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 labelled “(practice)” are included for those who want extra practice.

Submit your homework using the “submit” command. Please ensure your files execute correctly under Agda!

Good Scholarly Practice.

Please remember the University requirement as regards all assessed work. Details about this can be found at:

https://web.inf.ed.ac.uk/infweb/admin/policies/academic-misconduct

Furthermore, you are required to take reasonable measures to protect your assessed work from unauthorised access. For example, if you put any such work on a public repository then you must set access permissions appropriately (generally permitting access only to yourself, or your group in the case of group practicals).

Naturals

module Naturals where
  import Relation.Binary.PropositionalEquality as Eq
  open Eq using (_≡_; refl)
  open Eq.≡-Reasoning using (begin_; _≡⟨⟩_; _∎)
  import Data.Nat using (ℕ; zero; suc; _+_; _*_; _^_; _∸_)

Exercise seven (practice)

Write out 7 in longhand.

  -- Your code goes here

You will need to give both a type signature and definition for the variable seven. Type C-c C-l in Emacs to instruct Agda to re-load.

Exercise +-example (practice)

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

  -- Your code goes here

Exercise *-example (practice)

Compute 3 * 4, writing out your reasoning as a chain of equations, using the equations for *. (You do not need to step through the evaluation of +.)

  -- Your code goes here

Exercise _^_ (recommended)

Define exponentiation, which is given by the following equations:

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

Check that 3 ^ 4 is 81.

  -- Your code goes here

Exercise ∸-example₁ and ∸-example₂ (recommended)

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

  -- Your code goes here

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
    ⟨⟩ : Bin
    _O : Bin → Bin
    _I : Bin → Bin

For instance, the bitstring

1011

standing for the number eleven is encoded as

⟨⟩ I O I I

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

⟨⟩ O O I O I I

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 (⟨⟩ I O I I) ≡ ⟨⟩ I I O O

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 ⟨⟩ O. Confirm that these both give the correct answer for zero through four.

  -- Your code goes here

Induction

module Induction where
  import Data.Nat.Properties using (+-assoc; +-identityʳ; +-suc; +-comm)

Imports

We require equality as in the previous chapter, plus the naturals and some operations upon them. We also require a couple of new operations, cong, sym, and _≡⟨_⟩_, which are explained below:
  import Relation.Binary.PropositionalEquality as Eq
  open Eq using (_≡_; refl; cong; sym)
  open Eq.≡-Reasoning using (begin_; _≡⟨⟩_; step-≡; _∎)
  open import Data.Nat using (ℕ; zero; suc; _+_; _*_; _∸_)

(Importing step-≡ defines _≡⟨_⟩_.)

  open import plfa.part1.Induction
    hiding ()

Exercise operators (practice)

Give another example of a pair of operators that have an identity and are associative, commutative, and distribute over one another. (You do not have to prove these properties.)

Give an example of an operator that has an identity and is associative but is not commutative. (You do not have to prove these properties.)

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.

  -- Your code goes here

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.

  -- Your code goes here

Exercise *-distrib-+ (recommended)

Show multiplication distributes over addition, that is,

(m + n) * p ≡ m * p + n * p

for all naturals m, n, and p.

  -- Your code goes here

Exercise *-assoc (recommended)

Show multiplication is associative, that is,

(m * n) * p ≡ m * (n * p)

for all naturals m, n, and p.

  -- Your code goes here

Exercise *-comm (practice)

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.

  -- Your code goes here

Exercise 0∸n≡0 (practice)

Show

zero ∸ n ≡ zero

for all naturals n. Did your proof require induction?

  -- Your code goes here

Exercise ∸-+-assoc (practice)

Show that monus associates with addition, that is,

m ∸ n ∸ p ≡ m ∸ (n + p)

for all naturals m, n, and p.

  -- Your code goes here

Exercise +*^ (stretch)

Show the following three laws

 m ^ (n + p) ≡ (m ^ n) * (m ^ p)  (^-distribˡ-+-*)
 (m * n) ^ p ≡ (m ^ p) * (n ^ p)  (^-distribʳ-*)
 (m ^ n) ^ p ≡ m ^ (n * p)        (^-*-assoc)

for all m, n, and p.

  -- Your code goes here

Exercise Bin-laws (stretch)

Recall that Exercise 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 b over bitstrings:

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

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

  -- Your code goes here

Relations

module Relations where
  import Data.Nat using (_≤_; z≤n; s≤s)
  import Data.Nat.Properties using (≤-refl; ≤-trans; ≤-antisym; ≤-total;
                                    +-monoʳ-≤; +-monoˡ-≤; +-mono-≤)

Imports

  import Relation.Binary.PropositionalEquality as Eq
  open Eq using (_≡_; refl; cong)
  open import Data.Nat using (ℕ; zero; suc; _+_)
  open import Data.Nat.Properties using (+-comm; +-identityʳ)

Exercise orderings (practice)

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

  -- Your code goes here

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

  -- Your code goes here

Exercise ≤-antisym-cases (practice)

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

  -- Your code goes here

Exercise *-mono-≤ (stretch)

Show that multiplication is monotonic with regard to inequality.

  -- Your code goes here

Exercise <-trans (recommended)

Show that strict inequality is transitive. Use a direct proof. (A later exercise exploits the relation between < and ≤.)

  -- Your code goes here

Exercise trichotomy (practice)

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.)

  -- Your code goes here

Exercise +-mono-< (practice)

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

  -- Your code goes here

Exercise ≤-iff-< (recommended)

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

  -- Your code goes here

Exercise <-trans-revisited (practice)

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

  -- Your code goes here

Exercise o+o≡e (stretch)

Show that the sum of two odd numbers is even.

  -- Your code goes here

Exercise Bin-predicates (stretch)

Recall that Exercise 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:

⟨⟩ I O I I
⟨⟩ O O I O I I

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 bitstring 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 b
------------
Can (inc b)

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 b
---------------
to (from b) ≡ b

(Hint: For each of these, you may first need to prove related properties of One. Also, you may need to prove that if One b then 1 is less or equal to the result of from b.)

  -- Your code goes here