quox/lib/Quox/Syntax/Var.idr

module Quox.Syntax.Var

import public Quox.Loc
import public Quox.Name
import Quox.OPE

import Data.Nat
import Data.List
import Data.Vect
import public Quox.Decidable
import Data.Bool.Decidable
import Data.DPair

%default total


public export
data Var : Nat -> Type where
  VZ : Var (S n)
  VS : Var n -> Var (S n)
%name Var i, j
%builtin Natural Var

public export
(.nat) : Var n -> Nat
(VZ).nat   = 0
(VS i).nat = S i.nat
%transform "Var.(.nat)" Var.(.nat) i = believe_me i

public export %inline Cast (Var n) Nat     where cast = (.nat)
public export %inline Cast (Var n) Integer where cast = cast . cast {to = Nat}

export %inline Eq   (Var n) where i == j = i.nat == j.nat
export %inline Ord  (Var n) where compare i j = compare i.nat j.nat
export %inline Show (Var n) where showPrec d i = showCon d "V" $ showArg i.nat

public export %inline Injective VS where injective Refl = Refl

export Uninhabited (Var 0) where uninhabited _ impossible

export Uninhabited (VZ   = VS i) where uninhabited _ impossible
export Uninhabited (VS i = VZ)   where uninhabited _ impossible


public export
data Eqv : Var m -> Var n -> Type where
  EZ : VZ `Eqv` VZ
  ES : i `Eqv` j -> VS i `Eqv` VS j
%name Var.Eqv e

export
decEqv : Dec2 Eqv
decEqv VZ     VZ     = Yes EZ
decEqv VZ     (VS i) = No $ \case _ impossible
decEqv (VS i) VZ     = No $ \case _ impossible
decEqv (VS i) (VS j) =
  case decEqv i j of
    Yes y => Yes $ ES y
    No  n => No  $ \(ES y) => n y


private
lookupS : Nat -> SnocList a -> Maybe a
lookupS _     [<]       = Nothing
lookupS Z     (sx :< x) = Just x
lookupS (S i) (sx :< x) = lookupS i sx


public export
fromNatWith : (i : Nat) -> (0 p : i `LT` n) -> Var n
fromNatWith Z     (LTESucc _) = VZ
fromNatWith (S i) (LTESucc p) = VS $ fromNatWith i p
%transform "Var.fromNatWith" fromNatWith i p = believe_me i

public export %inline
V : (i : Nat) -> {auto 0 p : i `LT` n} -> Var n
V i = fromNatWith i p

export %inline
tryFromNat : Alternative f => (n : Nat) -> Nat -> f (Var n)
tryFromNat n i =
  case i `isLT` n of
       Yes p => pure $ fromNatWith i p
       No  _ => empty

export
0 toNatLT : (i : Var n) -> i.nat `LT` n
toNatLT VZ     = LTESucc LTEZero
toNatLT (VS i) = LTESucc $ toNatLT i

public export
toNatInj : {i, j : Var n} -> i.nat = j.nat -> i = j
toNatInj {i = VZ}     {j = VZ}     Refl = Refl
toNatInj {i = VZ}     {j = (VS i)} Refl impossible
toNatInj {i = (VS i)} {j = VZ}     Refl impossible
toNatInj {i = (VS i)} {j = (VS j)} prf  = cong VS $ toNatInj $ injective prf

public export %inline Injective (.nat) where injective = toNatInj

export
0 fromToNat : (i : Var n) -> (p : i.nat `LT` n) -> fromNatWith i.nat p = i
fromToNat VZ     (LTESucc p) = Refl
fromToNat (VS i) (LTESucc p) = rewrite fromToNat i p in Refl

export
0 toFromNat : (i : Nat) -> (p : i `LT` n) -> (fromNatWith i p).nat = i
toFromNat 0     (LTESucc x) = Refl
toFromNat (S k) (LTESucc x) = cong S $ toFromNat k x


-- not using %transform like other things because weakSpec requires the proof
-- to be relevant. but since only `LTESucc` is ever possible that seems
-- to be an instance of <https://github.com/idris-lang/Idris2/issues/1259>?
export
weak : (0 p : m `LTE` n) -> Var m -> Var n
weak p i = fromNatWith i.nat $ transitive (toNatLT i) p

public export
0 weakSpec : m `LTE` n -> Var m -> Var n
weakSpec LTEZero     _      impossible
weakSpec (LTESucc p) VZ     = VZ
weakSpec (LTESucc p) (VS i) = VS $ weakSpec p i

export
0 weakSpecCorrect : (p : m `LTE` n) -> (i : Var m) -> (weakSpec p i).nat = i.nat
weakSpecCorrect LTEZero     _      impossible
weakSpecCorrect (LTESucc x) VZ     = Refl
weakSpecCorrect (LTESucc x) (VS i) = cong S $ weakSpecCorrect x i

export
0 weakCorrect : (p : m `LTE` n) -> (i : Var m) -> (weak p i).nat = i.nat
weakCorrect LTEZero     _      impossible
weakCorrect (LTESucc p) VZ     = Refl
weakCorrect (LTESucc p) (VS i) = cong S $ weakCorrect p i

export
0 weakIsSpec : (p : m `LTE` n) -> (i : Var m) -> weak p i = weakSpec p i
weakIsSpec p i = toNatInj $ trans (weakCorrect p i) (sym $ weakSpecCorrect p i)


export
tabulateV : {0 tm : Nat -> Type} -> (forall n. Var n -> tm n) ->
            (n : Nat) -> Vect n (tm n)
tabulateV f 0     = []
tabulateV f (S n) = f VZ :: tabulateV (f . VS) n

export
allVars : (n : Nat) -> Vect n (Var n)
allVars n = tabulateV id n


public export
data LT : Rel (Var n) where
  LTZ : VZ `LT` VS i
  LTS : i `LT` j -> VS i `LT` VS j
%builtin Natural Var.LT
%name Var.LT lt

public export %inline
GT : Rel (Var n)
i `GT` j = j `LT` i

export
Transitive (Var n) LT where
  transitive LTZ     (LTS _) = LTZ
  transitive (LTS p) (LTS q) = LTS $ transitive p q

export Uninhabited (i    `Var.LT` i)  where uninhabited (LTS p) = uninhabited p
export Uninhabited (VS i `LT`     VZ) where uninhabited _ impossible


export
ltReflect : LT {n} `Reflects2` (<)
ltReflect VZ     VZ     = RFalse absurd
ltReflect VZ     (VS j) = RTrue LTZ
ltReflect (VS i) VZ     = RFalse absurd
ltReflect (VS i) (VS j) with (ltReflect i j) | (i < j)
  _ | RTrue  yes | True  = RTrue  $ LTS yes
  _ | RFalse no  | False = RFalse $ \case LTS p => no p

export
isLT : Dec2 Var.LT
isLT i j = reflectToDec $ ltReflect i j


public export
data Compare : (i, j : Var n) -> Ordering -> Type where
  IsLT : (lt : i `LT` j) -> Compare i j LT
  IsEQ :                    Compare i i EQ
  IsGT : (gt : i `GT` j) -> Compare i j GT
%name Compare cmp

export
compareS : Compare i j o -> Compare (VS i) (VS j) o
compareS (IsLT lt) = IsLT (LTS lt)
compareS IsEQ      = IsEQ
compareS (IsGT gt) = IsGT (LTS gt)

export
compareP : (i, j : Var n) -> Compare i j (compare i j)
compareP VZ     VZ     = IsEQ
compareP VZ     (VS j) = IsLT LTZ
compareP (VS i) VZ     = IsGT LTZ
compareP (VS i) (VS j) = compareS $ compareP i j

export
0 compare2 : Compare {n} i j o -> o = compare i j
compare2 (IsLT LTZ)      = Refl
compare2 (IsLT (LTS lt)) = compare2 (IsLT lt)
compare2 IsEQ            = sym $ compareNatDiag i.nat
compare2 (IsGT LTZ)      = Refl
compare2 (IsGT (LTS gt)) = compare2 $ IsGT gt
compare2 _ {n = 0}       = absurd i

export
0 compareSelf : Compare i i o -> o = EQ
compareSelf p = rewrite compare2 p in compareNatDiag i.nat

export
0 comparePSelf : (i : Var n) -> Compare i i EQ
comparePSelf i = rewrite sym $ compareNatDiag i.nat in compareP i i


public export
data LTE : Rel (Var n) where
  LTEZ : VZ `LTE` j
  LTES : i  `LTE` j -> VS i `LTE` VS j

export
Reflexive (Var n) LTE where
  reflexive {x = VZ}   = LTEZ
  reflexive {x = VS i} = LTES reflexive

export
Transitive (Var n) LTE where
  transitive LTEZ     q        = LTEZ
  transitive (LTES p) (LTES q) = LTES $ transitive p q

export
Antisymmetric (Var n) LTE where
  antisymmetric LTEZ     LTEZ     = Refl
  antisymmetric (LTES p) (LTES q) = cong VS $ antisymmetric p q

export
splitLTE : {j : Var n} -> i `LTE` j -> Either (i = j) (i `LT` j)
splitLTE {j = VZ}   LTEZ     = Left Refl
splitLTE {j = VS _} LTEZ     = Right LTZ
splitLTE            (LTES p) with (splitLTE p)
  _ | Left  eq = Left  $ cong VS eq
  _ | Right lt = Right $ LTS lt

export Uninhabited (VS i `LTE` VZ) where uninhabited _ impossible

export
lteReflect : (i, j : Var n) -> (LTE i j) `Reflects` (i <= j)
lteReflect VZ     VZ     = RTrue LTEZ
lteReflect VZ     (VS j) = RTrue LTEZ
lteReflect (VS i) VZ     = RFalse absurd
lteReflect (VS i) (VS j) with (lteReflect i j) | (i <= j)
  _ | RTrue  yes | True  = RTrue (LTES yes)
  _ | RFalse no  | False = RFalse $ \case LTES lte => no lte


public export
eqReflect : (i, j : Var n) -> (i = j) `Reflects` (i == j)
eqReflect VZ     VZ     = RTrue  Refl
eqReflect VZ     (VS i) = RFalse absurd
eqReflect (VS i) VZ     = RFalse absurd
eqReflect (VS i) (VS j) with (eqReflect i j) | (i == j)
  _ | RTrue  yes | True  = RTrue  $ cong VS yes
  _ | RFalse no  | False = RFalse $ no . injective

public export %inline
varDecEq : (i, j : Var n) -> Dec (i = j)
varDecEq i j = reflectToDec $ eqReflect i j

-- justified by eqReflect [citation needed]
private %inline
decEqFromBool : (i, j : Var n) -> Dec (i = j)
decEqFromBool i j =
  if i == j then Yes $ believe_me $ Refl {x = 0}
            else No  $ id . believe_me

%transform "Var.decEq" varDecEq = decEqFromBool

public export %inline DecEq (Var n) where decEq = varDecEq


export
Tighten Var where
  tighten Id       i      = Just i
  tighten (Drop p) VZ     = Nothing
  tighten (Drop p) (VS i) = tighten p i
  tighten (Keep p) VZ     = Just VZ
  tighten (Keep p) (VS i) = VS <$> tighten p i