184 lines
4.7 KiB
Idris
184 lines
4.7 KiB
Idris
module Quox.Syntax.DimEq
|
|
|
|
import public Quox.Syntax.Var
|
|
import public Quox.Syntax.Dim
|
|
import public Quox.Syntax.Subst
|
|
import public Quox.Context
|
|
|
|
import Data.Maybe
|
|
import Data.Nat
|
|
import Data.DPair
|
|
import Data.Fun.Graph
|
|
import Decidable.Decidable
|
|
import Decidable.Equality
|
|
|
|
%default total
|
|
|
|
|
|
public export
|
|
DimEq' : Nat -> Type
|
|
DimEq' = Context (Maybe . Dim)
|
|
|
|
|
|
public export
|
|
data DimEq : Nat -> Type where
|
|
ZeroIsOne : DimEq d
|
|
C : (eqs : DimEq' d) -> DimEq d
|
|
|
|
%name DimEq eqs
|
|
|
|
|
|
public export
|
|
data IfConsistent : DimEq d -> a -> Type where
|
|
Nothing : IfConsistent ZeroIsOne a
|
|
Just : a -> IfConsistent (C eqs) a
|
|
|
|
|
|
export %inline
|
|
zeroEq : DimEq 0
|
|
zeroEq = C [<]
|
|
|
|
export %inline
|
|
new' : {d : Nat} -> DimEq' d
|
|
new' {d = 0} = [<]
|
|
new' {d = S d} = new' :< Nothing
|
|
|
|
export %inline
|
|
new : {d : Nat} -> DimEq d
|
|
new = C new'
|
|
|
|
|
|
private %inline
|
|
shiftMay : Maybe (Dim from) -> Shift from to -> Maybe (Dim to)
|
|
shiftMay p by = map (// by) p
|
|
|
|
export %inline
|
|
get' : DimEq' d -> Var d -> Maybe (Dim d)
|
|
get' = getWith shiftMay
|
|
|
|
private %inline
|
|
getShift' : Shift len out -> DimEq' len -> Var len -> Maybe (Dim out)
|
|
getShift' = getShiftWith shiftMay
|
|
|
|
export %inline
|
|
get : DimEq' d -> Dim d -> Dim d
|
|
get _ (K e) = K e
|
|
get eqs (B i) = fromMaybe (B i) $ get' eqs i
|
|
|
|
|
|
export %inline
|
|
equal : DimEq d -> (p, q : Dim d) -> Bool
|
|
equal ZeroIsOne p q = True
|
|
equal (C eqs) p q = get eqs p == get eqs q
|
|
|
|
|
|
infixl 5 :<?
|
|
export %inline
|
|
(:<?) : DimEq d -> Maybe (Dim d) -> DimEq (S d)
|
|
ZeroIsOne :<? d = ZeroIsOne
|
|
C eqs :<? d = C $ eqs :< map (get eqs) d
|
|
|
|
|
|
private %inline
|
|
ifVar : Var d -> Dim d -> Maybe (Dim d) -> Maybe (Dim d)
|
|
ifVar i p = map $ \q => if q == B i then p else q
|
|
|
|
-- (using decEq instead of (==) because of the proofs below)
|
|
private %inline
|
|
checkConst : (e, f : DimConst) -> (eqs : Lazy (DimEq' d)) -> DimEq d
|
|
checkConst e f eqs = if isYes $ e `decEq` f then C eqs else ZeroIsOne
|
|
|
|
|
|
export
|
|
setConst : Var d -> DimConst -> DimEq' d -> DimEq d
|
|
setConst VZ e (eqs :< Nothing) = C $ eqs :< Just (K e)
|
|
setConst VZ e (eqs :< Just (K f)) = checkConst e f $ eqs :< Just (K f)
|
|
setConst VZ e (eqs :< Just (B i)) = setConst i e eqs :<? Just (K e)
|
|
setConst (VS i) e (eqs :< p) = setConst i e eqs :<? ifVar i (K e) p
|
|
|
|
mutual
|
|
private
|
|
setVar' : (i, j : Var d) -> i `LT` j -> DimEq' d -> DimEq d
|
|
setVar' VZ (VS i) LTZ (eqs :< Nothing) =
|
|
C $ eqs :< Just (B i)
|
|
setVar' VZ (VS i) LTZ (eqs :< Just (K e)) =
|
|
setConst i e eqs :<? Just (K e)
|
|
setVar' VZ (VS i) LTZ (eqs :< Just (B j)) =
|
|
setVar i j eqs :<? Just (B (max i j))
|
|
setVar' (VS i) (VS j) (LTS lt) (eqs :< p) =
|
|
setVar' i j lt eqs :<? ifVar i (B j) p
|
|
|
|
export %inline
|
|
setVar : (i, j : Var d) -> DimEq' d -> DimEq d
|
|
setVar i j eqs with (compareP i j) | (compare i.nat j.nat)
|
|
setVar i j eqs | IsLT lt | LT = setVar' i j lt eqs
|
|
setVar i i eqs | IsEQ | EQ = C eqs
|
|
setVar i j eqs | IsGT gt | GT = setVar' j i gt eqs
|
|
|
|
|
|
export %inline
|
|
set : (p, q : Dim d) -> DimEq d -> DimEq d
|
|
set _ _ ZeroIsOne = ZeroIsOne
|
|
set (K e) (K f) (C eqs) = checkConst e f eqs
|
|
set (K e) (B i) (C eqs) = setConst i e eqs
|
|
set (B i) (K e) (C eqs) = setConst i e eqs
|
|
set (B i) (B j) (C eqs) = setVar i j eqs
|
|
|
|
|
|
public export %inline
|
|
Split : Nat -> Type
|
|
Split d = (DimEq' d, DSubst (S d) d)
|
|
|
|
export %inline
|
|
split1 : DimConst -> DimEq' (S d) -> Maybe (Split d)
|
|
split1 e eqs = case setConst VZ e eqs of
|
|
ZeroIsOne => Nothing
|
|
C (eqs :< _) => Just (eqs, K e ::: id)
|
|
|
|
export %inline
|
|
split : DimEq' (S d) -> List (Split d)
|
|
split eqs = toList (split1 Zero eqs) <+> toList (split1 One eqs)
|
|
|
|
|
|
export
|
|
splits' : DimEq' d -> List (DSubst d 0)
|
|
splits' [<] = [id]
|
|
splits' eqs@(_ :< _) = [th . ph | (eqs', th) <- split eqs, ph <- splits' eqs']
|
|
|
|
export %inline
|
|
splits : DimEq d -> List (DSubst d 0)
|
|
splits ZeroIsOne = []
|
|
splits (C eqs) = splits' eqs
|
|
|
|
|
|
private
|
|
0 newGetShift : (d : Nat) -> (i : Var d) -> (by : Shift d d') ->
|
|
getShift' by (new' {d}) i = Nothing
|
|
newGetShift (S d) VZ by = Refl
|
|
newGetShift (S d) (VS i) by = newGetShift d i (ssDown by)
|
|
|
|
export
|
|
0 newGet' : (d : Nat) -> (i : Var d) -> get' (new' {d}) i = Nothing
|
|
newGet' d i = newGetShift d i SZ
|
|
|
|
export
|
|
0 newGet : (d : Nat) -> (p : Dim d) -> get (new' {d}) p = p
|
|
newGet d (K e) = Refl
|
|
newGet d (B i) = rewrite newGet' d i in Refl
|
|
|
|
|
|
export
|
|
0 setSelf : (p : Dim d) -> (eqs : DimEq d) -> set p p eqs = eqs
|
|
setSelf p ZeroIsOne = Refl
|
|
setSelf (K Zero) (C eqs) = Refl
|
|
setSelf (K One) (C eqs) = Refl
|
|
setSelf (B i) (C eqs) with (compareP i i) | (compare i.nat i.nat)
|
|
_ | IsLT lt | LT = absurd lt
|
|
_ | IsEQ | EQ = Refl
|
|
_ | IsGT gt | GT = absurd gt
|
|
|
|
|
|
-- [todo] "well formed" dimeqs
|
|
-- [todo] operations maintain well-formedness
|
|
-- [todo] if 'Wf eqs' then 'equal eqs' is an equivalence
|
|
-- [todo] 'set' never breaks existing equalities
|