OPE & scope stuff
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3 changed files with 370 additions and 169 deletions
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@ -55,3 +55,47 @@ parameters {base : Nat} {auto 0 _ : base `GTE` 2} (chars : Vect base Char)
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export
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showHex : Nat -> String
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showHex = showAtBase $ fromList $ unpack "0123456789ABCDEF"
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n2i = natToInteger
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i2n = fromInteger {ty = Nat}
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private partial %inline
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divNatViaInteger : (m, n : Nat) -> Nat
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divNatViaInteger m n = i2n $ n2i m `div` n2i n
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%transform "divNat" divNat = divNatViaInteger
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private %inline
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divNatViaIntegerNZ : (m, n : Nat) -> (0 _ : NonZero n) -> Nat
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divNatViaIntegerNZ m n _ = assert_total divNatViaInteger m n
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%transform "divNat" divNatNZ = divNatViaIntegerNZ
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private partial %inline
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modNatViaInteger : (m, n : Nat) -> Nat
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modNatViaInteger m n = i2n $ n2i m `mod` n2i n
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%transform "modNat" modNat = modNatViaInteger
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private %inline
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modNatViaIntegerNZ : (m, n : Nat) -> (0 _ : NonZero n) -> Nat
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modNatViaIntegerNZ m n _ = assert_total modNatViaInteger m n
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%transform "modNat" modNatNZ = modNatViaIntegerNZ
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public export
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data ViewLsb : Nat -> Type where
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Lsb0 : (n : Nat) -> ViewLsb (2 * n)
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Lsb1 : (n : Nat) -> ViewLsb (S (2 * n))
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private
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viewLsb' : (m, d : Nat) -> (0 _ : m `LT` 2) -> ViewLsb (m + 2 * d)
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viewLsb' 0 d p = Lsb0 d
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viewLsb' 1 d p = Lsb1 d
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viewLsb' (S (S _)) _ (LTESucc p) = void $ absurd p
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export
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viewLsb : (n : Nat) -> ViewLsb n
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viewLsb n =
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let nz = the (NonZero 2) %search in
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rewrite DivisionTheorem n 2 nz nz in
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rewrite multCommutative (divNatNZ n 2 nz) 2 in
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viewLsb' (modNatNZ n 2 nz) (divNatNZ n 2 nz) (boundModNatNZ n 2 nz)
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401
lib/Quox/OPE.idr
401
lib/Quox/OPE.idr
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@ -3,194 +3,257 @@
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module Quox.OPE
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import Quox.NatExtra
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import Data.Nat
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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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public export
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data OPE : Nat -> Nat -> Type where
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Id : OPE n n
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Drop : OPE m n -> OPE m (S n)
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Keep : OPE m n -> OPE (S m) (S n)
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%name OPE p, q
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public export %inline Injective Drop where injective Refl = Refl
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public export %inline Injective Keep where injective Refl = Refl
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public export
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opeZero : {n : Nat} -> OPE 0 n
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opeZero {n = 0} = Id
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opeZero {n = S n} = Drop opeZero
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public export
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(.) : OPE m n -> OPE n p -> OPE m p
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p . Id = p
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Id . q = q
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p . Drop q = Drop $ p . q
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Drop p . Keep q = Drop $ p . q
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Keep p . Keep q = Keep $ p . q
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public export
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toLTE : {m : Nat} -> OPE m n -> m `LTE` n
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toLTE Id = reflexive
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toLTE (Drop p) = lteSuccRight $ toLTE p
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toLTE (Keep p) = LTESucc $ toLTE p
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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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dropInner' : LTE' m n -> OPE m n
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dropInner' LTERefl = Id
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dropInner' (LTESuccR p) = Drop $ dropInner' $ force p
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Scope : Type -> Type
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Scope = SnocList
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public export
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dropInner : {n : Nat} -> LTE m n -> OPE m n
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dropInner = dropInner' . fromLte
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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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public export
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dropInnerN : (m : Nat) -> OPE n (m + n)
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dropInnerN 0 = Id
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dropInnerN (S m) = Drop $ dropInnerN m
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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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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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lteNilRight : xs `LTE` [<] -> xs = [<]
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lteNilRight End = Refl
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public export
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interface Tighten t where
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tighten : Alternative f => OPE m n -> t n -> f (t m)
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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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parameters {auto _ : Tighten t} {auto _ : Alternative f}
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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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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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private
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0 compExample :
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Comp {zs = [< a, b, c, d, e]}
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(Keep $ Drop $ Keep $ Drop $ Keep End)
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(Keep $ Drop $ Keep End)
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(Keep $ Drop $ Drop $ Drop $ Keep End)
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compExample = %search
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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 xs = Exists (`LTE` xs)
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public export
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SubMap : {xs, ys : Scope a} -> xs `LTE` zs -> ys `LTE` zs -> Type
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SubMap p q = DPair (xs `LTE` ys) (\r => Comp q r p)
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parameters (p : xs `LTE` ys)
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export
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tightenInner : {n : Nat} -> m `LTE` n -> t n -> f (t m)
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tightenInner = tighten . dropInner
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subId : SubMap p p
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subId = (id ** compIdRight p)
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export
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tightenN : (m : Nat) -> t (m + n) -> f (t n)
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tightenN m = tighten $ dropInnerN m
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subZero : SubMap OPE.zero p
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subZero = (zero ** compZero p)
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export
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tighten1 : t (S n) -> f (t n)
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tighten1 = tightenN 1
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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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-- [todo] can this be done with fancy nats too?
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-- with bitmasks sure but that might not be worth the effort
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-- [the next day] it probably isn't
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public export
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data LTEMaskView : xs `LTE` ys -> Nat -> Type where
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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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-- public export
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-- data OPE' : Nat -> Nat -> Type where
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-- None : OPE' 0 0
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-- Drop : OPE' m n -> OPE' m (S n)
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-- Keep : OPE' m n -> OPE' (S m) (S n)
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-- %name OPE' 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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-- public export %inline
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-- drop' : Integer -> Integer
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-- drop' n = n * 2
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-- public export %inline
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-- keep' : Integer -> Integer
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-- keep' n = 1 + 2 * n
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-- public export
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-- data IsOPE : Integer -> (OPE' m n) -> Type where
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-- IsNone : 0 `IsOPE` None
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-- IsDrop : (0 _ : m `IsOPE` q) -> drop' m `IsOPE` Drop q
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-- IsKeep : (0 _ : m `IsOPE` q) -> keep' m `IsOPE` Keep q
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-- %name IsOPE p
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-- public export
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-- record OPE m n where
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-- constructor MkOPE
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-- value : Integer
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-- 0 spec : OPE' m n
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-- 0 prf : value `IsOPE` spec
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-- 0 pos : So (value >= 0)
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-- private
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-- 0 idrisPleaseLearnAboutIntegers : {x, y : Integer} -> x = y
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-- idrisPleaseLearnAboutIntegers {x, y} = believe_me $ Refl {x}
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-- private
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-- 0 natIntPlus : (m, n : Nat) ->
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-- natToInteger (m + n) = natToInteger m + natToInteger n
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-- natIntPlus m n = idrisPleaseLearnAboutIntegers
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-- private
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-- 0 shiftTwice : (x : Integer) -> (m, n : Nat) ->
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-- x `shiftL` (m + n) = (x `shiftL` m) `shiftL` n
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-- shiftTwice x m n = idrisPleaseLearnAboutIntegers
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-- private
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-- 0 shift1 : (x : Integer) -> (x `shiftL` 1) = 2 * x
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-- shift1 x = idrisPleaseLearnAboutIntegers
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-- private
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-- 0 intPlusComm : (x, y : Integer) -> (x + y) = (y + x)
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-- intPlusComm x y = idrisPleaseLearnAboutIntegers
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-- private
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-- 0 intTimes2Minus1 : (x : Integer) -> 2 * x - 1 = 2 * (x - 1) + 1
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-- intTimes2Minus1 x = idrisPleaseLearnAboutIntegers
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-- private
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-- 0 intPosShift : So (x > 0) -> So (x `shiftL` i > 0)
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-- intPosShift p = believe_me Oh
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-- private
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-- 0 intNonnegDec : {x : Integer} -> So (x > 0) -> So (x - 1 >= 0)
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-- intNonnegDec p = believe_me Oh
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-- private
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-- 0 shiftSucc : (x : Integer) -> (n : Nat) ->
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-- x `shiftL` S n = 2 * (x `shiftL` n)
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-- shiftSucc x n = Calc $
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-- |~ x `shiftL` S n
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-- ~~ x `shiftL` (n + 1)
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-- ...(cong (x `shiftL`) $ sym $ plusCommutative {})
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-- ~~ (x `shiftL` n) `shiftL` 1
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-- ...(shiftTwice {})
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-- ~~ 2 * (x `shiftL` n)
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-- ...(shift1 {})
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-- private
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-- opeIdVal : (n : Nat) -> Integer
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-- opeIdVal n = (1 `shiftL` n) - 1
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-- private
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-- 0 opeIdValSpec : (n : Nat) -> Integer
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-- opeIdValSpec 0 = 0
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-- opeIdValSpec (S n) = keep' $ opeIdValSpec n
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||||
|
||||
-- private
|
||||
-- 0 opeIdValOk : (n : Nat) -> opeIdVal n = opeIdValSpec n
|
||||
-- opeIdValOk 0 = Refl
|
||||
-- opeIdValOk (S n) = Calc $
|
||||
-- |~ (1 `shiftL` S n) - 1
|
||||
-- ~~ 2 * (1 `shiftL` n) - 1 ...(cong (\x => x - 1) $ shiftSucc {})
|
||||
-- ~~ 2 * (1 `shiftL` n - 1) + 1 ...(intTimes2Minus1 {})
|
||||
-- ~~ 1 + 2 * (1 `shiftL` n - 1) ...(intPlusComm {})
|
||||
-- ~~ 1 + 2 * opeIdValSpec n ...(cong (\x => 1 + 2 * x) $ opeIdValOk {})
|
||||
|
||||
-- private
|
||||
-- 0 opeIdSpec : (n : Nat) -> OPE' n n
|
||||
-- opeIdSpec 0 = None
|
||||
-- opeIdSpec (S n) = Keep $ opeIdSpec n
|
||||
|
||||
-- private
|
||||
-- 0 opeIdProof' : (n : Nat) -> opeIdValSpec n `IsOPE` opeIdSpec n
|
||||
-- opeIdProof' 0 = IsNone
|
||||
-- opeIdProof' (S n) = IsKeep (opeIdProof' n)
|
||||
|
||||
-- private
|
||||
-- 0 opeIdProof : (n : Nat) -> opeIdVal n `IsOPE` opeIdSpec n
|
||||
-- opeIdProof n = rewrite opeIdValOk n in opeIdProof' n
|
||||
|
||||
-- export
|
||||
-- opeId : {n : Nat} -> OPE n n
|
||||
-- opeId {n} = MkOPE {prf = opeIdProof n, pos = intNonnegDec $ intPosShift Oh, _}
|
||||
export
|
||||
view : (sx : Length xs) => (sy : Length ys) =>
|
||||
(m : LTEMask xs ys) -> LTEMaskView m.lte m.mask
|
||||
|
|
|
|||
94
lib/Quox/Scope.idr
Normal file
94
lib/Quox/Scope.idr
Normal file
|
|
@ -0,0 +1,94 @@
|
|||
module Quox.Scope
|
||||
|
||||
import public Quox.OPE
|
||||
|
||||
|
||||
public export
|
||||
record Sub {a : Type} (t : Scoped a) (xs : Scope a) where
|
||||
constructor Su
|
||||
{0 support : Scope a}
|
||||
term : t support
|
||||
thin : support `LTE` xs
|
||||
|
||||
public export
|
||||
CoverS : Sub s xs -> Sub t xs -> Type
|
||||
CoverS ss tt = Cover ss.thin tt.thin
|
||||
|
||||
export %inline
|
||||
inj : Length xs => t xs -> Sub t xs
|
||||
inj x = Su x id
|
||||
|
||||
export %inline
|
||||
flat : Sub (Sub t) xs -> Sub t xs
|
||||
flat (Su (Su x q) p) = Su x (p . q)
|
||||
|
||||
export %inline
|
||||
thinSub : xs `LTE` ys -> Sub t xs -> Sub t ys
|
||||
thinSub p x = flat $ Su x p
|
||||
|
||||
export %inline
|
||||
map : (forall x. s x -> t x) -> Sub s x -> Sub t x
|
||||
map f = {term $= f}
|
||||
|
||||
public export
|
||||
data Unit : Scoped a where
|
||||
Nil : Unit [<]
|
||||
|
||||
export %inline
|
||||
unit : Length xs => Sub Unit xs
|
||||
unit = Su [] zero
|
||||
|
||||
public export
|
||||
record Coprod {a : Type} {xs, ys, zs : Scope a}
|
||||
(p : xs `LTE` zs) (q : ys `LTE` zs) where
|
||||
constructor Cop
|
||||
{0 scope : Scope a}
|
||||
{thin : scope `LTE` zs}
|
||||
{thinP : xs `LTE` scope}
|
||||
{thinQ : ys `LTE` scope}
|
||||
compP : Comp thin thinP p
|
||||
compQ : Comp thin thinQ q
|
||||
0 cover : Cover thinP thinQ
|
||||
|
||||
export
|
||||
coprod : (p : xs `LTE` zs) -> (q : ys `LTE` zs) -> Coprod p q
|
||||
coprod End End = %search
|
||||
coprod (Keep p) (Keep q) = let Cop {} = coprod p q in %search
|
||||
coprod (Keep p) (Drop q) = let Cop {} = coprod p q in %search
|
||||
coprod (Drop p) (Keep q) = let Cop {} = coprod p q in %search
|
||||
coprod (Drop p) (Drop q) =
|
||||
-- idk why this one is harder to solve
|
||||
let Cop compP compQ cover = coprod p q in
|
||||
Cop (CD0 compP) (CD0 compQ) cover
|
||||
|
||||
public export
|
||||
record Pair {a : Type} (s, t : Scoped a) (zs : Scope a) where
|
||||
constructor MkPair
|
||||
fst : Sub s zs
|
||||
snd : Sub t zs
|
||||
0 cover : CoverS fst snd
|
||||
|
||||
export
|
||||
pair : Sub s xs -> Sub t xs -> Sub (Pair s t) xs
|
||||
pair (Su x p) (Su y q) =
|
||||
let c = coprod p q in
|
||||
Su (MkPair (Su x c.thinP) (Su y c.thinQ) c.cover) c.thin
|
||||
|
||||
|
||||
public export
|
||||
data Var : a -> Scope a -> Type where
|
||||
Only : Var x [< x]
|
||||
|
||||
export %inline
|
||||
var : (0 x : a) -> [< x] `LTE` xs => Sub (Var x) xs
|
||||
var x @{p} = Su Only p
|
||||
|
||||
|
||||
infix 0 \\\, \\
|
||||
public export
|
||||
data Bind : (t : Scoped a) -> (new, old : Scope a) -> Type where
|
||||
(\\\) : new' `LTE` new -> t (old ++ new') -> Bind t new old
|
||||
|
||||
(\\) : (new : Scope a) -> Sub t (old ++ new) -> Sub (Bind t new) old
|
||||
vs \\ Su term p with (split vs p)
|
||||
vs \\ Su term (l ++ r) | MkSplit l r Refl Refl = Su (r \\\ term) l
|
||||
Loading…
Reference in a new issue