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wadler 2019-09-05 13:50:11 +02:00
parent 68a128e819
commit 0de4fad2c0
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2
courses/tspl/2018/Assignment3.lagda.md
View file

@ -275,7 +275,7 @@ plus′ : Term
plus′ = μ′ + ⇒ ƛ′ m ⇒ ƛ′ n ⇒ plus′ = μ′ + ⇒ ƛ′ m ⇒ ƛ′ n ⇒
case′ m case′ m
[zero⇒ n [zero⇒ n
|suc m ⇒ `suc (+ · m · n) ] |suc m ⇒ suc (+ · m · n) ]
where where
+ = ` "+" + = ` "+"
m = ` "m" m = ` "m"

104
courses/tspl/2018/Assignment4.lagda.md
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@ -104,7 +104,6 @@ Remember to indent all code by two spaces.
infix 5 μ_ infix 5 μ_
infixl 7 _·_ infixl 7 _·_
infixl 8 _`*_ infixl 8 _`*_
infix 8 `suc_
infix 9 `_ infix 9 `_
infix 9 S_ infix 9 S_
infix 9 #_ infix 9 #_
@ -174,11 +173,11 @@ Remember to indent all code by two spaces.
-- naturals -- naturals
`zero : ∀ {Γ} zero : ∀ {Γ}
------ ------
→ Γ ⊢ `ℕ → Γ ⊢ `ℕ
`suc_ : ∀ {Γ} suc : ∀ {Γ}
→ Γ ⊢ `ℕ → Γ ⊢ `ℕ
------ ------
→ Γ ⊢ `ℕ → Γ ⊢ `ℕ
@ -220,7 +219,7 @@ Remember to indent all code by two spaces.
-- products -- products
`⟨_,_⟩ : ∀ {Γ A B} ⟨_,_⟩ : ∀ {Γ A B}
→ Γ ⊢ A → Γ ⊢ A
→ Γ ⊢ B → Γ ⊢ B
----------- -----------
@ -276,14 +275,14 @@ Remember to indent all code by two spaces.
rename ρ (` x) = ` (ρ x) rename ρ (` x) = ` (ρ x)
rename ρ (ƛ N) = ƛ (rename (ext ρ) N) rename ρ (ƛ N) = ƛ (rename (ext ρ) N)
rename ρ (L · M) = (rename ρ L) · (rename ρ M) rename ρ (L · M) = (rename ρ L) · (rename ρ M)
rename ρ (`zero) = `zero rename ρ (zero) = zero
rename ρ (`suc M) = `suc (rename ρ M) 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 (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 (μ N) = μ (rename (ext ρ) N)
rename ρ (con n) = con n rename ρ (con n) = con n
rename ρ (M `* N) = rename ρ M `* rename ρ 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 (`let M N) = `let (rename ρ M) (rename (ext ρ) N)
rename ρ `⟨ M , N ⟩ = `⟨ rename ρ M , rename ρ N ⟩ rename ρ ⟨ M , N ⟩ = ⟨ rename ρ M , rename ρ N ⟩
rename ρ (`proj₁ L) = `proj₁ (rename ρ L) rename ρ (`proj₁ L) = `proj₁ (rename ρ L)
rename ρ (`proj₂ L) = `proj₂ (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) rename ρ (case× L M) = case× (rename ρ L) (rename (ext (ext ρ)) M)
@ -300,14 +299,14 @@ Remember to indent all code by two spaces.
subst σ (` k) = σ k subst σ (` k) = σ k
subst σ (ƛ N) = ƛ (subst (exts σ) N) subst σ (ƛ N) = ƛ (subst (exts σ) N)
subst σ (L · M) = (subst σ L) · (subst σ M) subst σ (L · M) = (subst σ L) · (subst σ M)
subst σ (`zero) = `zero subst σ (zero) = zero
subst σ (`suc M) = `suc (subst σ M) 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 (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 (μ N) = μ (subst (exts σ) N)
subst σ (con n) = con n subst σ (con n) = con n
subst σ (M `* N) = subst σ M `* subst σ 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 (`let M N) = `let (subst σ M) (subst (exts σ) N)
subst σ `⟨ M , N ⟩ = `⟨ subst σ M , subst σ N ⟩ subst σ ⟨ M , N ⟩ = ⟨ subst σ M , subst σ N ⟩
subst σ (`proj₁ L) = `proj₁ (subst σ L) subst σ (`proj₁ L) = `proj₁ (subst σ L)
subst σ (`proj₂ L) = `proj₂ (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) subst σ (case× L M) = case× (subst σ L) (subst (exts (exts σ)) M)
@ -356,12 +355,12 @@ Remember to indent all code by two spaces.
V-zero : ∀ {Γ} → V-zero : ∀ {Γ} →
----------------- -----------------
Value (`zero {Γ}) Value (zero {Γ})
V-suc_ : ∀ {Γ} {V : Γ ⊢ `ℕ} V-suc : ∀ {Γ} {V : Γ ⊢ `ℕ}
→ Value V → Value V
-------------- --------------
→ Value (`suc V) → Value (suc V)
-- primitives -- primitives
@ -375,7 +374,7 @@ Remember to indent all code by two spaces.
→ Value V → Value V
→ Value W → Value W
---------------- ----------------
→ Value `⟨ V , W ⟩ → Value ⟨ V , W ⟩
``` ```
Implicit arguments need to be supplied when they are Implicit arguments need to be supplied when they are
@ -411,7 +410,7 @@ not fixed by the given arguments.
ξ-suc : ∀ {Γ} {M M′ : Γ ⊢ `ℕ} ξ-suc : ∀ {Γ} {M M′ : Γ ⊢ `ℕ}
→ M —→ M′ → M —→ M′
----------------- -----------------
→ `suc M —→ `suc M′ → suc M —→ suc M′
ξ-case : ∀ {Γ A} {L L′ : Γ ⊢ `ℕ} {M : Γ ⊢ A} {N : Γ , `ℕ ⊢ A} ξ-case : ∀ {Γ A} {L L′ : Γ ⊢ `ℕ} {M : Γ ⊢ A} {N : Γ , `ℕ ⊢ A}
→ L —→ L′ → L —→ L′
@ -420,12 +419,12 @@ not fixed by the given arguments.
β-zero : ∀ {Γ A} {M : Γ ⊢ A} {N : Γ , `ℕ ⊢ A} β-zero : ∀ {Γ A} {M : Γ ⊢ A} {N : Γ , `ℕ ⊢ A}
------------------- -------------------
→ case `zero M N —→ M → case zero M N —→ M
β-suc : ∀ {Γ A} {V : Γ ⊢ `ℕ} {M : Γ ⊢ A} {N : Γ , `ℕ ⊢ A} β-suc : ∀ {Γ A} {V : Γ ⊢ `ℕ} {M : Γ ⊢ A} {N : Γ , `ℕ ⊢ A}
→ Value V → Value V
---------------------------- ----------------------------
→ case (`suc V) M N —→ N [ V ] → case (suc V) M N —→ N [ V ]
-- fixpoint -- fixpoint
@ -467,13 +466,13 @@ not fixed by the given arguments.
ξ-⟨,⟩₁ : ∀ {Γ A B} {M M′ : Γ ⊢ A} {N : Γ ⊢ B} ξ-⟨,⟩₁ : ∀ {Γ A B} {M M′ : Γ ⊢ A} {N : Γ ⊢ B}
→ M —→ M′ → M —→ M′
------------------------- -------------------------
→ `⟨ M , N ⟩ —→ `⟨ M′ , N ⟩ → ⟨ M , N ⟩ —→ ⟨ M′ , N ⟩
ξ-⟨,⟩₂ : ∀ {Γ A B} {V : Γ ⊢ A} {N N′ : Γ ⊢ B} ξ-⟨,⟩₂ : ∀ {Γ A B} {V : Γ ⊢ A} {N N′ : Γ ⊢ B}
→ Value V → Value V
→ N —→ N′ → N —→ N′
------------------------- -------------------------
→ `⟨ V , N ⟩ —→ `⟨ V , N′ ⟩ → ⟨ V , N ⟩ —→ ⟨ V , N′ ⟩
ξ-proj₁ : ∀ {Γ A B} {L L′ : Γ ⊢ A `× B} ξ-proj₁ : ∀ {Γ A B} {L L′ : Γ ⊢ A `× B}
→ L —→ L′ → L —→ L′
@ -489,13 +488,13 @@ not fixed by the given arguments.
→ Value V → Value V
→ Value W → Value W
---------------------- ----------------------
→ `proj₁ `⟨ V , W ⟩ —→ V → `proj₁ ⟨ V , W ⟩ —→ V
β-proj₂ : ∀ {Γ A B} {V : Γ ⊢ A} {W : Γ ⊢ B} β-proj₂ : ∀ {Γ A B} {V : Γ ⊢ A} {W : Γ ⊢ B}
→ Value V → Value V
→ Value W → Value W
---------------------- ----------------------
→ `proj₂ `⟨ V , W ⟩ —→ W → `proj₂ ⟨ V , W ⟩ —→ W
-- alternative formulation of products -- alternative formulation of products
@ -508,7 +507,7 @@ not fixed by the given arguments.
→ Value V → Value V
→ Value W → Value W
---------------------------------- ----------------------------------
→ case× `⟨ V , W ⟩ M —→ M [ V ][ W ] → case× ⟨ V , W ⟩ M —→ M [ V ][ W ]
``` ```
## Reflexive and transitive closure ## Reflexive and transitive closure
@ -581,8 +580,8 @@ not fixed by the given arguments.
... | done V-ƛ with progress M ... | done V-ƛ with progress M
... | step M—→M′ = step (ξ-·₂ V-ƛ M—→M′) ... | step M—→M′ = step (ξ-·₂ V-ƛ M—→M′)
... | done VM = step (β-ƛ VM) ... | done VM = step (β-ƛ VM)
progress (`zero) = done V-zero progress (zero) = done V-zero
progress (`suc M) with progress M progress (suc M) with progress M
... | step M—→M′ = step (ξ-suc M—→M′) ... | step M—→M′ = step (ξ-suc M—→M′)
... | done VM = done (V-suc VM) ... | done VM = done (V-suc VM)
progress (case L M N) with progress L progress (case L M N) with progress L
@ -599,7 +598,7 @@ not fixed by the given arguments.
progress (`let M N) with progress M progress (`let M N) with progress M
... | step M—→M′ = step (ξ-let M—→M′) ... | step M—→M′ = step (ξ-let M—→M′)
... | done VM = step (β-let VM) ... | done VM = step (β-let VM)
progress `⟨ M , N ⟩ with progress M progress ⟨ M , N ⟩ with progress M
... | step M—→M′ = step (ξ-⟨,⟩₁ M—→M′) ... | step M—→M′ = step (ξ-⟨,⟩₁ M—→M′)
... | done VM with progress N ... | done VM with progress N
... | step N—→N′ = step (ξ-⟨,⟩₂ VM N—→N′) ... | step N—→N′ = step (ξ-⟨,⟩₂ VM N—→N′)
@ -700,31 +699,31 @@ not fixed by the given arguments.
∎ ∎
swap× : ∀ {A B} → ∅ ⊢ A `× B ⇒ B `× A swap× : ∀ {A B} → ∅ ⊢ A `× B ⇒ B `× A
swap× = ƛ `⟨ `proj₂ (# 0) , `proj₁ (# 0) ⟩ swap× = ƛ ⟨ `proj₂ (# 0) , `proj₁ (# 0) ⟩
_ : swap× · `⟨ con 42 , `zero ⟩ —↠ `⟨ `zero , con 42 ⟩ _ : swap× · ⟨ con 42 , zero ⟩ —↠ ⟨ zero , con 42 ⟩
_ = _ =
begin begin
swap× · `⟨ con 42 , `zero ⟩ swap× · ⟨ con 42 , zero ⟩
—→⟨ β-ƛ V-⟨ V-con , V-zero ⟩ ⟩ —→⟨ β-ƛ V-⟨ V-con , V-zero ⟩ ⟩
`⟨ `proj₂ `⟨ con 42 , `zero ⟩ , `proj₁ `⟨ con 42 , `zero ⟩ ⟩ ⟨ `proj₂ ⟨ con 42 , zero ⟩ , `proj₁ ⟨ con 42 , zero ⟩ ⟩
—→⟨ ξ-⟨,⟩₁ (β-proj₂ V-con V-zero) ⟩ —→⟨ ξ-⟨,⟩₁ (β-proj₂ V-con V-zero) ⟩
`⟨ `zero , `proj₁ `⟨ con 42 , `zero ⟩ ⟩ ⟨ zero , `proj₁ ⟨ con 42 , zero ⟩ ⟩
—→⟨ ξ-⟨,⟩₂ V-zero (β-proj₁ V-con V-zero) ⟩ —→⟨ ξ-⟨,⟩₂ V-zero (β-proj₁ V-con V-zero) ⟩
`⟨ `zero , con 42 ⟩ ⟨ zero , con 42 ⟩
∎ ∎
swap×-case : ∀ {A B} → ∅ ⊢ A `× B ⇒ B `× A swap×-case : ∀ {A B} → ∅ ⊢ A `× B ⇒ B `× A
swap×-case = ƛ case× (# 0) `⟨ # 0 , # 1 ⟩ swap×-case = ƛ case× (# 0) ⟨ # 0 , # 1 ⟩
_ : swap×-case · `⟨ con 42 , `zero ⟩ —↠ `⟨ `zero , con 42 ⟩ _ : swap×-case · ⟨ con 42 , zero ⟩ —↠ ⟨ zero , con 42 ⟩
_ = _ =
begin begin
swap×-case · `⟨ con 42 , `zero ⟩ swap×-case · ⟨ con 42 , zero ⟩
—→⟨ β-ƛ V-⟨ V-con , V-zero ⟩ ⟩ —→⟨ β-ƛ V-⟨ V-con , V-zero ⟩ ⟩
case× `⟨ con 42 , `zero ⟩ `⟨ # 0 , # 1 ⟩ case× ⟨ con 42 , zero ⟩ ⟨ # 0 , # 1 ⟩
—→⟨ β-case× V-con V-zero ⟩ —→⟨ β-case× V-con V-zero ⟩
`⟨ `zero , con 42 ⟩ ⟨ zero , con 42 ⟩
∎ ∎
``` ```
@ -788,7 +787,6 @@ Remember to indent all code by two spaces.
infix 6 _↑ infix 6 _↑
infix 6 _↓_ infix 6 _↓_
infixl 7 _·_ infixl 7 _·_
infix 8 `suc_
infix 9 `_ infix 9 `_
``` ```
@ -820,9 +818,9 @@ Remember to indent all code by two spaces.
data Term⁻ where data Term⁻ where
ƛ_⇒_ : Id → Term⁻ → Term⁻ ƛ_⇒_ : Id → Term⁻ → Term⁻
`zero : Term⁻ zero : Term⁻
`suc_ : Term⁻ → Term⁻ suc : Term⁻ → Term⁻
`case_[zero⇒_|suc_⇒_] : Term⁺ → Term⁻ → Id → Term⁻ → Term⁻ case_[zero⇒_|suc_⇒_] : Term⁺ → Term⁻ → Id → Term⁻ → Term⁻
μ_⇒_ : Id → Term⁻ → Term⁻ μ_⇒_ : Id → Term⁻ → Term⁻
_↑ : Term⁺ → Term⁻ _↑ : Term⁺ → Term⁻
``` ```
@ -831,12 +829,12 @@ Remember to indent all code by two spaces.
``` ```
two : Term⁻ two : Term⁻
two = `suc (`suc `zero) two = suc (suc zero)
plus : Term⁺ plus : Term⁺
plus = (μ "p" ⇒ ƛ "m" ⇒ ƛ "n" ⇒ plus = (μ "p" ⇒ ƛ "m" ⇒ ƛ "n" ⇒
`case (` "m") [zero⇒ ` "n" ↑ case (` "m") [zero⇒ ` "n" ↑
|suc "m" ⇒ `suc (` "p" · (` "m" ↑) · (` "n" ↑) ↑) ]) |suc "m" ⇒ suc (` "p" · (` "m" ↑) · (` "n" ↑) ↑) ])
↓ `ℕ ⇒ `ℕ ⇒ `ℕ ↓ `ℕ ⇒ `ℕ ⇒ `ℕ
``` ```
@ -889,19 +887,19 @@ Remember to indent all code by two spaces.
⊢zero : ∀ {Γ} ⊢zero : ∀ {Γ}
-------------- --------------
→ Γ ⊢ `zero ↓ `ℕ → Γ ⊢ zero ↓ `ℕ
⊢suc : ∀ {Γ M} ⊢suc : ∀ {Γ M}
→ Γ ⊢ M ↓ `ℕ → Γ ⊢ M ↓ `ℕ
--------------- ---------------
→ Γ ⊢ `suc M ↓ `ℕ → Γ ⊢ suc M ↓ `ℕ
⊢case : ∀ {Γ L M x N A} ⊢case : ∀ {Γ L M x N A}
→ Γ ⊢ L ↑ `ℕ → Γ ⊢ L ↑ `ℕ
→ Γ ⊢ M ↓ A → Γ ⊢ M ↓ A
→ Γ , x ⦂ `ℕ ⊢ N ↓ A → Γ , x ⦂ `ℕ ⊢ N ↓ A
------------------------------------- -------------------------------------
→ Γ ⊢ `case L [zero⇒ M |suc x ⇒ N ] ↓ A → Γ ⊢ case L [zero⇒ M |suc x ⇒ N ] ↓ A
⊢μ : ∀ {Γ x N A} ⊢μ : ∀ {Γ x N A}
→ Γ , x ⦂ A ⊢ N ↓ A → Γ , x ⦂ A ⊢ N ↓ A
@ -1032,13 +1030,13 @@ Remember to indent all code by two spaces.
inherit Γ (ƛ x ⇒ N) (A ⇒ B) with inherit (Γ , x ⦂ A) N B inherit Γ (ƛ x ⇒ N) (A ⇒ B) with inherit (Γ , x ⦂ A) N B
... | no ¬⊢N = no (λ{ (⊢ƛ ⊢N) → ¬⊢N ⊢N }) ... | no ¬⊢N = no (λ{ (⊢ƛ ⊢N) → ¬⊢N ⊢N })
... | yes ⊢N = yes (⊢ƛ ⊢N) ... | yes ⊢N = yes (⊢ƛ ⊢N)
inherit Γ `zero `ℕ = yes ⊢zero inherit Γ zero `ℕ = yes ⊢zero
inherit Γ `zero (A ⇒ B) = no (λ()) inherit Γ zero (A ⇒ B) = no (λ())
inherit Γ (`suc M) `ℕ with inherit Γ M `ℕ inherit Γ (suc M) `ℕ with inherit Γ M `ℕ
... | no ¬⊢M = no (λ{ (⊢suc ⊢M) → ¬⊢M ⊢M }) ... | no ¬⊢M = no (λ{ (⊢suc ⊢M) → ¬⊢M ⊢M })
... | yes ⊢M = yes (⊢suc ⊢M) ... | yes ⊢M = yes (⊢suc ⊢M)
inherit Γ (`suc M) (A ⇒ B) = no (λ()) inherit Γ (suc M) (A ⇒ B) = no (λ())
inherit Γ (`case L [zero⇒ M |suc x ⇒ N ]) A with synthesize Γ L inherit Γ (case L [zero⇒ M |suc x ⇒ N ]) A with synthesize Γ L
... | no ¬∃ = no (λ{ (⊢case ⊢L _ _) → ¬∃ ⟨ `ℕ , ⊢L ⟩}) ... | no ¬∃ = no (λ{ (⊢case ⊢L _ _) → ¬∃ ⟨ `ℕ , ⊢L ⟩})
... | yes ⟨ _ ⇒ _ , ⊢L ⟩ = no (λ{ (⊢case ⊢L′ _ _) → ℕ≢⇒ (uniq-↑ ⊢L′ ⊢L) }) ... | yes ⟨ _ ⇒ _ , ⊢L ⟩ = no (λ{ (⊢case ⊢L′ _ _) → ℕ≢⇒ (uniq-↑ ⊢L′ ⊢L) })
... | yes ⟨ `ℕ , ⊢L ⟩ with inherit Γ M A ... | yes ⟨ `ℕ , ⊢L ⟩ with inherit Γ M A
@ -1079,8 +1077,8 @@ Remember to indent all code by two spaces.
∥ ⊢↓ ⊢M ∥⁺ = ∥ ⊢M ∥⁻ ∥ ⊢↓ ⊢M ∥⁺ = ∥ ⊢M ∥⁻
∥ ⊢ƛ ⊢N ∥⁻ = DB.ƛ ∥ ⊢N ∥⁻ ∥ ⊢ƛ ⊢N ∥⁻ = DB.ƛ ∥ ⊢N ∥⁻
∥ ⊢zero ∥⁻ = DB.`zero ∥ ⊢zero ∥⁻ = DB.zero
∥ ⊢suc ⊢M ∥⁻ = DB.`suc ∥ ⊢M ∥⁻ ∥ ⊢suc ⊢M ∥⁻ = DB.suc ∥ ⊢M ∥⁻
∥ ⊢case ⊢L ⊢M ⊢N ∥⁻ = DB.case ∥ ⊢L ∥⁺ ∥ ⊢M ∥⁻ ∥ ⊢N ∥⁻ ∥ ⊢case ⊢L ⊢M ⊢N ∥⁻ = DB.case ∥ ⊢L ∥⁺ ∥ ⊢M ∥⁻ ∥ ⊢N ∥⁻
∥ ⊢μ ⊢M ∥⁻ = DB.μ ∥ ⊢M ∥⁻ ∥ ⊢μ ⊢M ∥⁻ = DB.μ ∥ ⊢M ∥⁻
∥ ⊢↑ ⊢M refl ∥⁻ = ∥ ⊢M ∥⁺ ∥ ⊢↑ ⊢M refl ∥⁻ = ∥ ⊢M ∥⁺

1
src/plfa/dedication.md
View file

@ -9,4 +9,5 @@ next : /Preface/
<h2>To Wanda</h2> <h2>To Wanda</h2>
<h3><em>amor da minha vida</em></h3> <h3><em>amor da minha vida</em></h3>
<h4><em>knock knock knock</em></h4> <h4><em>knock knock knock</em></h4>
<h4><em>...</em></h4>
</center> </center>