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456 lines
14 KiB
456 lines
14 KiB
{-# LANGUAGE ViewPatterns #-}
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{-# LANGUAGE DeriveAnyClass #-}
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{-# LANGUAGE BlockArguments #-}
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{-# LANGUAGE LambdaCase #-}
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module Eval where
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import Syntax
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import qualified Data.Map.Strict as Map
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import Data.Foldable
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import Control.Exception
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import Data.Typeable
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import System.IO.Unsafe (unsafePerformIO)
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import Data.IORef
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import Systems
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import Presyntax (Formula)
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import qualified Presyntax as P
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import Data.Maybe
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import Debug.Trace
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import GHC.Stack
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iand :: Value -> Value -> Value
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iand = \case
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VI1 -> id
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VI0 -> const VI0
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x -> \case
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VI0 -> VI0
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VI1 -> x
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y -> VIAnd x y
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ior :: Value -> Value -> Value
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ior = \case
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VI0 -> id
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VI1 -> const VI1
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x -> \case
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VI1 -> VI1
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VI0 -> x
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y -> VIOr x y
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inot :: Value -> Value
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inot VI1 = VI0
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inot VI0 = VI1
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inot (VIOr x y) = iand (inot x) (inot y)
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inot (VIAnd x y) = ior (inot x) (inot y)
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inot (VINot x) = x
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inot x = VINot x
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(@@) :: Value -> Value -> Value
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VNe hd xs @@ vl = VNe hd (PApp vl:xs)
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VLam _ _ k @@ vl = k vl
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VEqGlued a b @@ vl = VEqGlued (a @@ vl) (b @@ vl)
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VOfSub a phi u0 x @@ vl = x @@ vl
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f @@ _ = error $ "can't apply argument to " ++ show f
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proj1 :: Value -> Value
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proj1 (VPair x _) = x
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proj1 (VEqGlued x y) = VEqGlued (proj1 x) (proj1 y)
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proj1 (VNe s xs) = VNe s (PProj1:xs)
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proj1 (VOfSub (VSigma _ d _) phi u0 x) = VOfSub d phi (proj1 u0) (proj1 x)
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proj1 x = error $ "can't proj1 " ++ show x
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proj2 :: Value -> Value
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proj2 (VPair _ y) = y
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proj2 (VEqGlued x y) = VEqGlued (proj1 x) (proj1 y)
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proj2 (VNe s xs) = VNe s (PProj2:xs)
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proj2 (VOfSub (VSigma _ d r) phi u0 x) =
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VOfSub (r (proj1 x)) phi (proj2 u0) (proj2 x)
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proj2 x = error $ "can't proj2 " ++ show x
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pathp :: Env -> Value -> Value -> Value -> Value -> Value -> Value
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pathp env p x y f@(VLine _a _x _y e) i =
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case reduceCube env i of
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Just P.Bot -> VEqGlued (e i) x
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Just P.Top -> VEqGlued (e i) y
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_ -> e i
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pathp env p x y (VEqGlued e e') i = VEqGlued (pathp env p x y e i) (pathp env p x y e' i)
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pathp env p x y (VNe hd sp) i =
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case reduceCube env i of
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Just P.Bot -> VEqGlued (VNe hd (PPathP p x y i:sp)) x
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Just P.Top -> VEqGlued (VNe hd (PPathP p x y i:sp)) y
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_ -> VNe hd (PPathP p x y i:sp)
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pathp env p x y (VOfSub _ _ _ v) i = pathp env p x y v i
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comp :: Env -> Value -> Formula -> Value -> Value -> Value
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comp env a@(VLam ivar VI fam) phi u a0 = go (fam undefined) phi u a0 where
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i = VVar ivar
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stuck :: Value
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stuck = maybeAddEq $ VComp a (toValue phi) u a0
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maybeAddEq :: Value -> Value
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maybeAddEq =
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if phi == P.Top
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then flip VEqGlued (u @@ VI1)
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else id
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go :: HasCallStack => Value -> Formula -> Value -> Value -> Value
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go VPi{} phi u a0 =
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let
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dom x = let VPi _ d _ = fam x in d
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rng x = let VPi _ d _ = fam x in d
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ai1 = dom VI0
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y' i y = fill env i (dom . inot . fam) P.Bot (VSystem emptySystem) y
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ybar i y = y' (inot i) y
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in VLam "x" ai1 \arg ->
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comp env
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(VLam ivar VI (\i -> rng (ybar i arg)))
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phi
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(VLam "i" VI \i -> mapVSystem (u @@ i) (@@ ybar i arg))
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(a0 @@ ybar VI0 arg)
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go VSigma{} phi u a0 =
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let
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dom x = let VSigma _ d _ = fam x in d
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rng x = let VSigma _ d _ = fam x in d
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a i = fill env i (dom . fam) phi (VLam "j" VI \v -> mapVSystem (u @@ v) proj1) (proj1 a0)
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c1 = comp env (VLam ivar VI (getd . fam)) phi (VLam "i" VI \v -> mapVSystem (u @@ v) proj1) (proj1 a0)
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c2 = comp env (VLam ivar VI (apr (a VI1) . fam)) phi (VLam "i" VI \v -> mapVSystem (u @@ v) proj2) (proj2 a0)
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getd (VSigma _ d _) = d
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apr x (VSigma _ _ r) = r x
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in VPair c1 c2
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go VPath{} phi p p0 =
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let
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~(VPath ai1 u1 v1) = fam VI1
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~(VPath ai0 u0 v0) = fam VI0
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getA (VPath a _ _) = a
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u' x = let ~(VPath _ u _) = fam x in u
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v' x = let ~(VPath _ _ v) = fam x in v
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in
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VLine (ai1 @@ VI1) u1 v1 \j ->
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let
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jc = reduceCube' env j
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in comp env (VLam ivar VI (getA . fam))
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(orFormula [phi, jc, notFormula jc])
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(VLam "j" VI \v ->
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let
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VSystem (FMap sys) = p @@ v
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sys' = fmap (flip (pathp env ai0 u0 v0) j) sys
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in mkVSystem $ Map.fromList [(phi, mapVSystem (p @@ v) (flip (pathp env ai0 u0 v0) j))
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, (notFormula jc, u' v), (jc, v' v)])
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(pathp env (ai0 @@ VI0) u0 v0 p0 j)
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go a P.Top u a0 = u @@ VI1
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go a phi u a0 = maybeAddEq stuck
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comp env va phi u a0 =
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if phi == P.Top
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then VEqGlued (VComp va phi' u a0) (u @@ VI1)
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else VComp va phi' u a0
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where
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phi' = toValue phi
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mkVSystem :: Map.Map Formula Value -> Value
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mkVSystem mp
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| Just e <- Map.lookup P.Top mp = e
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| otherwise = VSystem $ FMap $ Map.filterWithKey f mp
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where
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f P.Bot _ = False
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f _ _ = True
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reduceCube' :: Env -> Value -> Formula
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reduceCube' env = fromJust . reduceCube env
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mapVSystem :: Value -> (Value -> Value) -> Value
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mapVSystem (VSystem ss) f = VSystem (mapSystem ss f)
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mapVSystem x f = f x
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evalSystem :: Env -> Map.Map Formula Term -> Value
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evalSystem env face = mk . Map.mapMaybeWithKey go $ face where
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go :: Formula -> Term -> Maybe Value
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go face tm
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| VI0 <- toValue' env face = Nothing
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| otherwise = Just (eval env tm)
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differsFromEnv :: String -> Bool -> Bool
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differsFromEnv x True =
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case Map.lookup x (names env) of
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Just (VI, VI0) -> True
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_ -> False
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differsFromEnv x False =
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case Map.lookup x (names env) of
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Just (VI, VI1) -> True
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_ -> False
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mk x = case Map.toList x of
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[(_, x)] -> x
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_ -> mkVSystem x
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eval :: Env -> Term -> Value
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eval env = \case
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Var v ->
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case Map.lookup v (names env) of
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Just (_, vl) -> vl
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Nothing -> error $ "variable not in scope: " ++ show v
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App f x -> eval env f @@ eval env x
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Lam s d b ->
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let d' = eval env d
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in VLam s d' \a -> eval env{ names = Map.insert s (d', a) (names env) } b
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Let s t b d ->
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let b' = eval env b
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t' = eval env t
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in eval env{ names = Map.insert s (t', b') (names env) } d
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Pi s d r ->
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let d' = eval env d
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in VPi s d' \a -> eval env{ names = Map.insert s (d', a) (names env) } r
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Sigma s d r ->
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let d' = eval env d
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in VSigma s d' \a -> eval env{ names = Map.insert s (d', a) (names env) } r
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Pair a b -> VPair (eval env a) (eval env b)
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Proj1 x -> proj1 (eval env x)
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Proj2 y -> proj2 (eval env y)
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Type -> VType
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I -> VI
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I0 -> VI0
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I1 -> VI1
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Path p x y -> VPath (eval env p) (eval env x) (eval env y)
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Partial r a -> VPartial (eval env r) (eval env a)
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PathI p x y s e -> VLine (eval env p) (eval env x) (eval env y) (\ a -> eval env{ names = Map.insert s (VI, a) (names env) } e)
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PathP p x y f i -> pathp env (eval env p) (eval env x) (eval env y) (eval env f) (eval env i)
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Sub p x y -> VSub (eval env p) (eval env x) (eval env y)
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InclSub a phi u a0 -> VOfSub (eval env a) (eval env phi) (eval env u) (eval env a0)
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IAnd x y -> iand (eval env x) (eval env y)
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IOr x y -> ior (eval env x) (eval env y)
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INot x -> inot (eval env x)
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Comp a phi u a0 ->
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case reduceCube env (eval env phi) of
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Just formula -> comp env (eval env a) formula (eval env u) (eval env a0)
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Nothing -> VComp (eval env a) (eval env phi) (eval env u) (eval env a0)
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System fs -> evalSystem env (getSystem fs)
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data UnifyError
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= Mismatch Value Value
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| NotPiType Value
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| NotPartialType Formula Value
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| NotSigmaType Value
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deriving (Show, Typeable, Exception)
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unify :: Env -> Value -> Value -> IO ()
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unify env (VEqGlued a b) c =
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unify env a c `catch` \e -> const (unify env b c) (e :: UnifyError)
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unify env c (VEqGlued a b) =
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unify env c a `catch` \e -> const (unify env c b) (e :: UnifyError)
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unify env (VLine a x y f) e = unify env (f (VVar "i")) (pathp env a x y e (VVar "i"))
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unify env e (VLine a x y f) = unify env (f (VVar "i")) (pathp env a x y e (VVar "i"))
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unify env (VPartial r b) (VPartial r' b') = do
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unify env b b'
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case sameCube env r r' of
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Just True -> pure ()
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_ -> unify env r r'
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unify env (VPartial r b) x = do
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case sameCube env r VI1 of
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Just True -> pure ()
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_ -> unify env r VI1
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unify env b x
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unify env x (VPartial r b) = do
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case sameCube env r VI1 of
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Just True -> pure ()
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_ -> unify env r VI1
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unify env x b
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unify env (VSub a phi _u0) vl = unify env a vl
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unify env u1 (VOfSub _a phi u0 a) = do
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case sameCube env phi VI1 of
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Just True -> unify env u1 u0
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_ -> unify env u1 a
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unify env (VOfSub _a phi u0 a) u1 = do
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case sameCube env phi VI1 of
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Just True -> unify env u1 u0
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_ -> unify env u1 a
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unify env vl1@(VNe x sp) vl2@(VNe y sp')
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| x == y = traverse_ (uncurry unifySp) (zip sp sp')
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| otherwise = throwIO $ Mismatch vl1 vl2
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where
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unifySp (PApp x) (PApp y) = unify env x y
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unifySp (PPathP _a _x _y i) (PPathP _a' _x' _y' i') = unify env i i'
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unifySp PProj1 PProj1 = pure ()
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unifySp PProj2 PProj2 = pure ()
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unify env (VLam x _ k) e = unify env (k (VVar x)) (e @@ VVar x)
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unify env e (VLam x _ k) = unify env (e @@ VVar x) (k (VVar x))
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unify env (VPi x d r) (VPi _ d' r') = do
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unify env d d'
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unify env (r (VVar x)) (r' (VVar x))
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unify env (VSigma x d r) (VSigma _ d' r') = do
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unify env d d'
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unify env (r (VVar x)) (r' (VVar x))
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unify env VType VType = pure ()
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unify env VI VI = pure ()
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unify env (VPair a b) (VPair c d) = unify env a c *> unify env b d
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unify env (VPath a x y) (VPath a' x' y') = unify env a a' *> unify env x x' *> unify env y y'
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unify env (VSystem fs) vl
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| ((_, vl'):_) <- Map.toList (Map.filterWithKey (\f _ -> isTrue (toValue' env f)) (getSystem fs))
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= unify env vl' vl
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unify env vl (VSystem fs)
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| ((_, vl'):_) <- Map.toList (Map.filterWithKey (\f _ -> isTrue (toValue' env f)) (getSystem fs))
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= unify env vl' vl
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unify env x y =
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case sameCube env x y of
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Just True -> pure ()
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_ -> throwIO $ Mismatch x y
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reduceCube :: Env -> Value -> Maybe Formula
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reduceCube env x = fmap (toDNF . simplify) (toFormula x) where
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simplify :: Formula -> Formula
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simplify (P.Is0 x) =
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case Map.lookup x (names env) of
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Just (VI, VI0) -> P.Top
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Just (VI, VI1) -> P.Bot
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_ -> P.Is0 x
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simplify (P.Is1 x) =
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case Map.lookup x (names env) of
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Just (VI, VI1) -> P.Top
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Just (VI, VI0) -> P.Bot
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_ -> P.Is0 x
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simplify (P.And x y) = P.And (simplify x) (simplify y)
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simplify (P.Or x y) = P.Or (simplify x) (simplify y)
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simplify x = x
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sameCube :: Env -> Value -> Value -> Maybe Bool
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sameCube env x y =
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case (reduceCube env x, reduceCube env y) of
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(Just x, Just y) -> Just (x == y)
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_ -> Nothing
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toFormula :: Value -> Maybe Formula
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toFormula VI0 = Just P.Bot
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toFormula VI1 = Just P.Top
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toFormula (VNe x []) = Just (P.Is1 x)
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toFormula (VINot f) = notFormula <$> toFormula f
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toFormula (VIAnd x y) = do
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s <- toFormula y
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t <- toFormula x
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pure $ andFormula [s, t]
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toFormula (VIOr x y) = do
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s <- toFormula y
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t <- toFormula x
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pure $ orFormula [s, t]
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toFormula _ = Nothing
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faceInEnv :: Env -> Face -> Bool
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faceInEnv e f = Map.isSubmapOf (getFace f) (faceOfEnv (names e)) where
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faceOfEnv = Map.map (\(_, v) -> case v of { VI1 -> True; VEqGlued _ VI1 -> True; _ -> False }) . Map.filter (\(_, v) -> isI v)
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isI VI1 = True
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isI VI0 = True
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isI (VEqGlued _ x) = isI x
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isI _ = False
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isPiType :: Value -> IO (String, Value, Value -> Value)
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isPiType (VPi x d r) = pure (x, d, r)
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isPiType x = throwIO $ NotPiType x
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isSigmaType :: Value -> IO (String, Value, Value -> Value)
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isSigmaType (VSigma x d r) = pure (x, d, r)
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isSigmaType x = throwIO $ NotSigmaType x
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isPiOrPathType :: Value -> IO (Either (String, Value, Value -> Value) (Value, Value, Value))
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isPiOrPathType (VPi x d r) = pure (Left (x, d, r))
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isPiOrPathType (VPath x d r) = pure (Right (x, d, r))
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isPiOrPathType x = throwIO $ NotPiType x
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isPartialType :: Formula -> Value -> IO (Formula, Value)
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isPartialType f p@(VPartial x y) =
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case toFormula x of
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Just x -> pure (x, y)
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Nothing -> throwIO $ NotPartialType f p
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isPartialType f x = throwIO $ NotPartialType f x
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getVar :: IO Value
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getVar =
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do
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n <- atomicModifyIORef ref \x -> (x + 1, x)
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pure (VVar (show n))
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where
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ref :: IORef Int
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ref = unsafePerformIO (newIORef 0)
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{-# NOINLINE ref #-}
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fill :: Env
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-> Value
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-> (Value -> Value) -- (Γ i : I, A : Type)
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-> Formula -- (phi : I)
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-> Value -- (u : (i : I) -> Partial phi (A i))
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-> Value -- (Sub (A i0) phi (u i0))
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-> Value -- -> A i
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fill env i a phi u a0 =
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comp env
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(VLam "j" VI \j -> a (i `iand` j))
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(phi `P.Or` ifc)
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(VLam "j" VI \j ->
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mkVSystem (Map.fromList [ (phi, uiand j)
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, (notFormula ifc, a0) ]))
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a0
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where
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uiand j = u @@ (i `iand` j)
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ifc = fromJust (reduceCube env i)
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toValue :: Formula -> Value
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toValue P.Top = VI1
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toValue P.Bot = VI0
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toValue (P.And x y) = toValue x `iand` toValue y
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toValue (P.Or x y) = toValue x `ior` toValue y
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toValue (P.Is0 x) = inot (VVar x)
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toValue (P.Is1 x) = VVar x
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toValue' :: Env -> Formula -> Value
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toValue' env P.Top = VI1
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toValue' env P.Bot = VI0
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toValue' env (P.And x y) = toValue x `iand` toValue y
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toValue' env (P.Or x y) = toValue x `ior` toValue y
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toValue' env (P.Is0 x) =
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case Map.lookup x (names env) of
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Just (VI, VI0) -> VI1
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Just (VI, VI1) -> VI0
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Just (VI, x) -> inot x
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_ -> error $ "type error in toValue'"
|
|
toValue' env (P.Is1 x) =
|
|
case Map.lookup x (names env) of
|
|
Just (VI, x) -> x
|
|
_ -> error $ "type error in toValue'"
|
|
|
|
isTrue :: Value -> Bool
|
|
isTrue VI1 = True
|
|
isTrue _ = False
|