chld1, MkExtension (shp2 ** p2) chld2) = fromPair {c} {d = Derivative d} p in
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MkExtension ((shp1, shp2) ** Right p2) $ \ (p' ** neq) => case p' of
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Left p1' => chld1 p1'
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Right p2' => chld2 (p2' ** (\prf => neq $ cong Right prf))
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export
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fromPair : Extension (Derivative (Pair c d)) x ->
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Extension (Sum (Pair (Derivative c) d) (Pair c (Derivative d))) x
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fromPair (MkExtension ((shp1, shp2) ** p) chld) = case p of
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Left p1 => toSum {c = Pair (Derivative c) d} {d = Pair c (Derivative d)}
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(Left (MkExtension ((shp1 ** p1), shp2) $ either
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(\ p1' => chld (Left (DPair.fst p1') ** DPair.snd p1' . injective))
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(\ p2 => chld (Right p2 ** absurd))))
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Right p2 => toSum {c = Pair (Derivative c) d} {d = Pair c (Derivative d)}
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(Right (MkExtension (shp1, (shp2 ** p2)) $ either
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(\ p1 => chld (Left p1 ** absurd))
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(\ p2' => chld (Right (DPair.fst p2') ** DPair.snd p2' . injective))))
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export
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fromCompose : Extension (Derivative (Compose c d)) x ->
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Extension (Pair (Derivative d) (Compose (Derivative c) d)) x
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fromCompose (MkExtension (MkExtension shp1 shp2 ** (p1 ** p2)) chld)
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= toPair (left, right) where
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left : Extension (Derivative d) x
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left = MkExtension (shp2 p1 ** p2)
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$ \ (p2' ** neqp2) => chld ((p1 ** p2') ** neqp2 . mkDPairInjectiveSnd)
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right : Extension (Compose (Derivative c) d) x
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right = toCompose
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$ MkExtension (shp1 ** p1)
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$ \ (p1' ** neqp1) => MkExtension (shp2 p1')
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$ \ p2' => chld ((p1' ** p2') ** (\prf => neqp1 $ cong fst prf))
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export
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toCompose : ((s : _) -> DecEq (Position c s)) -> ((s : _) -> DecEq (Position d s)) ->
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Extension (Pair (Derivative d) (Compose (Derivative c) d)) x ->
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Extension (Derivative (Compose c d)) x
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toCompose dec1 dec2 v with (fromPair {c = Derivative d} {d = Compose (Derivative c) d} v)
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toCompose dec1 dec2 v | (MkExtension (shp2 ** p2) chld2, w) with (fromCompose w)
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toCompose dec1 dec2 v
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| (MkExtension (shp2 ** p2) chld2, w)
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| (MkExtension (shp1 ** p1) chld1)
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= MkExtension (MkExtension shp1 (\ p1' => shp2' p1' (decEq @{dec1 shp1} p1 p1')) **
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(p1 ** (p2' (decEq @{dec1 shp1} p1 p1))))
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$ \ ((p1' ** p2'') ** neq) => chld2' p1' p2'' neq
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where
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shp2' : (p1' : Position c shp1) -> Dec (p1 === p1') -> Shape d
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shp2' p1' (Yes eq) = shp2
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shp2' p1' (No neq) = shape (chld1 (p1' ** neq))
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p2' : (eq : Dec (p1 === p1)) -> Position d (shp2' p1 eq)
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p2' (Yes Refl) = p2
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p2' (No neq) = absurd (neq Refl)
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chld2' : (p1' : Position c shp1) ->
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(p2'' : Position d (shp2' p1' (decEq @{dec1 shp1} p1 p1'))) ->
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(neq : Not (MkDPair p1 (p2' (decEq @{dec1 shp1} p1 p1)) = MkDPair p1' p2'')) -> x
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chld2' p1' p2'' neq with (decEq @{dec1 shp1} p1 p1')
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chld2' p1' p2'' neq | No neq1 = payloads (chld1 (p1' ** neq1)) p2''
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chld2' _ p2'' neq | Yes Refl with (decEq @{dec1 shp1} p1 p1)
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chld2' _ p2'' neq | Yes Refl | No argh = absurd (argh Refl)
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chld2' _ p2'' neq | Yes Refl | Yes Refl with (decEq @{dec2 shp2} p2 p2'')
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chld2' _ p2'' neq | Yes Refl | Yes Refl | No neq2 = chld2 (p2'' ** neq2)
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chld2' _ _ neq | Yes Refl | Yes Refl | Yes Refl = absurd (neq Refl)
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