mirror of
https://github.com/idris-lang/Idris2.git
synced 2024-12-20 18:21:47 +03:00
54 lines
1.9 KiB
Idris
54 lines
1.9 KiB
Idris
module Decidable.Equality.Core
|
|
|
|
import Control.Function
|
|
|
|
%default total
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- Decidable equality
|
|
--------------------------------------------------------------------------------
|
|
|
|
||| Decision procedures for propositional equality
|
|
public export
|
|
interface DecEq t where
|
|
||| Decide whether two elements of `t` are propositionally equal
|
|
decEq : (x1 : t) -> (x2 : t) -> Dec (x1 = x2)
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- Utility lemmas
|
|
--------------------------------------------------------------------------------
|
|
|
|
||| The negation of equality is symmetric (follows from symmetry of equality)
|
|
export
|
|
negEqSym : Not (a = b) -> Not (b = a)
|
|
negEqSym p h = p (sym h)
|
|
|
|
||| Everything is decidably equal to itself
|
|
export
|
|
decEqSelfIsYes : DecEq a => {x : a} -> decEq x x = Yes Refl
|
|
decEqSelfIsYes with (decEq x x)
|
|
decEqSelfIsYes | Yes Refl = Refl
|
|
decEqSelfIsYes | No contra = absurd $ contra Refl
|
|
|
|
||| If you have a proof of inequality, you're sure that `decEq` would give a `No`.
|
|
export
|
|
decEqContraIsNo : DecEq a => {x, y : a} -> Not (x = y) -> (p ** decEq x y = No p)
|
|
decEqContraIsNo uxy with (decEq x y)
|
|
decEqContraIsNo uxy | Yes xy = absurd $ uxy xy
|
|
decEqContraIsNo _ | No uxy = (uxy ** Refl)
|
|
|
|
public export
|
|
decEqCong : (0 _ : Injective f) => Dec (x = y) -> Dec (f x = f y)
|
|
decEqCong $ Yes prf = Yes $ cong f prf
|
|
decEqCong $ No contra = No $ \c => contra $ inj f c
|
|
|
|
public export
|
|
decEqInj : (0 _ : Injective f) => Dec (f x = f y) -> Dec (x = y)
|
|
decEqInj $ Yes prf = Yes $ inj f prf
|
|
decEqInj $ No contra = No $ \c => contra $ cong f c
|
|
|
|
public export
|
|
decEqCong2 : (0 _ : Biinjective f) => Dec (x = y) -> Lazy (Dec (v = w)) -> Dec (f x v = f y w)
|
|
decEqCong2 (Yes Refl) s = decEqCong s @{FromBiinjectiveL}
|
|
decEqCong2 (No contra) _ = No $ \c => let (Refl, Refl) = biinj f c in contra Refl
|