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54 lines
1.9 KiB
Idris
54 lines
1.9 KiB
Idris
module Decidable.Equality.Core
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import Control.Function
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%default total
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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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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
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negEqSym : Not (a = b) -> Not (b = a)
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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
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decEqSelfIsYes : DecEq a => {x : a} -> decEq x x = Yes Refl
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decEqSelfIsYes with (decEq x x)
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decEqSelfIsYes | Yes Refl = Refl
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decEqSelfIsYes | No contra = absurd $ contra Refl
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||| If you have a proof of inequality, you're sure that `decEq` would give a `No`.
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export
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decEqContraIsNo : DecEq a => {x, y : a} -> Not (x = y) -> (p ** decEq x y = No p)
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decEqContraIsNo uxy with (decEq x y)
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decEqContraIsNo uxy | Yes xy = absurd $ uxy xy
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decEqContraIsNo _ | No uxy = (uxy ** Refl)
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public export
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decEqCong : (0 _ : Injective f) => Dec (x = y) -> Dec (f x = f y)
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decEqCong $ Yes prf = Yes $ cong f prf
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decEqCong $ No contra = No $ \c => contra $ inj f c
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public export
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decEqInj : (0 _ : Injective f) => Dec (f x = f y) -> Dec (x = y)
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decEqInj $ Yes prf = Yes $ inj f prf
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decEqInj $ No contra = No $ contra . cong f
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public export
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decEqCong2 : (0 _ : Biinjective f) => Dec (x = y) -> Lazy (Dec (v = w)) -> Dec (f x v = f y w)
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decEqCong2 (Yes Refl) s = decEqCong s @{FromBiinjectiveL}
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decEqCong2 (No contra) _ = No $ \c => let (Refl, Refl) = biinj f c in contra Refl
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