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147 lines
3.8 KiB
Idris
147 lines
3.8 KiB
Idris
||| The content of this module is based on the paper
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||| Applications of Applicative Proof Search
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||| by Liam O'Connor
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module Search.HDecidable
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import Data.List.Lazy
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import Data.List.Lazy.Quantifiers
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import Data.List.Quantifiers
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import Data.So
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import Search.Negation
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%default total
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------------------------------------------------------------------------
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-- Type, basic functions, and interface
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||| Half a decider: when the search succeeds we bother building the proof
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public export
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record HDec (a : Type) where
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constructor MkHDec
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isTrue : Bool
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evidence : So isTrue -> a
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||| Happy path: we have found a proof!
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public export
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yes : a -> HDec a
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yes = MkHDec True . const
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||| Giving up
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public export
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no : HDec a
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no = MkHDec False absurd
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public export
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fromDec : Dec a -> HDec a
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fromDec (Yes p) = yes p
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fromDec (No _) = no
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public export
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fromMaybe : Maybe a -> HDec a
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fromMaybe = maybe no yes
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public export
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toMaybe : HDec a -> Maybe a
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toMaybe (MkHDec True p) = Just (p Oh)
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toMaybe (MkHDec False _) = Nothing
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||| A type constructor satisfying AnHdec is morally an HDec i.e. we can
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||| turn values of this type constructor into half deciders
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||| It may be more powerful (like Dec) or more basic (like Maybe).
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public export
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interface AnHDec (0 t : Type -> Type) where
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toHDec : t a -> HDec a
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public export AnHDec Dec where toHDec = fromDec
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public export AnHDec HDec where toHDec = id
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public export AnHDec Maybe where toHDec = fromMaybe
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------------------------------------------------------------------------
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-- Implementations
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public export
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Functor HDec where
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map f (MkHDec b prf) = MkHDec b (f . prf)
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public export
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Applicative HDec where
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pure = yes
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MkHDec False prff <*> _ = MkHDec False absurd
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_ <*> MkHDec False _ = MkHDec False absurd
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MkHDec True prff <*> MkHDec True prfx
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= yes (prff Oh (prfx Oh))
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||| Lazy in the second argument
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public export
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Alternative HDec where
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empty = no
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p@(MkHDec True _) <|> _ = p
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_ <|> q = q
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public export
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Monad HDec where
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MkHDec True x >>= f = f (x Oh)
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_ >>= _ = no
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public export
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Show f => Show (HDec f) where
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show (MkHDec True p) = "True: " ++ show (p Oh)
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show _ = "False"
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------------------------------------------------------------------------
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-- Combinators
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||| Half deciders are closed under product
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public export
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(&&) : (AnHDec l, AnHDec r) => l a -> r b -> HDec (a, b)
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p && q = [| (toHDec p, toHDec q) |]
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||| Half deciders are closed under sum
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public export
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(||) : (AnHDec l, AnHDec r) => l a -> r b -> HDec (Either a b)
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p || q = [| Left (toHDec p) |] <|> [| Right (toHDec q) |]
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||| Half deciders are closed negation. Here we use the `Negates` interface
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||| so that we end up looking for *positive* evidence of something which is
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||| much easier to find than negative one.
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public export
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not : (AnHDec l, Negates na a) => l na -> HDec (Not a)
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not p = [| toNegation (toHDec p) |]
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infixr 3 ==>
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||| Half deciders are closed under implication
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public export
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(==>) : (AnHDec l, AnHDec r, Negates na a) => l na -> r b -> HDec (a -> b)
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p ==> q = [| contra (not p) |] <|> [| const (toHDec q) |] where
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contra : Not a -> a -> b
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contra na a = void (na a)
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namespace List
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||| Half deciders are closed under the list quantifier any
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public export
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any : AnHDec l => (xs : List a) -> ((x : a) -> l (p x)) -> HDec (Any p xs)
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any [] p = no
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any (x :: xs) p = [| Here (toHDec (p x)) |] <|> [| There (any xs p) |]
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||| Half deciders are closed under the list quantifier all
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public export
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all : AnHDec l => (xs : List a) -> ((x : a) -> l (p x)) -> HDec (All p xs)
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all [] p = yes []
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all (x :: xs) p = [| toHDec (p x) :: all xs p |]
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namespace LazyList
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||| Half deciders are closed under the lazy list quantifier any
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public export
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any : AnHDec l => (xs : LazyList a) -> ((x : a) -> l (p x)) -> HDec (Any p xs)
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any [] p = no
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any (x :: xs) p = [| Here (toHDec (p x)) |] <|> [| There (any xs p) |]
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