编程语言基础：Agda 描述

module Assignment4 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 Gradescope. Go to the course page under Learn. Select “Assessment”, then select “Assignment Submission”. Please ensure your files execute correctly under Agda!

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

-- begin
-- end

before and after code you add, to indicate your changes. Full credit may not be awarded if you fail to mark changes clearly.

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).

module More 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)

Syntax

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

  infixr 7 _⇒_
  infixr 9 _`×_

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

Types

  data Type : Set where
    `ℕ    : Type
    _⇒_   : Type → Type → Type
    Nat   : Type
    _`×_  : Type → 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

  length : Context → ℕ
  length ∅        =  zero
  length (Γ , _)  =  suc (length Γ)

  lookup : {Γ : Context} → {n : ℕ} → (p : n < length Γ) → Type
  lookup {(_ , A)} {zero}    (s≤s z≤n)  =  A
  lookup {(Γ , _)} {(suc n)} (s≤s p)    =  lookup p

  count : ∀ {Γ} → {n : ℕ} → (p : n < length Γ) → Γ ∋ lookup p
  count {_ , _} {zero}    (s≤s z≤n)  =  Z
  count {Γ , _} {(suc n)} (s≤s p)    =  S (count p)

  #_ : ∀ {Γ}
    → (n : ℕ)
    → {n∈Γ : True (suc n ≤? length Γ)}
      --------------------------------
    → Γ ⊢ lookup (toWitness n∈Γ)
  #_ n {n∈Γ}  =  ` count (toWitness n∈Γ)

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 , B ⊢ A
    → Γ ⊢ B
      ---------
    → Γ ⊢ A
  _[_] {Γ} {A} {B} N M =  subst {Γ , B} {Γ} σ {A} N
    where
    σ : ∀ {A} → Γ , B ∋ A → Γ ⊢ A
    σ Z      =  M
    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 (con {Γ} 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} 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

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

    step—→ : (L : Γ ⊢ A) {M N : Γ ⊢ A}
      → M —↠ N
      → L —→ M
        ------
      → L —↠ N

  pattern _—→⟨_⟩_ L L—→M M—↠N = step—→ L M—↠N L—→M

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

  record Gas : Set where
    constructor gas
    field
      amount : ℕ

  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 : {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 ⟩
     ∎

Exercise `More` (recommended and practice)

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

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

Please delimit any code you add as follows:

-- begin
-- end

Bisimulation

(No recommended exercises for this chapter.)

Exercise `~val⁻¹` (practice)

Show that this also holds in the reverse direction: if M ~ M† and Value M† then Value M.

-- Your code goes here

Exercise `sim⁻¹` (practice)

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

Exercise `products` (practice)

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

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.String using (String; _≟_)
  open import Data.Product using (_×_; ∃; ∃-syntax) renaming (_,_ to ⟨_,_⟩)
  open import Relation.Nullary using (¬_; Dec; yes; no)
  open import Relation.Nullary.Decidable using (False; toWitnessFalse)
  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   8  `suc_
  infix   9  `_

Identifiers, types, and contexts are as before:

  Id : Set
  Id = String

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

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

The syntax of terms is defined by mutual recursion.

  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⁻

Example 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)

  2+2 : Term⁺
  2+2 = plus · two · two

  Ch : Type
  Ch = (`ℕ ⇒ `ℕ) ⇒ `ℕ ⇒ `ℕ

  twoᶜ : Term⁻
  twoᶜ = (ƛ "s" ⇒ ƛ "z" ⇒ ` "s" · (` "s" · (` "z" ↑) ↑) ↑)

  plusᶜ : Term⁺
  plusᶜ = (ƛ "m" ⇒ ƛ "n" ⇒ ƛ "s" ⇒ ƛ "z" ⇒
             ` "m" · (` "s" ↑) · (` "n" · (` "s" ↑) · (` "z" ↑) ↑) ↑)
               ↓ (Ch ⇒ Ch ⇒ Ch)

  sucᶜ : Term⁻
  sucᶜ = ƛ "x" ⇒ `suc (` "x" ↑)

  2+2ᶜ : Term⁺
  2+2ᶜ = plusᶜ · twoᶜ · twoᶜ · sucᶜ · `zero

Bidirectional type checking

The typing rules for variables:

  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

The judgments for synthesizing and inheriting types are mutually recursive:

  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

Exercise `bidirectional-mul` (recommended)

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

  -- Your code goes here

Exercise `bidirectional-products` (recommended)

Extend the bidirectional type rules to include products from Chapter More. Please delimit any code you add as follows:

-- begin
-- end

Exercise `bidirectional-rest` (stretch)

Extend the bidirectional type rules to include the rest of the constructs from Chapter More. Please delimit any code you add as follows:

-- begin
-- end

Prerequisites

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

The domain and range of equal function types are equal:

  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

The types `ℕ and A ⇒ B are not equal:

  ℕ≢⇒ : ∀ {A B} → `ℕ ≢ A ⇒ B
  ℕ≢⇒ ()

Unique types

Looking up a type in the context is unique.

  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′

Synthesizing a type is also unique.

  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)

Synthesis.

  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 ⟩

Inheritance:

  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)

Testing the example terms

  S′ : ∀ {Γ x y A B}
     → {x≢y : False (x ≟ y)}
     → Γ ∋ x ⦂ A
       ------------------
     → Γ , y ⦂ B ∋ x ⦂ A

  S′ {x≢y = x≢y} x = S (toWitnessFalse x≢y) x

Typing two plus two on naturals:

  ⊢2+2 : ∅ ⊢ 2+2 ↑ `ℕ
  ⊢2+2 =
    (⊢↓
     (⊢μ
      (⊢ƛ
       (⊢ƛ
        (⊢case (⊢` (S′ Z)) (⊢↑ (⊢` Z) refl)
         (⊢suc
          (⊢↑
           (⊢`
            (S′
             (S′
              (S′ Z)))
            · ⊢↑ (⊢` Z) refl
            · ⊢↑ (⊢` (S′ Z)) refl)
           refl))))))
     · ⊢suc (⊢suc ⊢zero)
     · ⊢suc (⊢suc ⊢zero))

  _ : synthesize ∅ 2+2 ≡ yes ⟨ `ℕ , ⊢2+2 ⟩
  _ = refl

Typing two plus two with Church numerals:

  ⊢2+2ᶜ : ∅ ⊢ 2+2ᶜ ↑ `ℕ
  ⊢2+2ᶜ =
    ⊢↓
    (⊢ƛ
     (⊢ƛ
      (⊢ƛ
       (⊢ƛ
        (⊢↑
         (⊢`
          (S′
           (S′
            (S′ Z)))
          · ⊢↑ (⊢` (S′ Z)) refl
          ·
          ⊢↑
          (⊢`
           (S′
            (S′ Z))
           · ⊢↑ (⊢` (S′ Z)) refl
           · ⊢↑ (⊢` Z) refl)
          refl)
         refl)))))
    ·
    ⊢ƛ
    (⊢ƛ
     (⊢↑
      (⊢` (S′ Z) ·
       ⊢↑ (⊢` (S′ Z) · ⊢↑ (⊢` Z) refl)
       refl)
      refl))
    ·
    ⊢ƛ
    (⊢ƛ
     (⊢↑
      (⊢` (S′ Z) ·
       ⊢↑ (⊢` (S′ Z) · ⊢↑ (⊢` Z) refl)
       refl)
      refl))
    · ⊢ƛ (⊢suc (⊢↑ (⊢` Z) refl))
    · ⊢zero

  _ : synthesize ∅ 2+2ᶜ ≡ yes ⟨ `ℕ , ⊢2+2ᶜ ⟩
  _ = refl

Testing the error cases

Unbound variable:

  _ : synthesize ∅ ((ƛ "x" ⇒ ` "y" ↑) ↓ (`ℕ ⇒ `ℕ)) ≡ no _
  _ = refl

Argument in application is ill typed:

  _ : synthesize ∅ (plus · sucᶜ) ≡ no _
  _ = refl

Function in application is ill typed:

  _ : synthesize ∅ (plus · sucᶜ · two) ≡ no _
  _ = refl

Function in application has type natural:

  _ : synthesize ∅ ((two ↓ `ℕ) · two) ≡ no _
  _ = refl

Abstraction inherits type natural:

  _ : synthesize ∅ (twoᶜ ↓ `ℕ) ≡ no _
  _ = refl

Zero inherits a function type:

  _ : synthesize ∅ (`zero ↓ `ℕ ⇒ `ℕ) ≡ no _
  _ = refl

Successor inherits a function type:

  _ : synthesize ∅ (two ↓ `ℕ ⇒ `ℕ) ≡ no _
  _ = refl

Successor of an ill-typed term:

  _ : synthesize ∅ (`suc twoᶜ ↓ `ℕ) ≡ no _
  _ = refl

Case of a term with a function type:

  _ : synthesize ∅
        ((`case (twoᶜ ↓ Ch) [zero⇒ `zero |suc "x" ⇒ ` "x" ↑ ] ↓ `ℕ) ) ≡ no _
  _ = refl

Case of an ill-typed term:

  _ : synthesize ∅
        ((`case (twoᶜ ↓ `ℕ) [zero⇒ `zero |suc "x" ⇒ ` "x" ↑ ] ↓ `ℕ) ) ≡ no _
  _ = refl

Inherited and synthesised types disagree in a switch:

  _ : synthesize ∅ (((ƛ "x" ⇒ ` "x" ↑) ↓ `ℕ ⇒ (`ℕ ⇒ `ℕ))) ≡ no _
  _ = refl

Erasure

Erase a type:

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

Erase a context:

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

Erase a lookup judgment:

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

Erase a typing judgment.

  ∥_∥⁺ : ∀ {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 ∥⁺

Erasure of the type derivations in this chapter yield the corresponding intrinsically-typed terms from the earlier chapter:

  _ : ∥ ⊢2+2 ∥⁺ ≡ DB.2+2
  _ = refl

  _ : ∥ ⊢2+2ᶜ ∥⁺ ≡ DB.2+2ᶜ
  _ = refl

Exercise `inference-multiplication` (recommended)

Apply inference to your decorated definition of multiplication from exercise bidirectional-mul, and show that erasure of the inferred typing yields your definition of multiplication from Chapter DeBruijn.

-- Your code goes here

Exercise `inference-products` (recommended)

Using your rules from exercise bidirectional-products, extend bidirectional inference to include products. Also extend erasure. Please delimit any code you add as follows:

-- begin
-- end

Exercise `inference-rest` (stretch)

Using your rules from exercise bidirectional-rest, extend bidirectional inference to include the rest of the constructs from Chapter More. Also extend erasure. Please delimit any code you add as follows:

-- begin
-- end

Untyped

module Untyped where

Imports

  import Relation.Binary.PropositionalEquality as Eq
  open Eq using (_≡_; refl; sym; trans; cong)
  open import Data.Empty using (⊥; ⊥-elim)
  open import Data.Nat using (ℕ; zero; suc; _+_; _∸_)
  open import Data.Product using (_×_) renaming (_,_ to ⟨_,_⟩)
  open import Data.Unit using (⊤; tt)
  open import Function using (_∘_)
  open import Function.Equivalence using (_⇔_; equivalence)
  open import Relation.Nullary using (¬_; Dec; yes; no)
  open import Relation.Nullary.Decidable using (map)
  open import Relation.Nullary.Negation using (contraposition)
  open import Relation.Nullary.Product using (_×-dec_)

  open import plfa.part2.Untyped
    hiding ()

Exercise (`Type≃⊤`) (practice)

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

  -- Your code goes here

Exercise (`Context≃ℕ`) (practice)

Show that Context is isomorphic to ℕ.

  -- Your code goes here

Exercise (`variant-1`) (practice)

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.

  -- Your code goes here

Exercise (`variant-2`) (practice)

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 2+2ᶜ reduce to in this case?

  -- Your code goes here

Exercise `plus-eval` (practice)

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

  -- Your code goes here

Exercise `multiplication-untyped` (recommended)

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.

  -- Your code goes here

Exercise `encode-more` (stretch)

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

  -- Your code goes here

YOUR NAME AND EMAIL GOES HERE

Introduction

Good Scholarly Practice.

More

Imports

Syntax

Types

Contexts

Variables and the lookup judgment

Terms and the typing judgment

Abbreviating de Bruijn indices

Renaming

Simultaneous Substitution

Single and double substitution

Values

Reduction

Reflexive and transitive closure

Progress

Evaluation

Examples

Exercise More (recommended and practice)

Bisimulation

Exercise ~val⁻¹ (practice)

Exercise sim⁻¹ (practice)

Exercise products (practice)

Inference

Imports

Syntax

Example terms

Bidirectional type checking

Exercise bidirectional-mul (recommended)

Exercise bidirectional-products (recommended)

Exercise bidirectional-rest (stretch)

Prerequisites

Unique types

Lookup type of a variable in the context

Promoting negations

Synthesize and inherit types

Testing the example terms

Testing the error cases

Erasure

Exercise inference-multiplication (recommended)

Exercise inference-products (recommended)

Exercise inference-rest (stretch)

Untyped

Imports

Exercise (Type≃⊤) (practice)

Exercise (Context≃ℕ) (practice)

Exercise (variant-1) (practice)

Exercise (variant-2) (practice)

Exercise plus-eval (practice)

Exercise multiplication-untyped (recommended)

Exercise encode-more (stretch)

Exercise `More` (recommended and practice)

Exercise `~val⁻¹` (practice)

Exercise `sim⁻¹` (practice)

Exercise `products` (practice)

Exercise `bidirectional-mul` (recommended)

Exercise `bidirectional-products` (recommended)

Exercise `bidirectional-rest` (stretch)

Exercise `inference-multiplication` (recommended)

Exercise `inference-products` (recommended)

Exercise `inference-rest` (stretch)

Exercise (`Type≃⊤`) (practice)

Exercise (`Context≃ℕ`) (practice)

Exercise (`variant-1`) (practice)

Exercise (`variant-2`) (practice)

Exercise `plus-eval` (practice)

Exercise `multiplication-untyped` (recommended)

Exercise `encode-more` (stretch)