Good introductory book to type theory?
Here are some resources:
UniMath school teaching materials, and in particular:
Spartan type theory, an introduction to type theory (slides)
Introduction to Univalent Foundations of Mathematics with Agda by Martín Escardó.
Univalent Foundations programme: Homotopy Type Theory: Univalent Foundations of Mathematics
- Bengt Nordström, Kent Petersson, and Jan M. Smith: Programming in Martin-Löf's Type Theory
I am far from being an expert. I will make a few suggestions.
Per Martin-Löf. Intuitionistic type theory. (Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980). Napoli, Bibliopolis, 1984
T. Streicher (1991), Semantics of Type Theory: Correctness, Completeness, and Independence Results, Birkhäuser Boston.
Andre Joyal. Notes on Clans and Tribes.
Michael Shulman. Homotopy type theory: the logic of space.
Thorsten Altenkirch. Naive Type Theory.
It seems that the HoTT book and Vladimir Voevodsky’s program for Univalent Foundations of Mathematics is made for you !
You will find everything from here: https://homotopytypetheory.org/