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e6121e0935
This is the result of running the command:
$ find . -name '*.idr' -type f -exec sed -i -E 's/\s+$//' {} +
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formatting in documentation comments.
146 lines
5.5 KiB
Idris
146 lines
5.5 KiB
Idris
module Decidable.Equality
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import Data.Maybe
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import Data.Nat
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--------------------------------------------------------------------------------
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-- Decidable equality
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--------------------------------------------------------------------------------
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||| Decision procedures for propositional equality
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public export
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interface DecEq t where
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||| Decide whether two elements of `t` are propositionally equal
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total decEq : (x1 : t) -> (x2 : t) -> Dec (x1 = x2)
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--------------------------------------------------------------------------------
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-- Utility lemmas
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--------------------------------------------------------------------------------
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||| The negation of equality is symmetric (follows from symmetry of equality)
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export total
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negEqSym : forall a, b . (a = b -> Void) -> (b = a -> Void)
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negEqSym p h = p (sym h)
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||| Everything is decidably equal to itself
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export total
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decEqSelfIsYes : DecEq a => {x : a} -> decEq x x = Yes Refl
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decEqSelfIsYes {x} with (decEq x x)
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decEqSelfIsYes {x} | Yes Refl = Refl
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decEqSelfIsYes {x} | No contra = absurd $ contra Refl
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--------------------------------------------------------------------------------
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--- Unit
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--------------------------------------------------------------------------------
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export
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implementation DecEq () where
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decEq () () = Yes Refl
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--------------------------------------------------------------------------------
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-- Booleans
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--------------------------------------------------------------------------------
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total trueNotFalse : True = False -> Void
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trueNotFalse Refl impossible
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export
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implementation DecEq Bool where
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decEq True True = Yes Refl
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decEq False False = Yes Refl
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decEq True False = No trueNotFalse
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decEq False True = No (negEqSym trueNotFalse)
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--------------------------------------------------------------------------------
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-- Nat
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--------------------------------------------------------------------------------
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total ZnotS : Z = S n -> Void
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ZnotS Refl impossible
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export
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implementation DecEq Nat where
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decEq Z Z = Yes Refl
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decEq Z (S _) = No ZnotS
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decEq (S _) Z = No (negEqSym ZnotS)
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decEq (S n) (S m) with (decEq n m)
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decEq (S n) (S m) | Yes p = Yes $ cong S p
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decEq (S n) (S m) | No p = No $ \h : (S n = S m) => p $ succInjective n m h
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--------------------------------------------------------------------------------
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-- Maybe
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--------------------------------------------------------------------------------
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total nothingNotJust : {x : t} -> (Nothing {ty = t} = Just x) -> Void
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nothingNotJust Refl impossible
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export
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implementation (DecEq t) => DecEq (Maybe t) where
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decEq Nothing Nothing = Yes Refl
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decEq (Just x') (Just y') with (decEq x' y')
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decEq (Just x') (Just y') | Yes p = Yes $ cong Just p
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decEq (Just x') (Just y') | No p
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= No $ \h : Just x' = Just y' => p $ justInjective h
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decEq Nothing (Just _) = No nothingNotJust
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decEq (Just _) Nothing = No (negEqSym nothingNotJust)
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-- TODO: Other prelude data types
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-- For the primitives, we have to cheat because we don't have access to their
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-- internal implementations. We use believe_me for the inequality proofs
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-- because we don't them to reduce (and they should never be needed anyway...)
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-- A postulate would be better, but erasure analysis may think they're needed
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-- for computation in a higher order setting.
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--------------------------------------------------------------------------------
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-- Int
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--------------------------------------------------------------------------------
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export
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implementation DecEq Int where
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decEq x y = case x == y of -- Blocks if x or y not concrete
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True => Yes primitiveEq
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False => No primitiveNotEq
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where primitiveEq : forall x, y . x = y
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primitiveEq = believe_me (Refl {x})
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primitiveNotEq : forall x, y . x = y -> Void
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primitiveNotEq prf = believe_me {b = Void} ()
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--------------------------------------------------------------------------------
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-- Char
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--------------------------------------------------------------------------------
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export
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implementation DecEq Char where
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decEq x y = case x == y of -- Blocks if x or y not concrete
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True => Yes primitiveEq
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False => No primitiveNotEq
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where primitiveEq : forall x, y . x = y
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primitiveEq = believe_me (Refl {x})
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primitiveNotEq : forall x, y . x = y -> Void
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primitiveNotEq prf = believe_me {b = Void} ()
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--------------------------------------------------------------------------------
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-- Integer
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--------------------------------------------------------------------------------
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export
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implementation DecEq Integer where
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decEq x y = case x == y of -- Blocks if x or y not concrete
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True => Yes primitiveEq
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False => No primitiveNotEq
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where primitiveEq : forall x, y . x = y
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primitiveEq = believe_me (Refl {x})
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primitiveNotEq : forall x, y . x = y -> Void
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primitiveNotEq prf = believe_me {b = Void} ()
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--------------------------------------------------------------------------------
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-- String
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--------------------------------------------------------------------------------
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export
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implementation DecEq String where
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decEq x y = case x == y of -- Blocks if x or y not concrete
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True => Yes primitiveEq
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False => No primitiveNotEq
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where primitiveEq : forall x, y . x = y
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primitiveEq = believe_me (Refl {x})
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primitiveNotEq : forall x, y . x = y -> Void
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primitiveNotEq prf = believe_me {b = Void} ()
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