Lecture
This lecture introduces inductive propositions and their application in defining evaluation rules for arithmetic expressions in Coq. The instructor begins by discussing the structure of inductive propositions, emphasizing their role in establishing relationships between valuations, arithmetic expressions, and their evaluated results. The lecture covers various constructors for arithmetic operations, including constants, variables, addition, and multiplication, demonstrating how to define these rules in Coq. The instructor illustrates the process of proving properties of arithmetic expressions, such as the evaluation of expressions to specific values. Additionally, the lecture explores the concept of partial and non-deterministic functions, highlighting how inductive propositions can accommodate these definitions. The instructor also addresses the Collatz conjecture, showcasing how inductive propositions can be used to reason about non-terminating sequences. Overall, the lecture provides a comprehensive overview of inductive reasoning in Coq, equipping students with the tools to define and prove properties of arithmetic expressions effectively.