module Assignment4 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.

IMPORTANT For ease of marking, when modifying the given code please write

-- begin
-- end


## Imports

import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; sym; trans; cong; cong₂; _≢_)
open import Data.Empty using (⊥; ⊥-elim)
open import Data.Nat using (ℕ; zero; suc; _+_; _*_)
open import Data.Product using (_×_; ∃; ∃-syntax) renaming (_,_ to ⟨_,_⟩)
open import Data.String using (String; _≟_)
open import Relation.Nullary using (¬_; Dec; yes; no)


## DeBruijn

module DeBruijn where


Remember to indent all code by two spaces.

  open import plfa.part2.DeBruijn


Write out the definition of a lambda term that multiplies two natural numbers, now adapted to the inherently typed DeBruijn representation.

#### Exercise V¬—→

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

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

## More

module More where


Remember to indent all code by two spaces.

### Syntax

  infix  4 _⊢_
infix  4 _∋_
infixl 5 _,_

infixr 7 _⇒_
infixr 8 _⊎_
infixr 9 _×_

infix  5 ƛ_
infix  5 μ_
infixl 7 _·_
infixl 8 _*_
infix  9 _
infix  9 S_
infix  9 #_


### Types

  data Type : Set where
ℕ    : Type
_⇒_   : Type → Type → Type
Nat   : Type
_×_  : Type → Type → Type
_⊎_  : Type → Type → Type
⊤    : Type
⊥    : Type
List : Type → Type


### Contexts

  data Context : Set where
∅   : Context
_,_ : Context → Type → Context


### Variables and the lookup judgment

  data _∋_ : Context → Type → Set where

Z : ∀ {Γ A}
---------
→ Γ , A ∋ A

S_ : ∀ {Γ A B}
→ Γ ∋ B
---------
→ Γ , A ∋ B


### Terms and the typing judgment

  data _⊢_ : Context → Type → Set where

-- variables

_ : ∀ {Γ A}
→ Γ ∋ A
-----
→ Γ ⊢ A

-- functions

ƛ_  :  ∀ {Γ A B}
→ Γ , A ⊢ B
---------
→ Γ ⊢ A ⇒ B

_·_ : ∀ {Γ A B}
→ Γ ⊢ A ⇒ B
→ Γ ⊢ A
---------
→ Γ ⊢ B

-- naturals

zero : ∀ {Γ}
------
→ Γ ⊢ ℕ

suc : ∀ {Γ}
→ Γ ⊢ ℕ
------
→ Γ ⊢ ℕ

case : ∀ {Γ A}
→ Γ ⊢ ℕ
→ Γ ⊢ A
→ Γ , ℕ ⊢ A
-----
→ Γ ⊢ A

-- fixpoint

μ_ : ∀ {Γ A}
→ Γ , A ⊢ A
----------
→ Γ ⊢ A

-- primitive numbers

con : ∀ {Γ}
→ ℕ
-------
→ Γ ⊢ Nat

_*_ : ∀ {Γ}
→ Γ ⊢ Nat
→ Γ ⊢ Nat
-------
→ Γ ⊢ Nat

-- let

let : ∀ {Γ A B}
→ Γ ⊢ A
→ Γ , A ⊢ B
----------
→ Γ ⊢ B

-- products

⟨_,_⟩ : ∀ {Γ A B}
→ Γ ⊢ A
→ Γ ⊢ B
-----------
→ Γ ⊢ A × B

proj₁ : ∀ {Γ A B}
→ Γ ⊢ A × B
-----------
→ Γ ⊢ A

proj₂ : ∀ {Γ A B}
→ Γ ⊢ A × B
-----------
→ Γ ⊢ B

-- alternative formulation of products

case× : ∀ {Γ A B C}
→ Γ ⊢ A × B
→ Γ , A , B ⊢ C
--------------
→ Γ ⊢ C



### Abbreviating de Bruijn indices

  lookup : Context → ℕ → Type
lookup (Γ , A) zero     =  A
lookup (Γ , _) (suc n)  =  lookup Γ n
lookup ∅       _        =  ⊥-elim impossible
where postulate impossible : ⊥

count : ∀ {Γ} → (n : ℕ) → Γ ∋ lookup Γ n
count {Γ , _} zero     =  Z
count {Γ , _} (suc n)  =  S (count n)
count {∅}     _        =  ⊥-elim impossible
where postulate impossible : ⊥

#_ : ∀ {Γ} → (n : ℕ) → Γ ⊢ lookup Γ n
# n  =   count n


## Renaming

  ext : ∀ {Γ Δ} → (∀ {A} → Γ ∋ A → Δ ∋ A) → (∀ {A B} → Γ , A ∋ B → Δ , A ∋ B)
ext ρ Z      =  Z
ext ρ (S x)  =  S (ρ x)

rename : ∀ {Γ Δ} → (∀ {A} → Γ ∋ A → Δ ∋ A) → (∀ {A} → Γ ⊢ A → Δ ⊢ A)
rename ρ ( x)          =   (ρ x)
rename ρ (ƛ N)          =  ƛ (rename (ext ρ) N)
rename ρ (L · M)        =  (rename ρ L) · (rename ρ M)
rename ρ (zero)         =  zero
rename ρ (suc M)        =  suc (rename ρ M)
rename ρ (case L M N)   =  case (rename ρ L) (rename ρ M) (rename (ext ρ) N)
rename ρ (μ N)          =  μ (rename (ext ρ) N)
rename ρ (con n)        =  con n
rename ρ (M * N)       =  rename ρ M * rename ρ N
rename ρ (let M N)     =  let (rename ρ M) (rename (ext ρ) N)
rename ρ ⟨ M , N ⟩      =  ⟨ rename ρ M , rename ρ N ⟩
rename ρ (proj₁ L)     =  proj₁ (rename ρ L)
rename ρ (proj₂ L)     =  proj₂ (rename ρ L)
rename ρ (case× L M)    =  case× (rename ρ L) (rename (ext (ext ρ)) M)


## Simultaneous Substitution

  exts : ∀ {Γ Δ} → (∀ {A} → Γ ∋ A → Δ ⊢ A) → (∀ {A B} → Γ , A ∋ B → Δ , A ⊢ B)
exts σ Z      =   Z
exts σ (S x)  =  rename S_ (σ x)

subst : ∀ {Γ Δ} → (∀ {C} → Γ ∋ C → Δ ⊢ C) → (∀ {C} → Γ ⊢ C → Δ ⊢ C)
subst σ ( k)          =  σ k
subst σ (ƛ N)          =  ƛ (subst (exts σ) N)
subst σ (L · M)        =  (subst σ L) · (subst σ M)
subst σ (zero)         =  zero
subst σ (suc M)        =  suc (subst σ M)
subst σ (case L M N)   =  case (subst σ L) (subst σ M) (subst (exts σ) N)
subst σ (μ N)          =  μ (subst (exts σ) N)
subst σ (con n)        =  con n
subst σ (M * N)       =  subst σ M * subst σ N
subst σ (let M N)     =  let (subst σ M) (subst (exts σ) N)
subst σ ⟨ M , N ⟩      =  ⟨ subst σ M , subst σ N ⟩
subst σ (proj₁ L)     =  proj₁ (subst σ L)
subst σ (proj₂ L)     =  proj₂ (subst σ L)
subst σ (case× L M)    =  case× (subst σ L) (subst (exts (exts σ)) M)


## Single and double substitution

  _[_] : ∀ {Γ A B}
→ Γ , A ⊢ B
→ Γ ⊢ A
------------
→ Γ ⊢ B
_[_] {Γ} {A} N V =  subst {Γ , A} {Γ} σ N
where
σ : ∀ {B} → Γ , A ∋ B → Γ ⊢ B
σ Z      =  V
σ (S x)  =   x

_[_][_] : ∀ {Γ A B C}
→ Γ , A , B ⊢ C
→ Γ ⊢ A
→ Γ ⊢ B
---------------
→ Γ ⊢ C
_[_][_] {Γ} {A} {B} N V W =  subst {Γ , A , B} {Γ} σ N
where
σ : ∀ {C} → Γ , A , B ∋ C → Γ ⊢ C
σ Z          =  W
σ (S Z)      =  V
σ (S (S x))  =   x


## Values

  data Value : ∀ {Γ A} → Γ ⊢ A → Set where

-- functions

V-ƛ : ∀ {Γ A B} {N : Γ , A ⊢ B}
---------------------------
→ Value (ƛ N)

-- naturals

V-zero : ∀ {Γ} →
-----------------
Value (zero {Γ})

V-suc : ∀ {Γ} {V : Γ ⊢ ℕ}
→ Value V
--------------
→ Value (suc V)

-- primitives

V-con : ∀ {Γ n}
---------------------
→ Value {Γ = Γ} (con n)

-- products

V-⟨_,_⟩ : ∀ {Γ A B} {V : Γ ⊢ A} {W : Γ ⊢ B}
→ Value V
→ Value W
----------------
→ Value ⟨ V , W ⟩


Implicit arguments need to be supplied when they are not fixed by the given arguments.

## Reduction

  infix 2 _—→_

data _—→_ : ∀ {Γ A} → (Γ ⊢ A) → (Γ ⊢ A) → Set where

-- functions

ξ-·₁ : ∀ {Γ A B} {L L′ : Γ ⊢ A ⇒ B} {M : Γ ⊢ A}
→ L —→ L′
---------------
→ L · M —→ L′ · M

ξ-·₂ : ∀ {Γ A B} {V : Γ ⊢ A ⇒ B} {M M′ : Γ ⊢ A}
→ Value V
→ M —→ M′
---------------
→ V · M —→ V · M′

β-ƛ : ∀ {Γ A B} {N : Γ , A ⊢ B} {V : Γ ⊢ A}
→ Value V
--------------------
→ (ƛ N) · V —→ N [ V ]

-- naturals

ξ-suc : ∀ {Γ} {M M′ : Γ ⊢ ℕ}
→ M —→ M′
-----------------
→ suc M —→ suc M′

ξ-case : ∀ {Γ A} {L L′ : Γ ⊢ ℕ} {M : Γ ⊢ A} {N : Γ , ℕ ⊢ A}
→ L —→ L′
-------------------------
→ case L M N —→ case L′ M N

β-zero :  ∀ {Γ A} {M : Γ ⊢ A} {N : Γ , ℕ ⊢ A}
-------------------
→ case zero M N —→ M

β-suc : ∀ {Γ A} {V : Γ ⊢ ℕ} {M : Γ ⊢ A} {N : Γ , ℕ ⊢ A}
→ Value V
----------------------------
→ case (suc V) M N —→ N [ V ]

-- fixpoint

β-μ : ∀ {Γ A} {N : Γ , A ⊢ A}
----------------
→ μ N —→ N [ μ N ]

-- primitive numbers

ξ-*₁ : ∀ {Γ} {L L′ M : Γ ⊢ Nat}
→ L —→ L′
-----------------
→ L * M —→ L′ * M

ξ-*₂ : ∀ {Γ} {V M M′ : Γ ⊢ Nat}
→ Value V
→ M —→ M′
-----------------
→ V * M —→ V * M′

δ-* : ∀ {Γ c d}
-------------------------------------
→ con {Γ = Γ} c * con d —→ con (c * d)

-- let

ξ-let : ∀ {Γ A B} {M M′ : Γ ⊢ A} {N : Γ , A ⊢ B}
→ M —→ M′
---------------------
→ let M N —→ let M′ N

β-let : ∀ {Γ A B} {V : Γ ⊢ A} {N : Γ , A ⊢ B}
→ Value V
-------------------
→ let V N —→ N [ V ]

-- products

ξ-⟨,⟩₁ : ∀ {Γ A B} {M M′ : Γ ⊢ A} {N : Γ ⊢ B}
→ M —→ M′
-------------------------
→ ⟨ M , N ⟩ —→ ⟨ M′ , N ⟩

ξ-⟨,⟩₂ : ∀ {Γ A B} {V : Γ ⊢ A} {N N′ : Γ ⊢ B}
→ Value V
→ N —→ N′
-------------------------
→ ⟨ V , N ⟩ —→ ⟨ V , N′ ⟩

ξ-proj₁ : ∀ {Γ A B} {L L′ : Γ ⊢ A × B}
→ L —→ L′
---------------------
→ proj₁ L —→ proj₁ L′

ξ-proj₂ : ∀ {Γ A B} {L L′ : Γ ⊢ A × B}
→ L —→ L′
---------------------
→ proj₂ L —→ proj₂ L′

β-proj₁ : ∀ {Γ A B} {V : Γ ⊢ A} {W : Γ ⊢ B}
→ Value V
→ Value W
----------------------
→ proj₁ ⟨ V , W ⟩ —→ V

β-proj₂ : ∀ {Γ A B} {V : Γ ⊢ A} {W : Γ ⊢ B}
→ Value V
→ Value W
----------------------
→ proj₂ ⟨ V , W ⟩ —→ W

-- alternative formulation of products

ξ-case× : ∀ {Γ A B C} {L L′ : Γ ⊢ A × B} {M : Γ , A , B ⊢ C}
→ L —→ L′
-----------------------
→ case× L M —→ case× L′ M

β-case× : ∀ {Γ A B C} {V : Γ ⊢ A} {W : Γ ⊢ B} {M : Γ , A , B ⊢ C}
→ Value V
→ Value W
----------------------------------
→ case× ⟨ V , W ⟩ M —→ M [ V ][ W ]


## Reflexive and transitive closure

  infix  2 _—↠_
infix  1 begin_
infixr 2 _—→⟨_⟩_
infix  3 _∎

data _—↠_ : ∀ {Γ A} → (Γ ⊢ A) → (Γ ⊢ A) → Set where

_∎ : ∀ {Γ A} (M : Γ ⊢ A)
--------
→ M —↠ M

_—→⟨_⟩_ : ∀ {Γ A} (L : Γ ⊢ A) {M N : Γ ⊢ A}
→ L —→ M
→ M —↠ N
------
→ L —↠ N

begin_ : ∀ {Γ} {A} {M N : Γ ⊢ A}
→ M —↠ N
------
→ M —↠ N
begin M—↠N = M—↠N


## Values do not reduce

  V¬—→ : ∀ {Γ A} {M N : Γ ⊢ A}
→ Value M
----------
→ ¬ (M —→ N)
V¬—→ V-ƛ          ()
V¬—→ V-zero       ()
V¬—→ (V-suc VM)   (ξ-suc M—→M′)     =  V¬—→ VM M—→M′
V¬—→ V-con        ()
V¬—→ V-⟨ VM , _ ⟩ (ξ-⟨,⟩₁ M—→M′)    =  V¬—→ VM M—→M′
V¬—→ V-⟨ _ , VN ⟩ (ξ-⟨,⟩₂ _ N—→N′)  =  V¬—→ VN N—→N′


## Progress

  data Progress {A} (M : ∅ ⊢ A) : Set where

step : ∀ {N : ∅ ⊢ A}
→ M —→ N
----------
→ Progress M

done :
Value M
----------
→ Progress M

progress : ∀ {A}
→ (M : ∅ ⊢ A)
-----------
→ Progress M
progress ( ())
progress (ƛ N)                              =  done V-ƛ
progress (L · M) with progress L
...    | step L—→L′                         =  step (ξ-·₁ L—→L′)
...    | done V-ƛ with progress M
...        | step M—→M′                     =  step (ξ-·₂ V-ƛ M—→M′)
...        | done VM                        =  step (β-ƛ VM)
progress (zero)                             =  done V-zero
progress (suc M) with progress M
...    | step M—→M′                         =  step (ξ-suc M—→M′)
...    | done VM                            =  done (V-suc VM)
progress (case L M N) with progress L
...    | step L—→L′                         =  step (ξ-case L—→L′)
...    | done V-zero                        =  step β-zero
...    | done (V-suc VL)                    =  step (β-suc VL)
progress (μ N)                              =  step β-μ
progress (con n)                            =  done V-con
progress (L * M) with progress L
...    | step L—→L′                         =  step (ξ-*₁ L—→L′)
...    | done V-con with progress M
...        | step M—→M′                     =  step (ξ-*₂ V-con M—→M′)
...        | done V-con                     =  step δ-*
progress (let M N) with progress M
...    | step M—→M′                         =  step (ξ-let M—→M′)
...    | done VM                            =  step (β-let VM)
progress ⟨ M , N ⟩ with progress M
...    | step M—→M′                         =  step (ξ-⟨,⟩₁ M—→M′)
...    | done VM with progress N
...        | step N—→N′                     =  step (ξ-⟨,⟩₂ VM N—→N′)
...        | done VN                        =  done (V-⟨ VM , VN ⟩)
progress (proj₁ L) with progress L
...    | step L—→L′                         =  step (ξ-proj₁ L—→L′)
...    | done (V-⟨ VM , VN ⟩)               =  step (β-proj₁ VM VN)
progress (proj₂ L) with progress L
...    | step L—→L′                         =  step (ξ-proj₂ L—→L′)
...    | done (V-⟨ VM , VN ⟩)               =  step (β-proj₂ VM VN)
progress (case× L M) with progress L
...    | step L—→L′                         =  step (ξ-case× L—→L′)
...    | done (V-⟨ VM , VN ⟩)               =  step (β-case× VM VN)


## Evaluation

  data Gas : Set where
gas : ℕ → Gas

data Finished {Γ A} (N : Γ ⊢ A) : Set where

done :
Value N
----------
→ Finished N

out-of-gas :
----------
Finished N

data Steps : ∀ {A} → ∅ ⊢ A → Set where

steps : ∀ {A} {L N : ∅ ⊢ A}
→ L —↠ N
→ Finished N
----------
→ Steps L

eval : ∀ {A}
→ Gas
→ (L : ∅ ⊢ A)
-----------
→ Steps L
eval (gas zero)    L                     =  steps (L ∎) out-of-gas
eval (gas (suc m)) L with progress L
... | done VL                            =  steps (L ∎) (done VL)
... | step {M} L—→M with eval (gas m) M
...    | steps M—↠N fin                  =  steps (L —→⟨ L—→M ⟩ M—↠N) fin


## Examples

  cube : ∅ ⊢ Nat ⇒ Nat
cube = ƛ (# 0 * # 0 * # 0)

_ : cube · con 2 —↠ con 8
_ =
begin
cube · con 2
—→⟨ β-ƛ V-con ⟩
con 2 * con 2 * con 2
—→⟨ ξ-*₁ δ-* ⟩
con 4 * con 2
—→⟨ δ-* ⟩
con 8
∎

exp10 : ∅ ⊢ Nat ⇒ Nat
exp10 = ƛ (let (# 0 * # 0)
(let (# 0 * # 0)
(let (# 0 * # 2)
(# 0 * # 0))))

_ : exp10 · con 2 —↠ con 1024
_ =
begin
exp10 · con 2
—→⟨ β-ƛ V-con ⟩
let (con 2 * con 2) (let (# 0 * # 0) (let (# 0 * con 2) (# 0 * # 0)))
—→⟨ ξ-let δ-* ⟩
let (con 4) (let (# 0 * # 0) (let (# 0 * con 2) (# 0 * # 0)))
—→⟨ β-let V-con ⟩
let (con 4 * con 4) (let (# 0 * con 2) (# 0 * # 0))
—→⟨ ξ-let δ-* ⟩
let (con 16) (let (# 0 * con 2) (# 0 * # 0))
—→⟨ β-let V-con ⟩
let (con 16 * con 2) (# 0 * # 0)
—→⟨ ξ-let δ-* ⟩
let (con 32) (# 0 * # 0)
—→⟨ β-let V-con ⟩
con 32 * con 32
—→⟨ δ-* ⟩
con 1024
∎

swap× : ∀ {A B} → ∅ ⊢ A × B ⇒ B × A
swap× = ƛ ⟨ proj₂ (# 0) , proj₁ (# 0) ⟩

_ : swap× · ⟨ con 42 , zero ⟩ —↠ ⟨ zero , con 42 ⟩
_ =
begin
swap× · ⟨ con 42 , zero ⟩
—→⟨ β-ƛ V-⟨ V-con , V-zero ⟩ ⟩
⟨ proj₂ ⟨ con 42 , zero ⟩ , proj₁ ⟨ con 42 , zero ⟩ ⟩
—→⟨ ξ-⟨,⟩₁ (β-proj₂ V-con V-zero) ⟩
⟨ zero , proj₁ ⟨ con 42 , zero ⟩ ⟩
—→⟨ ξ-⟨,⟩₂ V-zero (β-proj₁ V-con V-zero) ⟩
⟨ zero , con 42 ⟩
∎

swap×-case : ∀ {A B} → ∅ ⊢ A × B ⇒ B × A
swap×-case = ƛ case× (# 0) ⟨ # 0 , # 1 ⟩

_ : swap×-case · ⟨ con 42 , zero ⟩ —↠ ⟨ zero , con 42 ⟩
_ =
begin
swap×-case · ⟨ con 42 , zero ⟩
—→⟨ β-ƛ V-⟨ V-con , V-zero ⟩ ⟩
case× ⟨ con 42 , zero ⟩ ⟨ # 0 , # 1 ⟩
—→⟨ β-case× V-con V-zero ⟩
⟨ zero , con 42 ⟩
∎


Formalise the remaining constructs defined in this chapter. Evaluate each example, applied to data as needed, to confirm it returns the expected answer.

• sums (recommended)
• unit type
• an alternative formulation of unit type
• empty type (recommended)
• lists

## Bisimulation

(No recommended exercises for this chapter.)

#### Exercise sim⁻¹

Show that we also have a simulation in the other direction, and hence that we have a bisimulation.

#### Exercise products

Show that the two formulations of products in Chapter [More][plfa.More] are in bisimulation. The only constructs you need to include are variables, and those connected to functions and products. In this case, the simulation is not lock-step.

## Inference

module Inference where


Remember to indent all code by two spaces.

### Imports

  import plfa.part2.More as DB


### Syntax

  infix   4  _∋_⦂_
infix   4  _⊢_↑_
infix   4  _⊢_↓_
infixl  5  _,_⦂_

infixr  7  _⇒_

infix   5  ƛ_⇒_
infix   5  μ_⇒_
infix   6  _↑
infix   6  _↓_
infixl  7  _·_
infix   9  _


### Identifiers, types, and contexts

  Id : Set
Id = String

data Type : Set where
ℕ    : Type
_⇒_   : Type → Type → Type

data Context : Set where
∅     : Context
_,_⦂_ : Context → Id → Type → Context


### Terms

  data Term⁺ : Set
data Term⁻ : Set

data Term⁺ where
_                        : Id → Term⁺
_·_                       : Term⁺ → Term⁻ → Term⁺
_↓_                       : Term⁻ → Type → Term⁺

data Term⁻ where
ƛ_⇒_                     : Id → Term⁻ → Term⁻
zero                     : Term⁻
suc                      : Term⁻ → Term⁻
case_[zero⇒_|suc_⇒_]     : Term⁺ → Term⁻ → Id → Term⁻ → Term⁻
μ_⇒_                     : Id → Term⁻ → Term⁻
_↑                       : Term⁺ → Term⁻


### Sample terms

  two : Term⁻
two = suc (suc zero)

plus : Term⁺
plus = (μ "p" ⇒ ƛ "m" ⇒ ƛ "n" ⇒
case ( "m") [zero⇒  "n" ↑
|suc "m" ⇒ suc ( "p" · ( "m" ↑) · ( "n" ↑) ↑) ])
↓ ℕ ⇒ ℕ ⇒ ℕ


### Lookup

  data _∋_⦂_ : Context → Id → Type → Set where

Z : ∀ {Γ x A}
--------------------
→ Γ , x ⦂ A ∋ x ⦂ A

S : ∀ {Γ x y A B}
→ x ≢ y
→ Γ ∋ x ⦂ A
-----------------
→ Γ , y ⦂ B ∋ x ⦂ A


### Bidirectional type checking

  data _⊢_↑_ : Context → Term⁺ → Type → Set
data _⊢_↓_ : Context → Term⁻ → Type → Set

data _⊢_↑_ where

⊢ : ∀ {Γ A x}
→ Γ ∋ x ⦂ A
-----------
→ Γ ⊢  x ↑ A

_·_ : ∀ {Γ L M A B}
→ Γ ⊢ L ↑ A ⇒ B
→ Γ ⊢ M ↓ A
-------------
→ Γ ⊢ L · M ↑ B

⊢↓ : ∀ {Γ M A}
→ Γ ⊢ M ↓ A
---------------
→ Γ ⊢ (M ↓ A) ↑ A

data _⊢_↓_ where

⊢ƛ : ∀ {Γ x N A B}
→ Γ , x ⦂ A ⊢ N ↓ B
-------------------
→ Γ ⊢ ƛ x ⇒ N ↓ A ⇒ B

⊢zero : ∀ {Γ}
--------------
→ Γ ⊢ zero ↓ ℕ

⊢suc : ∀ {Γ M}
→ Γ ⊢ M ↓ ℕ
---------------
→ Γ ⊢ suc M ↓ ℕ

⊢case : ∀ {Γ L M x N A}
→ Γ ⊢ L ↑ ℕ
→ Γ ⊢ M ↓ A
→ Γ , x ⦂ ℕ ⊢ N ↓ A
-------------------------------------
→ Γ ⊢ case L [zero⇒ M |suc x ⇒ N ] ↓ A

⊢μ : ∀ {Γ x N A}
→ Γ , x ⦂ A ⊢ N ↓ A
-----------------
→ Γ ⊢ μ x ⇒ N ↓ A

⊢↑ : ∀ {Γ M A B}
→ Γ ⊢ M ↑ A
→ A ≡ B
-------------
→ Γ ⊢ (M ↑) ↓ B


### Type equality

  _≟Tp_ : (A B : Type) → Dec (A ≡ B)
ℕ      ≟Tp ℕ              =  yes refl
ℕ      ≟Tp (A ⇒ B)         =  no λ()
(A ⇒ B) ≟Tp ℕ              =  no λ()
(A ⇒ B) ≟Tp (A′ ⇒ B′)
with A ≟Tp A′ | B ≟Tp B′
...  | no A≢    | _         =  no λ{refl → A≢ refl}
...  | yes _    | no B≢     =  no λ{refl → B≢ refl}
...  | yes refl | yes refl  =  yes refl


### Prerequisites

  dom≡ : ∀ {A A′ B B′} → A ⇒ B ≡ A′ ⇒ B′ → A ≡ A′
dom≡ refl = refl

rng≡ : ∀ {A A′ B B′} → A ⇒ B ≡ A′ ⇒ B′ → B ≡ B′
rng≡ refl = refl

ℕ≢⇒ : ∀ {A B} → ℕ ≢ A ⇒ B
ℕ≢⇒ ()


### Unique lookup

  uniq-∋ : ∀ {Γ x A B} → Γ ∋ x ⦂ A → Γ ∋ x ⦂ B → A ≡ B
uniq-∋ Z Z                 =  refl
uniq-∋ Z (S x≢y _)         =  ⊥-elim (x≢y refl)
uniq-∋ (S x≢y _) Z         =  ⊥-elim (x≢y refl)
uniq-∋ (S _ ∋x) (S _ ∋x′)  =  uniq-∋ ∋x ∋x′


### Unique synthesis

  uniq-↑ : ∀ {Γ M A B} → Γ ⊢ M ↑ A → Γ ⊢ M ↑ B → A ≡ B
uniq-↑ (⊢ ∋x) (⊢ ∋x′)       =  uniq-∋ ∋x ∋x′
uniq-↑ (⊢L · ⊢M) (⊢L′ · ⊢M′)  =  rng≡ (uniq-↑ ⊢L ⊢L′)
uniq-↑ (⊢↓ ⊢M) (⊢↓ ⊢M′)       =  refl


## Lookup type of a variable in the context

  ext∋ : ∀ {Γ B x y}
→ x ≢ y
→ ¬ ∃[ A ]( Γ ∋ x ⦂ A )
-----------------------------
→ ¬ ∃[ A ]( Γ , y ⦂ B ∋ x ⦂ A )
ext∋ x≢y _  ⟨ A , Z ⟩       =  x≢y refl
ext∋ _   ¬∃ ⟨ A , S _ ⊢x ⟩  =  ¬∃ ⟨ A , ⊢x ⟩

lookup : ∀ (Γ : Context) (x : Id)
-----------------------
→ Dec (∃[ A ](Γ ∋ x ⦂ A))
lookup ∅ x                        =  no  (λ ())
lookup (Γ , y ⦂ B) x with x ≟ y
... | yes refl                    =  yes ⟨ B , Z ⟩
... | no x≢y with lookup Γ x
...             | no  ¬∃          =  no  (ext∋ x≢y ¬∃)
...             | yes ⟨ A , ⊢x ⟩  =  yes ⟨ A , S x≢y ⊢x ⟩


### Promoting negations

  ¬arg : ∀ {Γ A B L M}
→ Γ ⊢ L ↑ A ⇒ B
→ ¬ Γ ⊢ M ↓ A
-------------------------
→ ¬ ∃[ B′ ](Γ ⊢ L · M ↑ B′)
¬arg ⊢L ¬⊢M ⟨ B′ , ⊢L′ · ⊢M′ ⟩ rewrite dom≡ (uniq-↑ ⊢L ⊢L′) = ¬⊢M ⊢M′

¬switch : ∀ {Γ M A B}
→ Γ ⊢ M ↑ A
→ A ≢ B
---------------
→ ¬ Γ ⊢ (M ↑) ↓ B
¬switch ⊢M A≢B (⊢↑ ⊢M′ A′≡B) rewrite uniq-↑ ⊢M ⊢M′ = A≢B A′≡B


## Synthesize and inherit types

  synthesize : ∀ (Γ : Context) (M : Term⁺)
-----------------------
→ Dec (∃[ A ](Γ ⊢ M ↑ A))

inherit : ∀ (Γ : Context) (M : Term⁻) (A : Type)
---------------
→ Dec (Γ ⊢ M ↓ A)

synthesize Γ ( x) with lookup Γ x
... | no  ¬∃              =  no  (λ{ ⟨ A , ⊢ ∋x ⟩ → ¬∃ ⟨ A , ∋x ⟩ })
... | yes ⟨ A , ∋x ⟩      =  yes ⟨ A , ⊢ ∋x ⟩
synthesize Γ (L · M) with synthesize Γ L
... | no  ¬∃              =  no  (λ{ ⟨ _ , ⊢L  · _  ⟩  →  ¬∃ ⟨ _ , ⊢L ⟩ })
... | yes ⟨ ℕ ,    ⊢L ⟩  =  no  (λ{ ⟨ _ , ⊢L′ · _  ⟩  →  ℕ≢⇒ (uniq-↑ ⊢L ⊢L′) })
... | yes ⟨ A ⇒ B , ⊢L ⟩ with inherit Γ M A
...    | no  ¬⊢M          =  no  (¬arg ⊢L ¬⊢M)
...    | yes ⊢M           =  yes ⟨ B , ⊢L · ⊢M ⟩
synthesize Γ (M ↓ A) with inherit Γ M A
... | no  ¬⊢M             =  no  (λ{ ⟨ _ , ⊢↓ ⊢M ⟩  →  ¬⊢M ⊢M })
... | yes ⊢M              =  yes ⟨ A , ⊢↓ ⊢M ⟩

inherit Γ (ƛ x ⇒ N) ℕ      =  no  (λ())
inherit Γ (ƛ x ⇒ N) (A ⇒ B) with inherit (Γ , x ⦂ A) N B
... | no ¬⊢N                =  no  (λ{ (⊢ƛ ⊢N)  →  ¬⊢N ⊢N })
... | yes ⊢N                =  yes (⊢ƛ ⊢N)
inherit Γ zero ℕ           =  yes ⊢zero
inherit Γ zero (A ⇒ B)      =  no  (λ())
inherit Γ (suc M) ℕ with inherit Γ M ℕ
... | no ¬⊢M                =  no  (λ{ (⊢suc ⊢M)  →  ¬⊢M ⊢M })
... | yes ⊢M                =  yes (⊢suc ⊢M)
inherit Γ (suc M) (A ⇒ B)   =  no  (λ())
inherit Γ (case L [zero⇒ M |suc x ⇒ N ]) A with synthesize Γ L
... | no ¬∃                 =  no  (λ{ (⊢case ⊢L  _ _) → ¬∃ ⟨ ℕ , ⊢L ⟩})
... | yes ⟨ _ ⇒ _ , ⊢L ⟩    =  no  (λ{ (⊢case ⊢L′ _ _) → ℕ≢⇒ (uniq-↑ ⊢L′ ⊢L) })
... | yes ⟨ ℕ ,    ⊢L ⟩ with inherit Γ M A
...    | no ¬⊢M             =  no  (λ{ (⊢case _ ⊢M _) → ¬⊢M ⊢M })
...    | yes ⊢M with inherit (Γ , x ⦂ ℕ) N A
...       | no ¬⊢N          =  no  (λ{ (⊢case _ _ ⊢N) → ¬⊢N ⊢N })
...       | yes ⊢N          =  yes (⊢case ⊢L ⊢M ⊢N)
inherit Γ (μ x ⇒ N) A with inherit (Γ , x ⦂ A) N A
... | no ¬⊢N                =  no  (λ{ (⊢μ ⊢N) → ¬⊢N ⊢N })
... | yes ⊢N                =  yes (⊢μ ⊢N)
inherit Γ (M ↑) B with synthesize Γ M
... | no  ¬∃                =  no  (λ{ (⊢↑ ⊢M _) → ¬∃ ⟨ _ , ⊢M ⟩ })
... | yes ⟨ A , ⊢M ⟩ with A ≟Tp B
...   | no  A≢B             =  no  (¬switch ⊢M A≢B)
...   | yes A≡B             =  yes (⊢↑ ⊢M A≡B)


### Erasure

  ∥_∥Tp : Type → DB.Type
∥ ℕ ∥Tp             =  DB.ℕ
∥ A ⇒ B ∥Tp          =  ∥ A ∥Tp DB.⇒ ∥ B ∥Tp

∥_∥Cx : Context → DB.Context
∥ ∅ ∥Cx              =  DB.∅
∥ Γ , x ⦂ A ∥Cx      =  ∥ Γ ∥Cx DB., ∥ A ∥Tp

∥_∥∋ : ∀ {Γ x A} → Γ ∋ x ⦂ A → ∥ Γ ∥Cx DB.∋ ∥ A ∥Tp
∥ Z ∥∋               =  DB.Z
∥ S x≢ ⊢x ∥∋         =  DB.S ∥ ⊢x ∥∋

∥_∥⁺ : ∀ {Γ M A} → Γ ⊢ M ↑ A → ∥ Γ ∥Cx DB.⊢ ∥ A ∥Tp
∥_∥⁻ : ∀ {Γ M A} → Γ ⊢ M ↓ A → ∥ Γ ∥Cx DB.⊢ ∥ A ∥Tp

∥ ⊢ ⊢x ∥⁺           =  DB. ∥ ⊢x ∥∋
∥ ⊢L · ⊢M ∥⁺         =  ∥ ⊢L ∥⁺ DB.· ∥ ⊢M ∥⁻
∥ ⊢↓ ⊢M ∥⁺           =  ∥ ⊢M ∥⁻

∥ ⊢ƛ ⊢N ∥⁻           =  DB.ƛ ∥ ⊢N ∥⁻
∥ ⊢zero ∥⁻           =  DB.zero
∥ ⊢suc ⊢M ∥⁻         =  DB.suc ∥ ⊢M ∥⁻
∥ ⊢case ⊢L ⊢M ⊢N ∥⁻  =  DB.case ∥ ⊢L ∥⁺ ∥ ⊢M ∥⁻ ∥ ⊢N ∥⁻
∥ ⊢μ ⊢M ∥⁻           =  DB.μ ∥ ⊢M ∥⁻
∥ ⊢↑ ⊢M refl ∥⁻      =  ∥ ⊢M ∥⁺


#### Exercise bidirectional-mul (recommended)

Rewrite your definition of multiplication from Chapter [Lambda][plfa.Lambda], decorated to support inference.

#### Exercise bidirectional-products (recommended)

Extend the bidirectional type rules to include products from Chapter [More][plfa.More].

#### Exercise bidirectional-rest (stretch)

Extend the bidirectional type rules to include the rest of the constructs from Chapter [More][plfa.More].

Rewrite your definition of multiplication from Chapter [Lambda][plfa.Lambda] decorated to support inference, and show that erasure of the inferred typing yields your definition of multiplication from Chapter [DeBruijn][plfa.DeBruijn].

Extend bidirectional inference to include products from Chapter [More][plfa.More].

#### Exercise inference-rest (stretch)

Extend bidirectional inference to include the rest of the constructs from Chapter [More][plfa.More].

## Untyped

#### Exercise (Type≃⊤)

Show that Type is isomorphic to ⊤, the unit type.

#### Exercise (Context≃ℕ)

Show that Context is isomorphic to ℕ.

#### Exercise (variant-1)

How would the rules change if we want call-by-value where terms normalise completely? Assume that β should not permit reduction unless both terms are in normal form.

#### Exercise (variant-2)

How would the rules change if we want call-by-value where terms do not reduce underneath lambda? Assume that β permits reduction when both terms are values (that is, lambda abstractions). What would plusᶜ · twoᶜ · twoᶜ reduce to in this case?

#### Exercise plus-eval

Use the evaluator to confirm that plus · two · two and four normalise to the same term.

Use the encodings above to translate your definition of multiplication from previous chapters with the Scott representation and the encoding of the fixpoint operator. Confirm that two times two is four.

#### Exercise encode-more` (stretch)

Along the lines above, encode all of the constructs of Chapter [More][plfa.More], save for primitive numbers, in the untyped lambda calculus.