1. Introduction to Proofs Proof Methods and Strategy
Lecture 5
Introduction to Proofs Proof Methods and Strategy

2. Introduction

3. Proof Terminology
theorem - a statement that can be shown to be true (commonly refers to major results only)
proposition - less important fact or result
lemma - a less important theorem that is used in a step in the proof of a major theorem
proof - method to demonstrate that a theorem or proposition
axiom - also called a postulate is a statement that is assumed to be true
corollary - a theorem that can be established directly from a theorem
conjecture - a statement that is being proposed to be true, but not yet proved

4. Direct Proof

5. Proof by Contraposition

6. Proof by Contradiction

7. Mistakes in Proofs

8. Circular Reasoning

9. Exhaustive Proof

10. Proof by Cases

11. Existence Proof

12. A Constructive Existence Proof Candidate for a Computer Program

13. A Nonconstructive Existence Proof

14. Uniqueness Proof

15. Candidate for a Computer Program
Counterexamples
Candidate for a Computer Program

16. Important Open Problems

17. Other Open Problems

18. Computer-Assisted Proofs
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer.
Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program; the Kepler conjecture followed in 1998.
Attempts have also been made in the area of artificial intelligence research to create smaller, explicit, new proofs of mathematical theorems from the bottom up using machine reasoning techniques such as heuristic search. Such automated theorem provers have proved a number of new results and found new proofs for known theorems. Additionally, interactive proof assistants allow mathematicians to develop human-readable proofs which are nonetheless formally verified for correctness. Since these proofs are generally human-surveyable (albeit with difficulty, as with the proof of the Robbins conjecture) they do not share the controversial implications of computer-aided proofs-by-exhaustion.

19. Four Color Theorem
In mathematics, the four color theorem, or the four color map theorem, states that given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color. Two regions are called adjacent only if they share a border segment, not just a point.
The four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer. Appel and Haken's approach started by showing there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem. Appel and Haken used a special-purpose computer program to check each of these maps had this property.
To dispel remaining doubt about the Appel–Haken proof, a simpler proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour, and Thomas. Additionally in 2005, the theorem was proven by Georges Gonthier with general purpose theorem proving software.

20. Kepler Conjecture
The Kepler conjecture, named after Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is slightly greater than 74%.
In 1998 Thomas Hales, following an approach suggested by Fejes Tóth (1953), announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving checking of many individual cases using complex computer calculations. Referees have said that they are "99% certain" of the correctness of Hales' proof. So the Kepler conjecture is now very close to being accepted as a theorem.

21. Automated Reasoning and Theorem Proving
Automated reasoning is an area of computer science dedicated to understanding different aspects of reasoning in a way that allows the creation of software which allows computers to reason completely or nearly completely automatically. As such, it is usually considered a subfield of artificial intelligence, but it also has strong connections to theoretical computer science and even philosophy.
The most developed subareas of automated reasoning probably are automated theorem proving (and the less automated but more pragmatic subfield of interactive theorem proving) and automated proof checking (viewed as guaranteed correct reasoning under fixed assumptions), but extensive work has also been done in reasoning by analogy induction and abduction.