211 lines
5.3 KiB
Idris
211 lines
5.3 KiB
Idris
module Quox.Syntax.DimEq
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import public Quox.Syntax.Var
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import public Quox.Syntax.Dim
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import public Quox.Syntax.Subst
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import public Quox.Context
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import Data.Maybe
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import Data.Nat
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import Data.DPair
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import Data.Fun.Graph
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import Decidable.Decidable
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import Decidable.Equality
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%default total
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public export
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DimEq' : Nat -> Type
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DimEq' = Context (Maybe . Dim)
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public export
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data DimEq : Nat -> Type where
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ZeroIsOne : DimEq d
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C : (eqs : DimEq' d) -> DimEq d
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%name DimEq eqs
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public export
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consistent : DimEq d -> Bool
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consistent ZeroIsOne = False
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consistent (C eqs) = True
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public export
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data IfConsistent : DimEq d -> Type -> Type where
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Nothing : IfConsistent ZeroIsOne a
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Just : a -> IfConsistent (C eqs) a
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export
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Functor (IfConsistent eqs) where
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map f Nothing = Nothing
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map f (Just x) = Just (f x)
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export
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Foldable (IfConsistent eqs) where
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foldr f z Nothing = z
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foldr f z (Just x) = f x z
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export
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Traversable (IfConsistent eqs) where
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traverse f Nothing = pure Nothing
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traverse f (Just x) = Just <$> f x
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public export
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ifConsistent : Applicative f => (eqs : DimEq d) -> f a -> f (IfConsistent eqs a)
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ifConsistent ZeroIsOne act = pure Nothing
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ifConsistent (C _) act = Just <$> act
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export %inline
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zeroEq : DimEq 0
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zeroEq = C [<]
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export %inline
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new' : {d : Nat} -> DimEq' d
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new' {d = 0} = [<]
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new' {d = S d} = new' :< Nothing
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export %inline
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new : {d : Nat} -> DimEq d
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new = C new'
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private %inline
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shiftMay : Maybe (Dim from) -> Shift from to -> Maybe (Dim to)
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shiftMay p by = map (// by) p
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export %inline
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get' : DimEq' d -> Var d -> Maybe (Dim d)
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get' = getWith shiftMay
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private %inline
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getShift' : Shift len out -> DimEq' len -> Var len -> Maybe (Dim out)
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getShift' = getShiftWith shiftMay
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export %inline
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get : DimEq' d -> Dim d -> Dim d
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get _ (K e) = K e
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get eqs (B i) = fromMaybe (B i) $ get' eqs i
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export %inline
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equal : DimEq d -> (p, q : Dim d) -> Bool
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equal ZeroIsOne p q = True
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equal (C eqs) p q = get eqs p == get eqs q
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infixl 7 :<?
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export %inline
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(:<?) : DimEq d -> Maybe (Dim d) -> DimEq (S d)
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ZeroIsOne :<? d = ZeroIsOne
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C eqs :<? d = C $ eqs :< map (get eqs) d
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private %inline
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ifVar : Var d -> Dim d -> Maybe (Dim d) -> Maybe (Dim d)
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ifVar i p = map $ \q => if q == B i then p else q
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-- (using decEq instead of (==) because of the proofs below)
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private %inline
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checkConst : (e, f : DimConst) -> (eqs : Lazy (DimEq' d)) -> DimEq d
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checkConst e f eqs = if isYes $ e `decEq` f then C eqs else ZeroIsOne
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export
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setConst : Var d -> DimConst -> DimEq' d -> DimEq d
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setConst VZ e (eqs :< Nothing) = C $ eqs :< Just (K e)
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setConst VZ e (eqs :< Just (K f)) = checkConst e f $ eqs :< Just (K f)
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setConst VZ e (eqs :< Just (B i)) = setConst i e eqs :<? Just (K e)
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setConst (VS i) e (eqs :< p) = setConst i e eqs :<? ifVar i (K e) p
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mutual
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private
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setVar' : (i, j : Var d) -> i `LT` j -> DimEq' d -> DimEq d
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setVar' VZ (VS i) LTZ (eqs :< Nothing) =
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C $ eqs :< Just (B i)
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setVar' VZ (VS i) LTZ (eqs :< Just (K e)) =
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setConst i e eqs :<? Just (K e)
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setVar' VZ (VS i) LTZ (eqs :< Just (B j)) =
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setVar i j eqs :<? Just (B (max i j))
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setVar' (VS i) (VS j) (LTS lt) (eqs :< p) =
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setVar' i j lt eqs :<? ifVar i (B j) p
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export %inline
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setVar : (i, j : Var d) -> DimEq' d -> DimEq d
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setVar i j eqs with (compareP i j) | (compare i.nat j.nat)
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setVar i j eqs | IsLT lt | LT = setVar' i j lt eqs
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setVar i i eqs | IsEQ | EQ = C eqs
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setVar i j eqs | IsGT gt | GT = setVar' j i gt eqs
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export %inline
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set : (p, q : Dim d) -> DimEq d -> DimEq d
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set _ _ ZeroIsOne = ZeroIsOne
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set (K e) (K f) (C eqs) = checkConst e f eqs
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set (K e) (B i) (C eqs) = setConst i e eqs
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set (B i) (K e) (C eqs) = setConst i e eqs
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set (B i) (B j) (C eqs) = setVar i j eqs
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public export %inline
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Split : Nat -> Type
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Split d = (DimEq' d, DSubst (S d) d)
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export %inline
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split1 : DimConst -> DimEq' (S d) -> Maybe (Split d)
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split1 e eqs = case setConst VZ e eqs of
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ZeroIsOne => Nothing
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C (eqs :< _) => Just (eqs, K e ::: id)
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export %inline
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split : DimEq' (S d) -> List (Split d)
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split eqs = toList (split1 Zero eqs) <+> toList (split1 One eqs)
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export
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splits' : DimEq' d -> List (DSubst d 0)
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splits' [<] = [id]
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splits' eqs@(_ :< _) = [th . ph | (eqs', th) <- split eqs, ph <- splits' eqs']
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export %inline
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splits : DimEq d -> List (DSubst d 0)
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splits ZeroIsOne = []
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splits (C eqs) = splits' eqs
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private
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0 newGetShift : (d : Nat) -> (i : Var d) -> (by : Shift d d') ->
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getShift' by (new' {d}) i = Nothing
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newGetShift (S d) VZ by = Refl
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newGetShift (S d) (VS i) by = newGetShift d i (ssDown by)
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export
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0 newGet' : (d : Nat) -> (i : Var d) -> get' (new' {d}) i = Nothing
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newGet' d i = newGetShift d i SZ
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export
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0 newGet : (d : Nat) -> (p : Dim d) -> get (new' {d}) p = p
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newGet d (K e) = Refl
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newGet d (B i) = rewrite newGet' d i in Refl
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export
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0 setSelf : (p : Dim d) -> (eqs : DimEq d) -> set p p eqs = eqs
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setSelf p ZeroIsOne = Refl
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setSelf (K Zero) (C eqs) = Refl
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setSelf (K One) (C eqs) = Refl
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setSelf (B i) (C eqs) with (compareP i i) | (compare i.nat i.nat)
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_ | IsLT lt | LT = absurd lt
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_ | IsEQ | EQ = Refl
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_ | IsGT gt | GT = absurd gt
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-- [todo] "well formed" dimeqs
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-- [todo] operations maintain well-formedness
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-- [todo] if 'Wf eqs' then 'equal eqs' is an equivalence
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-- [todo] 'set' never breaks existing equalities
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