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updated Maps with extensionality

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wadler 2017-06-26 13:16:13 +01:00
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85
src/Maps.lagda
View file

@ -86,6 +86,8 @@ y = id "y"
z = id "z" z = id "z"
\end{code} \end{code}
## Extensionality
## Total Maps ## Total Maps
Our main job in this chapter will be to build a definition of Our main job in this chapter will be to build a definition of
@ -219,8 +221,8 @@ back $$v$$:
update-eq′ : ∀ {A} (ρ : TotalMap A) (x : Id) (v : A) update-eq′ : ∀ {A} (ρ : TotalMap A) (x : Id) (v : A)
→ (ρ , x ↦ v) x ≡ v → (ρ , x ↦ v) x ≡ v
update-eq′ ρ x v with x ≟ x update-eq′ ρ x v with x ≟ x
... | yes x=x = refl ... | yes x≡x = refl
... | no x≠x = ⊥-elim (x≠x refl) ... | no x≢x = ⊥-elim (x≢x refl)
\end{code} \end{code}
</div> </div>
@ -237,6 +239,14 @@ the same result that $$m$$ would have given:
... | no _ = refl ... | no _ = refl
\end{code} \end{code}
For the following lemmas, since maps are represented by functions, to
show two maps equal we will need to postulate extensionality.
\begin{code}
postulate
extensionality : ∀ {A : Set} {ρ ρ′ : TotalMap A} → (∀ x → ρ x ≡ ρ′ x) → ρ ≡ ρ′
\end{code}
#### Exercise: 2 stars, optional (update-shadow) #### Exercise: 2 stars, optional (update-shadow)
If we update a map $$ρ$$ at a key $$x$$ with a value $$v$$ and then If we update a map $$ρ$$ at a key $$x$$ with a value $$v$$ and then
update again with the same key $$x$$ and another value $$w$$, the update again with the same key $$x$$ and another value $$w$$, the
@ -246,18 +256,20 @@ the second update on $$ρ$$:
\begin{code} \begin{code}
postulate postulate
update-shadow : ∀ {A} (ρ : TotalMap A) (x : Id) (v w : A) (y : Id) update-shadow : ∀ {A} (ρ : TotalMap A) (x : Id) (v w : A)
→ (ρ , x ↦ v , x ↦ w) y ≡ (ρ , x ↦ w) y → (ρ , x ↦ v , x ↦ w) ≡ (ρ , x ↦ w)
\end{code} \end{code}
<div class="hidden"> <div class="hidden">
\begin{code} \begin{code}
update-shadow′ : ∀ {A} (ρ : TotalMap A) (x : Id) (v w : A) (y : Id) update-shadow′ : ∀ {A} (ρ : TotalMap A) (x : Id) (v w : A)
→ ((ρ , x ↦ v) , x ↦ w) y ≡ (ρ , x ↦ w) y → ((ρ , x ↦ v) , x ↦ w) ≡ (ρ , x ↦ w)
update-shadow′ ρ x v w y update-shadow′ ρ x v w = extensionality lemma
with x ≟ y where
... | yes x≡y = refl lemma : ∀ y → ((ρ , x ↦ v) , x ↦ w) y ≡ (ρ , x ↦ w) y
... | no x≢y = update-neq ρ x v y x≢y lemma y with x ≟ y
... | yes refl = refl
... | no x≢y = update-neq ρ x v y x≢y
\end{code} \end{code}
</div> </div>
@ -268,16 +280,18 @@ result is equal to $$ρ$$:
\begin{code} \begin{code}
postulate postulate
update-same : ∀ {A} (ρ : TotalMap A) (x y : Id) → (ρ , x ↦ ρ x) y ≡ ρ y update-same : ∀ {A} (ρ : TotalMap A) (x : Id) → (ρ , x ↦ ρ x) ≡ ρ
\end{code} \end{code}
<div class="hidden"> <div class="hidden">
\begin{code} \begin{code}
update-same′ : ∀ {A} (ρ : TotalMap A) (x y : Id) → (ρ , x ↦ ρ x) y ≡ ρ y update-same′ : ∀ {A} (ρ : TotalMap A) (x : Id) → (ρ , x ↦ ρ x) ≡ ρ
update-same′ ρ x y update-same′ ρ x = extensionality lemma
with x ≟ y where
... | yes refl = refl lemma : ∀ y → (ρ , x ↦ ρ x) y ≡ ρ y
... | no x≢y = refl lemma y with x ≟ y
... | yes refl = refl
... | no x≢y = refl
\end{code} \end{code}
</div> </div>
@ -288,19 +302,22 @@ updates.
\begin{code} \begin{code}
postulate postulate
update-permute : ∀ {A} (ρ : TotalMap A) (x : Id) (v : A) (y : Id) (w : A) (z : Id) update-permute : ∀ {A} (ρ : TotalMap A) (x : Id) (v : A) (y : Id) (w : A)
→ x ≢ y → (ρ , x ↦ v , y ↦ w) z ≡ (ρ , y ↦ w , x ↦ v) z → x ≢ y → (ρ , x ↦ v , y ↦ w) ≡ (ρ , y ↦ w , x ↦ v)
\end{code} \end{code}
<div class="hidden"> <div class="hidden">
\begin{code} \begin{code}
update-permute′ : ∀ {A} (ρ : TotalMap A) (x : Id) (v : A) (y : Id) (w : A) (z : Id) update-permute′ : ∀ {A} (ρ : TotalMap A) (x : Id) (v : A) (y : Id) (w : A)
→ x ≢ y → (ρ , x ↦ v , y ↦ w) z ≡ (ρ , y ↦ w , x ↦ v) z → x ≢ y → (ρ , x ↦ v , y ↦ w) ≡ (ρ , y ↦ w , x ↦ v)
update-permute′ {A} ρ x v y w z x≢y with x ≟ z | y ≟ z update-permute′ {A} ρ x v y w x≢y = extensionality lemma
... | yes refl | yes refl = ⊥-elim (x≢y refl) where
... | no x≢z | yes refl = sym (update-eq′ ρ z w) lemma : ∀ z → (ρ , x ↦ v , y ↦ w) z ≡ (ρ , y ↦ w , x ↦ v) z
... | yes refl | no y≢z = update-eq′ ρ z v lemma z with x ≟ z | y ≟ z
... | no x≢z | no y≢z = trans (update-neq ρ x v z x≢z) (sym (update-neq ρ y w z y≢z)) ... | yes refl | yes refl = ⊥-elim (x≢y refl)
... | no x≢z | yes refl = sym (update-eq′ ρ z w)
... | yes refl | no y≢z = update-eq′ ρ z v
... | no x≢z | no y≢z = trans (update-neq ρ x v z x≢z) (sym (update-neq ρ y w z y≢z))
\end{code} \end{code}
And a slightly different version of the same proof. And a slightly different version of the same proof.
@ -373,20 +390,20 @@ We now lift all of the basic lemmas about total maps to partial maps.
\end{code} \end{code}
\begin{code} \begin{code}
update-shadow : ∀ {A} (ρ : PartialMap A) (x : Id) (v w : A) (y : Id) update-shadow : ∀ {A} (ρ : PartialMap A) (x : Id) (v w : A)
→ (ρ , x ↦ v , x ↦ w) y ≡ (ρ , x ↦ w) y → (ρ , x ↦ v , x ↦ w) ≡ (ρ , x ↦ w)
update-shadow ρ x v w y = TotalMap.update-shadow ρ x (just v) (just w) y update-shadow ρ x v w = TotalMap.update-shadow ρ x (just v) (just w)
\end{code} \end{code}
\begin{code} \begin{code}
update-same : ∀ {A} (ρ : PartialMap A) (x : Id) (v : A) (y : Id) update-same : ∀ {A} (ρ : PartialMap A) (x : Id) (v : A)
→ ρ x ≡ just v → ρ x ≡ just v
→ (ρ , x ↦ v) y ≡ ρ y → (ρ , x ↦ v) ≡ ρ
update-same ρ x v y ρx≡v rewrite sym ρx≡v = TotalMap.update-same ρ x y update-same ρ x v ρx≡v rewrite sym ρx≡v = TotalMap.update-same ρ x
\end{code} \end{code}
\begin{code} \begin{code}
update-permute : ∀ {A} (ρ : PartialMap A) (x : Id) (v : A) (y : Id) (w : A) (z : Id) update-permute : ∀ {A} (ρ : PartialMap A) (x : Id) (v : A) (y : Id) (w : A)
→ x ≢ y → (ρ , x ↦ v , y ↦ w) z ≡ (ρ , y ↦ w , x ↦ v) z → x ≢ y → (ρ , x ↦ v , y ↦ w) ≡ (ρ , y ↦ w , x ↦ v)
update-permute ρ x v y w z x≢y = TotalMap.update-permute ρ x (just v) y (just w) z x≢y update-permute ρ x v y w x≢y = TotalMap.update-permute ρ x (just v) y (just w) x≢y
\end{code} \end{code}

43
src/StlcProp.lagda
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@ -277,7 +277,7 @@ postulate
<div class="hidden"> <div class="hidden">
\begin{code} \begin{code}
contradiction : ∀ {A : Set} → ∀ {v : A} → ¬ (_≡_ {A = Maybe A} (just v) nothing) contradiction : ∀ {X : Set} → ∀ {x : X} → ¬ (_≡_ {A = Maybe X} (just x) nothing)
contradiction () contradiction ()
∅⊢-closed′ : ∀ {M A} → ∅ ⊢ M ∈ A → closed M ∅⊢-closed′ : ∀ {M A} → ∅ ⊢ M ∈ A → closed M
@ -396,8 +396,8 @@ $$Γ ⊢ (N [ x := V ]) ∈ B$$.
\begin{code} \begin{code}
preservation-[:=] : ∀ {Γ x A N B V} preservation-[:=] : ∀ {Γ x A N B V}
→ ∅ ⊢ V ∈ A
→ (Γ , x ↦ A) ⊢ N ∈ B → (Γ , x ↦ A) ⊢ N ∈ B
→ ∅ ⊢ V ∈ A
→ Γ ⊢ (N [ x := V ]) ∈ B → Γ ⊢ (N [ x := V ]) ∈ B
\end{code} \end{code}
@ -464,29 +464,32 @@ $$Γ \vdash N [ x := V ] ∈ B$$.
- The remaining cases are similar to the application case. - The remaining cases are similar to the application case.
For one case, we need to know that weakening applies to any closed term. We need a couple of lemmas. A closed term can be weakened to any context, and just is injective.
\begin{code} \begin{code}
weaken-closed : ∀ {P A Γ} → ∅ ⊢ P ∈ A → Γ ⊢ P ∈ A weaken-closed : ∀ {V A Γ} → ∅ ⊢ V ∈ A → Γ ⊢ V ∈ A
weaken-closed {P} {A} {Γ} ⊢P = weaken Γ~Γ′ ⊢P weaken-closed {V} {A} {Γ} ⊢V = weaken Γ~Γ′ ⊢V
where where
Γ~Γ′ : ∀ {x} → x FreeIn P → ∅ x ≡ Γ x Γ~Γ′ : ∀ {x} → x FreeIn V → ∅ x ≡ Γ x
Γ~Γ′ {x} x∈P = ⊥-elim (x∉P x∈P) Γ~Γ′ {x} x∈V = ⊥-elim (x∉V x∈V)
where where
x∉P : ¬ (x FreeIn P) x∉V : ¬ (x FreeIn V)
x∉P = ∅⊢-closed ⊢P {x} x∉V = ∅⊢-closed ⊢V {x}
just-injective : ∀ {X : Set} {x y : X} → _≡_ {A = Maybe X} (just x) (just y) → x ≡ y
just-injective refl = refl
\end{code} \end{code}
\begin{code} \begin{code}
preservation-[:=] {Γ} {y} {A} ⊢P (Ax {_} {x} {B} Γx≡justB) with x ≟ y preservation-[:=] {Γ} {x} {A} {varᵀ x′} {B} {V} (Ax {.(Γ , x ↦ A)} {.x′} {.B} Γx′≡B) ⊢V with x ≟ x′
...| yes x≡y = {!!} -- weaken-closed ⊢P ...| yes x≡x′ rewrite just-injective Γx′≡B = weaken-closed ⊢V
...| no x≢y = {!!} -- Ax {_} {x} Γx≡justB ...| no x≢x′ = Ax {Γ} {x′} {B} Γx′≡B
preservation-[:=] {Γ} {y} {A} ⊢P (⇒-I {_} {x} ⊢N) with x ≟ y preservation-[:=] {Γ} {x} {A} {λᵀ x′ ∈ A′ ⇒ N′} {.A′ ⇒ B′} {V} (⇒-I {.(Γ , x ↦ A)} {.x′} {.N′} {.A′} {.B′} ⊢N′) ⊢V with x ≟ x′
...| yes x≡y = {!!} -- ⇒-I {_} {x} ⊢N ...| yes x≡x′ = {!!} -- rewrite x≡x′ | update-shadow Γ x A A′ = ⇒-I ⊢N′
...| no x≢y = {!!} -- ⇒-I {_} {x} (preservation-[:=] {_} {y} {A} ⊢P ⊢N) ...| no x≢x′ rewrite update-permute Γ x A x′ A′ x≢x′ = ⇒-I {Γ} {x′} {N′} {A′} {B′} (preservation-[:=] {(Γ , x′ ↦ A′)} {x} {A} ⊢N′ ⊢V)
preservation-[:=] ⊢P (⇒-E ⊢L ⊢M) = ⇒-E (preservation-[:=] ⊢P ⊢L) (preservation-[:=] ⊢P ⊢M) preservation-[:=] (⇒-E ⊢L ⊢M) ⊢V = ⇒-E (preservation-[:=] ⊢L ⊢V) (preservation-[:=] ⊢M ⊢V)
preservation-[:=] ⊢P 𝔹-I₁ = 𝔹-I₁ preservation-[:=] 𝔹-I₁ ⊢V = 𝔹-I₁
preservation-[:=] ⊢P 𝔹-I₂ = 𝔹-I₂ preservation-[:=] 𝔹-I₂ ⊢V = 𝔹-I₂
preservation-[:=] ⊢P (𝔹-E ⊢L ⊢M ⊢N) = 𝔹-E (preservation-[:=] ⊢P ⊢L) (preservation-[:=] ⊢P ⊢M) (preservation-[:=] ⊢P ⊢N) preservation-[:=] (𝔹-E ⊢L ⊢M ⊢N) ⊢V = 𝔹-E (preservation-[:=] ⊢L ⊢V) (preservation-[:=] ⊢M ⊢V) (preservation-[:=] ⊢N ⊢V)
{- {-
@ -551,7 +554,7 @@ _Proof_: By induction on the derivation of $$\vdash t : T$$.
\begin{code} \begin{code}
preservation (Ax x₁) () preservation (Ax x₁) ()
preservation (⇒-I ⊢N) () preservation (⇒-I ⊢N) ()
preservation (⇒-E (⇒-I ⊢N) ⊢V) (β⇒ valueV) = preservation-[:=] ⊢V ⊢N preservation (⇒-E (⇒-I ⊢N) ⊢V) (β⇒ valueV) = preservation-[:=] ⊢N ⊢V
preservation (⇒-E ⊢L ⊢M) (γ⇒₁ L⟹L′) with preservation ⊢L L⟹L′ preservation (⇒-E ⊢L ⊢M) (γ⇒₁ L⟹L′) with preservation ⊢L L⟹L′
...| ⊢L′ = ⇒-E ⊢L′ ⊢M ...| ⊢L′ = ⇒-E ⊢L′ ⊢M
preservation (⇒-E ⊢L ⊢M) (γ⇒₂ valueL M⟹M′) with preservation ⊢M M⟹M′ preservation (⇒-E ⊢L ⊢M) (γ⇒₂ valueL M⟹M′) with preservation ⊢M M⟹M′