module Assignment3 where


## 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.

## Good Scholarly Practice.

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).

## Imports

import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; cong; sym)
open Eq.≡-Reasoning using (begin_; _≡⟨⟩_; step-≡; _∎)
open import Data.Bool.Base using (Bool; true; false; T; _∧_; _∨_; not)
open import Data.Nat using (ℕ; zero; suc; _+_; _*_; _∸_; _≤_; s≤s; z≤n)
open import Data.Nat.Properties using
(+-assoc; +-identityˡ; +-identityʳ; *-assoc; *-identityˡ; *-identityʳ)
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Data.Product using (_×_; ∃; ∃-syntax) renaming (_,_ to ⟨_,_⟩)
open import Data.Empty using (⊥; ⊥-elim)
open import Function using (_∘_)
open import Algebra.Structures using (IsMonoid)
open import Level using (Level)
open import Relation.Unary using (Decidable)
open import plfa.part1.Relations using (_<_; z<s; s<s)
open import plfa.part1.Isomorphism using (_≃_; ≃-sym; ≃-trans; _≲_; extensionality)
open plfa.part1.Isomorphism.≃-Reasoning
open import plfa.part1.Lists using (List; []; _∷_; [_]; [_,_]; [_,_,_]; [_,_,_,_];
_++_; reverse; map; foldr; sum; All; Any; here; there; _∈_)
open import plfa.part2.Lambda hiding (ƛ′_⇒_; case′_[zero⇒_|suc_⇒_]; μ′_⇒_; plus′)
open import plfa.part2.Properties hiding (value?; unstuck; preserves; wttdgs)


## Lists

Show that the reverse of one list appended to another is the reverse of the second appended to the reverse of the first:

reverse (xs ++ ys) ≡ reverse ys ++ reverse xs


A function is an involution if when applied twice it acts as the identity function. Show that reverse is an involution:

reverse (reverse xs) ≡ xs


#### Exercise map-compose (practice)

Prove that the map of a composition is equal to the composition of two maps:

map (g ∘ f) ≡ map g ∘ map f


The last step of the proof requires extensionality.

#### Exercise map-++-distribute (practice)

Prove the following relationship between map and append:

map f (xs ++ ys) ≡ map f xs ++ map f ys

#### Exercise map-Tree (practice)

Define a type of trees with leaves of type A and internal nodes of type B:

data Tree (A B : Set) : Set where
leaf : A → Tree A B
node : Tree A B → B → Tree A B → Tree A B


Define a suitable map operator over trees:

postulate
map-Tree : ∀ {A B C D : Set}
→ (A → C) → (B → D) → Tree A B → Tree C D


Use fold to define a function to find the product of a list of numbers. For example:

product [ 1 , 2 , 3 , 4 ] ≡ 24

-- Your code goes here


Show that fold and append are related as follows:

postulate
foldr-++ : ∀ {A B : Set} (_⊗_ : A → B → B) (e : B) (xs ys : List A) →
foldr _⊗_ e (xs ++ ys) ≡ foldr _⊗_ (foldr _⊗_ e ys) xs


#### Exercise map-is-foldr (practice)

Show that map can be defined using fold:

postulate
map-is-foldr : ∀ {A B : Set} {f : A → B} →
map f ≡ foldr (λ x xs → f x ∷ xs) []


This requires extensionality.

#### Exercise fold-Tree (practice)

Define a suitable fold function for the type of trees given earlier:

postulate
fold-Tree : ∀ {A B C : Set}
→ (A → C) → (C → B → C → C) → Tree A B → C

-- Your code goes here


#### Exercise map-is-fold-Tree (practice)

Demonstrate an analogue of map-is-foldr for the type of trees.

-- Your code goes here


#### Exercise sum-downFrom (stretch)

Define a function that counts down as follows:

downFrom : ℕ → List ℕ
downFrom zero     =  []
downFrom (suc n)  =  n ∷ downFrom n


For example:

_ : downFrom 3 ≡ [ 2 , 1 , 0 ]
_ = refl


Prove that the sum of the numbers (n - 1) + ⋯ + 0 is equal to n * (n ∸ 1) / 2:

postulate
sum-downFrom : ∀ (n : ℕ)
→ sum (downFrom n) * 2 ≡ n * (n ∸ 1)


#### Exercise foldl (practice)

Define a function foldl which is analogous to foldr, but where operations associate to the left rather than the right. For example:

foldr _⊗_ e [ x , y , z ]  =  x ⊗ (y ⊗ (z ⊗ e))
foldl _⊗_ e [ x , y , z ]  =  ((e ⊗ x) ⊗ y) ⊗ z

-- Your code goes here


#### Exercise foldr-monoid-foldl (practice)

Show that if _⊗_ and e form a monoid, then foldr _⊗_ e and foldl _⊗_ e always compute the same result.

-- Your code goes here


Prove a result similar to All-++-⇔, but with Any in place of All, and a suitable replacement for _×_. As a consequence, demonstrate an equivalence relating _∈_ and _++_.

-- Your code goes here


#### Exercise All-++-≃ (stretch)

Show that the equivalence All-++-⇔ can be extended to an isomorphism.

-- Your code goes here


Show that Any and All satisfy a version of De Morgan’s Law:

(¬_ ∘ Any P) xs ⇔ All (¬_ ∘ P) xs


(Can you see why it is important that here _∘_ is generalised to arbitrary levels, as described in the section on universe polymorphism?)

Do we also have the following?

(¬_ ∘ All P) xs ⇔ Any (¬_ ∘ P) xs


If so, prove; if not, explain why.

-- Your code goes here


#### Exercise ¬Any≃All¬ (stretch)

Show that the equivalence ¬Any⇔All¬ can be extended to an isomorphism. You will need to use extensionality.

-- Your code goes here


#### Exercise All-∀ (practice)

Show that All P xs is isomorphic to ∀ {x} → x ∈ xs → P x.

-- You code goes here


#### Exercise Any-∃ (practice)

Show that Any P xs is isomorphic to ∃[ x ] (x ∈ xs × P x).

-- You code goes here


#### Exercise any? (stretch)

Just as All has analogues all and All? which determine whether a predicate holds for every element of a list, so does Any have analogues any and Any? which determine whether a predicate holds for some element of a list. Give their definitions.

-- Your code goes here


#### Exercise filter? (stretch)

Define the following variant of the traditional filter function on lists, which given a decidable predicate and a list returns all elements of the list satisfying the predicate:

postulate
filter? : ∀ {A : Set} {P : A → Set}
→ (P? : Decidable P) → List A → ∃[ ys ]( All P ys )


## Lambda

Write out the definition of a lambda term that multiplies two natural numbers. Your definition may use plus as defined earlier.

-- Your code goes here


#### Exercise mulᶜ (practice)

Write out the definition of a lambda term that multiplies two natural numbers represented as Church numerals. Your definition may use plusᶜ as defined earlier (or may not — there are nice definitions both ways).

-- Your code goes here


#### Exercise primed (stretch)

Some people find it annoying to write  "x" instead of x. We can make examples with lambda terms slightly easier to write by adding the following definitions:

ƛ′_⇒_ : Term → Term → Term
ƛ′ ( x) ⇒ N  =  ƛ x ⇒ N
ƛ′ _ ⇒ _      =  ⊥-elim impossible
where postulate impossible : ⊥

case′_[zero⇒_|suc_⇒_] : Term → Term → Term → Term → Term
case′ L [zero⇒ M |suc ( x) ⇒ N ]  =  case L [zero⇒ M |suc x ⇒ N ]
case′ _ [zero⇒ _ |suc _ ⇒ _ ]      =  ⊥-elim impossible
where postulate impossible : ⊥

μ′_⇒_ : Term → Term → Term
μ′ ( x) ⇒ N  =  μ x ⇒ N
μ′ _ ⇒ _      =  ⊥-elim impossible
where postulate impossible : ⊥


We intend to apply the function only when the first term is a variable, which we indicate by postulating a term impossible of the empty type ⊥. If we use C-c C-n to normalise the term

ƛ′ two ⇒ two

Agda will return an answer warning us that the impossible has occurred:

⊥-elim (plfa.part2.Lambda.impossible (suc (suc zero)) (suc (suc zero)))

While postulating the impossible is a useful technique, it must be used with care, since such postulation could allow us to provide evidence of any proposition whatsoever, regardless of its truth.

The definition of plus can now be written as follows:

plus′ : Term
plus′ = μ′ + ⇒ ƛ′ m ⇒ ƛ′ n ⇒
case′ m
[zero⇒ n
|suc m ⇒ suc (+ · m · n) ]
where
+  =   "+"
m  =   "m"
n  =   "n"


Write out the definition of multiplication in the same style.

#### Exercise _[_:=_]′ (stretch)

The definition of substitution above has three clauses (ƛ, case, and μ) that invoke a with clause to deal with bound variables. Rewrite the definition to factor the common part of these three clauses into a single function, defined by mutual recursion with substitution.

-- Your code goes here


#### Exercise —↠≲—↠′ (practice)

Show that the first notion of reflexive and transitive closure above embeds into the second. Why are they not isomorphic?

-- Your code goes here


#### Exercise plus-example (practice)

Write out the reduction sequence demonstrating that one plus one is two.

-- Your code goes here


#### Exercise Context-≃ (practice)

Show that Context is isomorphic to List (Id × Type). For instance, the isomorphism relates the context

∅ , "s" ⦂ ℕ ⇒ ℕ , "z" ⦂ ℕ


to the list

[ ⟨ "z" , ℕ ⟩ , ⟨ "s" , ℕ ⇒ ℕ ⟩ ]

-- Your code goes here


Using the term mul you defined earlier, write out the derivation showing that it is well typed.

-- Your code goes here


#### Exercise mulᶜ-type (practice)

Using the term mulᶜ you defined earlier, write out the derivation showing that it is well typed.

-- Your code goes here


## Properties

#### Exercise Progress-≃ (practice)

Show that Progress M is isomorphic to Value M ⊎ ∃[ N ](M —→ N).

-- Your code goes here


#### Exercise progress′ (practice)

Write out the proof of progress′ in full, and compare it to the proof of progress above.

-- Your code goes here


#### Exercise value? (practice)

Combine progress and —→¬V to write a program that decides whether a well-typed term is a value:

postulate
value? : ∀ {A M} → ∅ ⊢ M ⦂ A → Dec (Value M)


#### Exercise subst′ (stretch)

Rewrite subst to work with the modified definition _[_:=_]′ from the exercise in the previous chapter. As before, this should factor dealing with bound variables into a single function, defined by mutual recursion with the proof that substitution preserves types.

-- Your code goes here


Using the evaluator, confirm that two times two is four.

-- Your code goes here


#### Exercise: progress-preservation (practice)

Without peeking at their statements above, write down the progress and preservation theorems for the simply typed lambda-calculus.

-- Your code goes here


#### Exercise subject_expansion (practice)

We say that M reduces to N if M —→ N, but we can also describe the same situation by saying that N expands to M. The preservation property is sometimes called subject reduction. Its opposite is subject expansion, which holds if M —→ N and ∅ ⊢ N ⦂ A imply ∅ ⊢ M ⦂ A. Find two counter-examples to subject expansion, one with case expressions and one not involving case expressions.

-- Your code goes here


#### Exercise stuck (practice)

Give an example of an ill-typed term that does get stuck.

-- Your code goes here


Provide proofs of the three postulates, unstuck, preserves, and wttdgs above.

-- Your code goes here