1. Lecture 13 CS 1813 – Discrete Mathematics
Lecture 16 - CS 1813 Discrete Math, University of Oklahoma
11/18/2018
Lecture 13 CS 1813 – Discrete Mathematics
Induction
Induction …
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

2. Concatenating Sequences
Lecture 16 - CS 1813 Discrete Math, University of Oklahoma
11/18/2018
Concatenating Sequences
(++) :: [a] -> [a] -> [a]
(x: xs) ++ ys = x: (xs ++ ys) (++).:
[ ] ++ ys = ys (++).[]
Proposition P(n) (universe of discourse: n N )
length xs = n  length(xs ++ ys) = length xs + length ys
P(0): length xs = 0  length(xs++ys) = length xs + length ys
length(xs ++ ys)
= length([ ] ++ ys) len 0  [ ]
= length ys (++).[]
= 0 + length ys 2nd-grade arithmetic
= length xs + length ys hypothesis of implication
Next, prove P(n)  P(n+1) … next slide
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

3. Concatenating Sequences the inductive case
Lecture 16 - CS 1813 Discrete Math, University of Oklahoma
11/18/2018
Concatenating Sequences the inductive case
P(n+1): length xs = n+1  length(xs++ys) = (length xs) + (length ys)
length(xs++ys)
= length((z:zs) ++ ys) xs = (z: zs) {see note}
= length(z: (zs ++ ys)) (++).:
= 1 + length(zs ++ ys) (length).:
= 1 + length zs + length ys P(n), since length zs = n
= 1 + n + length ys (length zs) = n {see note}
= n length ys + comm
= length xs + length ys hypothesis in P(n+1)
Note: z. zs. (xs = (z: zs))  ((length zs) = n) (:len) corollary
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

4. Concatenating Sequences applying the principle of induction
Lecture 16 - CS 1813 Discrete Math, University of Oklahoma
11/18/2018
Concatenating Sequences applying the principle of induction
Proved: P(0)
Proved: P(n)  P(n+1)
Conclude nN. P(n) — by the principle of induction
P(n): length xs = n  length(xs ++ ys) = length xs + length ys
qed
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

5. Concatenating a List of Sequences the big ++
Lecture 16 - CS 1813 Discrete Math, University of Oklahoma
11/18/2018
Concatenating a List of Sequences the big ++
concat :: [[a]] -> [a]
concat(xs: xss) = xs ++ concat xss (concat).:
concat[ ] = [ ] (concat).[]
Theorem: n N. P(n) where P(n) is defined as follows: P(n)  (k{1, 2, … n}. length(xsk)N)  length(concat [xs1, xs2, … xsn]) = sum[length xs1, length xs2, … length xsn]
List Comprehension — like set comprehension, but for sequences
[sequence-element | generator, optional-constraint]
[x - 2 | x <- [12, 9, 27, 19, 13]] = [10, 7, 25, 17, 11]
[x | x <- [12, 9, 27, 19, 13], x < 15] = [12, 9, 13]
[x + 3 | x <- [12, 9, 27, 19, 13], x < 15] = [15, 12, 16]
Proof
Induction on n
All this stuff is starting to look alike, isn’t it?
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

6. Lecture 16 - CS 1813 Discrete Math, University of Oklahoma
11/18/2018
shuffle
shuffle :: [a] -> [a] -> [a]
Examples
shuffle [1, 2, 3, 4, 5] [6, 7, 8, 9, 10] = [1, 6, 2, 7, 3, 8, 4, 9, 5, 10]
shuffle [1, 2, 3, 4, 5] [6, 7, 8] = [1, 6, 2, 7, 3, 8, 4, 5]
shuffle [1, 2, 3, 4] [6, 7, 8, 9, 10] = [1, 6, 2, 7, 3, 8, 4, 9, 10]
Pattern of computation
shuffle [x1, x2, … xn] [y1, y2, … yn] = [x1, y1, x2 , y2, … xn , yn]
Note: extra elements in either sequence appended to the end
Definition
shuffle (x: xs) (y: ys) =
[x, y] ++ shuffle xs ys
shuffle [ ] ys = ys
shuffle xs [ ] = xs
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

7. Lecture 16 - CS 1813 Discrete Math, University of Oklahoma
11/18/2018
shuffle Works
shuffle :: [a] -> [a] -> [a]
shuffle (x: xs) (y: ys) = [x, y] ++ shuffle xs ys (shf).:
shuffle [ ] ys = ys (shf).[]L
shuffle xs [ ] = xs (shf).[]R
Theorem (shuffle works)
nN. P(n) where P(n)  shuffle [x1, x2, … xn] [y1, y2, … yn] = [x1, y1, x2 , y2, … xn , yn]
P(0) = shuffle [ ] [ ] = [ ] — Why? Because (shf).[]L That’s why!
P(n+1)  shuffle [x1, x2, … xn+1] [y1, y2, … yn+1]=[x1, y1, x2 , y2, … xn+1 , yn+1]
Proof of P(n+1)
shuffle [x1, x2, … xn+1] [y1, y2, … yn+1]
= shuffle (x1 : [x2, x3, … xn+1] ) (y1 : [y2, y3, … yn+1] ) (: …) (twice)
= [x1, y1] ++ shuffle [x2, x3, … xn+1] [y2, y3, … yn+1] (shf).:
= [x1, y1] ++ [x2, y2, x3 , y3, … xn+1 , yn+1]
P(n)
= x1: (y1: [x2, y2, x3 , y3, … xn+1 , yn+1] ) (++).: (twice)
= [x1, y1, x2 , y2, … xn+1 , yn+1] (: …) (twice)
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

8. Lecture 16 - CS 1813 Discrete Math, University of Oklahoma
11/18/2018
Indexing (!!)
(!!):: [a] -> Int -> a
(x: xs) !! 0 = x (!!).0
(x: xs) !! (n+1) = xs !! n (!!).n Note: n N
N = {0, 1, 2, …}
Pattern of computation
[x0, x1, x2, …] !! k = xk
What do the (!!)-equations say about the following formula?
[ ] !! 0
How about this formula?
[ ] !! k
How about these?
(x: xs) !! (-1)
xs !! (-1)
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

9. shuffle Works – more formally
Lecture 16 - CS 1813 Discrete Math, University of Oklahoma
11/18/2018
shuffle Works – more formally
shuffle (x: xs) (y: ys) = [x, y] ++ shuffle xs ys (shf).:
Theorem (shuffle works)
nN. P(n) where
P(n)  ( ((length xs) > n)  ((length ys) > n) )  (k, 0  k  n. (((shuffle xs ys) !! (2*k)) = (xs !! k))  (((shuffle xs ys) !! (2*k + 1)) = (ys !! k)) )
P(0)  ( ((length xs) > 0)  ((length ys) > 0) )  (k, 0  k  0. (((shuffle xs ys) !! (2*k)) = (xs !! k))  (((shuffle xs ys) !! (2*k + 1)) = (ys !! k)) )
= ( ((length xs) > 0)  ((length ys) > 0) )  ( (((shuffle xs ys) !! 0) = (xs !! 0))  (((shuffle xs ys) !! 1) = (ys !! 0)) ) {finite -universe}
= ( (x. ws. xs = (x: ws))  (y. zs. ys = (y: zs)) )  ( (((shuffle (x: ws) (y: zs)) !! 0) = ((x: ws) !! 0))  (((shuffle (x: ws) (y: zs)) !! 1) = ((y: zs) !! 0)) ) {(:), len 0  [ ], arith}
= ( (x. ws. xs = (x: ws))  (y. zs. ys = (y: zs)) )  ( ((([x, y] ++ shuffle ws zs) !! 0) = ((x: ws) !! 0))  ((([x, y] ++ shuffle ws zs) !! 1) = ((y: zs) !! 0)) ) {(shf).:}
= ( (x. ws. xs = (x: ws))  (y. zs. ys = (y: zs)) )  ( (x = x )  (y = y) ) { (++).:, , (!!).n+1, (!!).0}
= True {= reflexive,  id, Thm: (a  True) = True }
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page
end of base case

10. shuffle Works – more formally inductive case
Lecture 16 - CS 1813 Discrete Math, University of Oklahoma
11/18/2018
shuffle Works – more formally inductive case
P(n+1)  ( ((length xs) > n+1)  ((length ys) > n+1) )  (k, 0  k  n+1. (((shuffle xs ys) !! (2*k)) = (xs !! k))  (((shuffle xs ys) !! (2*k + 1)) = (ys !! k)) )
= ( (x.ws.xs=(x:ws)((length ws) > n) )(y.zs.ys=(y:zs)((length zs) > n)) )  (k, 0  k  n+1. (((shuffle xs ys) !! (2*k)) = (xs !! k))  (((shuffle xs ys) !! (2*k + 1)) = (ys !! k)) ) {len 0  [ ], }
= ( (x.ws.xs=(x:ws)((length ws) > n) )(y.zs.ys=(y:zs)((length zs) > n)) )  ((k, 0  k  n.( (((shuffle xs ys) !! (2*k)) = (xs !! k))  (((shuffle xs ys) !! (2*k + 1)) = (ys !! k)) )  (((shuffle (x:ws) (y:zs))!!(2*(n+1))) = ((x:ws)!!(n+1)))  (((shuffle (x:ws) (y:zs))!!(2*(n+1)+1)) = ((y:zs)!!(n+1))) ) {subst, finite -univ}
= ( (x.ws.xs=(x: ws)((length ws) > n))  (y.zs.ys=(y: zs) ((length zs) > n)))  (k, 0  k  n.( (((shuffle xs ys) !! (2*k)) = (xs !! k))  (((shuffle xs ys) !! (2*k + 1)) = (ys !! k)) )  (((shuffle ws zs)!!(2*n)) = (ws!!n))  (((shuffle ws zs)!!(2*n+1)) = (zs!!n)) ) {(shf).:, (++).:, , (!!).n+1}
= ( (x.ws.xs=(x: ws)((length ws) > n))  (y.zs.ys=(y: zs)  ((length zs) > n)))  (True  True  True) {P(n)}
= True { id, Thm: (a  True) = True}
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page
qed

11. Lecture 16 - CS 1813 Discrete Math, University of Oklahoma
11/18/2018
End of Lecture
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page