What qualifies as a "constructor" and the omitted $\equiv$ rules in appendix A.2 about book

[mikeshulman]

I think yes, this should apply to X-ELIM-EQ as well. If so, I think you're right that the word "constructor" is confusing. I think the intent was that we often view the syntax as inductively generated at the meta-level, and in this case each type- and term-forming rule (including both introductions and eliminations) is a meta-level "constructor" of this inductive type of syntax. Would it be clearer to say something like "each type- and term-forming operator"?

[FernandoChu]

I see, thanks for clearing the use of the word "constructor". Yes, I think your proposed solution would work fine. Maybe just mention the full list for the Π-Types, to be on the safe side, I'll be making a PR.

[cangiuli]

Yes, this applies also to elimination rules; there is a PI-ELIM-EQ.

FWIW I'd say that PI-ELIM "constructs" a term of type B[a/x], but I see how this phrasing is confusing. +1 to Mike's suggestion "each type- and term-forming operator".