Proofs involving decidable equality

Question

I'm trying to prove some simple things about a function which uses decidable equality. Here is a much simplified example:

open import Relation.Nullary
open import Relation.Binary
open import Relation.Binary.PropositionalEquality

module Foo {c} {ℓ} (ds : DecSetoid c ℓ) where

open DecSetoid ds hiding (refl)

data Result : Set where
  something something-else : Result

check : Carrier → Carrier → Result
check x y with x ≟ y
... | yes _ = something
... | no  _ = something-else

Now, I'm trying to prove something like this where I've already shown that both sides of the _≟_ are identical.

check-same : ∀ x → check x x ≡ something
check-same x = {!!}

At this point, the current goal is (check ds x x | x ≟ x) ≡ something. If the x ≟ x was by itself, I would solve it by using something like refl, but in this situation the best I can come up with is something like this:

check-same x with x ≟ x
... | yes p = refl
... | no ¬p with ¬p (DecSetoid.refl ds) 
... | ()

By itself this isn't that bad, but in the middle of a larger proof it's a mess. Surely there must be a better way to do this?

Answer 1

Since my function, like the one in the example, doesn't care about the proof object returned by x ≟ y, I was able to replace it with ⌊ x ≟ y ⌋ which returns a boolean.

That allowed me write this lemma

≟-refl : ∀ x → ⌊ x ≟ x ⌋ ≡ true
≟-refl x with x ≟ x 
... | yes p = refl
... | no ¬p = ⊥-elim (¬p (DecSetoid.refl ds))

which I could then use with rewrite to simplify my proof to

check-same x rewrite ≟-refl x = refl