Introduction to Coq: Arithmetic Expressions and Evaluators

This lecture introduces the basics of Coq, focusing on defining arithmetic expressions using inductive types. The instructor begins by explaining how to create a simple arithmetic type with constants, addition, and multiplication. Examples are provided to illustrate how to construct expressions and evaluate them. The concepts of size and depth of expressions are introduced, along with recursive functions to compute these properties. The instructor demonstrates how to prove theorems about the relationship between depth and size using induction. The lecture progresses to include variables in arithmetic expressions, showcasing how to modify the evaluation function to accommodate this change. The instructor emphasizes the importance of proof automation in Coq, demonstrating various tactics to simplify proofs. The lecture concludes with a discussion on optimizations for expressions and the correctness of the evaluation function, reinforcing the connection between theory and practical implementation in Coq.