```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.

Submit your homework using Gradescope. Go to the course page under Learn. Select “Assessment”, then select “Assignment Submission”. Please ensure your files execute correctly under Agda!

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).

Lists

```module Lists where
```

Imports

```  import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; sym; trans; cong)
open Eq.≡-Reasoning
open import Data.Bool 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; *-distribʳ-+)
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Data.Product using (_×_; ∃; ∃-syntax) renaming (_,_ to ⟨_,_⟩)
open import Function using (_∘_)
open import Level using (Level)
open import plfa.part1.Isomorphism using (_≃_; _⇔_)
```
```  open import plfa.part1.Lists
hiding (downFrom; Tree; leaf; node; merge)
```

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.

```  -- Your code goes here
```

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

Prove the following relationship between map and append:

``map f (xs ++ ys) ≡ map f xs ++ map f ys``
```  -- Your code goes here
```

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:

``map-Tree : ∀ {A B C D : Set} → (A → C) → (B → D) → Tree A B → Tree C D``
```  -- Your code goes here
```

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:

``foldr _⊗_ e (xs ++ ys) ≡ foldr _⊗_ (foldr _⊗_ e ys) xs``
```  -- Your code goes here
```

Exercise `foldr-∷` (practice)

Show

``foldr _∷_ [] xs ≡ xs``

Show as a consequence of `foldr-++` above that

``xs ++ ys ≡ foldr _∷_ ys xs``
```  -- Your code goes here
```

Exercise `map-is-foldr` (practice)

Show that map can be defined using fold:

``map f ≡ foldr (λ x xs → f x ∷ xs) []``

The proof requires extensionality.

```  -- Your code goes here
```

Exercise `fold-Tree` (practice)

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

``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` (practice) (was 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`:

``sum (downFrom n) * 2 ≡ n * (n ∸ 1)``
```  -- Your code goes here
```

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-++-≃` (practice) (was 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¬` (practice) (was 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?` (practice) (was 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 `split` (practice) (was stretch)

The relation `merge` holds when two lists merge to give a third list.
```  data merge {A : Set} : (xs ys zs : List A) → Set where

[] :
--------------
merge [] [] []

left-∷ : ∀ {x xs ys zs}
→ merge xs ys zs
--------------------------
→ merge (x ∷ xs) ys (x ∷ zs)

right-∷ : ∀ {y xs ys zs}
→ merge xs ys zs
--------------------------
→ merge xs (y ∷ ys) (y ∷ zs)
```
For example,
```  _ : merge [ 1 , 4 ] [ 2 , 3 ] [ 1 , 2 , 3 , 4 ]
_ = left-∷ (right-∷ (right-∷ (left-∷ [])))

```

Given a decidable predicate and a list, we can split the list into two lists that merge to give the original list, where all elements of one list satisfy the predicate, and all elements of the other do not satisfy the predicate.

Define the following variant of the traditional `filter` function on lists, which given a decidable predicate and a list returns a list of elements that satisfy the predicate and a list of elements that don’t, with their corresponding proofs.

``````split : ∀ {A : Set} {P : A → Set} (P? : Decidable P) (zs : List A)
→ ∃[ xs ] ∃[ ys ] ( merge xs ys zs × All P xs × All (¬_ ∘ P) ys )``````
```  -- Your code goes here
```

Lambda

```module Lambda where
```

Imports

```  open import Data.Bool using (Bool; true; false; T; not)
open import Data.Empty using (⊥; ⊥-elim)
open import Data.List using (List; _∷_; [])
open import Data.Nat using (ℕ; zero; suc)
open import Data.Product using (∃-syntax; _×_)
open import Data.String using (String; _≟_)
open import Data.Unit using (tt)
open import Relation.Nullary using (Dec; yes; no; ¬_)
open import Relation.Nullary.Decidable using (False; toWitnessFalse)
open import Relation.Nullary.Negation using (¬?)
open import Relation.Binary.PropositionalEquality using (_≡_; _≢_; refl)
```
```  open import plfa.part2.Lambda
hiding (var?; ƛ′_⇒_; case′_[zero⇒_|suc_⇒_]; μ′_⇒_; plus′)
```

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:
```  var? : (t : Term) → Bool
var? (` _)  =  true
var? _      =  false

ƛ′_⇒_ : (t : Term) → {_ : T (var? t)} → Term → Term
ƛ′_⇒_ (` x) N = ƛ x ⇒ N

case′_[zero⇒_|suc_⇒_] : Term → Term → (t : Term) → {_ : T (var? t)} → Term → Term
case′ L [zero⇒ M |suc (` x) ⇒ N ]  =  case L [zero⇒ M |suc x ⇒ N ]

μ′_⇒_ : (t : Term) → {_ : T (var? t)} → Term → Term
μ′ (` x) ⇒ N  =  μ x ⇒ N
```

Recall that `T` is a function that maps from the computation world to the evidence world, as defined in Chapter Decidable. We ensure to use the primed functions only when the respective term argument is a variable, which we do by providing implicit evidence. For example, if we tried to define an abstraction term that binds anything but a variable:

``````_ : Term
_ = ƛ′ two ⇒ two``````

Agda would complain it cannot find a value of the bottom type for the implicit argument. Note the implicit argument’s type reduces to `⊥` when term `t` is anything but a variable.

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` (practice)

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

```  -- Your code goes here
```

Properties

```module Properties where
```

Imports

```  open import Relation.Binary.PropositionalEquality
using (_≡_; _≢_; refl; sym; cong; cong₂)
open import Data.String using (String; _≟_)
open import Data.Nat using (ℕ; zero; suc)
open import Data.Empty using (⊥; ⊥-elim)
open import Data.Product
using (_×_; proj₁; proj₂; ∃; ∃-syntax)
renaming (_,_ to ⟨_,_⟩)
open import Data.Sum using (_⊎_; inj₁; inj₂)
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Function using (_∘_)
open import plfa.part1.Isomorphism
open import plfa.part2.Lambda
```
```  open import plfa.part2.Properties
hiding (value?; Canonical_⦂_; unstuck; preserves; wttdgs)
-- open Lambda using (mul; ⊢mul)
```

Exercise `Canonical-≃` (practice)

Well-typed values must take one of a small number of canonical forms, which provide an analogue of the `Value` relation that relates values to their types. A lambda expression must have a function type, and a zero or successor expression must be a natural. Further, the body of a function must be well typed in a context containing only its bound variable, and the argument of successor must itself be canonical:
```  infix  4 Canonical_⦂_

data Canonical_⦂_ : Term → Type → Set where

C-ƛ : ∀ {x A N B}
→ ∅ , x ⦂ A ⊢ N ⦂ B
-----------------------------
→ Canonical (ƛ x ⇒ N) ⦂ (A ⇒ B)

C-zero :
--------------------
Canonical `zero ⦂ `ℕ

C-suc : ∀ {V}
→ Canonical V ⦂ `ℕ
---------------------
→ Canonical `suc V ⦂ `ℕ
```

Show that `Canonical V ⦂ A` is isomorphic to `(∅ ⊢ V ⦂ A) × (Value V)`, that is, the canonical forms are exactly the well-typed values.

```  -- Your code goes here
```

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
```

DeBruijn

```module DeBruijn where
```

Imports

```  import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl)
open import Data.Empty using (⊥; ⊥-elim)
open import Data.Nat using (ℕ; zero; suc; _<_; _≤?_; z≤n; s≤s)
open import Relation.Nullary using (¬_)
open import Relation.Nullary.Decidable using (True; toWitness)
```
```  open import plfa.part2.DeBruijn
hiding ()
```

Write out the definition of a lambda term that multiplies two natural numbers, now adapted to the intrinsically-typed de Bruijn representation.

```  -- Your code goes here
```

Exercise `V¬—→` (practice)

Following the previous development, show values do not reduce, and its corollary, terms that reduce are not values.

```  -- Your code goes here
```

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

```  -- Your code goes here
```