≡-Reasoning and 'with' patterns

Question

I was proving some properties of filter and map, everything went quite good until I stumbled on this property: filter p (map f xs) ≡ map f (f

Answer 1

The trouble with with-clauses is that Agda forgets the information it learned from pattern match unless you arrange beforehand for this information to be preserved.

More precisely, when Agda sees a with expression clause, it replaces all the occurences of expression in the current context and goal with a fresh variable w and then gives you that variable with updated context and goal into the with-clause, forgetting everything about its origin.

In your case, you write filter p (map f (x ∷ xs)) inside the with-block, so it goes into scope after Agda has performed the rewriting, so Agda has already forgotten the fact that p (f x) is true and does not reduce the term.

You can preserve the proof of equality by using one of the "Inspect"-patterns from the standard library, but I'm not sure how it can be useful in your case.