408 lines
12 KiB
Idris
408 lines
12 KiB
Idris
module Quox.OPE.Sub
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import Quox.OPE.Basics
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import Quox.OPE.Length
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import Quox.NatExtra
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import Data.DPair
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import Data.SnocList.Elem
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import Data.SnocList.Quantifiers
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%default total
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public export
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data Sub' : Scope a -> Scope a -> Type where
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End : [<] `Sub'` [<]
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Keep : xs `Sub'` ys -> xs :< z `Sub'` ys :< z
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Drop : xs `Sub'` ys -> xs `Sub'` ys :< z
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%name Sub' p, q
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export
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keepInjective : Keep p = Keep q -> p = q
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keepInjective Refl = Refl
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export
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dropInjective : Drop p = Drop q -> p = q
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dropInjective Refl = Refl
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-- these need to be `public export` so that
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-- `id`, `zero`, and maybe others can reduce
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public export %hint
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lengthLeft : xs `Sub'` 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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public export %hint
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lengthRight : xs `Sub'` 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 %inline
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dropLast' : (xs :< x) `Sub'` ys -> xs `Sub'` 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 `Sub'` [<]) where uninhabited _ impossible
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export
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Uninhabited (xs :< x `Sub'` 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 `Sub'` ys -> length xs `LTE` 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 `Sub'` [<] -> xs = [<]
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lteNilRight End = Refl
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public export
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id' : Length xs => xs `Sub'` xs
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id' @{Z} = End
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id' @{S s} = Keep id'
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public export
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zero' : Length xs => [<] `Sub'` xs
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zero' @{Z} = End
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zero' @{S s} = Drop zero'
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public export
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single' : Length xs => x `Elem` xs -> [< x] `Sub'` 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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public export
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trans' : ys `Sub'` zs -> xs `Sub'` ys -> xs `Sub'` zs
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trans' End End = End
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trans' (Keep p) (Keep q) = Keep $ trans' p q
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trans' (Keep p) (Drop q) = Drop $ trans' p q
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trans' (Drop p) q = Drop $ trans' p q
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public export
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app' : xs1 `Sub'` ys1 -> xs2 `Sub'` ys2 -> (xs1 ++ xs2) `Sub'` (ys1 ++ ys2)
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app' p End = p
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app' p (Keep q) = Keep $ app' p q
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app' p (Drop q) = Drop $ app' p q
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||| if `p` holds for all elements of the main list,
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||| it holds for all elements of the sublist
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public export
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subAll' : xs `Sub'` 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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||| if `p` holds for one element of the sublist,
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||| it holds for one element of the main list
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public export
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subAny' : xs `Sub'` ys -> Any p xs -> Any p ys
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subAny' (Keep q) (Here x) = Here x
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subAny' (Keep q) (There x) = There (subAny' q x)
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subAny' (Drop q) x = There (subAny' q x)
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public export
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data SubView : (lte : xs `Sub'` ys) -> (mask : Nat) -> Type where
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[search lte]
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END : SubView End 0
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KEEP : {n : Nat} -> {0 p : xs `Sub'` ys} ->
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(0 v : SubView p n) -> SubView (Keep {z} p) (S (2 * n))
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DROP : {n : Nat} -> {0 p : xs `Sub'` ys} ->
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(0 v : SubView p n) -> SubView (Drop {z} p) (2 * n)
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%name SubView v
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public export
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record Sub {a : Type} (xs, ys : Scope a) where
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constructor SubM
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mask : Nat
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0 lte : xs `Sub'` ys
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0 view0 : SubView lte mask
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%name Sub m
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export
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getMask : SubView lte mask -> Subset Nat (Equal mask)
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private
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0 ltemNilLeftZero' : SubView {xs = [<]} lte mask -> mask = 0
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ltemNilLeftZero' END = Refl
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ltemNilLeftZero' (DROP v) = cong (2 *) $ ltemNilLeftZero' v
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export
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ltemNilLeftZero : (0 _ : SubView {xs = [<]} lte mask) -> mask = 0
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ltemNilLeftZero v = irrelevantEq $ ltemNilLeftZero' v
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private
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0 lteNilLeftDrop0 : (p : [<] `Sub'` (xs :< x)) -> (q ** p = Drop q)
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lteNilLeftDrop0 (Drop q) = (q ** Refl)
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private
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lteNilLeftDrop : (0 p : [<] `Sub'` (xs :< x)) -> Exists (\q => p = Drop q)
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lteNilLeftDrop q =
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let 0 res = lteNilLeftDrop0 q in
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Evidence res.fst (irrelevantEq res.snd)
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private
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0 lteNil2End : (p : [<] `Sub'` [<]) -> p = End
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lteNil2End End = Refl
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private
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0 ltemEnd' : SubView p n -> p = End -> n = 0
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ltemEnd' END Refl = Refl
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private
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0 ltemEven' : {p : xs `Sub'` (ys :< y)} ->
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n = 2 * n' -> SubView p n -> (q ** p = Drop q)
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ltemEven' eq (KEEP q) = absurd $ lsbMutex' eq Refl
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ltemEven' eq (DROP q) = (_ ** Refl)
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private
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ltemEven : {0 p : xs `Sub'` (ys :< y)} ->
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(0 _ : SubView p (2 * n)) -> Exists (\q => p = Drop q)
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ltemEven q =
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let 0 res = ltemEven' 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 `Sub'` ys} -> n = 2 * n' ->
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SubView (Drop lte) n -> SubView 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 ltemOdd' : {p : (xs :< x) `Sub'` (ys :< x)} -> {n' : Nat} ->
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n = S (2 * n') -> SubView p n -> (q ** p = Keep q)
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ltemOdd' eq (KEEP q) = (_ ** Refl)
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ltemOdd' eq (DROP q) = absurd $ lsbMutex' Refl eq
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ltemOdd' eq END impossible
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private
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ltemOdd : (0 _ : SubView p (S (2 * n))) -> Exists (\q => p = Keep q)
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ltemOdd q =
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let 0 res = ltemOdd' Refl q in
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Evidence res.fst (irrelevantEq res.snd)
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private
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0 ltemOddHead' : {p : (xs :< x) `Sub'` (ys :< y)} -> {n' : Nat} ->
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n = S (2 * n') -> SubView p n -> x = y
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ltemOddHead' eq (KEEP q) = Refl
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ltemOddHead' eq (DROP q) = absurd $ lsbMutex' Refl eq
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ltemOddHead' eq END impossible
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private
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ltemOddHead : {0 p : (xs :< x) `Sub'` (ys :< y)} ->
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(0 _ : SubView p (S (2 * n))) -> x = y
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ltemOddHead q = irrelevantEq $ ltemOddHead' Refl q
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private
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0 fromKEEP' : {lte : xs `Sub'` ys} -> n = S (2 * n') ->
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SubView (Keep lte) n -> SubView 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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export
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view : Length xs => Length ys =>
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(m : Sub xs ys) -> SubView m.lte m.mask
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view @{Z} @{Z} (SubM {lte, view0, _}) =
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rewrite lteNil2End lte in
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rewrite ltemEnd' view0 (lteNil2End lte) in
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END
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view @{S _} @{Z} (SubM {lte, _}) = void $ absurd lte
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view @{Z} @{S sy} (SubM mask lte view0) with (ltemNilLeftZero view0)
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view @{Z} @{S sy} (SubM 0 lte view0)
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| Refl with (lteNilLeftDrop lte)
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view @{Z} @{S sy} (SubM 0 (Drop lte) view0)
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| Refl | Evidence lte Refl =
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DROP {n = 0} $ let DROP {n = 0} p = view0 in p
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view @{S sx} @{S sy} (SubM mask lte view0) with (viewLsb mask)
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view @{S sx} @{S sy} (SubM (2 * n) lte view0)
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| Evidence Even (Lsb0 n) with (ltemEven view0)
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view @{S sx} @{S sy} (SubM (2 * m) (Drop lte) view0)
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| Evidence Even (Lsb0 m) | Evidence lte Refl =
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DROP $ fromDROP' Refl view0
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view @{S sx} @{S sy} (SubM (S (2 * n)) lte view0)
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| Evidence Odd (Lsb1 n) with (ltemOddHead view0)
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view @{S sx} @{S sy} (SubM (S (2 * n)) lte view0)
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| Evidence Odd (Lsb1 n) | Refl with (ltemOdd view0)
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view @{S sx} @{S sy} (SubM (S (2 * n)) (Keep lte) view0)
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| Evidence Odd (Lsb1 n) | Refl | Evidence lte Refl =
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KEEP $ fromKEEP' Refl view0
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export
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(.view) : Length xs => Length ys =>
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(m : Sub xs ys) -> SubView m.lte m.mask
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(.view) = view
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export
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ltemLen : Length xs => Length ys =>
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xs `Sub` ys -> length xs `LTE` length ys
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ltemLen @{sx} @{sy} sub@(SubM m l _) with (sub.view)
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ltemLen @{sx} @{sy} sub@(SubM 0 End _) | END = LTEZero
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ltemLen @{S sx} @{S sy} sub@(SubM (S (2 * n)) (Keep p) _) | (KEEP q) =
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LTESucc $ ltemLen $ SubM n p q
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ltemLen @{sx} @{S sy} sub@(SubM (2 * n) (Drop p) _) | (DROP q) =
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lteSuccRight $ ltemLen $ SubM n p q
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export
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ltemNilRight : xs `Sub` [<] -> xs = [<]
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ltemNilRight m = irrelevantEq $ lteNilRight m.lte
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public export %inline
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end : [<] `Sub` [<]
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end = SubM 0 End END
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public export %inline
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keep : xs `Sub` ys -> xs :< z `Sub` ys :< z
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keep (SubM mask lte view0) = SubM (S (2 * mask)) (Keep lte) (KEEP view0)
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public export %inline
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drop : xs `Sub` ys -> xs `Sub` ys :< z
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drop (SubM mask lte view0) = SubM (2 * mask) (Drop lte) (DROP view0)
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export %inline
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dropLast : Length xs => Length ys =>
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(xs :< x) `Sub` ys -> xs `Sub` ys
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dropLast @{sx} @{sy} sub@(SubM mask lte _) with (sub.view)
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dropLast sub@(SubM (S (2 * n)) (Keep p) _) | (KEEP v) =
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SubM (2 * n) (Drop p) (DROP v)
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dropLast @{_} @{S sy} sub@(SubM (2 * n) (Drop p) _) | DROP v =
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drop $ dropLast $ SubM n p v
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export
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Uninhabited (xs :< x `Sub` [<]) where
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uninhabited sub = void $ uninhabited sub.lte
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export
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Length xs => Uninhabited (xs :< x `Sub` xs) where
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uninhabited @{sx} sub@(SubM mask lte view0) with (sub.view)
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uninhabited @{S sx} sub@(SubM (S (2 * n)) (Keep p) _) | KEEP v =
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uninhabited $ SubM n p v
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uninhabited @{S sx} sub@(SubM (2 * n) (Drop p) _) | DROP v =
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uninhabited $ dropLast $ SubM n p v
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export
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refl : Length xs => xs `Sub` xs
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refl @{Z} = end
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refl @{S s} = keep refl
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export Reflexive (Scope a) Sub where reflexive = refl
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mutual
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private
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antisym_ : Length xs => Length ys =>
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{0 p : xs `Sub'` ys} -> {0 q : ys `Sub'` xs} ->
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SubView p m1 -> SubView q m2 -> xs = ys
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antisym_ END END = Refl
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antisym_ (KEEP v1) (KEEP v2 {z}) @{S sx} @{S sy} =
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cong (:< z) $ antisym (SubM _ _ v1) (SubM _ _ v2)
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antisym_ (KEEP v1) (DROP v2) {p = Keep p} {q = Drop q} =
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void $ succNotLTEpred $ lteLen q `transitive` lteLen p
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antisym_ (DROP v1) (KEEP v2) {p = Drop p} {q = Keep q} =
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void $ succNotLTEpred $ lteLen p `transitive` lteLen q
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antisym_ (DROP v1) (DROP v2) {p = Drop p} {q = Drop q} =
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void $ succNotLTEpred $ lteLen p `transitive` lteSuccLeft (lteLen q)
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export
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antisym : Length xs => Length ys => xs `Sub` ys -> ys `Sub` xs -> xs = ys
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antisym p q = antisym_ p.view q.view
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export
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Antisymmetric (Scope a) Sub where
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antisymmetric p q = antisym p q
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mutual
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private
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trans_ : Length xs => Length ys => Length zs =>
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{0 p : xs `Sub'` ys} -> {0 q : ys `Sub'` zs} ->
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SubView p m1 -> SubView q m2 -> xs `Sub` zs
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trans_ END END = end
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trans_ (KEEP v1) (KEEP v2) @{S sx} @{S sy} @{S sz} =
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keep $ SubM _ _ v1 `trans` SubM _ _ v2
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trans_ (DROP v1) (KEEP v2) @{sx} @{S sy} @{S sz} =
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drop $ SubM _ _ v1 `trans` SubM _ _ v2
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trans_ v1 (DROP v2) @{sx} @{sy} @{S sz} =
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let Element m1' eq = getMask v1 in
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drop $ SubM m1' _ (rewrite sym eq in v1) `trans` SubM _ _ v2
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export
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trans : Length xs => Length ys => Length zs =>
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xs `Sub` ys -> ys `Sub` zs -> xs `Sub` zs
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trans p q = trans_ p.view q.view
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export
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(.) : Length xs => Length ys => Length zs =>
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xs `Sub` ys -> ys `Sub` zs -> xs `Sub` zs
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(.) = trans
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export
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Transitive (Scope a) Sub where
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transitive p q = trans p q
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export
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zero : Length xs => [<] `Sub` 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] `Sub` xs
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single @{S sx} Here = keep zero
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single @{S sx} (There p) = drop $ single p
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export
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(++) : Length xs2 => Length ys2 =>
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xs1 `Sub` ys1 -> xs2 `Sub` ys2 -> (xs1 ++ xs2) `Sub` (ys1 ++ ys2)
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(++) sub1 sub2@(SubM {}) @{sx2} @{sy2} with (sub2.view)
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(++) sub1 sub2@(SubM {}) | END = sub1
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(++) sub1 sub2@(SubM {}) @{S sx2} @{S sy2} | KEEP v = keep $ sub1 ++ SubM _ _ v
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(++) sub1 sub2@(SubM {}) @{sx2} @{S sy2} | DROP v = drop $ sub1 ++ SubM _ _ v
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||| if `p` holds for all elements of the main list,
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||| it holds for all elements of the sublist
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export
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subAll : Length xs => Length ys =>
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xs `Sub` ys -> All prop ys -> All prop xs
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subAll sub@(SubM {}) ps @{sx} @{sy} with (sub.view)
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subAll sub@(SubM {}) [<] | END = [<]
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subAll sub@(SubM {}) (ps :< p) @{S sx} @{S sy} | KEEP v =
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subAll (SubM _ _ v) ps :< p
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subAll sub@(SubM {}) (ps :< p) @{sx} @{S sy} | DROP v =
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subAll (SubM _ _ v) ps
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||| if `p` holds for one element of the sublist,
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||| it holds for one element of the main list
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export
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subAny : Length xs => Length ys =>
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xs `Sub` ys -> Any prop xs -> Any prop ys
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subAny sub@(SubM {}) p @{sx} @{sy} with (sub.view)
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subAny sub@(SubM {}) p | END impossible
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subAny sub@(SubM {}) (Here p) | KEEP v = Here p
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subAny sub@(SubM {}) (There p) @{S sx} @{S sy} | KEEP v =
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There $ subAny (SubM _ _ v) p
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subAny sub@(SubM {}) p @{sx} @{S sy} | DROP v =
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There $ subAny (SubM _ _ v) p
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