Created
August 19, 2020 08:46
-
-
Save L-TChen/73754ecd5c305b93e6749076bf9c3070 to your computer and use it in GitHub Desktop.
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
open import Data.Nat | |
open import Data.Empty | |
hiding (⊥-elim) | |
open import Relation.Nullary | |
open import Relation.Binary.PropositionalEquality | |
hiding ([_]) | |
infix 3 _⊢_ _=β_ | |
infixr 7 _→̇_ | |
infixr 5 ƛ_ | |
infixl 7 _·_ | |
infixl 8 _[_] _⟪_⟫ | |
infixr 9 ᵒ_ `_ #_ | |
data Type : Set where | |
⋆ : Type | |
_→̇_ : Type → Type → Type | |
infixl 7 _⧺_ | |
infixl 6 _,_ | |
infix 4 _∋_ | |
data Context : Set where | |
∅ : Context | |
_,_ : Context → Type → Context | |
variable | |
Γ Δ Ξ : Context | |
A B C : Type | |
data _∋_ : Context → Type → Set where | |
Z --------- | |
: Γ , A ∋ A | |
S_ : Γ ∋ A | |
--------- | |
→ Γ , B ∋ A | |
variable | |
x : Γ ∋ A | |
_ : ∅ , A , B ∋ A | |
_ = S Z | |
⊥-elim : ∀ {T : Set} → ⊥ → T | |
⊥-elim () | |
lookup : Context → ℕ → Type | |
lookup (Γ , A) zero = A | |
lookup (Γ , B) (suc n) = lookup Γ n | |
lookup ∅ _ = ⊥-elim impossible | |
where postulate impossible : ⊥ | |
count : (n : ℕ) → Γ ∋ lookup Γ n | |
count {Γ = Γ , _} zero = Z | |
count {Γ = Γ , _} (suc n) = S (count n) | |
count {Γ = ∅ } _ = ⊥-elim impossible | |
where postulate impossible : ⊥ | |
_ : ∅ , A , B ∋ A | |
_ = count 1 | |
_ : ∅ , B , A ∋ A | |
_ = count 0 | |
ext | |
: (∀ {A} → Γ ∋ A → Δ ∋ A) | |
--------------------------------- | |
→ (∀ {A B} → Γ , B ∋ A → Δ , B ∋ A) | |
ext ρ Z = Z | |
ext ρ (S x) = S (ρ x) | |
_⧺_ : Context → Context → Context | |
Γ ⧺ ∅ = Γ | |
Γ ⧺ (Δ , x) = Γ ⧺ Δ , x | |
data _⊢_ (Γ : Context) : Type → Set where | |
`_ : Γ ∋ A | |
----- | |
→ Γ ⊢ A | |
_·_ : Γ ⊢ A →̇ B | |
→ Γ ⊢ A | |
----- | |
→ Γ ⊢ B | |
ƛ_ | |
: Γ , A ⊢ B | |
--------- | |
→ Γ ⊢ A →̇ B | |
variable | |
M N L M′ N′ L′ : Γ ⊢ A | |
#_ : (n : ℕ) → Γ ⊢ lookup Γ n | |
# n = ` count n | |
nat : Type → Type | |
nat A = (A →̇ A) →̇ A →̇ A | |
c₀ : ∀ {A} → ∅ ⊢ nat A | |
c₀ = ƛ ƛ # 0 | |
c₁ : ∀ {A} → ∅ ⊢ nat A | |
c₁ = ƛ ƛ # 1 · # 0 | |
add : ∀ {A} → ∅ ⊢ nat A →̇ nat A →̇ nat A | |
add = ƛ ƛ ƛ ƛ # 3 · # 1 · (# 2 · # 1 · # 0) | |
id : ∅ ⊢ A →̇ A | |
id = ƛ # 0 | |
id' : ∅ , A ⊢ A →̇ A | |
id' = ƛ # 1 | |
fst : ∅ ⊢ A →̇ B →̇ A | |
fst = ƛ ƛ # 1 | |
bool : Type → Type | |
bool A = A →̇ A →̇ A | |
if : ∅ ⊢ bool A →̇ A →̇ A →̇ A | |
if = ƛ ƛ ƛ # 2 · # 1 · # 0 | |
succ : ∅ ⊢ nat A →̇ nat A | |
succ = ƛ ƛ ƛ # 1 · (# 2 · # 1 · # 0) | |
Rename : Context → Context → Set | |
Rename Γ Δ = ∀ {A} → Γ ∋ A → Δ ∋ A | |
Subst : Context → Context → Set | |
Subst Γ Δ = ∀ {A} → Γ ∋ A → Δ ⊢ A | |
rename : Rename Γ Δ | |
→ (Γ ⊢ A) | |
→ (Δ ⊢ A) | |
rename ρ (` x) = ` ρ x | |
rename ρ (M · N) = rename ρ M · rename ρ N | |
rename ρ (ƛ M) = ƛ rename (ext ρ) M | |
exts : Subst Γ Δ → Subst (Γ , A) (Δ , A) | |
exts σ Z = ` Z | |
exts σ (S p) = rename S_ (σ p) | |
_⟪_⟫ | |
: Γ ⊢ A | |
→ Subst Γ Δ | |
→ Δ ⊢ A | |
(` x) ⟪ σ ⟫ = σ x | |
(M · N) ⟪ σ ⟫ = M ⟪ σ ⟫ · N ⟪ σ ⟫ | |
(ƛ M) ⟪ σ ⟫ = ƛ M ⟪ exts σ ⟫ | |
subst-zero : {B : Type} | |
→ Γ ⊢ B | |
→ Subst (Γ , B) Γ | |
subst-zero N Z = N | |
subst-zero _ (S x) = ` x | |
_[_] : Γ , B ⊢ A | |
→ Γ ⊢ B | |
--------- | |
→ Γ ⊢ A | |
_[_] N M = N ⟪ subst-zero M ⟫ | |
infix 3 _-→_ | |
data _-→_ {Γ} : (M N : Γ ⊢ A) → Set where | |
β-ƛ· | |
: (ƛ M) · N -→ M [ N ] | |
ξ-ƛ | |
: M -→ M′ | |
→ ƛ M -→ ƛ M′ | |
ξ-·ₗ | |
: L -→ L′ | |
--------------- | |
→ L · M -→ L′ · M | |
ξ-·ᵣ | |
: M -→ M′ | |
--------------- | |
→ L · M -→ L · M′ | |
data _-↠_ {Γ A} : (M N : Γ ⊢ A) → Set where | |
_∎ : (M : Γ ⊢ A) | |
→ M -↠ M -- empty list | |
_-→⟨_⟩_ | |
: ∀ L -- this can usually be inferred by the following reduction | |
→ L -→ M -- the head of a list | |
→ M -↠ N -- the tail | |
------- | |
→ L -↠ N | |
infix 2 _-↠_ | |
infixr 2 _-→⟨_⟩_ | |
infix 3 _∎ | |
_ : (ƛ (ƛ # 0) · # 1) · # 0 -↠ (ƛ # 1) · # 0 | |
_ = (ƛ (ƛ # 0) · # 1) · # 0 | |
-→⟨ ξ-·ₗ (ξ-ƛ β-ƛ·) ⟩ | |
(ƛ # 0 [ # 1 ]) · # 0 | |
∎ | |
_-↠⟨_⟩_ : ∀ L | |
→ L -↠ M → M -↠ N | |
----------------- | |
→ L -↠ N | |
L -↠⟨ M ∎ ⟩ M-↠N = M-↠N | |
L -↠⟨ L -→⟨ L→M′ ⟩ M′-↠M ⟩ M-↠N = L -→⟨ L→M′ ⟩ (_ -↠⟨ M′-↠M ⟩ M-↠N) | |
infixr 2 _-↠⟨_⟩_ | |
ƛ-↠ : M -↠ M′ | |
----------- | |
→ ƛ M -↠ ƛ M′ | |
ƛ-↠ (M ∎) = ƛ M ∎ | |
ƛ-↠ (M -→⟨ M→N ⟩ N-↠M′) = ƛ M -→⟨ ξ-ƛ M→N ⟩ ƛ-↠ N-↠M′ | |
·ᵣ-↠ : N -↠ N′ | |
→ M · N -↠ M · N′ | |
·ᵣ-↠ {M = M} (N ∎) = M · N ∎ | |
·ᵣ-↠ {M = M} (N -→⟨ N→M ⟩ M-↠N′) = M · N -→⟨ ξ-·ᵣ N→M ⟩ (·ᵣ-↠ M-↠N′) | |
·ₗ-↠ : M -↠ M′ | |
→ M · N -↠ M′ · N | |
·ₗ-↠ {M = M} {N = N} (M ∎) = M · N ∎ | |
·ₗ-↠ {M = M} {N = N} (M -→⟨ M→M₁ ⟩ M₁-↠M′) = | |
M · N -→⟨ ξ-·ₗ M→M₁ ⟩ ·ₗ-↠ M₁-↠M′ | |
·-↠ : M -↠ M′ | |
→ N -↠ N′ | |
→ M · N -↠ M′ · N′ | |
·-↠ {M = M} {M′ = M′} {N = N} {N′ = N′} M-↠M′ N-↠N′ = | |
M · N | |
-↠⟨ ·ₗ-↠ M-↠M′ ⟩ | |
M′ · N | |
-↠⟨ ·ᵣ-↠ N-↠N′ ⟩ | |
M′ · N′ | |
∎ | |
data _=β_ {Γ : Context} : Γ ⊢ A → Γ ⊢ A → Set where | |
=β-beta | |
: M -→ N → M =β N | |
=β-refl | |
: M =β M | |
=β-sym | |
: N =β M → M =β N | |
=β-trans | |
: L =β M → M =β N | |
→ L =β N | |
HW2 : M -↠ N → (∀ {N} → (M -→ N) → ⊥) → M ≡ N | |
HW2 (M ∎) M↛ = refl | |
HW2 (M -→⟨ M→N ⟩ _) M↛ = ⊥-elim (M↛ M→N) | |
data Neutral : Γ ⊢ A → Set | |
data Normal : Γ ⊢ A → Set | |
data Neutral where | |
`_ : (x : Γ ∋ A) | |
→ Neutral (` x) | |
_·_ : Neutral L | |
→ Normal M | |
→ Neutral (L · M) | |
data Normal where | |
ᵒ_ : Neutral M → Normal M | |
ƛ_ : Normal M → Normal (ƛ M) | |
normal-soundness : Normal M → ¬ (M -→ N) | |
neutral-soundness : Neutral M → ¬ (M -→ M′) | |
normal-soundness (ᵒ M↓) M→N = neutral-soundness M↓ M→N | |
normal-soundness (ƛ M↓) (ξ-ƛ M→N) = normal-soundness M↓ M→N | |
neutral-soundness (` x) () | |
neutral-soundness (L↓ · M↓) (ξ-·ₗ L→L′) = neutral-soundness L↓ L→L′ | |
neutral-soundness (L↓ · M↓) (ξ-·ᵣ M→M′) = normal-soundness M↓ M→M′ | |
normal-completeness | |
: (M : Γ ⊢ A) → (∀ N → ¬ (M -→ N)) | |
→ Normal M | |
normal-completeness (` x) M↛ = ᵒ ` x | |
normal-completeness (ƛ M) ƛM↛ with normal-completeness M M↛ | |
where M↛ : ∀ N → ¬ (M -→ N) | |
M↛ N M→N = ƛM↛ (ƛ N) (ξ-ƛ M→N) | |
... | M↓ = ƛ M↓ | |
normal-completeness (M · N) MN↛ with normal-completeness M M↛ | normal-completeness N N↛ | |
where M↛ : ∀ M′ → ¬ (M -→ M′) | |
M↛ M′ M↛ = MN↛ (M′ · N) (ξ-·ₗ M↛) | |
N↛ : ∀ N′ → ¬ (N -→ N′) | |
N↛ N′ N↛ = MN↛ (M · N′) (ξ-·ᵣ N↛) | |
... | ᵒ M↓ | N↓ = ᵒ (M↓ · N↓) | |
... | ƛ M↓ | N↓ = ⊥-elim (MN↛ _ β-ƛ· ) | |
data Progress (M : Γ ⊢ A) : Set where | |
step | |
: M -→ N | |
---------- | |
→ Progress M | |
done : Normal M | |
→ Progress M | |
progress : (M : Γ ⊢ A) | |
→ Progress M | |
progress (` x) = done (ᵒ ` x) | |
progress (ƛ M) with progress M | |
... | step r = step (ξ-ƛ r) | |
... | done M↓ = done (ƛ M↓) | |
progress (M · N) with progress M | progress N | |
... | step r | _ = step (ξ-·ₗ r) | |
... | done x | step r = step (ξ-·ᵣ r) | |
... | done (ᵒ M↓) | done N↓ = done (ᵒ (M↓ · N↓)) | |
... | done (ƛ M↓) | done N↓ = step β-ƛ· | |
data Progress′ (M : Γ ⊢ A) : Set where | |
step | |
: (r : M -→ N) | |
---------- | |
→ Progress′ M | |
done : (M↓ : (N : Γ ⊢ A) → M -→ N → ⊥) | |
→ Progress′ M | |
--progress′ : (M : Γ ⊢ A) → Progress′ M | |
--progress′ M = {!!} |
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment