370 lines
10 KiB
Idris
370 lines
10 KiB
Idris
||| "order preserving embeddings", for recording a correspondence between
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||| a smaller scope and part of a larger one.
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module Quox.OPE
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import Quox.NatExtra
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import public Data.DPair
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import public Data.SnocList
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import public Data.SnocList.Elem
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%default total
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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
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subAll End [<] = [<]
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subAll (Keep q) (ps :< x) = subAll q ps :< x
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subAll (Drop q) (ps :< x) = subAll q ps
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public export
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data Cover_ : (overlap : Bool) -> xs `LTE` zs -> ys `LTE` zs -> Type where
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CE : Cover_ ov End End
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CL : Cover_ ov p q -> Cover_ ov (Keep p) (Drop q)
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CR : Cover_ ov p q -> Cover_ ov (Drop p) (Keep q)
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C2 : Cover_ ov p q -> Cover_ True (Keep p) (Keep q)
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public export
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Cover : xs `LTE` zs -> ys `LTE` zs -> Type
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Cover = Cover_ True
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public export
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Partition : xs `LTE` zs -> ys `LTE` zs -> Type
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Partition = Cover_ False
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