Copyright © 2007-2014 Curt Hill Proofs An Introduction.

Copyright © 2007-2014 Curt Hill

Proofs

An Introduction

What is a Proof?• A valid argument• Shows the correctness of a

statement• Uses axioms, theorems and any

hypotheses of the problem• May be informal or formal• Informal uses some shortcuts

– Multiple transformations per step

• Formal tend to be publication quality– Long and difficult



Terminology• Axioms

– An unproven or unprovable given statement– These are self obvious or definitional

• Theorems– Statements of importance that have been proven

• Lemmas– Theorem of lesser importance– Often only used in the course of a proof

• Corollary – Theorem easy to prove from this one

• Conjectures or Suppositions– Statements we wish to prove– Today’s conjectures proven are tomorrow’s

theorems

Types of Proofs

• Two basic forms, direct and indirect

• Direct– Start with hypotheses and argue

towards the result

• Indirect– Any other direction– Two examples are proof by

contraposition and proof by contradiction



What constitutes a proof?• The usual proof is a series of steps• Each step consists of three pieces:

– An original statement– A transformed statement– The justification of the transformation

• The transformed statement is the original for the next step

• The justification must cite axioms or theorems

• The first two may be on one line or two


Proofs of Equivalence• Conjectures are very often stated in

terms of one thing equivalent to another

• In this case we can argue from one side to the other in either direction– Usually easier to go from more complicated

to simpler– It reduces the combinatorics

• We may also consider the entire expression as the supposition– Then show that it is a tautology by reducing

it to another tautology, that is an axiom or theorem

Implication Proofs

• Here the conjecture is an implication– Most common in Rosen

• We usually assert the antecedent • Then use a series of steps to prove

the consequent• What happens if the antecedent is

false?


Direct Example• Prove that the sum of two odd

integers is even• We will need the definition of odd and

even:– An integer n is even if there exists an

integer k such that 2k = n– An integer n is odd if there exists an

integer k such that 2k + 1 = n– Odd and even are opposites

• Assume that the propositions E(n) is true if n is even and O(n) is true if n is odd


Example Continues

• The statement of the conjecture is:– (O(n) O(m)) E(n+m)

• More formally: n,m(nI mI O(n)O(m)) E(n+m)


Sum of odds is even

• Let n be 2i+1 and m be 2j+1– The definition of odd

• Let p = n+m• p = 2i+1 + 2j+1• p= 2i+2j+1 + 1

– Algebraic commutativity

• p=2(i+j+1)– Algebraic distributivity

• p is even from definition of evenCopyright © 2007-2014 Curt Hill

Contrapositive Proofs

• Recall that if p q then q p– This is the contrapositive

• A proof by contraposition uses this as the starting point

• Instead of starting with p q, instead start with q p

• Sometimes this easier• However, it is an indirect proof since

we did not start with the hypothesis


Commentary

• The distinction between direct and indirect gets a little fuzzy here

• Start with the hypothesis– That would be a direct proof

• As a first step apply the contrapositive

• Is it now direct or indirect?• Lets do #17 or #18 on the board



Positive and Negative• Proofs usually assume the theorem is

true and argue it from there• Recall sum of odds is even theorem

n,m(nI mI O(n)O(m)) E(n+m)

• I could disprove this by showing only one pair of n and m that made the antecedent true and the consequent false

• Thus if I prove n,m(nI mI O(n)O(m)) O(n+m)

then I have disproved original

Proof by Contradiction

• Can also be quite effective but often easier

• State the theorem as false and show that there is a contradiction by reasoning to a known fallacy

• This fallacy is usually the opposite of a theorem or axiom

• This form is almost always backwards


Halting Problem• It is impossible to code a

procedure that will input an arbitrary procedure and its input and determine if it halts

• Its output will be true if the program will halt with this data

• It will output false if the program has an infinite loop with this data

• This is a specific case of static analysis


Proof 1 of 2• Suppose that a procedure exists

that flawlessly solves the halting problem– Call this procedure Q(P,I) where P is

the input procedure and I is the input

• We construct a procedure R as follows:– if(Q(R,R))– while(true)– println(“oops”)– else println(“It will halt”);


Proof 2 of 2• What is the result when R is fed

into Q?• If Q says R halts, then R infinitely

loops• If Q says R loops, then R halts• Since Q cannot give the right

answer for this example, there is a contradiction

• Thus no procedure that can solve the halting problem


Proof Commentary• This proof was done in 1936 by

Alan Turing– The proof by contradiction is simple– What would a direct proof look like?

• What does this theorem not say?– It does not say that static analysis is

completely impossible– It says that static analysis is

impossible to complete

• Most compiler writers use this as an excuse to do little static analysis


How many primes?• Can we prove that there are an

infinite number of primes?– Yes, by contradiction

• Assume that there are a finite number of primes

• Take these primes and multiply all of them together

• Add 1• The resulting number is not divisible

by any of the primes– Thus there are an infinite number of

primesCopyright © 2007-2014 Curt Hill

Almost Finally

• The only thing easier to mess up than a program is a proof– The text lists several– You will likely find others

• This presentation is not the end all on proofs– Rather it is just the beginning– We enlarge our tool set in the next

one


Exercises

• 3, 9, 27, 39


Copyright © 2007-2014 Curt Hill Proofs An Introduction.

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