univalence

Now, the issue is that currently we don't actually have a computational interpretation for the univalence axiom, so the above magic doesn't yet work.

What the univalence axiom says is that you can treat types you have proven isomorphic as equal.

The univalence axiom states that in this new constructive mathematical framework, 'equality' and 'isomorphism' are exactly the same thing.

The general form of excluded middle is inconsistent, because it would collapse the structures on which univalence is built.

What is the state of the art of the research on the computational interpretation of univalence?

This is the same Voevodsky that came up with the univalence axiom[1] that is foundational in HoTT[2] (homotopy type theory)?

The novelty of HoTT, and only extension over intensional Martin-Loef type-theory, is the univalence axiom.

Much of current research on HoTT is about the consequences of the univalence axiom and the similarity between logic and geometry/topology.

Some of those classically trivial equivalences might be recovered though via the univalence axiom.

When you think about it we do have a computational meaning and interpretation for the univalence axiom.

Sorry if I wasn't clear about this, but you can embed classical mathematics in a type theory with univalence.

So let's say I am writing a theorem proving system based on HoTT (with univalence).

What's not to love about: The univalence axiom states: (A = B) ≃ (A ≃ B) "In other words, identity is equivalent to equivalence.

The catch is that univalence is inconsistent with LEM at h-levels greater than -1, but assuming it is perfectly consistent for -1 types, which can be thought of as the "at most true" propositions of classical logic.

Homotopy Type Theory uses the idea of "univalence", which basically says "if you can transform back and forth between two things, then you can use them interchangably".

They found that in this model, "equality is equivalent to equivalence" (Voevodsky's Univalence Axiom), or in other words, the kind of trivial equality one deals with in Coq is generalized as just a very low-dimensional instance of topological and categorical equivalence.

Univalence is unrelated to the halting problem.