module Assignment1 where

YOUR NAME AND EMAIL GOES HERE

Introduction

You must do all the exercises labelled “(recommended)”.

Exercises labelled “(stretch)” are there to provide an extra challenge. You don’t need to do all of these, but should attempt at least a few.

Exercises without a label are optional, and may be done if you want some extra practice.

Please ensure your files execute correctly under Agda!

Imports

import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; cong; sym)
open Eq.≡-Reasoning using (begin_; _≡⟨⟩_; _≡⟨_⟩_; _∎)
open import Data.Nat using (ℕ; zero; suc; _+_; _*_; _∸_; _≤_; z≤n; s≤s)
open import Data.Nat.Properties using (+-assoc; +-identityʳ; +-suc; +-comm;
  ≤-refl; ≤-trans; ≤-antisym; ≤-total; +-monoʳ-≤; +-monoˡ-≤; +-mono-≤)
open import plfa.part1.Relations using (_<_; z<s; s<s; zero; suc; even; odd)

Naturals

Exercise seven

Write out 7 in longhand.

Exercise +-example

Compute 3 + 4, writing out your reasoning as a chain of equations.

Exercise *-example

Compute 3 * 4, writing out your reasoning as a chain of equations.

Exercise _^_ (recommended)

Define exponentiation, which is given by the following equations.

n ^ 0        =  1
n ^ (1 + m)  =  n * (n ^ m)

Check that 3 ^ 4 is 81.

Exercise ∸-examples (recommended)

Compute 5 ∸ 3 and 3 ∸ 5, writing out your reasoning as a chain of equations.

Exercise Bin (stretch)

A more efficient representation of natural numbers uses a binary rather than a unary system. We represent a number as a bitstring.

data Bin : Set where
  nil : Bin
  x0_ : Bin → Bin
  x1_ : Bin → Bin

For instance, the bitstring

1011

standing for the number eleven is encoded, right to left, as

x1 x1 x0 x1 nil

Representations are not unique due to leading zeros. Hence, eleven is also represented by 001011, encoded as

x1 x1 x0 x1 x0 x0 nil

Define a function

inc : Bin → Bin

that converts a bitstring to the bitstring for the next higher number. For example, since 1100 encodes twelve, we should have

inc (x1 x1 x0 x1 nil) ≡ x0 x0 x1 x1 nil

Confirm that this gives the correct answer for the bitstrings encoding zero through four.

Using the above, define a pair of functions to convert between the two representations.

to   : ℕ → Bin
from : Bin → ℕ

For the former, choose the bitstring to have no leading zeros if it represents a positive natural, and represent zero by x0 nil. Confirm that these both give the correct answer for zero through four.

Induction

Exercise operators

Give another example of a pair of operators that have an identity and are associative, commutative, and distribute over one another.

Give an example of an operator that has an identity and is associative but is not commutative.

Exercise finite-|-assoc (stretch)

Write out what is known about associativity of addition on each of the first four days using a finite story of creation, as [earlier][plfa.Naturals#finite-creation]

Exercise +-swap (recommended)

Show

m + (n + p) ≡ n + (m + p)

for all naturals m, n, and p. No induction is needed, just apply the previous results which show addition is associative and commutative. You may need to use the following function from the standard library:

sym : ∀ {m n : ℕ} → m ≡ n → n ≡ m

Exercise *-distrib-+ (recommended)

Show multiplication distributes over addition, that is,

(m + n) * p ≡ m * p + n * p

for all naturals m, n, and p.

Exercise *-assoc (recommended)

Show multiplication is associative, that is,

(m * n) * p ≡ m * (n * p)

for all naturals m, n, and p.

Exercise *-comm

Show multiplication is commutative, that is,

m * n ≡ n * m

for all naturals m and n. As with commutativity of addition, you will need to formulate and prove suitable lemmas.

Exercise 0∸n≡0

Show

zero ∸ n ≡ zero

for all naturals n. Did your proof require induction?

Exercise ∸-|-assoc

Show that monus associates with addition, that is,

m ∸ n ∸ p ≡ m ∸ (n + p)

for all naturals m, n, and p.

Exercise Bin-laws (stretch)

Recall that Exercise [Bin][plfa.Naturals#Bin] defines a datatype Bin of bitstrings representing natural numbers and asks you to define functions

inc   : Bin → Bin
to    : ℕ → Bin
from  : Bin → ℕ

Consider the following laws, where n ranges over naturals and x over bitstrings.

from (inc x) ≡ suc (from x)
to (from n) ≡ n
from (to x) ≡ x

For each law: if it holds, prove; if not, give a counterexample.

Relations

Exercise orderings

Give an example of a preorder that is not a partial order.

Give an example of a partial order that is not a preorder.

Exercise ≤-antisym-cases

The above proof omits cases where one argument is z≤n and one argument is s≤s. Why is it ok to omit them?

Exercise *-mono-≤ (stretch)

Show that multiplication is monotonic with regard to inequality.

Exercise <-trans (recommended)

Show that strict inequality is transitive.

Exercise trichotomy

Show that strict inequality satisfies a weak version of trichotomy, in the sense that for any m and n that one of the following holds:

  • m < n,
  • m ≡ n, or
  • m > n Define m > n to be the same as n < m. You will need a suitable data declaration, similar to that used for totality. (We will show that the three cases are exclusive after we introduce [negation][plfa.Negation].)

Exercise +-mono-<

Show that addition is monotonic with respect to strict inequality. As with inequality, some additional definitions may be required.

Exercise ≤-iff-< (recommended)

Show that suc m ≤ n implies m < n, and conversely.

Exercise <-trans-revisited

Give an alternative proof that strict inequality is transitive, using the relating between strict inequality and inequality and the fact that inequality is transitive.

Exercise o+o≡e (stretch)

Show that the sum of two odd numbers is even.

Exercise Bin-predicates (stretch)

Recall that Exercise [Bin][plfa.Naturals#Bin] defines a datatype Bin of bitstrings representing natural numbers. Representations are not unique due to leading zeros. Hence, eleven may be represented by both of the following

x1 x1 x0 x1 nil
x1 x1 x0 x1 x0 x0 nil

Define a predicate

Can : Bin → Set

over all bitstrings that holds if the bitstring is canonical, meaning it has no leading zeros; the first representation of eleven above is canonical, and the second is not. To define it, you will need an auxiliary predicate

One : Bin → Set

that holds only if the bistring has a leading one. A bitstring is canonical if it has a leading one (representing a positive number) or if it consists of a single zero (representing zero).

Show that increment preserves canonical bitstrings.

Can x
------------
Can (inc x)

Show that converting a natural to a bitstring always yields a canonical bitstring.

----------
Can (to n)

Show that converting a canonical bitstring to a natural and back is the identity.

Can x
---------------
to (from x) ≡ x

(Hint: For each of these, you may first need to prove related properties of One.)