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I thought about the exact meaning of "isomorphism" in our context a bit more. I had already posed the question what the actual structure is that is being preserved.

Lectures on CHI seems to stop at establishing the existence of a bijection, say F, between the set of lambda terms and the set of proofs. Proofs and Types (on page 20) mentions the fact that what is established at first is only this bijection. They name three relations – conversion, normality and reduction for the respective sets – that are invariant under the bijection.

While I think there is nothing wrong with that, I would assume the basic result to be that functional abstraction and function application are invariant under F. That is, letting a be functional abstraction and i be implication introduction, F(a(t)) = i(F(t)). Equivalently for function application and modus ponens.

If we want to present this result, it would be handy to have defined the term language in terms of a set and closure under operations on the set as you did. Additionally, I should possibly rewrite the presentation of ->Npi to introduce (only or additionally) a sequent style natural deduction calculus, although I am not yet sure about this, as it is only a different way of representing the tree structures.

What do you think? Also, please sanity check my reasoning. After all, I am deviating from the presentation in the book(s).