bit mask OPE stuff
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@ -84,58 +84,85 @@ modNatViaIntegerNZ m n _ = assert_total modNatViaInteger m n
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public export
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data Parity = Even | Odd
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public export
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data ViewLsb : Nat -> Parity -> Type where
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Lsb0 : (n : Nat) -> ViewLsb (2 * n) Even
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Lsb1 : (n : Nat) -> ViewLsb (S (2 * n)) Odd
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data ViewLsb : Nat -> Type where
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Lsb0 : (n : Nat) -> (0 eq : n' = 2 * n) -> ViewLsb n'
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Lsb1 : (n : Nat) -> (0 eq : n' = S (2 * n)) -> ViewLsb n'
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%name ViewLsb p, q
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export
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fromLsb0 : ViewLsb n Even -> Subset Nat (\n' => n = 2 * n')
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fromLsb0 (Lsb0 n') = Element n' Refl
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public export data IsLsb0 : ViewLsb n -> Type where ItIsLsb0 : IsLsb0 (Lsb0 n eq)
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public export data IsLsb1 : ViewLsb n -> Type where ItIsLsb1 : IsLsb1 (Lsb1 n eq)
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export
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fromLsb1 : ViewLsb n Odd -> Subset Nat (\n' => n = S (2 * n'))
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fromLsb1 (Lsb1 n') = Element n' Refl
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private
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viewLsb' : (m, d : Nat) -> (0 _ : m `LT` 2) -> Exists $ ViewLsb (m + 2 * d)
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viewLsb' 0 d p = Evidence Even (Lsb0 d)
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viewLsb' 1 d p = Evidence Odd (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) -> Exists $ ViewLsb n
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viewLsb n =
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let 0 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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export
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0 lsbMutex' : n = (2 * a) -> n = S (2 * b) -> Void
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lsbMutex' ev od {n = 0} impossible
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lsbMutex' ev od {n = 1} {a = S a} {b = 0} =
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0 lsbMutex : n = (2 * a) -> n = S (2 * b) -> Void
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lsbMutex ev od {n = 0} impossible
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lsbMutex ev od {n = 1} {a = S a} {b = 0} =
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let ev = injective ev in
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let s = sym $ plusSuccRightSucc a (a + 0) in
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absurd $ trans ev s
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lsbMutex' ev od {n = S (S n)} {a = S a} {b = S b} =
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lsbMutex ev od {n = S (S n)} {a = S a} {b = S b} =
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let ev = injective $
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trans (injective ev) (sym $ plusSuccRightSucc a (a + 0)) in
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let od = trans (injective $ injective od)
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(sym $ plusSuccRightSucc b (b + 0)) in
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lsbMutex' ev od
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lsbMutex ev od
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public export %hint
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doubleIsLsb0 : (p : ViewLsb (2 * n)) -> IsLsb0 p
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doubleIsLsb0 (Lsb0 k eq) = ItIsLsb0
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doubleIsLsb0 (Lsb1 k eq) = void $ absurd $ lsbMutex Refl eq
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public export %hint
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sDoubleIsLsb1 : (p : ViewLsb (S (2 * n))) -> IsLsb1 p
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sDoubleIsLsb1 (Lsb1 k eq) = ItIsLsb1
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sDoubleIsLsb1 (Lsb0 k eq) = void $ absurd $ lsbMutex Refl (sym eq)
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export
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0 lsbMutex : ViewLsb n Even -> ViewLsb n Odd -> Void
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lsbMutex p q = lsbMutex' (fromLsb0 p).snd (fromLsb1 q).snd
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fromLsb0 : (p : ViewLsb n) -> (0 _ : IsLsb0 p) =>
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Subset Nat (\n' => n = 2 * n')
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fromLsb0 (Lsb0 n' eq) @{ItIsLsb0} = Element n' eq
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export
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doubleInj : {m, n : Nat} -> 2 * m = 2 * n -> m = n
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fromLsb1 : (p : ViewLsb n) -> (0 _ : IsLsb1 p) =>
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Subset Nat (\n' => n = S (2 * n'))
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fromLsb1 (Lsb1 n' eq) @{ItIsLsb1} = Element n' eq
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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 Refl
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viewLsb' 1 d p = Lsb1 d Refl
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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 0 nz : 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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export
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0 doubleInj : {m, n : Nat} -> 2 * m = 2 * n -> m = n
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doubleInj eq =
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multRightCancel m n 2 %search $
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trans (multCommutative m 2) $ trans eq (multCommutative 2 n)
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export
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0 sDoubleInj : {m, n : Nat} -> S (2 * m) = S (2 * n) -> m = n
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sDoubleInj eq = doubleInj $ injective eq
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export
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0 lsbOdd : (n : Nat) -> (eq ** viewLsb (S (2 * n)) = Lsb1 n eq)
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lsbOdd n with (viewLsb (S (2 * n)))
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_ | Lsb0 _ eq = void $ absurd $ lsbMutex Refl (sym eq)
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_ | Lsb1 n' eq with (sDoubleInj eq)
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lsbOdd n | (Lsb1 n eq) | Refl = (eq ** Refl)
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export
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0 lsbEven : (n : Nat) -> (eq ** viewLsb (2 * n) = Lsb0 n eq)
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lsbEven n with (viewLsb (2 * n))
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_ | Lsb1 _ eq = void $ absurd $ lsbMutex Refl eq
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_ | Lsb0 n' eq with (doubleInj eq)
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lsbEven n | Lsb0 n eq | Refl = (eq ** Refl)
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@ -5,67 +5,124 @@ import Quox.OPE.Length
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import Quox.OPE.Sub
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import Data.DPair
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import Control.Function
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import Data.Nat
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%default total
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public export
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data Comp : ys `Sub` zs -> xs `Sub` ys -> xs `Sub` 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 `Sub` zs) -> (q : xs `Sub` 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 `Sub` ys) -> Comp p (Sub.zero @{sx}) (Sub.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 `Sub` ys) -> Comp (Sub.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 `Sub` ys) -> Comp p (Sub.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 `Sub` zs) -> (q : xs `Sub` ys) -> (r : ws `Sub` 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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data Comp' : ys `Sub'` zs -> xs `Sub'` ys -> xs `Sub'` 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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public export
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Subscope : Scope a -> Type
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record Comp {a : Type} {xs, ys, zs : Scope a}
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(p : ys `Sub` zs) (q : xs `Sub` ys) (pq : xs `Sub` zs) where
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constructor MkComp
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0 comp : Comp' p.lte q.lte pq.lte
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export
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compPrf' : (p : ys `Sub'` zs) -> (q : xs `Sub'` ys) -> Comp' p q (p . q)
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compPrf' End End = CEE
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compPrf' (Keep p) (Keep q) = CKK $ compPrf' p q
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compPrf' (Keep p) (Drop q) = CKD $ compPrf' p q
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compPrf' (Drop p) q = CD0 $ compPrf' 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 %inline
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compPrf : (0 sy : Length ys) => (0 sz : Length zs) =>
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(p : ys `Sub` zs) -> (q : xs `Sub` ys) ->
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Comp p q ((p . q) @{sy} @{sz})
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compPrf p q = MkComp $
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replace {p = Comp' p.lte q.lte} (sym $ compLte p q) $
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compPrf' p.lte q.lte
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export
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compZero' : (sx : Length xs) => (sy : Length ys) =>
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(p : xs `Sub'` ys) -> Comp' p (zero' @{sx}) (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 %inline
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compZero : (sx : Length xs) => (sy : Length ys) =>
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(p : xs `Sub` ys) -> Comp p (zero @{sx}) (zero @{sy})
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compZero p = MkComp $
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rewrite zeroLte {sx} in
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rewrite zeroLte {sx = sy} in
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compZero' {}
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export
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compIdLeft' : (sy : Length ys) =>
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(p : xs `Sub'` ys) -> Comp' (refl' @{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 %inline
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compIdLeft : (sy : Length ys) =>
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(p : xs `Sub` ys) -> Comp (refl @{sy}) p p
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compIdLeft {sy} p = MkComp $
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rewrite reflLte {sx = sy} in compIdLeft' {}
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export
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compIdRight' : (sx : Length xs) =>
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(p : xs `Sub'` ys) -> Comp' p (refl' @{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 %inline
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compIdRight : (sx : Length xs) =>
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(p : xs `Sub` ys) -> Comp p (refl @{sx}) p
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compIdRight {sx} p = MkComp $ rewrite reflLte {sx} in compIdRight' {}
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export
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0 compAssoc' : (p : ys `Sub'` zs) -> (q : xs `Sub'` ys) -> (r : ws `Sub'` 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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export %inline
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0 compAssoc : (sx : Length xs) => (sy : Length ys) => (sz : Length zs) =>
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(p : ys `Sub` zs) -> (q : xs `Sub` ys) -> (r : ws `Sub` xs) ->
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comp @{sy} @{sz} p (comp @{sx} @{sy} q r) =
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comp @{sx} @{sz} (comp @{sy} @{sz} p q) r
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compAssoc p q r = lteEq $
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trans (transLte {}) $
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trans (cong (p.lte .) (transLte {})) $
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sym $
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trans (transLte {}) $
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trans (cong (. r.lte) (transLte {})) $
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sym $ compAssoc' {}
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public export
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0 Subscope : Scope a -> Type
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Subscope ys = Exists (`Sub` 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 `Sub'` zs) (q : ys `Sub'` zs) where
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constructor SM'
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thin : xs `Sub'` ys
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0 comp : Comp' q thin p
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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 `Sub` zs) (q : ys `Sub` zs) where
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@ -73,10 +130,23 @@ record SubMap {a : Type} {xs, ys, zs : Scope a}
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thin : xs `Sub` ys
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0 comp : Comp q thin p
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parameters (p : xs `Sub` ys)
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export
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0 submap' : SubMap p q -> SubMap' p.lte q.lte
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submap' (SM thin comp) = SM' {thin = thin.lte, comp = comp.comp}
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parameters (p : xs `Sub'` ys)
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export
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subId' : SubMap' p p
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subId' = SM' refl' (compIdRight' p)
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export
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subZero' : SubMap' Sub.zero' p
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subZero' = SM' zero' (compZero' p)
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parameters {auto sx : Length xs} (p : xs `Sub` 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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subId = SM refl (compIdRight p)
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export
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subZero : SubMap Sub.zero p
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@ -8,17 +8,83 @@ import Quox.OPE.Sub
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public export
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data Cover_ : (overlap : Bool) -> xs `Sub` zs -> ys `Sub` 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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data Cover'_ : (overlap : Bool) -> xs `Sub'` zs -> ys `Sub'` 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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parameters (p : xs `Sub'` zs) (q : ys `Sub'` zs)
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public export
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0 Cover' : Type
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Cover' = Cover'_ True p q
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public export
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0 Partition' : Type
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Partition' = Cover'_ False p q
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public export
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Cover : xs `Sub` zs -> ys `Sub` zs -> Type
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Cover = Cover_ True
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record Cover_ {a : Type} {xs, ys, zs : Scope a} (overlap : Bool)
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(p : xs `Sub` zs) (q : ys `Sub` zs) where
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constructor MkCover
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0 cover : Cover'_ overlap p.lte q.lte
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public export
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Partition : xs `Sub` zs -> ys `Sub` zs -> Type
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Partition = Cover_ False
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parameters (p : xs `Sub` zs) (q : ys `Sub` zs)
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public export
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0 Cover : Type
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Cover = Cover_ True p q
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public export
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0 Partition : Type
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Partition = Cover_ False p q
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private %inline
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covCast' : {0 p, p' : xs `Sub'` zs} -> {0 q, q' : ys `Sub'` zs} ->
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(0 pe : p = p') -> (0 qe : q = q') ->
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Cover'_ overlap p' q' -> Cover'_ overlap p q
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covCast' pe qe c = replace {p = id} (sym $ cong2 (Cover'_ overlap) pe qe) c
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parameters {0 overlap : Bool} {0 p : xs `Sub` zs} {0 q : ys `Sub` zs}
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export %inline
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left : Cover_ overlap p q -> Cover_ overlap (keep p) (drop q)
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left (MkCover cover) = MkCover $ covCast' (keepLte p) (dropLte q) $ CL cover
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export %inline
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right : Cover_ overlap p q -> Cover_ overlap (drop p) (keep q)
|
||||
right (MkCover cover) = MkCover $ covCast' (dropLte p) (keepLte q) $ CR cover
|
||||
|
||||
export %inline
|
||||
both : Cover_ overlap p q -> Cover (keep p) (keep q)
|
||||
both (MkCover cover) = MkCover $ covCast' (keepLte p) (keepLte q) $ C2 cover
|
||||
|
||||
|
||||
export
|
||||
allLeft' : (sx : Length xs) => Partition' (refl' @{sx}) (zero' @{sx})
|
||||
allLeft' {sx = Z} = CE
|
||||
allLeft' {sx = S s} = CL allLeft'
|
||||
|
||||
export
|
||||
allRight' : (sx : Length xs) => Partition' (zero' @{sx}) (refl' @{sx})
|
||||
allRight' {sx = Z} = CE
|
||||
allRight' {sx = S s} = CR allRight'
|
||||
|
||||
export
|
||||
both' : (sx : Length xs) => Cover' (refl' @{sx}) (refl' @{sx})
|
||||
both' {sx = Z} = CE
|
||||
both' {sx = S s} = C2 both'
|
||||
|
||||
|
||||
export %inline
|
||||
allLeft : (sx : Length xs) => Partition (refl @{sx}) (zero @{sx})
|
||||
allLeft = MkCover $ rewrite reflLte {sx} in rewrite zeroLte {sx} in allLeft'
|
||||
|
||||
export %inline
|
||||
allRight : (sx : Length xs) => Partition (zero @{sx}) (refl @{sx})
|
||||
allRight = MkCover $ rewrite reflLte {sx} in rewrite zeroLte {sx} in allRight'
|
||||
|
||||
export %inline
|
||||
allBoth : (sx : Length xs) => Cover (refl @{sx}) (refl @{sx})
|
||||
allBoth = MkCover $ rewrite reflLte {sx} in both'
|
||||
|
|
|
|||
|
|
@ -27,3 +27,13 @@ public export %hint
|
|||
toLength : (xs : Scope a) -> Length xs
|
||||
toLength [<] = Z
|
||||
toLength (sx :< x) = S (toLength sx)
|
||||
|
||||
public export %hint
|
||||
lengthApp : Length xs -> Length ys -> Length (xs ++ ys)
|
||||
lengthApp sx Z = sx
|
||||
lengthApp sx (S sy) = S (lengthApp sx sy)
|
||||
|
||||
public export
|
||||
0 lengthIrrel : (sx1, sx2 : Length xs) -> sx1 = sx2
|
||||
lengthIrrel Z Z = Refl
|
||||
lengthIrrel (S sx1) (S sx2) = cong S $ lengthIrrel sx1 sx2
|
||||
|
|
|
|||
|
|
@ -17,11 +17,21 @@ record Split {a : Type} (xs, ys, zs : Scope a) (p : xs `Sub` ys ++ zs) where
|
|||
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
|
||||
split : Length ys =>
|
||||
(zs : Scope a) -> (p : xs `Sub` ys ++ zs) -> Split xs ys zs p
|
||||
split [<] p = MkSplit p zero Refl (endRight p)
|
||||
split (zs :< z) p @{ys} with (p.view @{S (lengthApp ys %search)})
|
||||
split (zs :< z) (SubM (S (2 * n)) (Keep p) v0) | (KEEP v Refl) =
|
||||
case split zs (sub v) of
|
||||
MkSplit l r Refl t =>
|
||||
MkSplit l (keep r) Refl $
|
||||
rewrite viewIrrel v0 (KEEP v Refl) in
|
||||
trans (cong keep {a = sub v} t) $
|
||||
sym $ keepAppRight l r
|
||||
split (zs :< z) (SubM (2 * n) (Drop p) v0) | (DROP v Refl) =
|
||||
case split zs (sub v) of
|
||||
MkSplit l r Refl t =>
|
||||
MkSplit l (drop r) Refl $
|
||||
rewrite viewIrrel v0 (DROP v Refl) in
|
||||
trans (cong drop {a = sub v} t) $
|
||||
sym $ dropAppRight l r
|
||||
|
|
|
|||
|
|
@ -9,6 +9,8 @@ import Data.SnocList.Quantifiers
|
|||
|
||||
%default total
|
||||
|
||||
infix 0 `Sub'`, `Sub`
|
||||
|
||||
public export
|
||||
data Sub' : Scope a -> Scope a -> Type where
|
||||
End : [<] `Sub'` [<]
|
||||
|
|
@ -24,7 +26,6 @@ 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
|
||||
|
|
@ -67,9 +68,9 @@ lteNilRight End = Refl
|
|||
|
||||
|
||||
public export
|
||||
id' : Length xs => xs `Sub'` xs
|
||||
id' @{Z} = End
|
||||
id' @{S s} = Keep id'
|
||||
refl' : Length xs => xs `Sub'` xs
|
||||
refl' @{Z} = End
|
||||
refl' @{S s} = Keep refl'
|
||||
|
||||
public export
|
||||
zero' : Length xs => [<] `Sub'` xs
|
||||
|
|
@ -82,12 +83,21 @@ single' @{S _} Here = Keep zero'
|
|||
single' @{S _} (There p) = Drop $ single' p
|
||||
|
||||
|
||||
namespace Sub
|
||||
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
|
||||
trans' : ys `Sub'` zs -> xs `Sub'` ys -> xs `Sub'` zs
|
||||
trans' End End = End
|
||||
trans' (Keep p) (Keep q) = Keep $ trans' p q
|
||||
trans' (Keep p) (Drop q) = Drop $ trans' p q
|
||||
trans' (Drop p) q = Drop $ trans' p q
|
||||
comp' : ys `Sub'` zs -> xs `Sub'` ys -> xs `Sub'` zs
|
||||
comp' = (.)
|
||||
|
||||
public export %inline
|
||||
trans' : xs `Sub'` ys -> ys `Sub'` zs -> xs `Sub'` zs
|
||||
trans' = flip (.)
|
||||
|
||||
public export
|
||||
app' : xs1 `Sub'` ys1 -> xs2 `Sub'` ys2 -> (xs1 ++ xs2) `Sub'` (ys1 ++ ys2)
|
||||
|
|
@ -116,11 +126,13 @@ subAny' (Drop q) x = There (subAny' q x)
|
|||
public export
|
||||
data SubView : (lte : xs `Sub'` ys) -> (mask : Nat) -> Type where
|
||||
[search lte]
|
||||
END : SubView End 0
|
||||
END : (0 eq : n = 0) -> SubView End n
|
||||
KEEP : {n : Nat} -> {0 p : xs `Sub'` ys} ->
|
||||
(0 v : SubView p n) -> SubView (Keep {z} p) (S (2 * n))
|
||||
(0 v : SubView p n) -> (0 eq : n' = S (2 * n)) ->
|
||||
SubView (Keep {z} p) n'
|
||||
DROP : {n : Nat} -> {0 p : xs `Sub'` ys} ->
|
||||
(0 v : SubView p n) -> SubView (Drop {z} p) (2 * n)
|
||||
(0 v : SubView p n) -> (0 eq : n' = 2 * n) ->
|
||||
SubView (Drop {z} p) n'
|
||||
%name SubView v
|
||||
|
||||
public export
|
||||
|
|
@ -129,162 +141,214 @@ record Sub {a : Type} (xs, ys : Scope a) where
|
|||
mask : Nat
|
||||
0 lte : xs `Sub'` ys
|
||||
0 view0 : SubView lte mask
|
||||
%name Sub m
|
||||
%name Sub sub
|
||||
|
||||
public export %inline
|
||||
sub : {mask : Nat} -> {0 lte : xs `Sub'` ys} -> (0 view0 : SubView lte mask) ->
|
||||
xs `Sub` ys
|
||||
sub v = SubM _ _ v
|
||||
|
||||
|
||||
private
|
||||
0 uip : (p, q : a = b) -> p = q
|
||||
uip Refl Refl = Refl
|
||||
|
||||
export
|
||||
0 viewIrrel : Length ys =>
|
||||
{m : Nat} -> {lte : xs `Sub'` ys} ->
|
||||
(v1, v2 : SubView lte m) -> v1 === v2
|
||||
viewIrrel (END eq1) (END eq2) = rewrite uip eq1 eq2 in Refl
|
||||
viewIrrel (KEEP v1 eq1) (KEEP v2 eq2) with (sDoubleInj $ trans (sym eq1) eq2)
|
||||
_ | Refl = rewrite viewIrrel v1 v2 in rewrite uip eq1 eq2 in Refl
|
||||
viewIrrel (DROP v1 eq1) (DROP v2 eq2) with (doubleInj $ trans (sym eq1) eq2)
|
||||
_ | Refl = rewrite viewIrrel v1 v2 in rewrite uip eq1 eq2 in Refl
|
||||
|
||||
export
|
||||
0 viewLteEq : {p, q : xs `Sub'` ys} -> SubView p m -> SubView q m -> p = q
|
||||
viewLteEq (END _) (END _) = Refl
|
||||
viewLteEq (KEEP {n = m} pv pe) (KEEP {n} qv qe) =
|
||||
let eq = sDoubleInj {m, n} $ trans (sym pe) qe in
|
||||
rewrite viewLteEq pv (rewrite eq in qv) in Refl
|
||||
viewLteEq (KEEP pv pe) (DROP qv qe) = absurd $ lsbMutex qe pe
|
||||
viewLteEq (DROP pv pe) (KEEP qv qe) = absurd $ lsbMutex pe qe
|
||||
viewLteEq (DROP {n = m} pv pe) (DROP {n} qv qe) =
|
||||
let eq = doubleInj {m, n} $ trans (sym pe) qe in
|
||||
rewrite viewLteEq pv (rewrite eq in qv) in Refl
|
||||
|
||||
private
|
||||
0 lteEq' : (p : xs `Sub'` ys) ->
|
||||
(v1 : SubView p m1) -> (v2 : SubView p m2) ->
|
||||
SubM m1 p v1 = SubM m2 p v2
|
||||
lteEq' End (END Refl) (END Refl) = Refl
|
||||
lteEq' {m1 = S (2 * n)} {m2 = S (2 * n)} (Keep p) (KEEP v1 Refl) (KEEP v2 Refl) =
|
||||
cong (\v => SubM (S (2 * n)) (Keep p) (KEEP v Refl)) $
|
||||
case lteEq' p v1 v2 of Refl => Refl
|
||||
lteEq' {m1 = 2 * n} {m2 = 2 * n} (Drop p) (DROP v1 Refl) (DROP v2 Refl) =
|
||||
cong (\v => SubM (2 * n) (Drop p) (DROP v Refl)) $
|
||||
case lteEq' p v1 v2 of Refl => Refl
|
||||
lteEq' {m1 = _} {m2 = S _} _ _ (END _) impossible
|
||||
lteEq' {m1 = Z} {m2 = _} _ (KEEP _ _) _ impossible
|
||||
lteEq' {m1 = _} {m2 = Z} _ _ (KEEP _ _) impossible
|
||||
|
||||
export
|
||||
0 lteEq : {p, q : xs `Sub` ys} -> p.lte = q.lte -> p = q
|
||||
lteEq {p = SubM pm pl pv} {q = SubM qm ql qv} eq =
|
||||
rewrite eq in lteEq' ql (rewrite sym eq in pv) qv
|
||||
|
||||
|
||||
public export %inline
|
||||
end : [<] `Sub` [<]
|
||||
end = SubM 0 End (END Refl)
|
||||
|
||||
public export %inline
|
||||
keep : xs `Sub` ys -> xs :< z `Sub` ys :< z
|
||||
keep (SubM mask lte view0) = SubM (S (2 * mask)) (Keep lte) (KEEP view0 Refl)
|
||||
|
||||
public export %inline
|
||||
drop : xs `Sub` ys -> xs `Sub` ys :< z
|
||||
drop (SubM mask lte view0) = SubM (2 * mask) (Drop lte) (DROP view0 Refl)
|
||||
|
||||
|
||||
export
|
||||
0 keepLte : (p : xs `Sub` ys) -> (keep p).lte = Keep p.lte
|
||||
keepLte (SubM mask lte view0) = Refl
|
||||
|
||||
export
|
||||
0 dropLte : (p : xs `Sub` ys) -> (drop p).lte = Drop p.lte
|
||||
dropLte (SubM mask lte view0) = Refl
|
||||
|
||||
|
||||
public export
|
||||
data Eqv : {xs, ys : Scope a} -> (p, q : xs `Sub` ys) -> Type where
|
||||
EQV : {0 p, q : xs `Sub` ys} -> p.mask = q.mask -> p `Eqv` q
|
||||
%name Eqv eqv
|
||||
|
||||
export
|
||||
Reflexive (xs `Sub` ys) Eqv where
|
||||
reflexive = EQV Refl
|
||||
|
||||
export
|
||||
Symmetric (xs `Sub` ys) Eqv where
|
||||
symmetric (EQV eq) = EQV $ sym eq
|
||||
|
||||
export
|
||||
Transitive (xs `Sub` ys) Eqv where
|
||||
transitive (EQV eq1) (EQV eq2) = EQV $ trans eq1 eq2
|
||||
|
||||
export
|
||||
0 eqvToEq : {p, q : xs `Sub` ys} -> p `Eqv` q -> p = q
|
||||
eqvToEq {p = SubM _ _ pv} {q = SubM _ _ qv} (EQV Refl) with (viewLteEq pv qv)
|
||||
_ | Refl = rewrite viewIrrel pv qv in Refl
|
||||
|
||||
export
|
||||
0 eqToEqv : {p, q : xs `Sub` ys} -> p = q -> p `Eqv` q
|
||||
eqToEqv Refl = reflexive
|
||||
|
||||
|
||||
export
|
||||
getMask : SubView lte mask -> Subset Nat (Equal mask)
|
||||
|
||||
private
|
||||
0 ltemNilLeftZero' : SubView {xs = [<]} lte mask -> mask = 0
|
||||
ltemNilLeftZero' END = Refl
|
||||
ltemNilLeftZero' (DROP v) = cong (2 *) $ ltemNilLeftZero' v
|
||||
|
||||
export
|
||||
ltemNilLeftZero : (0 _ : SubView {xs = [<]} lte mask) -> mask = 0
|
||||
ltemNilLeftZero v = irrelevantEq $ ltemNilLeftZero' v
|
||||
getMask (END Refl) = Element 0 Refl
|
||||
getMask (KEEP v eq) = Element _ eq
|
||||
getMask (DROP v eq) = Element _ eq
|
||||
|
||||
|
||||
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)
|
||||
splitExEq : {0 f, g : a -> b} ->
|
||||
(0 p : (x : a ** f x = g x)) -> Exists (\x => f x = g x)
|
||||
splitExEq p = Evidence p.fst (irr p.snd)
|
||||
where irr : forall a, b. (0 eq : a = b) -> a = b
|
||||
irr Refl = Refl
|
||||
|
||||
private
|
||||
0 lteNil2End : (p : [<] `Sub'` [<]) -> p = End
|
||||
lteNil2End End = Refl
|
||||
|
||||
private
|
||||
0 ltemEnd' : SubView p n -> p = End -> n = 0
|
||||
ltemEnd' END Refl = Refl
|
||||
0 ltemEnd : SubView p n -> p = End -> n = 0
|
||||
ltemEnd (END eq) Refl = eq
|
||||
|
||||
private
|
||||
0 ltemEven' : {p : xs `Sub'` (ys :< y)} ->
|
||||
n = 2 * n' -> SubView p n -> (q ** p = Drop q)
|
||||
ltemEven' eq (KEEP q) = absurd $ lsbMutex' eq Refl
|
||||
ltemEven' eq (DROP q) = (_ ** Refl)
|
||||
0 ltemEvenDrop : {p : xs `Sub'` (ys :< y)} ->
|
||||
n = 2 * n' -> SubView p n -> (q ** p = Drop q)
|
||||
ltemEvenDrop eq (KEEP _ eq') = absurd $ lsbMutex eq eq'
|
||||
ltemEvenDrop eq (DROP {}) = (_ ** Refl)
|
||||
|
||||
private
|
||||
ltemEven : {0 p : xs `Sub'` (ys :< y)} ->
|
||||
(0 _ : SubView 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' ->
|
||||
SubView (Drop lte) n -> SubView lte n'
|
||||
fromDROP' eq (DROP {n} p) =
|
||||
0 fromDROP : {lte : xs `Sub'` ys :< z} -> n = 2 * n' -> lte = Drop lte' ->
|
||||
SubView lte n -> SubView lte' n'
|
||||
fromDROP eq Refl (DROP {n} p Refl) =
|
||||
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} ->
|
||||
0 ltemOddKeep : {p : (xs :< x) `Sub'` (ys :< x)} -> {n' : Nat} ->
|
||||
n = S (2 * n') -> SubView p n -> (q ** p = Keep q)
|
||||
ltemOdd' eq (KEEP q) = (_ ** Refl)
|
||||
ltemOdd' eq (DROP q) = absurd $ lsbMutex' Refl eq
|
||||
ltemOdd' eq END impossible
|
||||
ltemOddKeep eq (KEEP {}) = (_ ** Refl)
|
||||
ltemOddKeep eq (DROP _ eq') = absurd $ lsbMutex eq' eq
|
||||
ltemOddKeep eq (END _) impossible
|
||||
|
||||
private
|
||||
ltemOdd : (0 _ : SubView p (S (2 * n))) -> Exists (\q => p = Keep q)
|
||||
ltemOdd q =
|
||||
let 0 res = ltemOdd' Refl q in
|
||||
Evidence res.fst (irrelevantEq res.snd)
|
||||
0 ltemOddLeft : {0 n, n' : Nat} -> {lte : xs `Sub'` ys :< z} ->
|
||||
n = S (2 * n') -> SubView lte n ->
|
||||
(xs' ** xs = xs' :< z)
|
||||
ltemOddLeft eq {lte = Keep p} _ = (_ ** Refl)
|
||||
ltemOddLeft eq {lte = Drop p} (DROP n' eq') = void $ lsbMutex eq' eq
|
||||
|
||||
private
|
||||
0 ltemOddHead' : {p : (xs :< x) `Sub'` (ys :< y)} -> {n' : Nat} ->
|
||||
n = S (2 * n') -> SubView 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 _ : SubView p (S (2 * n))) -> x = y
|
||||
ltemOddHead q = irrelevantEq $ ltemOddHead' Refl q
|
||||
|
||||
private
|
||||
0 fromKEEP' : {lte : xs `Sub'` ys} -> n = S (2 * n') ->
|
||||
SubView (Keep lte) n -> SubView lte n'
|
||||
fromKEEP' eq (KEEP {n} p) =
|
||||
let eq = doubleInj (injective eq) {m = n, n = n'} in
|
||||
0 fromKEEP : {lte : xs :< z `Sub'` ys :< z} ->
|
||||
n = S (2 * n') -> lte = Keep lte' ->
|
||||
SubView lte n -> SubView lte' n'
|
||||
fromKEEP eq Refl (KEEP {n} p Refl) =
|
||||
let eq = sDoubleInj eq {m = n, n = n'} in
|
||||
rewrite sym eq in p
|
||||
|
||||
export
|
||||
view : Length xs => Length ys =>
|
||||
(m : Sub xs ys) -> SubView 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 : Sub xs ys) -> SubView m.lte m.mask
|
||||
view : Length ys => (m : Sub xs ys) -> SubView m.lte m.mask
|
||||
view @{Z} (SubM mask lte view0) with 0 (lteNilRight lte)
|
||||
_ | Refl =
|
||||
rewrite lteNil2End lte in
|
||||
rewrite ltemEnd view0 (lteNil2End lte) in
|
||||
END Refl
|
||||
view @{S sy} (SubM mask lte view0) {ys = ys :< z} with (viewLsb mask)
|
||||
_ | Lsb0 n eqn with (splitExEq $ ltemEvenDrop eqn view0)
|
||||
_ | Evidence lte' eql =
|
||||
rewrite eqn in rewrite eql in
|
||||
DROP (fromDROP eqn eql view0) Refl
|
||||
_ | Lsb1 n eqn with (splitExEq $ ltemOddLeft eqn view0)
|
||||
_ | Evidence xs' Refl =
|
||||
let Evidence q eqq = splitExEq $ ltemOddKeep eqn view0 in
|
||||
rewrite eqn in rewrite eqq in
|
||||
KEEP (fromKEEP eqn eqq view0) Refl
|
||||
|
||||
public export %inline
|
||||
(.view) : Length ys => (m : Sub xs ys) -> SubView m.lte m.mask
|
||||
(.view) = view
|
||||
|
||||
public export %inline
|
||||
review : Length ys => {m : Nat} -> {0 lte : xs `Sub'` ys} ->
|
||||
(0 _ : SubView lte m) -> SubView lte m
|
||||
review v = view $ sub v
|
||||
|
||||
|
||||
|
||||
export
|
||||
ltemLen : Length xs => Length ys =>
|
||||
xs `Sub` ys -> length xs `LTE` length ys
|
||||
ltemLen @{sx} @{sy} sub@(SubM m l _) with (sub.view)
|
||||
ltemLen @{sx} @{sy} sub@(SubM 0 End _) | END = LTEZero
|
||||
ltemLen @{S sx} @{S sy} sub@(SubM (S (2 * n)) (Keep p) _) | (KEEP q) =
|
||||
LTESucc $ ltemLen $ SubM n p q
|
||||
ltemLen @{sx} @{S sy} sub@(SubM (2 * n) (Drop p) _) | (DROP q) =
|
||||
lteSuccRight $ ltemLen $ SubM n p q
|
||||
ltemLen : (sy : Length ys) => xs `Sub` ys -> length xs `LTE` sy.nat
|
||||
ltemLen (SubM mask lte view0) {sy} with (review view0)
|
||||
ltemLen (SubM _ _ _) | END eq = LTEZero
|
||||
ltemLen (SubM _ _ _) {sy = S _} | KEEP v eq = LTESucc $ ltemLen $ sub v
|
||||
ltemLen (SubM _ _ _) {sy = S _} | DROP v eq = lteSuccRight $ ltemLen $ sub v
|
||||
|
||||
export
|
||||
ltemNilRight : xs `Sub` [<] -> xs = [<]
|
||||
ltemNilRight m = irrelevantEq $ lteNilRight m.lte
|
||||
|
||||
|
||||
public export %inline
|
||||
end : [<] `Sub` [<]
|
||||
end = SubM 0 End END
|
||||
|
||||
public export %inline
|
||||
keep : xs `Sub` ys -> xs :< z `Sub` ys :< z
|
||||
keep (SubM mask lte view0) = SubM (S (2 * mask)) (Keep lte) (KEEP view0)
|
||||
|
||||
public export %inline
|
||||
drop : xs `Sub` ys -> xs `Sub` ys :< z
|
||||
drop (SubM mask lte view0) = SubM (2 * mask) (Drop lte) (DROP view0)
|
||||
|
||||
|
||||
export %inline
|
||||
dropLast : Length xs => Length ys =>
|
||||
(xs :< x) `Sub` ys -> xs `Sub` ys
|
||||
dropLast @{sx} @{sy} sub@(SubM mask lte _) with (sub.view)
|
||||
dropLast sub@(SubM (S (2 * n)) (Keep p) _) | (KEEP v) =
|
||||
SubM (2 * n) (Drop p) (DROP v)
|
||||
dropLast @{_} @{S sy} sub@(SubM (2 * n) (Drop p) _) | DROP v =
|
||||
drop $ dropLast $ SubM n p v
|
||||
dropLast : Length ys => (xs :< x) `Sub` ys -> xs `Sub` ys
|
||||
dropLast (SubM _ _ view0) @{sy} with (review view0)
|
||||
dropLast (SubM _ _ _) | KEEP v _ = drop $ sub v
|
||||
dropLast (SubM _ _ _) @{S _} | DROP v _ = drop $ dropLast $ sub v
|
||||
|
||||
|
||||
export
|
||||
|
|
@ -293,11 +357,9 @@ Uninhabited (xs :< x `Sub` [<]) where
|
|||
|
||||
export
|
||||
Length xs => Uninhabited (xs :< x `Sub` xs) where
|
||||
uninhabited @{sx} sub@(SubM mask lte view0) with (sub.view)
|
||||
uninhabited @{S sx} sub@(SubM (S (2 * n)) (Keep p) _) | KEEP v =
|
||||
uninhabited $ SubM n p v
|
||||
uninhabited @{S sx} sub@(SubM (2 * n) (Drop p) _) | DROP v =
|
||||
uninhabited $ dropLast $ SubM n p v
|
||||
uninhabited (SubM _ _ view0) @{sx} with (review view0)
|
||||
uninhabited (SubM _ _ _) @{S _} | KEEP v _ = uninhabited $ sub v
|
||||
uninhabited (SubM _ _ _) @{S _} | DROP v _ = uninhabited $ dropLast $ sub v
|
||||
|
||||
|
||||
export
|
||||
|
|
@ -305,103 +367,206 @@ refl : Length xs => xs `Sub` xs
|
|||
refl @{Z} = end
|
||||
refl @{S s} = keep refl
|
||||
|
||||
export Reflexive (Scope a) Sub where reflexive = refl
|
||||
export
|
||||
0 reflLte : {sx : Length xs} -> (refl @{sx}).lte = refl' @{sx}
|
||||
reflLte {sx = Z} = Refl
|
||||
reflLte {sx = S s} = trans (keepLte _) $ cong Keep reflLte
|
||||
|
||||
public export %inline
|
||||
Reflexive (Scope a) Sub where reflexive = refl
|
||||
|
||||
mutual
|
||||
private
|
||||
antisym_ : Length xs => Length ys =>
|
||||
{0 p : xs `Sub'` ys} -> {0 q : ys `Sub'` xs} ->
|
||||
SubView p m1 -> SubView q m2 -> xs = ys
|
||||
antisym_ END END = Refl
|
||||
antisym_ (KEEP v1) (KEEP v2 {z}) @{S sx} @{S sy} =
|
||||
antisym_ (END _) (END _) = Refl
|
||||
antisym_ (KEEP v1 _) (KEEP v2 _ {z}) @{S sx} @{S sy} =
|
||||
cong (:< z) $ antisym (SubM _ _ v1) (SubM _ _ v2)
|
||||
antisym_ (KEEP v1) (DROP v2) {p = Keep p} {q = Drop q} =
|
||||
antisym_ (KEEP {}) (DROP {}) {p = Keep p} {q = Drop q} =
|
||||
void $ succNotLTEpred $ lteLen q `transitive` lteLen p
|
||||
antisym_ (DROP v1) (KEEP v2) {p = Drop p} {q = Keep q} =
|
||||
antisym_ (DROP {}) (KEEP {}) {p = Drop p} {q = Keep q} =
|
||||
void $ succNotLTEpred $ lteLen p `transitive` lteLen q
|
||||
antisym_ (DROP v1) (DROP v2) {p = Drop p} {q = Drop q} =
|
||||
antisym_ (DROP {}) (DROP {}) {p = Drop p} {q = Drop q} =
|
||||
void $ succNotLTEpred $ lteLen p `transitive` lteSuccLeft (lteLen q)
|
||||
|
||||
export
|
||||
antisym : Length xs => Length ys => xs `Sub` ys -> ys `Sub` xs -> xs = ys
|
||||
antisym p q = antisym_ p.view q.view
|
||||
antisym v1 v2 = antisym_ v1.view v2.view
|
||||
|
||||
export
|
||||
public export %inline
|
||||
Antisymmetric (Scope a) Sub where
|
||||
antisymmetric p q = antisym p q
|
||||
|
||||
|
||||
mutual
|
||||
private
|
||||
trans_ : Length xs => Length ys => Length zs =>
|
||||
trans_ : Length ys => Length zs =>
|
||||
{0 p : xs `Sub'` ys} -> {0 q : ys `Sub'` zs} ->
|
||||
SubView p m1 -> SubView q m2 -> xs `Sub` zs
|
||||
trans_ END END = end
|
||||
trans_ (KEEP v1) (KEEP v2) @{S sx} @{S sy} @{S sz} =
|
||||
keep $ SubM _ _ v1 `trans` SubM _ _ v2
|
||||
trans_ (DROP v1) (KEEP v2) @{sx} @{S sy} @{S sz} =
|
||||
drop $ SubM _ _ v1 `trans` SubM _ _ v2
|
||||
trans_ v1 (DROP v2) @{sx} @{sy} @{S sz} =
|
||||
trans_ (END _) (END _) = end
|
||||
trans_ (KEEP v1 _) (KEEP v2 _) @{S sy} @{S sz} = keep $ sub v1 `trans` sub v2
|
||||
trans_ (DROP v1 _) (KEEP v2 _) @{S sy} @{S sz} = drop $ sub v1 `trans` sub v2
|
||||
trans_ v1 (DROP v2 _) @{sy} @{S sz} =
|
||||
let Element m1' eq = getMask v1 in
|
||||
drop $ SubM m1' _ (rewrite sym eq in v1) `trans` SubM _ _ v2
|
||||
drop $ SubM m1' _ (rewrite sym eq in v1) `trans` sub v2
|
||||
|
||||
export
|
||||
trans : Length xs => Length ys => Length zs =>
|
||||
xs `Sub` ys -> ys `Sub` zs -> xs `Sub` zs
|
||||
trans : Length ys => Length zs => xs `Sub` ys -> ys `Sub` zs -> xs `Sub` zs
|
||||
trans p q = trans_ p.view q.view
|
||||
|
||||
export
|
||||
(.) : Length xs => Length ys => Length zs =>
|
||||
xs `Sub` ys -> ys `Sub` zs -> xs `Sub` zs
|
||||
(.) = trans
|
||||
public export %inline
|
||||
(.) : Length ys => Length zs => ys `Sub` zs -> xs `Sub` ys -> xs `Sub` zs
|
||||
(.) = flip trans
|
||||
|
||||
export
|
||||
public export %inline
|
||||
comp : Length ys => Length zs => ys `Sub` zs -> xs `Sub` ys -> xs `Sub` zs
|
||||
comp = (.)
|
||||
|
||||
public export %inline
|
||||
Transitive (Scope a) Sub where
|
||||
transitive p q = trans p q
|
||||
transitive p q = q . p
|
||||
|
||||
|
||||
export
|
||||
public export
|
||||
zero : Length xs => [<] `Sub` xs
|
||||
zero @{Z} = end
|
||||
zero @{S s} = drop zero
|
||||
|
||||
export
|
||||
0 zeroLte : {sx : Length xs} -> (zero @{sx}).lte = zero' @{sx}
|
||||
zeroLte {sx = Z} = Refl
|
||||
zeroLte {sx = S s} = trans (dropLte zero) $ cong Drop zeroLte
|
||||
|
||||
public export
|
||||
single : Length xs => x `Elem` xs -> [< x] `Sub` xs
|
||||
single @{S sx} Here = keep zero
|
||||
single @{S sx} (There p) = drop $ single p
|
||||
|
||||
export
|
||||
0 singleLte : {sx : Length xs} -> (p : x `Elem` xs) ->
|
||||
(single p @{sx}).lte = single' p @{sx}
|
||||
singleLte {sx = S s} Here = trans (keepLte zero) $ cong Keep zeroLte
|
||||
singleLte {sx = S s} (There p) = trans (dropLte _) $ cong Drop $ singleLte p
|
||||
|
||||
|
||||
mutual
|
||||
private
|
||||
app0 : Length ys2 =>
|
||||
(l : xs1 `Sub` ys1) ->
|
||||
{rm : Nat} -> {0 rl : xs2 `Sub'` ys2} -> (rv : SubView rl rm) ->
|
||||
(xs1 ++ xs2) `Sub` (ys1 ++ ys2)
|
||||
app0 l (END eq) = l
|
||||
app0 l (KEEP v Refl) @{S sy} = keep $ l ++ sub v
|
||||
app0 l (DROP v Refl) @{S sy} = drop $ l ++ sub v
|
||||
|
||||
export
|
||||
app : Length ys2 =>
|
||||
xs1 `Sub` ys1 -> xs2 `Sub` ys2 -> (xs1 ++ xs2) `Sub` (ys1 ++ ys2)
|
||||
app l r = app0 l r.view
|
||||
|
||||
public export %inline
|
||||
(++) : Length ys2 =>
|
||||
xs1 `Sub` ys1 -> xs2 `Sub` ys2 -> (xs1 ++ xs2) `Sub` (ys1 ++ ys2)
|
||||
(++) = app
|
||||
|
||||
export
|
||||
(++) : Length xs2 => Length ys2 =>
|
||||
xs1 `Sub` ys1 -> xs2 `Sub` ys2 -> (xs1 ++ xs2) `Sub` (ys1 ++ ys2)
|
||||
(++) sub1 sub2@(SubM {}) @{sx2} @{sy2} with (sub2.view)
|
||||
(++) sub1 sub2@(SubM {}) | END = sub1
|
||||
(++) sub1 sub2@(SubM {}) @{S sx2} @{S sy2} | KEEP v = keep $ sub1 ++ SubM _ _ v
|
||||
(++) sub1 sub2@(SubM {}) @{sx2} @{S sy2} | DROP v = drop $ sub1 ++ SubM _ _ v
|
||||
0 appLte : {sy : Length ys2} ->
|
||||
(l : xs1 `Sub` ys1) -> (r : xs2 `Sub` ys2) ->
|
||||
(app l r @{sy}).lte = app' l.lte r.lte
|
||||
appLte l r@(SubM 0 End (END Refl)) =
|
||||
cong (\v => (app0 l v).lte) (viewIrrel (view end) (END Refl))
|
||||
appLte l r@(SubM (S (2 * n)) (Keep p) (KEEP v Refl)) {sy = S sy} =
|
||||
trans (cong (\v => (app0 l v @{S sy}).lte) (viewIrrel _ (KEEP v Refl))) $
|
||||
trans (keepLte _) $
|
||||
cong Keep $ appLte l (sub v)
|
||||
appLte l r@(SubM 0 (Drop p) (DROP {n = 0} v Refl)) {sy = S sy} =
|
||||
trans (cong (\v => (app0 l v @{S sy}).lte) (viewIrrel _ (DROP v Refl))) $
|
||||
trans (dropLte _) $
|
||||
cong Drop $ appLte l (sub v)
|
||||
appLte l r@(SubM (2 * S n) (Drop p) (DROP {n = S n} v Refl)) {sy = S sy} =
|
||||
trans (cong (\v => (app0 l v @{S sy}).lte) (viewIrrel _ (DROP v Refl))) $
|
||||
trans (dropLte _) $
|
||||
cong Drop $ appLte l (sub v)
|
||||
appLte {xs2 = _ :< _, ys2 = [<]} _ (SubM _ _ _) impossible
|
||||
|
||||
export
|
||||
0 keepAppRight : {sy : Length ys2} ->
|
||||
(l : xs1 `Sub` ys1) -> (r : xs2 `Sub` ys2) ->
|
||||
(l ++ keep r) @{S sy} = keep ((l ++ r) @{sy})
|
||||
keepAppRight l (SubM mask _ _) = rewrite (lsbOdd mask).snd in Refl
|
||||
|
||||
export
|
||||
0 dropAppRight : (l : xs1 `Sub` ys1) -> (r : xs2 `Sub` ys2) ->
|
||||
(l ++ drop r) @{S sy} = drop ((l ++ r) @{sy})
|
||||
dropAppRight l r = lteEq $
|
||||
trans (appLte l (drop r)) $
|
||||
trans (cong (app' l.lte) (dropLte r)) $
|
||||
trans (cong Drop (sym (appLte l r))) $
|
||||
sym $ dropLte (l ++ r)
|
||||
|
||||
|
||||
export
|
||||
0 endRight : {xs, ys : Scope a} -> (sub : xs `Sub` ys) -> sub = (sub ++ Sub.end)
|
||||
endRight sub = Refl
|
||||
|
||||
|
||||
private
|
||||
0 transDrop_ : {p : xs `Sub'` ys} -> {q : ys `Sub'` zs} ->
|
||||
(pv : SubView p m1) -> (qv : SubView q m2) ->
|
||||
trans_ pv (DROP qv qe) @{sy} @{S sz} =
|
||||
drop (trans_ pv qv @{sy} @{sz})
|
||||
transDrop_ {sy = Z} {sz = sz} (END Refl) qv =
|
||||
cong (\v => drop $ trans_ {m2, q} (END Refl) v) $ viewIrrel {}
|
||||
transDrop_ {sy = S sy} {sz = sz} (KEEP pv Refl) qv {p = Keep p} =
|
||||
cong2 (\x, y => drop (trans_ @{S sy} x y)) (viewIrrel {}) (viewIrrel {})
|
||||
transDrop_ {sy = S sy} {sz = S sz} (DROP pv Refl) qv =
|
||||
cong2 (\x, y => drop (trans_ @{S sy} @{S sz} x y))
|
||||
(viewIrrel {}) (viewIrrel {})
|
||||
transDrop_ {ys = _ :< _, zs = [<]} _ _ impossible
|
||||
|
||||
private
|
||||
0 transLte_ : {p : xs `Sub'` ys} -> {q : ys `Sub'` zs} ->
|
||||
(pv : SubView p m1) -> (qv : SubView q m2) ->
|
||||
(trans_ pv qv @{sy} @{sz}).lte = trans' p q
|
||||
transLte_ (END _) (END _) = Refl
|
||||
transLte_ (KEEP pv pe) (KEEP qv qe) {sy = S _} {sz = S _} =
|
||||
trans (keepLte _) $ cong Keep $ transLte_ {}
|
||||
transLte_ (DROP pv pe) (KEEP qv qe) {sy = S _} {sz = S _} =
|
||||
trans (dropLte _) $ cong Drop $ transLte_ {}
|
||||
transLte_ pv (DROP qv qe) {sz = S _} =
|
||||
trans (cong lte $ transDrop_ pv qv) $
|
||||
trans (dropLte _) $ cong Drop $ transLte_ pv qv
|
||||
transLte_ {ys = _ :< _, zs = [<]} _ _ impossible
|
||||
|
||||
export
|
||||
0 transLte : {sy : Length ys} -> {sz : Length zs} ->
|
||||
(p : xs `Sub` ys) -> (q : ys `Sub` zs) ->
|
||||
(trans p q @{sy} @{sz}).lte = trans' p.lte q.lte
|
||||
transLte p q = transLte_ p.view q.view
|
||||
|
||||
public export
|
||||
0 compLte : {sy : Length ys} -> {sz : Length zs} ->
|
||||
(q : ys `Sub` zs) -> (p : xs `Sub` ys) ->
|
||||
((q . p) @{sy} @{sz}).lte = q.lte . p.lte
|
||||
compLte q p = transLte p q
|
||||
|
||||
|
||||
||| if `p` holds for all elements of the main list,
|
||||
||| it holds for all elements of the sublist
|
||||
export
|
||||
subAll : Length xs => Length ys =>
|
||||
xs `Sub` ys -> All prop ys -> All prop xs
|
||||
subAll sub@(SubM {}) ps @{sx} @{sy} with (sub.view)
|
||||
subAll sub@(SubM {}) [<] | END = [<]
|
||||
subAll sub@(SubM {}) (ps :< p) @{S sx} @{S sy} | KEEP v =
|
||||
subAll (SubM _ _ v) ps :< p
|
||||
subAll sub@(SubM {}) (ps :< p) @{sx} @{S sy} | DROP v =
|
||||
subAll (SubM _ _ v) ps
|
||||
subAll : Length ys => xs `Sub` ys -> All prop ys -> All prop xs
|
||||
subAll (SubM _ _ v) ps @{sy} with (review v)
|
||||
subAll (SubM _ _ _) [<] | END _ = [<]
|
||||
subAll (SubM _ _ _) (ps :< p) @{S sy} | KEEP v _ = subAll (sub v) ps :< p
|
||||
subAll (SubM _ _ _) (ps :< p) @{S sy} | DROP v _ = subAll (sub v) ps
|
||||
|
||||
|
||||
||| if `p` holds for one element of the sublist,
|
||||
||| it holds for one element of the main list
|
||||
export
|
||||
subAny : Length xs => Length ys =>
|
||||
xs `Sub` ys -> Any prop xs -> Any prop ys
|
||||
subAny sub@(SubM {}) p @{sx} @{sy} with (sub.view)
|
||||
subAny sub@(SubM {}) p | END impossible
|
||||
subAny sub@(SubM {}) (Here p) | KEEP v = Here p
|
||||
subAny sub@(SubM {}) (There p) @{S sx} @{S sy} | KEEP v =
|
||||
There $ subAny (SubM _ _ v) p
|
||||
subAny sub@(SubM {}) p @{sx} @{S sy} | DROP v =
|
||||
There $ subAny (SubM _ _ v) p
|
||||
subAny : Length ys => xs `Sub` ys -> Any prop xs -> Any prop ys
|
||||
subAny (SubM _ _ v) p @{sy} with (review v)
|
||||
subAny (SubM _ _ _) (Here p) | KEEP v _ = Here p
|
||||
subAny (SubM _ _ _) (There p) @{S sy} | KEEP v _ = There $ subAny (sub v) p
|
||||
subAny (SubM _ _ _) p @{S sy} | DROP v _ = There $ subAny (sub v) p
|
||||
|
|
|
|||
Loading…
Reference in New Issue