2020-12-04 04:45:27 +03:00
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module Decidable.Equality.Core
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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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2021-02-16 14:56:43 +03:00
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negEqSym : Not (a = b) -> Not (b = a)
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2020-12-04 04:45:27 +03:00
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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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2021-02-16 14:56:43 +03:00
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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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2021-02-16 13:47:28 +03:00
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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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