I am proving a kind of structure invariance principle for magmas in Cubical Type Theory with the Agda/Cubical library. This is done by constructing a path between two simple magmas and then transporting proofs of simple properties about this path. I have already obtained most of the proof (see my code repository) but did not manage yet to complete the following lemma.
At some point in the proof I have the following given:
- A bijection between types:
f : ℕ → ℕ₀
- The isomorphism constructed with
fIso : Iso ℕ ℕ₀
- A function that gives the inverse isomorphism of an isomorphism:
invIso : Iso A B → Iso B A
Now, I would like to prove that:
sym (ua (isoToEquiv fIso)) ≡ ua (isoToEquiv (invIso fIso))
There are two parts to my question:
- Is this a valid theorem in HoTT? Although this statement seems valid, maybe I have produced a false statement?
- Are there built-in functions in the Agda/Cubical that may help in the proof?