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The reverse function in problem 3 incorrectly defined the recursive case

using the append operator @. This case should instead use the append function defined in

that problem. This has been fixed below.

Question 1: Power function, over natural numbers

Recall the OCaml power function over natural numbers, shown below:

let rec power n x =

match n with

| 0 -> 1.0

| _ -> x *. power (n-1) x

Using induction over natural numbers show that

power n x = x

n

.

Your proof must explicitly and clearly indicate the base case you prove, the inductive case you

prove and what the inductive hypothesis provides in the proof.

Each step in your proof must be accompanied by a justification describing why that step could be

taken.

Question 2: Power over structured numbers

Recall from lecture the OCaml type nat, the function toInt, and the power function working over

this representation of natural numbers:

type nat = Zero | Succ of nat

let toInt = function

| Zero -> 0

| Succ n -> toInt n + 1

let rec power n x = match n with

| Zero -> 1.0

| Succ n’-> x *. power n’ x

What is the principle of induction for the type nat?

Using induction over nat values show that

power n x = x

toInt(n)

Your proof must explicitly and clearly indicate the base case you prove, the inductive case you

prove and what the inductive hypothesis provides in the proof.

Each step in your proof must be accompanied by a justification describing why that step could be

taken.

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Question 3: List reverse and append

Consider the following definition of append:

let rec reverse l = match l with

| [] -> []

| (h::t) -> append (reverse t) [h]

let rec append l1 l2 = match l1 with

| [] -> l2

| (h::t) -> h :: (append t l2)

Using the definition of reverse and the definition of append above, show, using induction, that

reverse (append l1 l2) = append (reverse l2) (reverse l1)

Your proof must explicitly and clearly indicate the base case you prove, the inductive case you

prove and what the inductive hypothesis provides in the proof.

Each step in your proof must be accompanied by a justification describing why that step could be

taken.

Question 4: List processing

Consider the following OCaml function definitions.

let isupper c = Char.code c >= Char.code ’A’ &&

Char.code c <= Char.code ’Z’
let rec someupper lst = match lst with
| [] -> false

| x::xs -> isupper x || someupper xs

Using the definition above, show using induction that

someupper (l1 @ l2) = someupper l1 || someupper l2

Your proof must explicitly and clearly indicate the base case you prove, the inductive case you

prove and what the inductive hypothesis provides in the proof.

Each step in your proof must be accompanied by a justification describing why that step could be

taken.

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Question 5: List processing and folds Below we show again the functions defined in the

previous problem and a new alternative implementation of the recursive function defined in Question

4. This new version uses foldr. The definition of foldr is the same as you’ve seen before.

let isupper c = Char.code c >= Char.code ’A’ &&

Char.code c <= Char.code ’Z’
let rec someupper lst = match lst with
| [] -> false

| x::xs -> isupper x || someupper xs

let rec foldr (f:’a -> ’b -> ’b) (l:’a list) (v:’b) : ’b =

match l with

| [] -> v

| x::xs -> f x (foldr f xs v)

let upperor c b = isupper c || b

let foldupper lst = foldr upperor lst false

By induction, show that

someupper chs = foldupper chs

Your proof must explicitly and clearly indicate the base case you prove, the inductive case you

prove and what the inductive hypothesis provides in the proof.

Each step in your proof must be accompanied by a justification describing why that step could be

taken.

3

Question 6: Tree processing

Consider the following OCaml definitions of a tree type and a few functions over values of this type.

type ’a tree = Leaf of ’a

| Branch of ’a tree * ’a tree

let min x y = if x < y then x else y
let rec mintree t = match t with
| Leaf v -> v

| Branch (t1, t2) -> min (mintree t1) (mintree t2)

let rec tfold (l:’a -> ’b) (f: ’b -> ’b -> ’b) (t: ’a tree) : ’b = match t with

| Leaf v -> l v

| Branch (t1, t2) -> f (tfold l f t1) (tfold l f t2)

let fold_mintree t = tfold (fun x -> x) min t

Prove using induction that for any tree t of type int tree

mintree t = fold mintree t

Your proof must explicitly and clearly indicate the base case you prove, the inductive case you

prove and what the inductive hypothesis provides in the proof.

Each step in your proof must be accompanied by a justification describing why that step could be

taken.

Submission instructions: Writing proofs such as these requires a bit of clear thinking and it is

important to check your work.

Checking your work means you need to be able to read it. And to assess it, we need to be able

to read it as well.

Thus, we are requiring your solutions be electronically generated. You may turn your work in

using any of the following forms.

1. A PDF file – named hwk 03.pdf.

You may use LaTeX, enscript, or even MS Word to generate a PDF file that contains your

solutions.

2. A Markdown file – named hwk 03.md.

This is used for the lab and other homework specifications and makes it easy to see your

solution in GitHub.

3. A text file – named hwk 03.txt.

We’ve written proofs in text files in class and examples can be found in the Notes directory

of the public class repository.

Scanned or photographed versions of hand-written solutions will not be accepted.

This work is to be submitted via GitHub in a folder named Hwk 03.

The work is due by 5:00pm on Wednesday, March 7.

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