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.
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 8 `suc_ 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 8 `suc_ 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.