quox/lib/Quox/Context.idr

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module Quox.Context
import Quox.Syntax.Shift
import Quox.Pretty
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import public Quox.NatExtra
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import Data.DPair
import Data.Nat
import Data.SnocList
import Control.Monad.Identity
%default total
infixl 5 :<
||| a sequence of bindings under an existing context. each successive element
||| has one more bound variable, which correspond to all previous elements
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||| as well as the surrounding context.
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public export
data Telescope : (tm : Nat -> Type) -> (from, to : Nat) -> Type where
Lin : Telescope tm from from
(:<) : Telescope tm from to -> tm to -> Telescope tm from (S to)
%name Telescope tel
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public export
Telescope' : (a : Type) -> (from, to : Nat) -> Type
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Telescope' a = Telescope (\_ => a)
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||| a top level context is actually just a telescope over no existing bindings
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public export
Context : (tm : Nat -> Type) -> (len : Nat) -> Type
Context tm len = Telescope tm 0 len
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public export
Context' : (a : Type) -> (len : Nat) -> Type
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Context' a = Context (\_ => a)
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public export
tail : Context tm (S n) -> Context tm n
tail (tel :< _) = tel
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export
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toSnocList : Telescope tm _ _ -> SnocList (Exists tm)
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toSnocList [<] = [<]
toSnocList (tel :< t) = toSnocList tel :< Evidence _ t
private
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toListAcc : Telescope tm _ _ -> List (Exists tm) -> List (Exists tm)
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toListAcc [<] acc = acc
toListAcc (tel :< t) acc = toListAcc tel (Evidence _ t :: acc)
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export %inline
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toList : Telescope tm _ _ -> List (Exists tm)
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toList tel = toListAcc tel []
export %inline
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toSnocList' : Telescope' a _ _ -> SnocList a
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toSnocList' = map snd . toSnocList
export %inline
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toList' : Telescope' a _ _ -> List a
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toList' = map snd . toList
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infixl 9 .
public export
(.) : Telescope tm from mid -> Telescope tm mid to -> Telescope tm from to
tel1 . [<] = tel1
tel1 . (tel2 :< s) = (tel1 . tel2) :< s
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public export
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getShiftWith : (forall from, to. tm from -> Shift from to -> tm to) ->
Shift len out -> Context tm len -> Var len -> tm out
getShiftWith shft by (ctx :< t) VZ = t `shft` ssDown by
getShiftWith shft by (ctx :< t) (VS i) = getShiftWith shft (ssDown by) ctx i
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public export %inline
getShift : CanShift tm => Shift len out -> Context tm len -> Var len -> tm out
getShift = getShiftWith (//)
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public export %inline
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getWith : (forall from, to. tm from -> Shift from to -> tm to) ->
Context tm len -> Var len -> tm len
getWith shft = getShiftWith shft SZ
infixl 8 !!
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public export %inline
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(!!) : CanShift tm => Context tm len -> Var len -> tm len
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(!!) = getWith (//)
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infixl 8 !!!
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public export %inline
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(!!!) : Context' tm len -> Var len -> tm
(!!!) = getWith const
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||| a triangle of bindings. each type binding in a context counts the ues of
||| others in its type, and all of these together form a triangle.
public export
Triangle : (tm : Nat -> Type) -> (len : Nat) -> Type
Triangle = Context . Context
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public export
Triangle' : Type -> (len : Nat) -> Type
Triangle' a = Context $ Context (\_ => a)
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export
0 telescopeLTE : Telescope _ from to -> from `LTE` to
telescopeLTE [<] = reflexive
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telescopeLTE (tel :< _) = lteSuccRight $ telescopeLTE tel
export
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(gt : from `GT` to) => Uninhabited (Telescope _ from to) where
uninhabited tel = void $ LTEImpliesNotGT (telescopeLTE tel) gt
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export %hint
0 succGT : S n `GT` n
succGT = LTESucc reflexive
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parameters {auto _ : Applicative f}
export
traverse : (forall n. tm1 n -> f (tm2 n)) ->
Telescope tm1 from to -> f (Telescope tm2 from to)
traverse f [<] = pure [<]
traverse f (tel :< x) = [|traverse f tel :< f x|]
infixl 3 `app`
||| like `(<*>)` but with effects
export
app : Telescope (\n => tm1 n -> f (tm2 n)) from to ->
Telescope tm1 from to -> f (Telescope tm2 from to)
app [<] [<] = pure [<]
app (ftel :< f) (xtel :< x) = [|app ftel xtel :< f x|]
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app [<] (xtel :< _) = void $ uninhabited xtel
app (ftel :< _) [<] = void $ uninhabited ftel
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export %inline
sequence : Telescope (f . tm) from to -> f (Telescope tm from to)
sequence = traverse id
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parameters {0 tm1, tm2 : Nat -> Type}
(f : forall n. tm1 n -> tm2 n)
export %inline
map : Telescope tm1 from to -> Telescope tm2 from to
map = runIdentity . traverse (pure . f)
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export %inline
(<$>) : Telescope tm1 from to -> Telescope tm2 from to
(<$>) = map
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export %inline
(<*>) : Telescope (\n => tm1 n -> tm2 n) from to ->
Telescope tm1 from to -> Telescope tm2 from to
ftel <*> xtel = runIdentity $ (pure .) <$> ftel `app` xtel
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-- ...but can't write pure without `from,to` being ω, so no idiom brackets ☹
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export %inline
(<$) : (forall n. tm1 n) -> Telescope tm2 from to -> Telescope tm1 from to
x <$ tel = const x <$> tel
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export
teleLte' : Telescope tm from to -> from `LTE'` to
teleLte' [<] = LTERefl
teleLte' (tel :< _) = LTESuccR (teleLte' tel)
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export
tabulate : ((n : Nat) -> tm n) ->
(from, to : Nat) -> from `LTE'` to => Telescope tm from to
tabulate f from from @{LTERefl} = [<]
tabulate f from (S to) @{LTESuccR _} = tabulate f from to :< f to
export
tabulate0 : ((n : Nat) -> tm n) -> (n : Nat) -> Context tm n
tabulate0 f n = tabulate f 0 n
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export
pure : from `LTE'` to => a -> Telescope' a from to
pure @{LTERefl} x = [<]
pure @{LTESuccR _} x = pure x :< x
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export %inline
zipWith : (forall n. tm1 n -> tm2 n -> tm3 n) ->
Telescope tm1 from to -> Telescope tm2 from to ->
Telescope tm3 from to
zipWith f tel1 tel2 = f <$> tel1 <*> tel2
export %inline
zipWith3 : (forall n. tm1 n -> tm2 n -> tm3 n -> tm4 n) ->
Telescope tm1 from to ->
Telescope tm2 from to ->
Telescope tm3 from to ->
Telescope tm4 from to
zipWith3 f tel1 tel2 tel3 = f <$> tel1 <*> tel2 <*> tel3
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export %inline
zipWithLazy : forall tm1, tm2, tm3.
(forall n. tm1 n -> tm2 n -> tm3 n) ->
Telescope tm1 from to -> Telescope tm2 from to ->
Telescope (\n => Lazy (tm3 n)) from to
zipWithLazy f = zipWith $ delay .: f
export %inline
zipWith3Lazy : forall tm1, tm2, tm3, tm4.
(forall n. tm1 n -> tm2 n -> tm3 n -> tm4 n) ->
Telescope tm1 from to ->
Telescope tm2 from to ->
Telescope tm3 from to ->
Telescope (\n => Lazy (tm4 n)) from to
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zipWith3Lazy f = zipWith3 $ \x, y, z => delay $ f x y z
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export
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lengthPrf : Telescope _ from to -> Subset Nat (\len => len + from = to)
lengthPrf [<] = Element 0 Refl
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lengthPrf (tel :< _) =
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let len = lengthPrf tel in Element (S len.fst) (cong S len.snd)
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export
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lengthPrf0 : Context _ to -> Subset Nat (\len => len = to)
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lengthPrf0 ctx =
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let len = lengthPrf ctx in
Element len.fst (rewrite sym $ plusZeroRightNeutral len.fst in len.snd)
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public export %inline
length : Telescope {} -> Nat
length = fst . lengthPrf
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export
foldl : {0 acc : Nat -> Type} ->
(f : forall n. acc n -> tm (n + from) -> acc (S n)) ->
(z : acc 0) -> (tel : Telescope tm from to) -> acc (length tel)
foldl f z [<] = z
foldl f z (tel :< t) = f (foldl f z tel) (rewrite (lengthPrf tel).snd in t)
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export %inline
foldMap : Monoid a => (forall n. tm n -> a) -> Telescope tm from to -> a
foldMap f = foldl (\acc, tm => acc <+> f tm) neutral
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export %inline
fold : Monoid a => Telescope' a from to -> a
fold = foldMap id
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||| like `fold` but calculate the elements only when actually appending
export %inline
foldLazy : Monoid a => Telescope' (Lazy a) from to -> a
foldLazy = foldMap force
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export %inline
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and : Telescope' (Lazy Bool) _ _ -> Bool
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and = force . fold @{All}
export %inline
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all : (forall n. tm n -> Bool) -> Telescope tm _ _ -> Bool
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all p = and . map (delay . p)
export %inline
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all2 : (forall n. tm n -> tm2 n -> Bool) ->
Telescope tm from to -> Telescope tm2 from to -> Bool
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all2 p = and .: zipWithLazy p
export %inline
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or : Telescope' (Lazy Bool) _ _ -> Bool
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or = force . fold @{Any}
export %inline
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any : (forall n. tm n -> Bool) -> Telescope tm _ _ -> Bool
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any p = or . map (delay . p)
export %inline
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any2 : (forall n. tm1 n -> tm2 n -> Bool) ->
Telescope tm1 from to -> Telescope tm2 from to -> Bool
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any2 p = or .: zipWithLazy p
export %inline
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(forall n. Eq (tm n)) => Eq (Telescope tm from to) where
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(==) = all2 (==)
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export %inline
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(forall n. Ord (tm n)) => Ord (Telescope tm from to) where
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compare = foldLazy .: zipWithLazy compare
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export %inline
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(forall n. PrettyHL (tm n)) => PrettyHL (Telescope tm from to) where
prettyM tel = separate (hl Delim ";") <$> traverse prettyM (toList tel)