305 lines
8.5 KiB
Idris
305 lines
8.5 KiB
Idris
module Quox.Syntax.Var
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import public Quox.Loc
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import public Quox.Name
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import Quox.OPE
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import Data.Nat
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import Data.List
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import Data.Vect
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import public Quox.Decidable
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import Data.Bool.Decidable
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import Data.DPair
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%default total
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public export
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data Var : Nat -> Type where
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VZ : Var (S n)
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VS : Var n -> Var (S n)
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%name Var i, j
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%builtin Natural Var
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public export
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(.nat) : Var n -> Nat
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(VZ).nat = 0
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(VS i).nat = S i.nat
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%transform "Var.(.nat)" Var.(.nat) i = believe_me i
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public export %inline Cast (Var n) Nat where cast = (.nat)
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public export %inline Cast (Var n) Integer where cast = cast . cast {to = Nat}
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export %inline Eq (Var n) where i == j = i.nat == j.nat
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export %inline Ord (Var n) where compare i j = compare i.nat j.nat
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export %inline Show (Var n) where showPrec d i = showCon d "V" $ showArg i.nat
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public export %inline Injective VS where injective Refl = Refl
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export Uninhabited (Var 0) where uninhabited _ impossible
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export Uninhabited (VZ = VS i) where uninhabited _ impossible
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export Uninhabited (VS i = VZ) where uninhabited _ impossible
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public export
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data Eqv : Var m -> Var n -> Type where
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EZ : VZ `Eqv` VZ
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ES : i `Eqv` j -> VS i `Eqv` VS j
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%name Var.Eqv e
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export
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decEqv : Dec2 Eqv
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decEqv VZ VZ = Yes EZ
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decEqv VZ (VS i) = No $ \case _ impossible
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decEqv (VS i) VZ = No $ \case _ impossible
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decEqv (VS i) (VS j) =
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case decEqv i j of
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Yes y => Yes $ ES y
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No n => No $ \(ES y) => n y
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private
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lookupS : Nat -> SnocList a -> Maybe a
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lookupS _ [<] = Nothing
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lookupS Z (sx :< x) = Just x
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lookupS (S i) (sx :< x) = lookupS i sx
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public export
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fromNatWith : (i : Nat) -> (0 p : i `LT` n) -> Var n
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fromNatWith Z (LTESucc _) = VZ
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fromNatWith (S i) (LTESucc p) = VS $ fromNatWith i p
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%transform "Var.fromNatWith" fromNatWith i p = believe_me i
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public export %inline
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V : (i : Nat) -> {auto 0 p : i `LT` n} -> Var n
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V i = fromNatWith i p
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export %inline
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tryFromNat : Alternative f => (n : Nat) -> Nat -> f (Var n)
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tryFromNat n i =
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case i `isLT` n of
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Yes p => pure $ fromNatWith i p
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No _ => empty
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export
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0 toNatLT : (i : Var n) -> i.nat `LT` n
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toNatLT VZ = LTESucc LTEZero
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toNatLT (VS i) = LTESucc $ toNatLT i
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public export
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toNatInj : {i, j : Var n} -> i.nat = j.nat -> i = j
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toNatInj {i = VZ} {j = VZ} Refl = Refl
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toNatInj {i = VZ} {j = (VS i)} Refl impossible
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toNatInj {i = (VS i)} {j = VZ} Refl impossible
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toNatInj {i = (VS i)} {j = (VS j)} prf = cong VS $ toNatInj $ injective prf
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public export %inline Injective (.nat) where injective = toNatInj
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export
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0 fromToNat : (i : Var n) -> (p : i.nat `LT` n) -> fromNatWith i.nat p = i
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fromToNat VZ (LTESucc p) = Refl
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fromToNat (VS i) (LTESucc p) = rewrite fromToNat i p in Refl
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export
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0 toFromNat : (i : Nat) -> (p : i `LT` n) -> (fromNatWith i p).nat = i
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toFromNat 0 (LTESucc x) = Refl
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toFromNat (S k) (LTESucc x) = cong S $ toFromNat k x
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-- not using %transform like other things because weakSpec requires the proof
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-- to be relevant. but since only `LTESucc` is ever possible that seems
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-- to be an instance of <https://github.com/idris-lang/Idris2/issues/1259>?
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export
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weak : (0 p : m `LTE` n) -> Var m -> Var n
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weak p i = fromNatWith i.nat $ transitive (toNatLT i) p
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public export
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0 weakSpec : m `LTE` n -> Var m -> Var n
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weakSpec LTEZero _ impossible
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weakSpec (LTESucc p) VZ = VZ
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weakSpec (LTESucc p) (VS i) = VS $ weakSpec p i
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export
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0 weakSpecCorrect : (p : m `LTE` n) -> (i : Var m) -> (weakSpec p i).nat = i.nat
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weakSpecCorrect LTEZero _ impossible
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weakSpecCorrect (LTESucc x) VZ = Refl
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weakSpecCorrect (LTESucc x) (VS i) = cong S $ weakSpecCorrect x i
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export
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0 weakCorrect : (p : m `LTE` n) -> (i : Var m) -> (weak p i).nat = i.nat
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weakCorrect LTEZero _ impossible
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weakCorrect (LTESucc p) VZ = Refl
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weakCorrect (LTESucc p) (VS i) = cong S $ weakCorrect p i
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export
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0 weakIsSpec : (p : m `LTE` n) -> (i : Var m) -> weak p i = weakSpec p i
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weakIsSpec p i = toNatInj $ trans (weakCorrect p i) (sym $ weakSpecCorrect p i)
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public export
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interface FromVar f where %inline fromVarLoc : Var n -> Loc -> f n
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public export %inline
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fromVar : FromVar f => Var n -> {default noLoc loc : Loc} -> f n
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fromVar x = fromVarLoc x loc
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public export FromVar Var where fromVarLoc x _ = x
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export
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tabulateV : {0 tm : Nat -> Type} -> (forall n. Var n -> tm n) ->
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(n : Nat) -> Vect n (tm n)
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tabulateV f 0 = []
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tabulateV f (S n) = f VZ :: tabulateV (f . VS) n
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export
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allVars : (n : Nat) -> Vect n (Var n)
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allVars n = tabulateV id n
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public export
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data LT : Rel (Var n) where
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LTZ : VZ `LT` VS i
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LTS : i `LT` j -> VS i `LT` VS j
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%builtin Natural Var.LT
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%name Var.LT lt
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public export %inline
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GT : Rel (Var n)
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i `GT` j = j `LT` i
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export
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Transitive (Var n) LT where
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transitive LTZ (LTS _) = LTZ
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transitive (LTS p) (LTS q) = LTS $ transitive p q
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export Uninhabited (i `Var.LT` i) where uninhabited (LTS p) = uninhabited p
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export Uninhabited (VS i `LT` VZ) where uninhabited _ impossible
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export
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ltReflect : LT {n} `Reflects2` (<)
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ltReflect VZ VZ = RFalse absurd
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ltReflect VZ (VS j) = RTrue LTZ
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ltReflect (VS i) VZ = RFalse absurd
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ltReflect (VS i) (VS j) with (ltReflect i j) | (i < j)
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_ | RTrue yes | True = RTrue $ LTS yes
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_ | RFalse no | False = RFalse $ \case LTS p => no p
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export
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isLT : Dec2 Var.LT
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isLT i j = reflectToDec $ ltReflect i j
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public export
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data Compare : (i, j : Var n) -> Ordering -> Type where
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IsLT : (lt : i `LT` j) -> Compare i j LT
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IsEQ : Compare i i EQ
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IsGT : (gt : i `GT` j) -> Compare i j GT
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%name Compare cmp
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export
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compareS : Compare i j o -> Compare (VS i) (VS j) o
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compareS (IsLT lt) = IsLT (LTS lt)
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compareS IsEQ = IsEQ
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compareS (IsGT gt) = IsGT (LTS gt)
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export
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compareP : (i, j : Var n) -> Compare i j (compare i j)
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compareP VZ VZ = IsEQ
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compareP VZ (VS j) = IsLT LTZ
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compareP (VS i) VZ = IsGT LTZ
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compareP (VS i) (VS j) = compareS $ compareP i j
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export
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0 compare2 : Compare {n} i j o -> o = compare i j
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compare2 (IsLT LTZ) = Refl
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compare2 (IsLT (LTS lt)) = compare2 (IsLT lt)
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compare2 IsEQ = sym $ compareNatDiag i.nat
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compare2 (IsGT LTZ) = Refl
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compare2 (IsGT (LTS gt)) = compare2 $ IsGT gt
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compare2 _ {n = 0} = absurd i
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export
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0 compareSelf : Compare i i o -> o = EQ
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compareSelf p = rewrite compare2 p in compareNatDiag i.nat
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export
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0 comparePSelf : (i : Var n) -> Compare i i EQ
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comparePSelf i = rewrite sym $ compareNatDiag i.nat in compareP i i
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public export
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data LTE : Rel (Var n) where
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LTEZ : VZ `LTE` j
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LTES : i `LTE` j -> VS i `LTE` VS j
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export
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Reflexive (Var n) LTE where
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reflexive {x = VZ} = LTEZ
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reflexive {x = VS i} = LTES reflexive
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export
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Transitive (Var n) LTE where
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transitive LTEZ q = LTEZ
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transitive (LTES p) (LTES q) = LTES $ transitive p q
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export
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Antisymmetric (Var n) LTE where
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antisymmetric LTEZ LTEZ = Refl
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antisymmetric (LTES p) (LTES q) = cong VS $ antisymmetric p q
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export
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splitLTE : {j : Var n} -> i `LTE` j -> Either (i = j) (i `LT` j)
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splitLTE {j = VZ} LTEZ = Left Refl
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splitLTE {j = VS _} LTEZ = Right LTZ
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splitLTE (LTES p) with (splitLTE p)
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_ | Left eq = Left $ cong VS eq
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_ | Right lt = Right $ LTS lt
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export Uninhabited (VS i `LTE` VZ) where uninhabited _ impossible
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export
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lteReflect : (i, j : Var n) -> (LTE i j) `Reflects` (i <= j)
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lteReflect VZ VZ = RTrue LTEZ
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lteReflect VZ (VS j) = RTrue LTEZ
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lteReflect (VS i) VZ = RFalse absurd
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lteReflect (VS i) (VS j) with (lteReflect i j) | (i <= j)
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_ | RTrue yes | True = RTrue (LTES yes)
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_ | RFalse no | False = RFalse $ \case LTES lte => no lte
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public export
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eqReflect : (i, j : Var n) -> (i = j) `Reflects` (i == j)
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eqReflect VZ VZ = RTrue Refl
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eqReflect VZ (VS i) = RFalse absurd
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eqReflect (VS i) VZ = RFalse absurd
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eqReflect (VS i) (VS j) with (eqReflect i j) | (i == j)
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_ | RTrue yes | True = RTrue $ cong VS yes
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_ | RFalse no | False = RFalse $ no . injective
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public export %inline
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varDecEq : (i, j : Var n) -> Dec (i = j)
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varDecEq i j = reflectToDec $ eqReflect i j
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-- justified by eqReflect [citation needed]
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private %inline
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decEqFromBool : (i, j : Var n) -> Dec (i = j)
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decEqFromBool i j =
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if i == j then Yes $ believe_me $ Refl {x = 0}
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else No $ id . believe_me
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%transform "Var.decEq" varDecEq = decEqFromBool
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public export %inline DecEq (Var n) where decEq = varDecEq
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export
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Tighten Var where
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tighten Id i = Just i
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tighten (Drop p) VZ = Nothing
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tighten (Drop p) (VS i) = tighten p i
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tighten (Keep p) VZ = Just VZ
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tighten (Keep p) (VS i) = VS <$> tighten p i
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