Lecture Notes on Proofs & Arithmetic

Lecture 8 on Events & Delays discussed and developed two models for the ping pong ball whose control was a ping pong paddle. First an event-driven controller and then a time-triggered controller. Invariants have been identified in Lecture 8 but not proved. This lecture will study proofs. This lecture discusses: • What is a proof? • How does arithmetic happen in a proof? • Arithmetic hiding to weaken arithmetic • Instantiating arithmetic quantifiers to the extreme • Updates as delayed substitutions/assignments • Creative cuts for arithmetic • Substituting equations • If-then-else proofs There are many more interesting things to be discussed about the mysteries of arithmetic and how real arithmetic properties themselves can be proved [PQR09, Pla10, Appendix D]. That is a topic for a later lecture, though. This lecture focuses on how arithmetic interfaces with proofs.