||| "order preserving embeddings", for recording a correspondence between
||| a smaller scope and part of a larger one.
module Quox.OPE

import Quox.NatExtra

import Data.Nat


public export
data OPE : Nat -> Nat -> Type where
  Id   : OPE n n
  Drop : OPE m n -> OPE m     (S n)
  Keep : OPE m n -> OPE (S m) (S n)
%name OPE p, q

public export %inline Injective Drop where injective Refl = Refl
public export %inline Injective Keep where injective Refl = Refl

public export
opeZero : {n : Nat} -> OPE 0 n
opeZero {n = 0}   = Id
opeZero {n = S n} = Drop opeZero

public export
(.) : OPE m n -> OPE n p -> OPE m p
p      . Id     = p
Id     . q      = q
p      . Drop q = Drop $ p . q
Drop p . Keep q = Drop $ p . q
Keep p . Keep q = Keep $ p . q

public export
toLTE : {m : Nat} -> OPE m n -> m `LTE` n
toLTE Id       = reflexive
toLTE (Drop p) = lteSuccRight $ toLTE p
toLTE (Keep p) = LTESucc      $ toLTE p


public export
dropInner' : LTE' m n -> OPE m n
dropInner' LTERefl      = Id
dropInner' (LTESuccR p) = Drop $ dropInner' $ force p

public export
dropInner : {n : Nat} -> LTE m n -> OPE m n
dropInner = dropInner' . fromLTE

public export
dropInnerN : (m : Nat) -> OPE n (m + n)
dropInnerN 0     = Id
dropInnerN (S m) = Drop $ dropInnerN m



-- [todo] can this be done with fancy nats too?
-- with bitmasks sure but that might not be worth the effort
-- [the next day] it probably isn't

-- public export
-- data OPE' : Nat -> Nat -> Type where
--   None : OPE' 0 0
--   Drop : OPE' m n -> OPE' m (S n)
--   Keep : OPE' m n -> OPE' (S m) (S n)
-- %name OPE' q


-- public export %inline
-- drop' : Integer -> Integer
-- drop' n = n * 2

-- public export %inline
-- keep' : Integer -> Integer
-- keep' n = 1 + 2 * n

-- public export
-- data IsOPE : Integer -> (OPE' m n) -> Type where
--   IsNone : 0 `IsOPE` None
--   IsDrop : (0 _ : m `IsOPE` q) -> drop' m `IsOPE` Drop q
--   IsKeep : (0 _ : m `IsOPE` q) -> keep' m `IsOPE` Keep q
-- %name IsOPE p


-- public export
-- record OPE m n where
--   constructor MkOPE
--   value  : Integer
--   0 spec : OPE' m n
--   0 prf  : value `IsOPE` spec
--   0 pos  : So (value >= 0)


-- private
-- 0 idrisPleaseLearnAboutIntegers : {x, y : Integer} -> x = y
-- idrisPleaseLearnAboutIntegers {x, y} = believe_me $ Refl {x}

-- private
-- 0 natIntPlus : (m, n : Nat) ->
--                natToInteger (m + n) = natToInteger m + natToInteger n
-- natIntPlus m n = idrisPleaseLearnAboutIntegers

-- private
-- 0 shiftTwice : (x : Integer) -> (m, n : Nat) ->
--                x `shiftL` (m + n) = (x `shiftL` m) `shiftL` n
-- shiftTwice x m n = idrisPleaseLearnAboutIntegers

-- private
-- 0 shift1 : (x : Integer) -> (x `shiftL` 1) = 2 * x
-- shift1 x = idrisPleaseLearnAboutIntegers

-- private
-- 0 intPlusComm : (x, y : Integer) -> (x + y) = (y + x)
-- intPlusComm x y = idrisPleaseLearnAboutIntegers

-- private
-- 0 intTimes2Minus1 : (x : Integer) -> 2 * x - 1 = 2 * (x - 1) + 1
-- intTimes2Minus1 x = idrisPleaseLearnAboutIntegers

-- private
-- 0 intPosShift : So (x > 0) -> So (x `shiftL` i > 0)
-- intPosShift p = believe_me Oh

-- private
-- 0 intNonnegDec : {x : Integer} -> So (x > 0) -> So (x - 1 >= 0)
-- intNonnegDec p = believe_me Oh


-- private
-- 0 shiftSucc : (x : Integer) -> (n : Nat) ->
--               x `shiftL` S n = 2 * (x `shiftL` n)
-- shiftSucc x n = Calc $
--   |~  x `shiftL` S n
--   ~~  x `shiftL` (n + 1)
--         ...(cong (x `shiftL`) $ sym $ plusCommutative {})
--   ~~  (x `shiftL` n) `shiftL` 1
--         ...(shiftTwice {})
--   ~~  2 * (x `shiftL` n)
--         ...(shift1 {})


-- private
-- opeIdVal : (n : Nat) -> Integer
-- opeIdVal n = (1 `shiftL` n) - 1

-- private
-- 0 opeIdValSpec : (n : Nat) -> Integer
-- opeIdValSpec 0     = 0
-- opeIdValSpec (S n) = keep' $ opeIdValSpec n

-- 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, _}