split OPE stuff into modules
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lib/Quox/OPE.idr
371
lib/Quox/OPE.idr
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@ -2,368 +2,9 @@
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||| a smaller scope and part of a larger one.
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module Quox.OPE
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module Quox.OPE
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import Quox.NatExtra
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import public Quox.OPE.Basics
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import public Data.DPair
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import public Quox.OPE.Length
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import public Data.SnocList
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import public Quox.OPE.Sub
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import public Data.SnocList.Elem
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import public Quox.OPE.Split
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import public Quox.OPE.Comp
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%default total
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import public Quox.OPE.Cover
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LTE_n = Nat.LTE
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%hide Nat.LTE
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public export
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Scope : Type -> Type
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Scope = SnocList
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public export
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data LTE : Scope a -> Scope a -> Type where
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End : [<] `LTE` [<]
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Keep : xs `LTE` ys -> xs :< z `LTE` ys :< z
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Drop : xs `LTE` ys -> xs `LTE` ys :< z
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%name LTE p, q
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-- [todo] bitmask representation???
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export
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dropLast : (xs :< x) `LTE` ys -> xs `LTE` ys
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dropLast (Keep p) = Drop p
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dropLast (Drop p) = Drop $ dropLast p
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export
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Uninhabited (xs :< x `LTE` [<]) where uninhabited _ impossible
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export
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Uninhabited (xs :< x `LTE` xs) where
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uninhabited (Keep p) = uninhabited p
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uninhabited (Drop p) = uninhabited $ dropLast p
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export
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0 lteLen : xs `LTE` ys -> length xs `LTE_n` length ys
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lteLen End = LTEZero
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lteLen (Keep p) = LTESucc $ lteLen p
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lteLen (Drop p) = lteSuccRight $ lteLen p
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export
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0 lteNilRight : xs `LTE` [<] -> xs = [<]
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lteNilRight End = Refl
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export
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0 lteNilLeftDrop : (p : [<] `LTE` (xs :< x)) -> Exists (\q => p = Drop q)
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lteNilLeftDrop (Drop q) = Evidence q Refl
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export
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0 lteNil2End : (p : [<] `LTE` [<]) -> p = End
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lteNil2End End = Refl
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public export
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data Length : Scope a -> Type where
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Z : Length [<]
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S : (s : Length xs) -> Length (xs :< x)
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%name Length s
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%builtin Natural Length
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namespace Length
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public export
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(.nat) : Length xs -> Nat
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(Z).nat = Z
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(S s).nat = S s.nat
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%transform "Length.nat" Length.(.nat) xs = believe_me xs
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export
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0 lengthOk : (s : Length xs) -> s.nat = length xs
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lengthOk Z = Refl
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lengthOk (S s) = cong S $ lengthOk s
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export %hint
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lengthLeft : xs `LTE` ys -> Length xs
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lengthLeft End = Z
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lengthLeft (Keep p) = S (lengthLeft p)
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lengthLeft (Drop p) = lengthLeft p
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export %hint
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lengthRight : xs `LTE` ys -> Length ys
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lengthRight End = Z
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lengthRight (Keep p) = S (lengthRight p)
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lengthRight (Drop p) = S (lengthRight p)
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export
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id : Length xs => xs `LTE` xs
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id @{Z} = End
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id @{S s} = Keep id
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export
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zero : Length xs => [<] `LTE` xs
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zero @{Z} = End
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zero @{S s} = Drop zero
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export
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single : Length xs => x `Elem` xs -> [< x] `LTE` xs
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single @{S _} Here = Keep zero
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single @{S _} (There p) = Drop $ single p
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export
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(.) : ys `LTE` zs -> xs `LTE` ys -> xs `LTE` zs
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End . End = End
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Keep p . Keep q = Keep (p . q)
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Keep p . Drop q = Drop (p . q)
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Drop p . q = Drop (p . q)
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export
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(++) : xs1 `LTE` ys1 -> xs2 `LTE` ys2 -> (xs1 ++ xs2) `LTE` (ys1 ++ ys2)
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p ++ End = p
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p ++ Keep q = Keep (p ++ q)
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p ++ Drop q = Drop (p ++ q)
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public export
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data LTEMaskView : (lte : xs `LTE` ys) -> (mask : Nat) -> Type where
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[search lte]
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END : LTEMaskView End 0
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KEEP : (0 _ : LTEMaskView p n) -> LTEMaskView (Keep p) (S (2 * n))
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DROP : (0 _ : LTEMaskView p n) -> LTEMaskView (Drop p) (2 * n)
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%name LTEMaskView p, q
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record LTEMask {a : Type} (xs, ys : Scope a) where
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constructor LTEM
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mask : Nat
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0 lte : xs `LTE` ys
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0 view0 : LTEMaskView lte mask
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%name LTEMask m
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namespace View
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private
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0 lteMaskEnd' : LTEMaskView p n -> p = End -> n = 0
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lteMaskEnd' END Refl = Refl
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private
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0 lteMaskDrop' : LTEMaskView p n -> p = Drop q -> (n' ** n = 2 * n')
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lteMaskDrop' (DROP p {n = n'}) Refl = (n' ** Refl)
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private
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0 lteMaskEven' : {p : xs `LTE` (ys :< y)} ->
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n = 2 * n' -> LTEMaskView p n -> (q ** p = Drop q)
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lteMaskEven' eq (KEEP q) = absurd $ lsbMutex' eq Refl
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lteMaskEven' eq (DROP q) = (_ ** Refl)
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private
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lteMaskEven : {0 p : xs `LTE` (ys :< y)} ->
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(0 _ : LTEMaskView p (2 * n)) -> Exists (\q => p = Drop q)
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lteMaskEven q =
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let 0 res = lteMaskEven' Refl q in
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Evidence res.fst (irrelevantEq res.snd)
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private
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0 fromDROP' : {lte : xs `LTE` ys} -> n = 2 * n' ->
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LTEMaskView (Drop lte) n -> LTEMaskView lte n'
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fromDROP' eq (DROP {n} p) =
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let eq = doubleInj eq {m = n, n = n'} in
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rewrite sym eq in p
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private
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0 fromDROP : LTEMaskView (Drop lte) (2 * n) -> LTEMaskView lte n
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fromDROP = fromDROP' Refl
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private
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0 lteMaskOdd' : {p : (xs :< x) `LTE` (ys :< x)} -> {n' : Nat} ->
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n = S (2 * n') -> LTEMaskView p n -> (q ** p = Keep q)
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lteMaskOdd' eq (KEEP q) = (_ ** Refl)
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lteMaskOdd' eq (DROP q) = absurd $ lsbMutex' Refl eq
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lteMaskOdd' _ END impossible
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private
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lteMaskOdd : (0 _ : LTEMaskView p (S (2 * n))) -> Exists (\q => p = Keep q)
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lteMaskOdd q =
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let 0 res = lteMaskOdd' Refl q in
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Evidence res.fst (irrelevantEq res.snd)
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private
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0 lteMaskOddHead' : {p : (xs :< x) `LTE` (ys :< y)} -> {n' : Nat} ->
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n = S (2 * n') -> LTEMaskView p n -> x = y
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lteMaskOddHead' eq (KEEP q) = Refl
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lteMaskOddHead' eq (DROP q) = absurd $ lsbMutex' Refl eq
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lteMaskOddHead' eq END impossible
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private
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lteMaskOddHead : {0 p : (xs :< x) `LTE` (ys :< y)} ->
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(0 _ : LTEMaskView p (S (2 * n))) -> x = y
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lteMaskOddHead q = irrelevantEq $ lteMaskOddHead' Refl q
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private
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0 fromKEEP' : {lte : xs `LTE` ys} -> n = S (2 * n') ->
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LTEMaskView (Keep lte) n -> LTEMaskView lte n'
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fromKEEP' eq (KEEP {n} p) =
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let eq = doubleInj (injective eq) {m = n, n = n'} in
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rewrite sym eq in p
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private
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0 fromKEEP : LTEMaskView (Keep lte) (S (2 * n)) -> LTEMaskView lte n
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fromKEEP = fromKEEP' Refl
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export
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view : (sx : Length xs) => (sy : Length ys) =>
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(m : LTEMask xs ys) -> LTEMaskView m.lte m.mask
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view @{Z} @{Z} (LTEM {lte, view0, _}) =
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rewrite lteNil2End lte in
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rewrite lteMaskEnd' view0 (lteNil2End lte) in
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END
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view @{S _} @{Z} (LTEM {lte, _}) = void $ absurd lte
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view @{Z} @{S sy} (LTEM mask lte view0) =
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rewrite (lteNilLeftDrop lte).snd in
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rewrite (lteMaskDrop' view0 (lteNilLeftDrop lte).snd).snd in
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DROP $ let DROP p = view0 in p
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view @{S sx} @{S sy} (LTEM mask lte view0) with (viewLsb mask)
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view @{S sx} @{S sy} (LTEM (2 * n) lte view0)
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| Evidence Even (Lsb0 n) with (lteMaskEven view0)
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view @{S sx} @{S sy} (LTEM (2 * m) (Drop lte) view0)
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| Evidence Even (Lsb0 m) | Evidence lte Refl =
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DROP $ fromDROP view0
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view @{S sx} @{S sy} (LTEM (S (2 * n)) lte view0)
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| Evidence Odd (Lsb1 n) with (lteMaskOddHead view0)
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view @{S sx} @{S sy} (LTEM (S (2 * n)) lte view0)
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| Evidence Odd (Lsb1 n) | Refl with (lteMaskOdd view0)
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view @{S sx} @{S sy} (LTEM (S (2 * n)) (Keep lte) view0)
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| Evidence Odd (Lsb1 n) | Refl | Evidence lte Refl =
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KEEP $ fromKEEP view0
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public export
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record Split {a : Type} (xs, ys, zs : Scope a) (p : xs `LTE` ys ++ zs) where
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constructor MkSplit
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{0 leftSub, rightSub : Scope a}
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leftThin : leftSub `LTE` ys
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rightThin : rightSub `LTE` zs
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0 eqScope : xs = leftSub ++ rightSub
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0 eqThin : p ~=~ leftThin ++ rightThin
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export
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split : (zs : Scope a) -> (p : xs `LTE` ys ++ zs) -> Split xs ys zs p
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split [<] p = MkSplit p zero Refl Refl
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split (zs :< z) (Keep p) with (split zs p)
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split (zs :< z) (Keep (l ++ r)) | MkSplit l r Refl Refl =
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MkSplit l (Keep r) Refl Refl
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split (zs :< z) (Drop p) {xs} with (split zs p)
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split (zs :< z) (Drop (l ++ r)) {xs = _} | MkSplit l r Refl Refl =
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MkSplit l (Drop r) Refl Refl
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public export
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data Comp : ys `LTE` zs -> xs `LTE` ys -> xs `LTE` zs -> Type where
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CEE : Comp End End End
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CKK : Comp p q pq -> Comp (Keep p) (Keep q) (Keep pq)
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CKD : Comp p q pq -> Comp (Keep p) (Drop q) (Drop pq)
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CD0 : Comp p q pq -> Comp (Drop p) q (Drop pq)
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export
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comp : (p : ys `LTE` zs) -> (q : xs `LTE` ys) -> Comp p q (p . q)
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comp End End = CEE
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comp (Keep p) (Keep q) = CKK (comp p q)
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comp (Keep p) (Drop q) = CKD (comp p q)
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comp (Drop p) q = CD0 (comp p q)
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export
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0 compOk : Comp p q r -> r = (p . q)
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compOk CEE = Refl
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compOk (CKK z) = cong Keep $ compOk z
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compOk (CKD z) = cong Drop $ compOk z
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compOk (CD0 z) = cong Drop $ compOk z
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export
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compZero : (sx : Length xs) => (sy : Length ys) =>
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(p : xs `LTE` ys) -> Comp p (OPE.zero @{sx}) (OPE.zero @{sy})
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compZero {sx = Z, sy = Z} End = CEE
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compZero {sx = S _, sy = S _} (Keep p) = CKD (compZero p)
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compZero {sy = S _} (Drop p) = CD0 (compZero p)
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export
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compIdLeft : (sy : Length ys) =>
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(p : xs `LTE` ys) -> Comp (OPE.id @{sy}) p p
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compIdLeft {sy = Z} End = CEE
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compIdLeft {sy = S _} (Keep p) = CKK (compIdLeft p)
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compIdLeft {sy = S _} (Drop p) = CKD (compIdLeft p)
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export
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compIdRight : (sx : Length xs) =>
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(p : xs `LTE` ys) -> Comp p (OPE.id @{sx}) p
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compIdRight {sx = Z} End = CEE
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compIdRight {sx = S _} (Keep p) = CKK (compIdRight p)
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compIdRight (Drop p) = CD0 (compIdRight p)
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export
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0 compAssoc : (p : ys `LTE` zs) -> (q : xs `LTE` ys) -> (r : ws `LTE` xs) ->
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p . (q . r) = (p . q) . r
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compAssoc End End End = Refl
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compAssoc (Keep p) (Keep q) (Keep r) = cong Keep $ compAssoc p q r
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compAssoc (Keep p) (Keep q) (Drop r) = cong Drop $ compAssoc p q r
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compAssoc (Keep p) (Drop q) r = cong Drop $ compAssoc p q r
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compAssoc (Drop p) q r = cong Drop $ compAssoc p q r
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compAssoc End (Drop _) _ impossible
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public export
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Scoped : Type -> Type
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Scoped a = Scope a -> Type
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public export
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Subscope : Scope a -> Type
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Subscope ys = Exists (`LTE` ys)
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public export
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record SubMap {a : Type} {xs, ys, zs : Scope a}
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(p : xs `LTE` zs) (q : ys `LTE` zs) where
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constructor SM
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thin : xs `LTE` ys
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0 comp : Comp q thin p
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parameters (p : xs `LTE` ys)
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export
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subId : SubMap p p
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subId = SM id (compIdRight p)
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export
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subZero : SubMap OPE.zero p
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subZero = SM zero (compZero p)
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public export
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data All : (a -> Type) -> Scoped a where
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Lin : All p [<]
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(:<) : All p xs -> p x -> All p (xs :< x)
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%name OPE.All ps, qs
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export
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mapAll : (forall x. p x -> q x) -> All p xs -> All q xs
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mapAll f [<] = [<]
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mapAll f (x :< y) = mapAll f x :< f y
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export
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|
||||||
subAll : xs `LTE` ys -> All p ys -> All p xs
|
|
||||||
subAll End [<] = [<]
|
|
||||||
subAll (Keep q) (ps :< x) = subAll q ps :< x
|
|
||||||
subAll (Drop q) (ps :< x) = subAll q ps
|
|
||||||
|
|
||||||
|
|
||||||
public export
|
|
||||||
data Cover_ : (overlap : Bool) -> xs `LTE` zs -> ys `LTE` zs -> Type where
|
|
||||||
CE : Cover_ ov End End
|
|
||||||
CL : Cover_ ov p q -> Cover_ ov (Keep p) (Drop q)
|
|
||||||
CR : Cover_ ov p q -> Cover_ ov (Drop p) (Keep q)
|
|
||||||
C2 : Cover_ ov p q -> Cover_ True (Keep p) (Keep q)
|
|
||||||
|
|
||||||
public export
|
|
||||||
Cover : xs `LTE` zs -> ys `LTE` zs -> Type
|
|
||||||
Cover = Cover_ True
|
|
||||||
|
|
||||||
public export
|
|
||||||
Partition : xs `LTE` zs -> ys `LTE` zs -> Type
|
|
||||||
Partition = Cover_ False
|
|
||||||
|
|
||||||
|
|
|
||||||
|
|
@ -0,0 +1,22 @@
|
||||||
|
module Quox.OPE.Basics
|
||||||
|
|
||||||
|
%default total
|
||||||
|
|
||||||
|
public export
|
||||||
|
Scope : Type -> Type
|
||||||
|
Scope = SnocList
|
||||||
|
|
||||||
|
public export
|
||||||
|
Scoped : Type -> Type
|
||||||
|
Scoped a = Scope a -> Type
|
||||||
|
|
||||||
|
public export
|
||||||
|
data All : (a -> Type) -> Scoped a where
|
||||||
|
Lin : All p [<]
|
||||||
|
(:<) : All p xs -> p x -> All p (xs :< x)
|
||||||
|
%name OPE.Basics.All ps, qs
|
||||||
|
|
||||||
|
public export
|
||||||
|
mapAll : (forall x. p x -> q x) -> All p xs -> All q xs
|
||||||
|
mapAll f [<] = [<]
|
||||||
|
mapAll f (x :< y) = mapAll f x :< f y
|
||||||
|
|
@ -0,0 +1,83 @@
|
||||||
|
module Quox.OPE.Comp
|
||||||
|
|
||||||
|
import Quox.OPE.Basics
|
||||||
|
import Quox.OPE.Length
|
||||||
|
import Quox.OPE.Sub
|
||||||
|
import Data.DPair
|
||||||
|
|
||||||
|
%default total
|
||||||
|
|
||||||
|
|
||||||
|
public export
|
||||||
|
data Comp : ys `Sub` zs -> xs `Sub` ys -> xs `Sub` zs -> Type where
|
||||||
|
CEE : Comp End End End
|
||||||
|
CKK : Comp p q pq -> Comp (Keep p) (Keep q) (Keep pq)
|
||||||
|
CKD : Comp p q pq -> Comp (Keep p) (Drop q) (Drop pq)
|
||||||
|
CD0 : Comp p q pq -> Comp (Drop p) q (Drop pq)
|
||||||
|
|
||||||
|
export
|
||||||
|
comp : (p : ys `Sub` zs) -> (q : xs `Sub` ys) -> Comp p q (p . q)
|
||||||
|
comp End End = CEE
|
||||||
|
comp (Keep p) (Keep q) = CKK (comp p q)
|
||||||
|
comp (Keep p) (Drop q) = CKD (comp p q)
|
||||||
|
comp (Drop p) q = CD0 (comp p q)
|
||||||
|
|
||||||
|
export
|
||||||
|
0 compOk : Comp p q r -> r = (p . q)
|
||||||
|
compOk CEE = Refl
|
||||||
|
compOk (CKK z) = cong Keep $ compOk z
|
||||||
|
compOk (CKD z) = cong Drop $ compOk z
|
||||||
|
compOk (CD0 z) = cong Drop $ compOk z
|
||||||
|
|
||||||
|
export
|
||||||
|
compZero : (sx : Length xs) => (sy : Length ys) =>
|
||||||
|
(p : xs `Sub` ys) -> Comp p (Sub.zero @{sx}) (Sub.zero @{sy})
|
||||||
|
compZero {sx = Z, sy = Z} End = CEE
|
||||||
|
compZero {sx = S _, sy = S _} (Keep p) = CKD (compZero p)
|
||||||
|
compZero {sy = S _} (Drop p) = CD0 (compZero p)
|
||||||
|
|
||||||
|
export
|
||||||
|
compIdLeft : (sy : Length ys) =>
|
||||||
|
(p : xs `Sub` ys) -> Comp (Sub.id @{sy}) p p
|
||||||
|
compIdLeft {sy = Z} End = CEE
|
||||||
|
compIdLeft {sy = S _} (Keep p) = CKK (compIdLeft p)
|
||||||
|
compIdLeft {sy = S _} (Drop p) = CKD (compIdLeft p)
|
||||||
|
|
||||||
|
export
|
||||||
|
compIdRight : (sx : Length xs) =>
|
||||||
|
(p : xs `Sub` ys) -> Comp p (Sub.id @{sx}) p
|
||||||
|
compIdRight {sx = Z} End = CEE
|
||||||
|
compIdRight {sx = S _} (Keep p) = CKK (compIdRight p)
|
||||||
|
compIdRight (Drop p) = CD0 (compIdRight p)
|
||||||
|
|
||||||
|
|
||||||
|
export
|
||||||
|
0 compAssoc : (p : ys `Sub` zs) -> (q : xs `Sub` ys) -> (r : ws `Sub` xs) ->
|
||||||
|
p . (q . r) = (p . q) . r
|
||||||
|
compAssoc End End End = Refl
|
||||||
|
compAssoc (Keep p) (Keep q) (Keep r) = cong Keep $ compAssoc p q r
|
||||||
|
compAssoc (Keep p) (Keep q) (Drop r) = cong Drop $ compAssoc p q r
|
||||||
|
compAssoc (Keep p) (Drop q) r = cong Drop $ compAssoc p q r
|
||||||
|
compAssoc (Drop p) q r = cong Drop $ compAssoc p q r
|
||||||
|
compAssoc End (Drop _) _ impossible
|
||||||
|
|
||||||
|
|
||||||
|
public export
|
||||||
|
Subscope : Scope a -> Type
|
||||||
|
Subscope ys = Exists (`Sub` ys)
|
||||||
|
|
||||||
|
public export
|
||||||
|
record SubMap {a : Type} {xs, ys, zs : Scope a}
|
||||||
|
(p : xs `Sub` zs) (q : ys `Sub` zs) where
|
||||||
|
constructor SM
|
||||||
|
thin : xs `Sub` ys
|
||||||
|
0 comp : Comp q thin p
|
||||||
|
|
||||||
|
parameters (p : xs `Sub` ys)
|
||||||
|
export
|
||||||
|
subId : SubMap p p
|
||||||
|
subId = SM id (compIdRight p)
|
||||||
|
|
||||||
|
export
|
||||||
|
subZero : SubMap Sub.zero p
|
||||||
|
subZero = SM zero (compZero p)
|
||||||
|
|
@ -0,0 +1,24 @@
|
||||||
|
module Quox.OPE.Cover
|
||||||
|
|
||||||
|
import Quox.OPE.Basics
|
||||||
|
import Quox.OPE.Length
|
||||||
|
import Quox.OPE.Sub
|
||||||
|
|
||||||
|
%default total
|
||||||
|
|
||||||
|
|
||||||
|
public export
|
||||||
|
data Cover_ : (overlap : Bool) -> xs `Sub` zs -> ys `Sub` zs -> Type where
|
||||||
|
CE : Cover_ ov End End
|
||||||
|
CL : Cover_ ov p q -> Cover_ ov (Keep p) (Drop q)
|
||||||
|
CR : Cover_ ov p q -> Cover_ ov (Drop p) (Keep q)
|
||||||
|
C2 : Cover_ ov p q -> Cover_ True (Keep p) (Keep q)
|
||||||
|
|
||||||
|
public export
|
||||||
|
Cover : xs `Sub` zs -> ys `Sub` zs -> Type
|
||||||
|
Cover = Cover_ True
|
||||||
|
|
||||||
|
public export
|
||||||
|
Partition : xs `Sub` zs -> ys `Sub` zs -> Type
|
||||||
|
Partition = Cover_ False
|
||||||
|
|
||||||
|
|
@ -0,0 +1,24 @@
|
||||||
|
module Quox.OPE.Length
|
||||||
|
|
||||||
|
import Quox.OPE.Basics
|
||||||
|
|
||||||
|
%default total
|
||||||
|
|
||||||
|
|
||||||
|
public export
|
||||||
|
data Length : Scope a -> Type where
|
||||||
|
Z : Length [<]
|
||||||
|
S : (s : Length xs) -> Length (xs :< x)
|
||||||
|
%name Length s
|
||||||
|
%builtin Natural Length
|
||||||
|
|
||||||
|
public export
|
||||||
|
(.nat) : Length xs -> Nat
|
||||||
|
(Z).nat = Z
|
||||||
|
(S s).nat = S s.nat
|
||||||
|
%transform "Length.nat" Length.(.nat) xs = believe_me xs
|
||||||
|
|
||||||
|
export
|
||||||
|
0 ok : (s : Length xs) -> s.nat = length xs
|
||||||
|
ok Z = Refl
|
||||||
|
ok (S s) = cong S $ ok s
|
||||||
|
|
@ -0,0 +1,27 @@
|
||||||
|
module Quox.OPE.Split
|
||||||
|
|
||||||
|
import Quox.OPE.Basics
|
||||||
|
import Quox.OPE.Length
|
||||||
|
import Quox.OPE.Sub
|
||||||
|
|
||||||
|
%default total
|
||||||
|
|
||||||
|
|
||||||
|
public export
|
||||||
|
record Split {a : Type} (xs, ys, zs : Scope a) (p : xs `Sub` ys ++ zs) where
|
||||||
|
constructor MkSplit
|
||||||
|
{0 leftSub, rightSub : Scope a}
|
||||||
|
leftThin : leftSub `Sub` ys
|
||||||
|
rightThin : rightSub `Sub` zs
|
||||||
|
0 eqScope : xs = leftSub ++ rightSub
|
||||||
|
0 eqThin : p ~=~ leftThin ++ rightThin
|
||||||
|
|
||||||
|
export
|
||||||
|
split : (zs : Scope a) -> (p : xs `Sub` ys ++ zs) -> Split xs ys zs p
|
||||||
|
split [<] p = MkSplit p zero Refl Refl
|
||||||
|
split (zs :< z) (Keep p) with (split zs p)
|
||||||
|
split (zs :< z) (Keep (l ++ r)) | MkSplit l r Refl Refl =
|
||||||
|
MkSplit l (Keep r) Refl Refl
|
||||||
|
split (zs :< z) (Drop p) {xs} with (split zs p)
|
||||||
|
split (zs :< z) (Drop (l ++ r)) {xs = _} | MkSplit l r Refl Refl =
|
||||||
|
MkSplit l (Drop r) Refl Refl
|
||||||
|
|
@ -0,0 +1,254 @@
|
||||||
|
module Quox.OPE.Sub
|
||||||
|
|
||||||
|
import Quox.OPE.Basics
|
||||||
|
import Quox.OPE.Length
|
||||||
|
import Quox.NatExtra
|
||||||
|
import Data.DPair
|
||||||
|
import Data.SnocList.Elem
|
||||||
|
|
||||||
|
%default total
|
||||||
|
|
||||||
|
public export
|
||||||
|
data Sub : Scope a -> Scope a -> Type where
|
||||||
|
End : [<] `Sub` [<]
|
||||||
|
Keep : xs `Sub` ys -> xs :< z `Sub` ys :< z
|
||||||
|
Drop : xs `Sub` ys -> xs `Sub` ys :< z
|
||||||
|
%name Sub p, q
|
||||||
|
|
||||||
|
export
|
||||||
|
keepInjective : Keep p = Keep q -> p = q
|
||||||
|
keepInjective Refl = Refl
|
||||||
|
|
||||||
|
export
|
||||||
|
dropInjective : Drop p = Drop q -> p = q
|
||||||
|
dropInjective Refl = Refl
|
||||||
|
|
||||||
|
|
||||||
|
-- these need to be `public export` so that
|
||||||
|
-- `id`, `zero`, and maybe others can reduce
|
||||||
|
public export %hint
|
||||||
|
lengthLeft : xs `Sub` ys -> Length xs
|
||||||
|
lengthLeft End = Z
|
||||||
|
lengthLeft (Keep p) = S (lengthLeft p)
|
||||||
|
lengthLeft (Drop p) = lengthLeft p
|
||||||
|
|
||||||
|
public export %hint
|
||||||
|
lengthRight : xs `Sub` ys -> Length ys
|
||||||
|
lengthRight End = Z
|
||||||
|
lengthRight (Keep p) = S (lengthRight p)
|
||||||
|
lengthRight (Drop p) = S (lengthRight p)
|
||||||
|
|
||||||
|
|
||||||
|
export
|
||||||
|
dropLast : (xs :< x) `Sub` ys -> xs `Sub` ys
|
||||||
|
dropLast (Keep p) = Drop p
|
||||||
|
dropLast (Drop p) = Drop $ dropLast p
|
||||||
|
|
||||||
|
export
|
||||||
|
Uninhabited (xs :< x `Sub` [<]) where uninhabited _ impossible
|
||||||
|
|
||||||
|
export
|
||||||
|
Uninhabited (xs :< x `Sub` xs) where
|
||||||
|
uninhabited (Keep p) = uninhabited p
|
||||||
|
uninhabited (Drop p) = uninhabited $ dropLast p
|
||||||
|
|
||||||
|
|
||||||
|
export
|
||||||
|
0 lteLen : xs `Sub` ys -> length xs `LTE` length ys
|
||||||
|
lteLen End = LTEZero
|
||||||
|
lteLen (Keep p) = LTESucc $ lteLen p
|
||||||
|
lteLen (Drop p) = lteSuccRight $ lteLen p
|
||||||
|
|
||||||
|
export
|
||||||
|
0 lteNilRight : xs `Sub` [<] -> xs = [<]
|
||||||
|
lteNilRight End = Refl
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
public export
|
||||||
|
id : Length xs => xs `Sub` xs
|
||||||
|
id @{Z} = End
|
||||||
|
id @{S s} = Keep id
|
||||||
|
|
||||||
|
public export
|
||||||
|
zero : Length xs => [<] `Sub` xs
|
||||||
|
zero @{Z} = End
|
||||||
|
zero @{S s} = Drop zero
|
||||||
|
|
||||||
|
public export
|
||||||
|
single : Length xs => x `Elem` xs -> [< x] `Sub` xs
|
||||||
|
single @{S _} Here = Keep zero
|
||||||
|
single @{S _} (There p) = Drop $ single p
|
||||||
|
|
||||||
|
|
||||||
|
public export
|
||||||
|
(.) : ys `Sub` zs -> xs `Sub` ys -> xs `Sub` zs
|
||||||
|
End . End = End
|
||||||
|
Keep p . Keep q = Keep (p . q)
|
||||||
|
Keep p . Drop q = Drop (p . q)
|
||||||
|
Drop p . q = Drop (p . q)
|
||||||
|
|
||||||
|
public export
|
||||||
|
(++) : xs1 `Sub` ys1 -> xs2 `Sub` ys2 -> (xs1 ++ xs2) `Sub` (ys1 ++ ys2)
|
||||||
|
p ++ End = p
|
||||||
|
p ++ Keep q = Keep (p ++ q)
|
||||||
|
p ++ Drop q = Drop (p ++ q)
|
||||||
|
|
||||||
|
export
|
||||||
|
0 appZeroRight : (p : xs `Sub` ys) -> p ++ zero @{len} {xs = [<]} = p
|
||||||
|
appZeroRight {len = Z} p = Refl
|
||||||
|
|
||||||
|
|
||||||
|
public export
|
||||||
|
subAll : xs `Sub` ys -> All p ys -> All p xs
|
||||||
|
subAll End [<] = [<]
|
||||||
|
subAll (Keep q) (ps :< x) = subAll q ps :< x
|
||||||
|
subAll (Drop q) (ps :< x) = subAll q ps
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
public export
|
||||||
|
data SubMaskView : (lte : xs `Sub` ys) -> (mask : Nat) -> Type where
|
||||||
|
[search lte]
|
||||||
|
END : SubMaskView End 0
|
||||||
|
KEEP : {n : Nat} -> {0 p : xs `Sub` ys} ->
|
||||||
|
(0 v : SubMaskView p n) -> SubMaskView (Keep {z} p) (S (2 * n))
|
||||||
|
DROP : {n : Nat} -> {0 p : xs `Sub` ys} ->
|
||||||
|
(0 v : SubMaskView p n) -> SubMaskView (Drop {z} p) (2 * n)
|
||||||
|
%name SubMaskView v
|
||||||
|
|
||||||
|
public export
|
||||||
|
record SubMask {a : Type} (xs, ys : Scope a) where
|
||||||
|
constructor SubM
|
||||||
|
mask : Nat
|
||||||
|
0 lte : xs `Sub` ys
|
||||||
|
0 view0 : SubMaskView lte mask
|
||||||
|
%name SubMask m
|
||||||
|
|
||||||
|
private
|
||||||
|
0 ltemNilLeftZero' : SubMaskView {xs = [<]} lte mask -> mask = 0
|
||||||
|
ltemNilLeftZero' END = Refl
|
||||||
|
ltemNilLeftZero' (DROP v) = cong (2 *) $ ltemNilLeftZero' v
|
||||||
|
|
||||||
|
export
|
||||||
|
ltemNilLeftZero : (0 _ : SubMaskView {xs = [<]} lte mask) -> mask = 0
|
||||||
|
ltemNilLeftZero v = irrelevantEq $ ltemNilLeftZero' v
|
||||||
|
|
||||||
|
|
||||||
|
private
|
||||||
|
0 lteNilLeftDrop0 : (p : [<] `Sub` (xs :< x)) -> (q ** p = Drop q)
|
||||||
|
lteNilLeftDrop0 (Drop q) = (q ** Refl)
|
||||||
|
|
||||||
|
private
|
||||||
|
lteNilLeftDrop : (0 p : [<] `Sub` (xs :< x)) -> Exists (\q => p = Drop q)
|
||||||
|
lteNilLeftDrop q =
|
||||||
|
let 0 res = lteNilLeftDrop0 q in
|
||||||
|
Evidence res.fst (irrelevantEq res.snd)
|
||||||
|
|
||||||
|
private
|
||||||
|
0 lteNil2End : (p : [<] `Sub` [<]) -> p = End
|
||||||
|
lteNil2End End = Refl
|
||||||
|
|
||||||
|
private
|
||||||
|
0 ltemEnd' : SubMaskView p n -> p = End -> n = 0
|
||||||
|
ltemEnd' END Refl = Refl
|
||||||
|
|
||||||
|
private
|
||||||
|
0 ltemEven' : {p : xs `Sub` (ys :< y)} ->
|
||||||
|
n = 2 * n' -> SubMaskView p n -> (q ** p = Drop q)
|
||||||
|
ltemEven' eq (KEEP q) = absurd $ lsbMutex' eq Refl
|
||||||
|
ltemEven' eq (DROP q) = (_ ** Refl)
|
||||||
|
|
||||||
|
private
|
||||||
|
ltemEven : {0 p : xs `Sub` (ys :< y)} ->
|
||||||
|
(0 _ : SubMaskView p (2 * n)) -> Exists (\q => p = Drop q)
|
||||||
|
ltemEven q =
|
||||||
|
let 0 res = ltemEven' Refl q in
|
||||||
|
Evidence res.fst (irrelevantEq res.snd)
|
||||||
|
|
||||||
|
private
|
||||||
|
0 fromDROP' : {lte : xs `Sub` ys} -> n = 2 * n' ->
|
||||||
|
SubMaskView (Drop lte) n -> SubMaskView lte n'
|
||||||
|
fromDROP' eq (DROP {n} p) =
|
||||||
|
let eq = doubleInj eq {m = n, n = n'} in
|
||||||
|
rewrite sym eq in p
|
||||||
|
|
||||||
|
private
|
||||||
|
0 ltemOdd' : {p : (xs :< x) `Sub` (ys :< x)} -> {n' : Nat} ->
|
||||||
|
n = S (2 * n') -> SubMaskView p n -> (q ** p = Keep q)
|
||||||
|
ltemOdd' eq (KEEP q) = (_ ** Refl)
|
||||||
|
ltemOdd' eq (DROP q) = absurd $ lsbMutex' Refl eq
|
||||||
|
ltemOdd' eq END impossible
|
||||||
|
|
||||||
|
private
|
||||||
|
ltemOdd : (0 _ : SubMaskView p (S (2 * n))) -> Exists (\q => p = Keep q)
|
||||||
|
ltemOdd q =
|
||||||
|
let 0 res = ltemOdd' Refl q in
|
||||||
|
Evidence res.fst (irrelevantEq res.snd)
|
||||||
|
|
||||||
|
private
|
||||||
|
0 ltemOddHead' : {p : (xs :< x) `Sub` (ys :< y)} -> {n' : Nat} ->
|
||||||
|
n = S (2 * n') -> SubMaskView p n -> x = y
|
||||||
|
ltemOddHead' eq (KEEP q) = Refl
|
||||||
|
ltemOddHead' eq (DROP q) = absurd $ lsbMutex' Refl eq
|
||||||
|
ltemOddHead' eq END impossible
|
||||||
|
|
||||||
|
private
|
||||||
|
ltemOddHead : {0 p : (xs :< x) `Sub` (ys :< y)} ->
|
||||||
|
(0 _ : SubMaskView p (S (2 * n))) -> x = y
|
||||||
|
ltemOddHead q = irrelevantEq $ ltemOddHead' Refl q
|
||||||
|
|
||||||
|
private
|
||||||
|
0 fromKEEP' : {lte : xs `Sub` ys} -> n = S (2 * n') ->
|
||||||
|
SubMaskView (Keep lte) n -> SubMaskView lte n'
|
||||||
|
fromKEEP' eq (KEEP {n} p) =
|
||||||
|
let eq = doubleInj (injective eq) {m = n, n = n'} in
|
||||||
|
rewrite sym eq in p
|
||||||
|
|
||||||
|
export
|
||||||
|
view : Length xs => Length ys =>
|
||||||
|
(m : SubMask xs ys) -> SubMaskView m.lte m.mask
|
||||||
|
view @{Z} @{Z} (SubM {lte, view0, _}) =
|
||||||
|
rewrite lteNil2End lte in
|
||||||
|
rewrite ltemEnd' view0 (lteNil2End lte) in
|
||||||
|
END
|
||||||
|
view @{S _} @{Z} (SubM {lte, _}) = void $ absurd lte
|
||||||
|
view @{Z} @{S sy} (SubM mask lte view0) with (ltemNilLeftZero view0)
|
||||||
|
view @{Z} @{S sy} (SubM 0 lte view0)
|
||||||
|
| Refl with (lteNilLeftDrop lte)
|
||||||
|
view @{Z} @{S sy} (SubM 0 (Drop lte) view0)
|
||||||
|
| Refl | Evidence lte Refl =
|
||||||
|
DROP {n = 0} $ let DROP {n = 0} p = view0 in p
|
||||||
|
view @{S sx} @{S sy} (SubM mask lte view0) with (viewLsb mask)
|
||||||
|
view @{S sx} @{S sy} (SubM (2 * n) lte view0)
|
||||||
|
| Evidence Even (Lsb0 n) with (ltemEven view0)
|
||||||
|
view @{S sx} @{S sy} (SubM (2 * m) (Drop lte) view0)
|
||||||
|
| Evidence Even (Lsb0 m) | Evidence lte Refl =
|
||||||
|
DROP $ fromDROP' Refl view0
|
||||||
|
view @{S sx} @{S sy} (SubM (S (2 * n)) lte view0)
|
||||||
|
| Evidence Odd (Lsb1 n) with (ltemOddHead view0)
|
||||||
|
view @{S sx} @{S sy} (SubM (S (2 * n)) lte view0)
|
||||||
|
| Evidence Odd (Lsb1 n) | Refl with (ltemOdd view0)
|
||||||
|
view @{S sx} @{S sy} (SubM (S (2 * n)) (Keep lte) view0)
|
||||||
|
| Evidence Odd (Lsb1 n) | Refl | Evidence lte Refl =
|
||||||
|
KEEP $ fromKEEP' Refl view0
|
||||||
|
|
||||||
|
export
|
||||||
|
(.view) : Length xs => Length ys =>
|
||||||
|
(m : SubMask xs ys) -> SubMaskView m.lte m.mask
|
||||||
|
(.view) = view
|
||||||
|
|
||||||
|
|
||||||
|
export
|
||||||
|
ltemLen : Length xs => Length ys =>
|
||||||
|
xs `SubMask` ys -> length xs `LTE` length ys
|
||||||
|
ltemLen @{sx} @{sy} lte@(SubM m l _) with (lte.view)
|
||||||
|
ltemLen @{sx} @{sy} lte@(SubM 0 End _) | END = LTEZero
|
||||||
|
ltemLen @{S sx} @{S sy} lte@(SubM (S (2 * n)) (Keep p) _) | (KEEP q) =
|
||||||
|
LTESucc $ ltemLen $ SubM n p q
|
||||||
|
ltemLen @{sx} @{S sy} lte@(SubM (2 * n) (Drop p) _) | (DROP q) =
|
||||||
|
lteSuccRight $ ltemLen $ SubM n p q
|
||||||
|
|
||||||
|
export
|
||||||
|
ltemNilRight : xs `SubMask` [<] -> xs = [<]
|
||||||
|
ltemNilRight m = irrelevantEq $ lteNilRight m.lte
|
||||||
|
|
@ -2,7 +2,6 @@ module Quox.Syntax.Var
|
||||||
|
|
||||||
import Quox.Name
|
import Quox.Name
|
||||||
import Quox.Pretty
|
import Quox.Pretty
|
||||||
import Quox.OPE
|
|
||||||
|
|
||||||
import Data.Nat
|
import Data.Nat
|
||||||
import Data.List
|
import Data.List
|
||||||
|
|
|
||||||
|
|
@ -11,6 +11,12 @@ modules =
|
||||||
Quox.NatExtra,
|
Quox.NatExtra,
|
||||||
Quox.Unicode,
|
Quox.Unicode,
|
||||||
Quox.OPE,
|
Quox.OPE,
|
||||||
|
Quox.OPE.Basics,
|
||||||
|
Quox.OPE.Length,
|
||||||
|
Quox.OPE.Sub,
|
||||||
|
Quox.OPE.Split,
|
||||||
|
Quox.OPE.Comp,
|
||||||
|
Quox.OPE.Cover,
|
||||||
Quox.Pretty,
|
Quox.Pretty,
|
||||||
Quox.Syntax,
|
Quox.Syntax,
|
||||||
Quox.Syntax.Dim,
|
Quox.Syntax.Dim,
|
||||||
|
|
|
||||||
Loading…
Reference in New Issue