Assignment4: TSPL Assignment 4

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

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.

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

Exercise (`mul`) (recommended)

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.

Exercise `mul-eval` (recommended)

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

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 ⟩
     ∎

Exercise `More` (recommended in part)

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].

Exercise `inference-mul` (recommended)

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].

Exercise `inference-products` (recommended)

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.

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.

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.

YOUR NAME AND EMAIL GOES HERE

Introduction

Imports

DeBruijn

Exercise (mul) (recommended)

Exercise V¬—→

Exercise mul-eval (recommended)

More

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

Values do not reduce

Progress

Evaluation

Examples

Exercise More (recommended in part)

Bisimulation

Exercise sim⁻¹

Exercise products

Inference

Imports

Syntax

Identifiers, types, and contexts

Terms

Sample terms

Lookup

Bidirectional type checking

Type equality

Prerequisites

Unique lookup

Unique synthesis

Lookup type of a variable in the context

Promoting negations

Synthesize and inherit types

Erasure

Exercise bidirectional-mul (recommended)

Exercise bidirectional-products (recommended)

Exercise bidirectional-rest (stretch)

Exercise inference-mul (recommended)

Exercise inference-products (recommended)

Exercise inference-rest (stretch)

Untyped

Exercise (Type≃⊤)

Exercise (Context≃ℕ)

Exercise (variant-1)

Exercise (variant-2)

Exercise plus-eval

Exercise multiplication-untyped (recommended)

Exercise encode-more (stretch)

Exercise (`mul`) (recommended)

Exercise `V¬—→`

Exercise `mul-eval` (recommended)

Exercise `More` (recommended in part)

Exercise `sim⁻¹`

Exercise `products`

Exercise `bidirectional-mul` (recommended)

Exercise `bidirectional-products` (recommended)

Exercise `bidirectional-rest` (stretch)

Exercise `inference-mul` (recommended)

Exercise `inference-products` (recommended)

Exercise `inference-rest` (stretch)

Exercise (`Type≃⊤`)

Exercise (`Context≃ℕ`)

Exercise (`variant-1`)

Exercise (`variant-2`)

Exercise `plus-eval`

Exercise `multiplication-untyped` (recommended)

Exercise `encode-more` (stretch)