mirror of
https://github.com/edwinb/Idris2-boot.git
synced 2024-12-22 04:11:31 +03:00
e6121e0935
This is the result of running the command:
$ find . -name '*.idr' -type f -exec sed -i -E 's/\s+$//' {} +
I confirmed before running it that this would not affect any markdown
formatting in documentation comments.
146 lines
5.5 KiB
Idris
146 lines
5.5 KiB
Idris
module Decidable.Equality
|
|
|
|
import Data.Maybe
|
|
import Data.Nat
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- Decidable equality
|
|
--------------------------------------------------------------------------------
|
|
|
|
||| Decision procedures for propositional equality
|
|
public export
|
|
interface DecEq t where
|
|
||| Decide whether two elements of `t` are propositionally equal
|
|
total decEq : (x1 : t) -> (x2 : t) -> Dec (x1 = x2)
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- Utility lemmas
|
|
--------------------------------------------------------------------------------
|
|
|
|
||| The negation of equality is symmetric (follows from symmetry of equality)
|
|
export total
|
|
negEqSym : forall a, b . (a = b -> Void) -> (b = a -> Void)
|
|
negEqSym p h = p (sym h)
|
|
|
|
||| Everything is decidably equal to itself
|
|
export total
|
|
decEqSelfIsYes : DecEq a => {x : a} -> decEq x x = Yes Refl
|
|
decEqSelfIsYes {x} with (decEq x x)
|
|
decEqSelfIsYes {x} | Yes Refl = Refl
|
|
decEqSelfIsYes {x} | No contra = absurd $ contra Refl
|
|
|
|
--------------------------------------------------------------------------------
|
|
--- Unit
|
|
--------------------------------------------------------------------------------
|
|
|
|
export
|
|
implementation DecEq () where
|
|
decEq () () = Yes Refl
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- Booleans
|
|
--------------------------------------------------------------------------------
|
|
total trueNotFalse : True = False -> Void
|
|
trueNotFalse Refl impossible
|
|
|
|
export
|
|
implementation DecEq Bool where
|
|
decEq True True = Yes Refl
|
|
decEq False False = Yes Refl
|
|
decEq True False = No trueNotFalse
|
|
decEq False True = No (negEqSym trueNotFalse)
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- Nat
|
|
--------------------------------------------------------------------------------
|
|
|
|
total ZnotS : Z = S n -> Void
|
|
ZnotS Refl impossible
|
|
|
|
export
|
|
implementation DecEq Nat where
|
|
decEq Z Z = Yes Refl
|
|
decEq Z (S _) = No ZnotS
|
|
decEq (S _) Z = No (negEqSym ZnotS)
|
|
decEq (S n) (S m) with (decEq n m)
|
|
decEq (S n) (S m) | Yes p = Yes $ cong S p
|
|
decEq (S n) (S m) | No p = No $ \h : (S n = S m) => p $ succInjective n m h
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- Maybe
|
|
--------------------------------------------------------------------------------
|
|
|
|
total nothingNotJust : {x : t} -> (Nothing {ty = t} = Just x) -> Void
|
|
nothingNotJust Refl impossible
|
|
|
|
export
|
|
implementation (DecEq t) => DecEq (Maybe t) where
|
|
decEq Nothing Nothing = Yes Refl
|
|
decEq (Just x') (Just y') with (decEq x' y')
|
|
decEq (Just x') (Just y') | Yes p = Yes $ cong Just p
|
|
decEq (Just x') (Just y') | No p
|
|
= No $ \h : Just x' = Just y' => p $ justInjective h
|
|
decEq Nothing (Just _) = No nothingNotJust
|
|
decEq (Just _) Nothing = No (negEqSym nothingNotJust)
|
|
|
|
-- TODO: Other prelude data types
|
|
|
|
-- For the primitives, we have to cheat because we don't have access to their
|
|
-- internal implementations. We use believe_me for the inequality proofs
|
|
-- because we don't them to reduce (and they should never be needed anyway...)
|
|
-- A postulate would be better, but erasure analysis may think they're needed
|
|
-- for computation in a higher order setting.
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- Int
|
|
--------------------------------------------------------------------------------
|
|
export
|
|
implementation DecEq Int where
|
|
decEq x y = case x == y of -- Blocks if x or y not concrete
|
|
True => Yes primitiveEq
|
|
False => No primitiveNotEq
|
|
where primitiveEq : forall x, y . x = y
|
|
primitiveEq = believe_me (Refl {x})
|
|
primitiveNotEq : forall x, y . x = y -> Void
|
|
primitiveNotEq prf = believe_me {b = Void} ()
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- Char
|
|
--------------------------------------------------------------------------------
|
|
export
|
|
implementation DecEq Char where
|
|
decEq x y = case x == y of -- Blocks if x or y not concrete
|
|
True => Yes primitiveEq
|
|
False => No primitiveNotEq
|
|
where primitiveEq : forall x, y . x = y
|
|
primitiveEq = believe_me (Refl {x})
|
|
primitiveNotEq : forall x, y . x = y -> Void
|
|
primitiveNotEq prf = believe_me {b = Void} ()
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- Integer
|
|
--------------------------------------------------------------------------------
|
|
export
|
|
implementation DecEq Integer where
|
|
decEq x y = case x == y of -- Blocks if x or y not concrete
|
|
True => Yes primitiveEq
|
|
False => No primitiveNotEq
|
|
where primitiveEq : forall x, y . x = y
|
|
primitiveEq = believe_me (Refl {x})
|
|
primitiveNotEq : forall x, y . x = y -> Void
|
|
primitiveNotEq prf = believe_me {b = Void} ()
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- String
|
|
--------------------------------------------------------------------------------
|
|
export
|
|
implementation DecEq String where
|
|
decEq x y = case x == y of -- Blocks if x or y not concrete
|
|
True => Yes primitiveEq
|
|
False => No primitiveNotEq
|
|
where primitiveEq : forall x, y . x = y
|
|
primitiveEq = believe_me (Refl {x})
|
|
primitiveNotEq : forall x, y . x = y -> Void
|
|
primitiveNotEq prf = believe_me {b = Void} ()
|
|
|