Question

I am having problems with rewriting using the setoid_rewrite tactic. In the following instance declaration, I expect that setoid_rewrite fmapComp would rewrite fmap iso ∘ fmap inv to fmap (iso ∘ inv). However, Coq reports that “no progress was made” during rewriting:

Instance functorsPreserveIsomorphisms
 `{C : Cat o η} `{D : Cat u ρ}
   {a b : o} {φ : o → u} (F : Functor C D φ) (I : a ≅ b) : φ a ≅ φ b.
Proof.
  apply (Build_Isomorphism _ _ _ (φ a) (φ b) (fmap iso) (fmap inv)).

 o : Type
 η : o → o → Type
 C : Cat o η
 u : Type
 ρ : u → u → Type
 D : Cat u ρ
 a : o
 b : o
 φ : o → u
 F : Functor C D φ
 I : a ≅ b
 ============================
  fmap iso ∘ fmap inv ≡ id (φ a)

I don’t understand why setoid_rewrite fails. For reference, the same command works in other contexts using the same terms. For example, it rewrites either side of the following goal to the other:

Lemma worksotherwise
 `{C : Cat o η} `{D : Cat u ρ}
   {a b : o} {φ : o → u} (F : Functor C D φ) (I : a ≅ b) :
     fmap iso ∘ fmap inv ≡ fmap (iso ∘ inv)

 o : Type
 η : o → o → Type
 C : Cat o η
 u : Type
 ρ : u → u → Type
 D : Cat u ρ
 a : o
 b : o
 φ : o → u
 F : Functor C D φ
 I : a ≅ b
 ============================
  fmap iso ∘ fmap inv ≡ fmap (iso ∘ inv)

It’s unclear why setoid_rewrite works in the second case but not the first. For reference, ≡ is equiv and fmap is Proper. Other than that ≅, Functor and Cat are classes, I don’t see any other relevant facts. Also, setoid_replace works fine.

Answer 1

This is a shot in the dark without seeing the whole development, but sometimes, when you can’t see a difference, it means that there’s a difference in a part you don’t see. Namely, implicit arguments.

For example, you might have an implicit argument somewhere that appears identically in two locations in the working proof attempt, and that appears in two distinct but interconvertible (or even merely equal) in the non-working proof attempt. Occasionally tactics require identical terms to fire up, whereas interconvertible terms would suffice with the same proof tree, and equal terms would suffice with judicious introduction of eq_refl. When you’re working with high-level tactics such as the setoid tactics, it can be difficult to understand what’s going on under the hood.

Try comparing the situations under Set Printing Implicit or Set Printing All, or working with No Strict Implicit or No Implicit Arguments for a small part of the proof.