1. A Note on Proofs Juan Carlos Guzmán CS 6413 Theory of Computation Southern Polytechnic State University

2. Formal Proofs • Proof [http://www.m-w.com] • the cogency of evidence that compels acceptance by the mind of a truth or a fact • the process or an instance of establishing the validity of a statement especially by derivation from other statements in accordance with principles of reasoning • Proof [http://www.wikipedia.com] • In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true

3. Deductive Proofs A set of hypotheses A set of steps statements that logically follow previous steps, hypotheses, or otherwise valid theorems A conclusion the last step The conclusion follows from the hypotheses Inductive Proofs Deal with recursively defined objects Induction on integers Structural induction Inductive hypothesis Base case A proof that the inductive hypothesis holds for the simplest (atomic) elements Induction A proof that the inductive hypothesis holds for an object if it holds for its components Formal Proofs

4. Deductive Proofs • Direct Proofs • Prove the conclusion from the hypotheses • Prove the contrapositive • Assume the negation of the conclusion, prove the negation of the hypotheses • Proof by contradiction • Assume the hypotheses and the negation of the conclusion, reach a contradiction • Exhibit a counterexample that satisfies all hypotheses, but does not satisfy the conclusion • Proof by cases • All cases must prove the same conclusion

5. Deductive Proofs • Proof for “A if and only if B” (AB) • (AB) Prove B from hypothesis A • (BA) Prove A from hypothesis B • Proof of set equality “A=B” • Prove AB and BA • x xA  xB • x xB  xA

6. Induction on Natural Numbers • To prove a property on natural numbers • Prove the property on the basis (usually 0 or 1) • Assume the property (inductive hypothesis) valid for some integer n • Prove that n+1 also satisfies the hypothesis • Note • You can also assume the property valid for all integers less than or equal to n • This generalizes nicely to other domains isomorphic to naturals • Integers • Odd numbers

7. Structural Induction • Important for our course • Natural numbers is a structured domain! • 0 is the basis • 1 is built from 0 via a successor constructor s(0) • 3 is shorthand for s(s(s(0)))

8. Structural Induction • Structured recursive domains can have • Several bases • Several recursive constructors • Must • Prove hypothesis for all bases • Assume hypothesis valid for a group of objects • Prove that all objects created from this “selected few” also satisfy the hypothesis