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

## Imports

import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; cong; sym)
open Eq.≡-Reasoning using (begin_; _≡⟨⟩_; _≡⟨_⟩_; _∎)
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)


## Lambda

Write out the definition of a lambda term that multiplies two natural numbers.

#### Exercise primed (stretch)

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 : ⊥


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.

#### Exercise —↠≲—↠′

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

#### Exercise plus-example

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

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

## Properties

#### Exercise Progress-≃

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

#### Exercise progress′

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

#### Exercise value?

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.

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

#### Exercise: progress-preservation

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

#### Exercise subject-expansion

We say that M reduces to N if M —→ N, and conversely that M expands to N if N —→ 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.

#### Exercise stuck

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

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