module Decidable.Equality.Core

%default total

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-- 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)

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-- Utility lemmas
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||| The negation of equality is symmetric (follows from symmetry of equality)
export
negEqSym : forall a, b . (a = b -> Void) -> (b = a -> Void)
negEqSym p h = p (sym h)

||| Everything is decidably equal to itself
export
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