1. Lecture 12 CS 1813 – Discrete Mathematics
Lecture 15 - CS 1813 Discrete Math, University of Oklahoma
9/22/2018
Lecture 12 CS 1813 – Discrete Mathematics
The Principle of
Mathematical Induction
ok … now we’re cooking with gas …
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

2. A Little Theorem about Sequences
Lecture 15 - CS 1813 Discrete Math, University of Oklahoma
9/22/2018
A Little Theorem about Sequences
Algebraic laws of sequence construction
(x: []) = [x] :[]
(xs  [ ] ) = (x. ys. xs = (x: ys) ) -- (:)
Informally
( x : [x1, x2, …] ) = [x, x1, x2, …] -- (: …)
Algebraic laws of concatenation
(++) :: [a] -> [a] -> [a]
([ ] ++ ys) = ys (++).[ ]
((x : xs) ++ ys) = (x : (xs ++ ys)) (++).:
An equational argument
Assume x :: a and xs :: [a]
[x] ++ xs
= (x : [ ]) ++ xs :[]
= x : ([ ] ++ xs) (++).:
= (x : xs) (++).[ ]
What did this prove?
([x] ++ xs) = (x : xs) ()
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

3. Software Equations = Algebraic Laws
Lecture 15 - CS 1813 Discrete Math, University of Oklahoma
9/22/2018
Software Equations = Algebraic Laws
foldr (the big picture)
foldr () z [x1, x2, …, xn] =
x1  (x2  … (xn-1  (xn  z)) … )
Algebraic laws of foldr
foldr :: (a -> b -> b) -> b -> [a] -> b
foldr () z [ ] = z (foldr).[]
foldr () z (x : xs) = x  (foldr () z xs) (foldr).:
The big or
(\/) :: Bool -> Bool -> Bool -- “little or” – satisfies Boolean laws for 
or :: [Bool] -> Bool “big or”
or = foldr (\/) False (or)
Theorem (or1). or ([True] ++ xs) = True
or ([True] ++ xs)
= or (True : xs) ()
= foldr (\/) False (True : xs) (or)
= True \/ (foldr (\/) False xs) (foldr).:
= (foldr (\/) False xs) \/ True ( comm)
= True ( null)
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page
qed

4. Theorem — or/singleton True
Lecture 15 - CS 1813 Discrete Math, University of Oklahoma
9/22/2018
Theorem — or/singleton True
Corollary (or1c): or [True] = True
Proof
True
= or ( [True] ++ [ ] ) (or1)
= or ( True : [ ] ) ()
= or ( [True] ) :[]
qed
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

5. Theorem — or[x,True, …] = True
Lecture 15 - CS 1813 Discrete Math, University of Oklahoma
9/22/2018
Theorem — or[x,True, …] = True
Theorem (or2): or ([x] ++ ([True] ++ xs)) = True
Proof
or([x] ++ ([True] ++ xs))
= or((x:[]) ++ ([True] ++ xs)) (:)
= or(x:([] ++ ([True] ++ xs))) (++).:
= foldr (\/) False (x:([] ++ ([True] ++ xs))) (or)
= x \/ (foldr (\/) False ([] ++ ([True] ++ xs))) (foldr).:
= x \/ (or([] ++ ([True] ++ xs))) (or)
= x \/ (or([True] ++ xs)) (++).[]
= x \/ True (or1)
= True  null
qed
Theorem (or1): or ([True] ++ xs) = True
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

6. Theorem — or[x, y,True, …] = True
Lecture 15 - CS 1813 Discrete Math, University of Oklahoma
9/22/2018
Theorem — or[x, y,True, …] = True
Theorem (or3): or ([x, y] ++ ([True] ++ xs)) = True
Proof
or([x, y] ++ ([True] ++ xs))
= or((x:[y]) ++ ([True] ++ xs)) (:)
= or(x:([y] ++ ([True] ++ xs))) (++).:
= foldr (\/) False (x:([y] ++ ([True] ++ xs))) (or)
= x \/(foldr (\/) False ([y] ++ ([True] ++ xs))) (foldr).:
= x \/ (or([y] ++ ([True] ++ xs))) (or)
= x \/ True (or2)
= True  null
qed
Theorem (or2): or ([x] ++ [True] ++ xs) = True
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

7. Theorem — or[x, y, z,True, …] = True
Lecture 15 - CS 1813 Discrete Math, University of Oklahoma
9/22/2018
Theorem — or[x, y, z,True, …] = True
Theorem (or4): or ([x, y, z] ++ ([True] ++ xs)) = True
Proof
or([x, y, z] ++ ([True] ++ xs))
= or((x:[y, z]) ++ ([True] ++ xs)) (:)
= or(x:([y, z] ++ ([True] ++ xs))) (++).:
= foldr (\/) False (x:([y,z] ++ ([True] ++ xs))) (or)
= x \/(foldr (\/) False ([y,z] ++ ([True] ++ xs))) (foldr).:
= x \/ (or([y,z] ++ ([True] ++ xs))) (or)
= x \/ True (or3)
= True  null
qed
Theorem (or3): or ([x, y] ++ [True] ++ xs) = True
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

8. Theorem — or[x, y, z, w,True, …] = True
Lecture 15 - CS 1813 Discrete Math, University of Oklahoma
9/22/2018
Theorem — or[x, y, z, w,True, …] = True
Theorem (or5): or ([x, y, z, w] ++ ([True] ++ xs)) = True
Proof
You can do this one, right?
or([x, y, z, w] ++ ([True] ++ xs))
= or((x:[y, z, w]) ++ ([True] ++ xs)) (:)
= or(x:([y, z, w] ++ ([True] ++ xs))) (++).:
= foldr (\/) False (x:([y,z,w] ++ ([True] ++ xs))) (or)
= x \/(foldr (\/) False ([y,z,w] ++ ([True] ++ xs))) (foldr).:
= x \/ (or([y,z,w] ++ ([True] ++ xs))) (or)
= x \/ True (or4)
= True  null
The last three proofs are all the same, except that they each cite a different theorem in the 5th line.
Theorem (or4): or ([x, y, z] ++ [True] ++ xs) = True
The proof of (orn+1) cites (orn)
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

9. Principle of Mathematical Induction another way to skin a cat
Lecture 15 - CS 1813 Discrete Math, University of Oklahoma
9/22/2018
Principle of Mathematical Induction another way to skin a cat
{I} — an inference rule
with n. P(n) as it’s conclusion
One way to use {I}
Prove P(0)
Prove P(n +1) for arbitrary n
Takes care of P(1), P(2), P(3), …
P(n) {n arbitrary}
{I}
n. P(n)
 Introduction
P(0) n.P(n)P(n+1)
{Ind}
n. P(n)
Induction
Mathematical induction makes it easier
Proof of P(n +1) can cite P(n) as a reason
If you cite P(n) as a reason in proof of P(n+1), your proof relies on mathematical induction
If you don’t, your proof relies on {I}
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

10. Theorem — or[x, y, …,True, …] = True
Lecture 15 - CS 1813 Discrete Math, University of Oklahoma
9/22/2018
Theorem — or[x, y, …,True, …] = True
Theorem (orn) nN. length ys = n  or (ys ++ ([True] ++ xs)) = True
Proof
P(n)  length ys = n  or (ys ++ ([True] ++ xs)) = True
Base case: P(0)  length ys = 0  or(ys ++ ([True] ++ xs)) = True
length ys = 0
 ys = [ ] zero len theorem
 or(ys ++ ([True] ++ xs)) = or([ ] ++ ([True] ++ xs)) substitution
= or ([True] ++ xs) []
= True or1
Inductive case: P(n)  P(n+1)
… next slide
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

11. Theorem — or[x, y, …,True, …] = True Inductive Case
Lecture 15 - CS 1813 Discrete Math, University of Oklahoma
9/22/2018
Theorem — or[x, y, …,True, …] = True Inductive Case
Predicate to prove for inductive case
P(n+1)  length ys = n+1  or (ys ++ ([True] ++ xs)) = True
length ys = n+1
 ys  [ ] zero len theorem
 ys = y: zs  length zs = n :len corollary
 or(ys ++ ([True] ++ xs)) = or((y:zs) ++ ([True] ++ xs)) subst  length zs = n
= or(y:(zs ++ ([True] ++ xs)))  length zs = n (++).:
= (y \/ (or(zs ++ ([True] ++ xs))))  length zs = n (foldr).:
= (y \/ True)  length zs = n P(n)
induction hypothesis
= (y \/ True)  True subst
= True  id,  null
Conclude: nN. P(n) principle of induction
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page
qed

12. Lecture 15 - CS 1813 Discrete Math, University of Oklahoma
9/22/2018
End of Lecture
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page