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53 lines
1.3 KiB
Idris
53 lines
1.3 KiB
Idris
module Decidable.Decidable
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import Data.Rel
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import Data.Fun
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%default total
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public export
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isNo : Dec a -> Bool
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isNo (Yes _) = False
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isNo (No _) = True
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public export
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isYes : Dec a -> Bool
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isYes (Yes _) = True
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isYes (No _) = False
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||| Proof that some `Dec` is actually `Yes`
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public export
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data IsYes : Dec a -> Type where
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ItIsYes : IsYes (Yes prf)
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public export
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Uninhabited (IsYes (No contra)) where
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uninhabited ItIsYes impossible
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||| Decide whether a 'Dec' is 'Yes'
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public export
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isItYes : (v : Dec a) -> Dec (IsYes v)
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isItYes (Yes _) = Yes ItIsYes
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isItYes (No _) = No absurd
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||| An n-ary relation is decidable if we can make a `Dec`
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||| of its result type for each combination of inputs
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public export
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IsDecidable : (k : Nat) -> (ts : Vect k Type) -> Rel ts -> Type
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IsDecidable k ts p = liftRel (the (Vect k Type) ts) (the (Rel ts) p) Dec
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||| Interface for decidable n-ary Relations
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public export
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interface Decidable k ts p where
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total decide : IsDecidable k ts p
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||| Given a `Decidable` n-ary relation, provides a decision procedure for
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||| this relation.
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decision : (ts : Vect k Type) -> (p : Rel ts) -> (Decidable k ts p) => liftRel ts p Dec
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decision ts p = decide {ts} {p}
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using (a : Type, x : a)
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public export
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data Given : Dec a -> Type where
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Always : Given (Yes x)
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