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backup after crash

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wadler 2018-04-16 18:11:24 -03:00
parent 6fad419599
commit 1bb5412c8d
1 changed files with 38 additions and 34 deletions
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68
src/Typed.lagda
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@ -244,12 +244,12 @@ xs \\ x = filter (¬? ∘ (x ≟_)) xs
∷⊆∷ xs⊆ (here refl) = here refl ∷⊆∷ xs⊆ (here refl) = here refl
∷⊆∷ xs⊆ (there ∈xs) = there (xs⊆ ∈xs) ∷⊆∷ xs⊆ (there ∈xs) = there (xs⊆ ∈xs)
[_]⊆ : ∀ {x xs} → [ x ] ⊆ xs → x ∈ xs []⊆ : ∀ {x xs} → [ x ] ⊆ xs → x ∈ xs
[_]⊆ ⊆xs = ⊆xs (here refl) []⊆ ⊆xs = ⊆xs (here refl)
⊆[_] : ∀ {x xs} → x ∈ xs → [ x ] ⊆ xs ⊆[] : ∀ {x xs} → x ∈ xs → [ x ] ⊆ xs
⊆[_] x∈ (here refl) = x∈ ⊆[] x∈ (here refl) = x∈
⊆[_] x∈ (there ()) ⊆[] x∈ (there ())
bind : ∀ {x xs} → xs \\ x ⊆ xs bind : ∀ {x xs} → xs \\ x ⊆ xs
bind {x} {[]} () bind {x} {[]} ()
@ -311,14 +311,14 @@ _,_↦_ : (Id → Term) → Id → Term → (Id → Term)
\begin{code} \begin{code}
subst : List Id → (Id → Term) → Term → Term subst : List Id → (Id → Term) → Term → Term
subst xs ρ ⌊ x ⌋ = ρ x subst ys ρ ⌊ x ⌋ = ρ x
subst xs ρ (ƛ x ⦂ A ⇒ N) = ƛ y ⦂ A ⇒ subst (y ∷ xs) (ρ , x ↦ ⌊ y ⌋) N subst ys ρ (ƛ x ⦂ A ⇒ N) = ƛ y ⦂ A ⇒ subst (y ∷ ys) (ρ , x ↦ ⌊ y ⌋) N
where where
y = fresh xs y = fresh ys
subst xs ρ (L · M) = subst xs ρ L · subst xs ρ M subst ys ρ (L · M) = subst ys ρ L · subst ys ρ M
_[_:=_] : Term → Id → Term → Term _[_:=_] : Term → Id → Term → Term
N [ x := M ] = subst (free M ∪ free N) (∅ , x ↦ M) N N [ x := M ] = subst (free M ∪ (free N \\ x)) (∅ , x ↦ M) N
\end{code} \end{code}
@ -470,7 +470,7 @@ j {x} {xs} {ys} ⊆ys {w} w∈ with x ≟ w
(∀ {M A} → free M ⊆ xs → Γ ⊢ M ⦂ A → Δ ⊢ M ⦂ A) (∀ {M A} → free M ⊆ xs → Γ ⊢ M ⦂ A → Δ ⊢ M ⦂ A)
⊢rename ⊢σ ⊆xs (⌊ ⊢x ⌋) = ⌊ ⊢σ ∈xs ⊢x ⌋ ⊢rename ⊢σ ⊆xs (⌊ ⊢x ⌋) = ⌊ ⊢σ ∈xs ⊢x ⌋
where where
∈xs = [_]⊆ ⊆xs ∈xs = []⊆ ⊆xs
⊢rename {Γ} {Δ} {xs} ⊢σ ⊆xs (ƛ_ {x = x} {A = A} {N = N} ⊢N) ⊢rename {Γ} {Δ} {xs} ⊢σ ⊆xs (ƛ_ {x = x} {A = A} {N = N} ⊢N)
= ƛ (⊢rename {Γ′} {Δ′} {xs′} ⊢σ′ ⊆xs′ ⊢N) -- ⊆xs : free N \\ x ⊆ xs = ƛ (⊢rename {Γ′} {Δ′} {xs′} ⊢σ′ ⊆xs′ ⊢N) -- ⊆xs : free N \\ x ⊆ xs
where where
@ -493,28 +493,29 @@ j {x} {xs} {ys} ⊆ys {w} w∈ with x ≟ w
### Substitution preserves types ### Substitution preserves types
\begin{code} \begin{code}
⊢subst : ∀ {Γ Δ xs ρ} → ⊢subst : ∀ {Γ Δ xs ys ρ} →
(∀ {x} → x ∈ xs → free (ρ x) ⊆ xs) → (∀ {x} → x ∈ xs → free (ρ x) ⊆ ys) →
(∀ {x A} → x ∈ xs → Γ ∋ x ⦂ A → Δ ⊢ ρ x ⦂ A) → (∀ {x A} → x ∈ xs → Γ ∋ x ⦂ A → Δ ⊢ ρ x ⦂ A) →
(∀ {M A} → free M ⊆ xs → Γ ⊢ M ⦂ A → Δ ⊢ subst xs ρ M ⦂ A) (∀ {M A} → free M ⊆ xs → Γ ⊢ M ⦂ A → Δ ⊢ subst ys ρ M ⦂ A)
⊢subst Σ ⊢ρ ⊆xs ⌊ ⊢x ⌋ = ⊢ρ (⊆xs (here refl)) ⊢x ⊢subst Σ ⊢ρ ⊆xs ⌊ ⊢x ⌋ = ⊢ρ (⊆xs (here refl)) ⊢x
⊢subst {Γ} {Δ} {xs} {ρ} Σ ⊢ρ ⊆xs (ƛ_ {x = x} {A = A} {N = N} ⊢N) ⊢subst {Γ} {Δ} {xs} {ys} {ρ} Σ ⊢ρ ⊆xs (ƛ_ {x = x} {A = A} {N = N} ⊢N)
= ƛ ⊢subst {Γ′} {Δ′} {xs′} {ρ′} Σ′ ⊢ρ′ ⊆xs′ ⊢N = ƛ_ {x = y} {A = A} (⊢subst {Γ′} {Δ′} {xs′} {ys′} {ρ′} Σ′ ⊢ρ′ ⊆xs′ ⊢N)
where where
y = fresh xs y = fresh ys
Γ′ = Γ , x ⦂ A Γ′ = Γ , x ⦂ A
Δ′ = Δ , y ⦂ A Δ′ = Δ , y ⦂ A
xs′ = y ∷ xs xs′ = x ∷ xs
ys′ = y ∷ ys
ρ′ = ρ , x ↦ ⌊ y ⌋ ρ′ = ρ , x ↦ ⌊ y ⌋
Σ′ : ∀ {z} → z ∈ xs′ → free (ρ′ z) ⊆ xs′ Σ′ : ∀ {z} → z ∈ xs′ → free (ρ′ z) ⊆ ys′
Σ′ (here refl) = {!!} Σ′ (here refl) = {!⊆[] here!}
Σ′ (there x∈) = {!!} Σ′ (there x∈) = {!there ∘ (Σ x∈)!}
⊆xs′ : free N ⊆ xs′ ⊆xs′ : free N ⊆ xs′
⊆xs′ = {!!} ⊆xs′ = {!!}
⊢σ : ∀ {z C} → z ∈ xs → Δ ∋ z ⦂ C → Δ′ ∋ z ⦂ C ⊢σ : ∀ {z C} → z ∈ ys → Δ ∋ z ⦂ C → Δ′ ∋ z ⦂ C
⊢σ z∈ ⊢z = S (fresh-lemma z∈) ⊢z ⊢σ z∈ ⊢z = S (fresh-lemma z∈) ⊢z
⊢ρ′ : ∀ {z C} → z ∈ xs′ → Γ′ ∋ z ⦂ C → Δ′ ⊢ ρ′ z ⦂ C ⊢ρ′ : ∀ {z C} → z ∈ xs′ → Γ′ ∋ z ⦂ C → Δ′ ⊢ ρ′ z ⦂ C
@ -523,14 +524,14 @@ j {x} {xs} {ys} ⊆ys {w} w∈ with x ≟ w
... | no x≢x = ⊥-elim (x≢x refl) ... | no x≢x = ⊥-elim (x≢x refl)
⊢ρ′ {z} z∈ (S x≢z ⊢z) with x ≟ z ⊢ρ′ {z} z∈ (S x≢z ⊢z) with x ≟ z
... | yes x≡z = ⊥-elim (x≢z x≡z) ... | yes x≡z = ⊥-elim (x≢z x≡z)
... | no _ = ⊢rename {Δ} {Δ′} {xs} ⊢σ (Σ (prev {!!} z∈)) (⊢ρ {!!} ⊢z) ... | no _ = ⊢rename {Δ} {Δ′} {ys} ⊢σ {!!} (⊢ρ {!!} ⊢z)
⊢subst {xs = xs} Σ ⊢ρ {L · M} ⊆xs (⊢L · ⊢M) = ⊢subst Σ ⊢ρ L⊆xs ⊢L · ⊢subst Σ ⊢ρ M⊆xs ⊢M ⊢subst {xs = xs} Σ ⊢ρ {L · M} ⊆xs (⊢L · ⊢M) = ⊢subst Σ ⊢ρ L⊆xs ⊢L · ⊢subst Σ ⊢ρ M⊆xs ⊢M
where where
L⊆xs : free L ⊆ xs L⊆xs : free L ⊆ xs
L⊆xs = {!!} L⊆xs = ⊆xs ∘ left
M⊆xs : free M ⊆ xs M⊆xs : free M ⊆ xs
M⊆xs = {!!} M⊆xs = ⊆xs ∘ right
⊢substitution : ∀ {Γ x A N B M} → ⊢substitution : ∀ {Γ x A N B M} →
Γ , x ⦂ A ⊢ N ⦂ B → Γ , x ⦂ A ⊢ N ⦂ B →
@ -538,16 +539,19 @@ j {x} {xs} {ys} ⊆ys {w} w∈ with x ≟ w
-------------------- --------------------
Γ ⊢ N [ x := M ] ⦂ B Γ ⊢ N [ x := M ] ⦂ B
⊢substitution {Γ} {x} {A} {N} {B} {M} ⊢N ⊢M = ⊢substitution {Γ} {x} {A} {N} {B} {M} ⊢N ⊢M =
⊢subst {Γ′} {Γ} {xs} {ρ} Σ ⊢ρ {N} {B} ⊆xs ⊢N ⊢subst {Γ′} {Γ} {xs} {ys} {ρ} Σ ⊢ρ {N} {B} ⊆xs ⊢N
where where
Γ′ = Γ , x ⦂ A Γ′ = Γ , x ⦂ A
xs = free M ∪ free N xs = free N
ys = free M ∪ (free N \\ x)
ρ = ∅ , x ↦ M ρ = ∅ , x ↦ M
Σ : ∀ {x} → x ∈ xs → free (ρ x) ⊆ xs Σ : ∀ {w} → w ∈ xs → free (ρ w) ⊆ ys
Σ {y} y∈ with x ≟ y Σ {w} w∈ y∈ with x ≟ w
... | no _ = ⊆[_] y∈ ... | yes _ = left y∈
... | yes _ = {!!} -- left ... | no x≢w with y∈
... | here refl = right (i {w} {x} {free N} w∈ x≢w)
... | there ()
⊢ρ : ∀ {z C} → z ∈ xs → Γ′ ∋ z ⦂ C → Γ ⊢ ρ z ⦂ C ⊢ρ : ∀ {z C} → z ∈ xs → Γ′ ∋ z ⦂ C → Γ ⊢ ρ z ⦂ C
⊢ρ {.x} z∈ Z with x ≟ x ⊢ρ {.x} z∈ Z with x ≟ x
@ -558,7 +562,7 @@ j {x} {xs} {ys} ⊆ys {w} w∈ with x ≟ w
... | no _ = ⌊ ⊢z ⌋ ... | no _ = ⌊ ⊢z ⌋
⊆xs : free N ⊆ xs ⊆xs : free N ⊆ xs
⊆xs = {!!} ⊆xs x∈ = x∈
\end{code} \end{code}
Can I falsify the theorem? Consider the case where the renamed variable Can I falsify the theorem? Consider the case where the renamed variable