halfway through preservation of substitution
This commit is contained in:
wadler
parent
fe5bf01e6b
commit
6ee9d8dd40
1 changed files with 56 additions and 54 deletions
|
|
@ -104,19 +104,19 @@ _Proof_: By induction on the derivation of $$\vdash t : A$$.
|
|||
\begin{code}
|
||||
progress (Ax ())
|
||||
progress (⇒-I ⊢N) = inj₁ value-λᵀ
|
||||
progress (⇒-E ⊢L ⊢M) with progress ⊢L
|
||||
... | inj₂ (_ , L⟹L′) = inj₂ (_ , γ⇒₁ L⟹L′)
|
||||
progress (⇒-E {Γ} {L} {M} {A} {B} ⊢L ⊢M) with progress ⊢L
|
||||
... | inj₂ (L′ , L⟹L′) = inj₂ (L′ ·ᵀ M , γ⇒₁ L⟹L′)
|
||||
... | inj₁ valueL with progress ⊢M
|
||||
... | inj₂ (_ , M⟹M′) = inj₂ (_ , γ⇒₂ valueL M⟹M′)
|
||||
... | inj₂ (M′ , M⟹M′) = inj₂ (L ·ᵀ M′ , γ⇒₂ valueL M⟹M′)
|
||||
... | inj₁ valueM with canonicalFormsLemma ⊢L valueL
|
||||
... | canonical-λᵀ = inj₂ (_ , β⇒ valueM)
|
||||
... | canonical-λᵀ {x} {.A} {N} {.B} = inj₂ ((N [ x := M ]) , β⇒ valueM)
|
||||
progress 𝔹-I₁ = inj₁ value-trueᵀ
|
||||
progress 𝔹-I₂ = inj₁ value-falseᵀ
|
||||
progress (𝔹-E ⊢L ⊢M ⊢N) with progress ⊢L
|
||||
... | inj₂ (_ , L⟹L′) = inj₂ (_ , γ𝔹 L⟹L′)
|
||||
progress (𝔹-E {Γ} {L} {M} {N} {A} ⊢L ⊢M ⊢N) with progress ⊢L
|
||||
... | inj₂ (L′ , L⟹L′) = inj₂ ((ifᵀ L′ then M else N) , γ𝔹 L⟹L′)
|
||||
... | inj₁ valueL with canonicalFormsLemma ⊢L valueL
|
||||
... | canonical-trueᵀ = inj₂ (_ , β𝔹₁)
|
||||
... | canonical-falseᵀ = inj₂ (_ , β𝔹₂)
|
||||
... | canonical-trueᵀ = inj₂ (M , β𝔹₁)
|
||||
... | canonical-falseᵀ = inj₂ (N , β𝔹₂)
|
||||
\end{code}
|
||||
|
||||
#### Exercise: 3 stars, optional (progress_from_term_ind)
|
||||
|
|
@ -417,73 +417,70 @@ _Lemma_: If $$\Gamma,x:U \vdash t : T$$ and $$\vdash v : U$$, then
|
|||
$$\Gamma \vdash [x:=v]t : T$$.
|
||||
|
||||
\begin{code}
|
||||
{-
|
||||
[:=]-preserves-⊢ : ∀ {Γ x A t v B}
|
||||
→ ∅ ⊢ v ∶ A
|
||||
→ Γ , x ∶ A ⊢ t ∶ B
|
||||
→ Γ , x ∶ A ⊢ [ x := v ] t ∶ B
|
||||
-}
|
||||
[:=]-preserves-⊢ : ∀ {Γ x A N P B}
|
||||
→ ∅ ⊢ P ∈ A
|
||||
→ Γ , x ↦ A ⊢ N ∈ B
|
||||
→ Γ , x ↦ A ⊢ N [ x := P ] ∈ B
|
||||
\end{code}
|
||||
|
||||
One technical subtlety in the statement of the lemma is that
|
||||
we assign $$v$$ the type $$U$$ in the _empty_ context---in other
|
||||
words, we assume $$v$$ is closed. This assumption considerably
|
||||
simplifies the $$abs$$ case of the proof (compared to assuming
|
||||
$$\Gamma \vdash v : U$$, which would be the other reasonable assumption
|
||||
we assign $$P$$ the type $$A$$ in the _empty_ context---in other
|
||||
words, we assume $$P$$ is closed. This assumption considerably
|
||||
simplifies the $$λᵀ$$ case of the proof (compared to assuming
|
||||
$$Γ ⊢ P ∈ A$$, which would be the other reasonable assumption
|
||||
at this point) because the context invariance lemma then tells us
|
||||
that $$v$$ has type $$U$$ in any context at all---we don't have to
|
||||
worry about free variables in $$v$$ clashing with the variable being
|
||||
introduced into the context by $$abs$$.
|
||||
that $$P$$ has type $$A$$ in any context at all---we don't have to
|
||||
worry about free variables in $$P$$ clashing with the variable being
|
||||
introduced into the context by $$λᵀ$$.
|
||||
|
||||
The substitution lemma can be viewed as a kind of "commutation"
|
||||
property. Intuitively, it says that substitution and typing can
|
||||
be done in either order: we can either assign types to the terms
|
||||
$$t$$ and $$v$$ separately (under suitable contexts) and then combine
|
||||
$$N$$ and $$P$$ separately (under suitable contexts) and then combine
|
||||
them using substitution, or we can substitute first and then
|
||||
assign a type to $$ $$x:=v$$ t $$---the result is the same either
|
||||
assign a type to $$N [ x := P ]$$---the result is the same either
|
||||
way.
|
||||
|
||||
_Proof_: We show, by induction on $$t$$, that for all $$T$$ and
|
||||
$$\Gamma$$, if $$\Gamma,x:U \vdash t : T$$ and $$\vdash v : U$$, then $$\Gamma
|
||||
\vdash $$x:=v$$t : T$$.
|
||||
_Proof_: We show, by induction on $$N$$, that for all $$A$$ and
|
||||
$$Γ$$, if $$Γ , x ↦ A \vdash N ∈ B$$ and $$∅ ⊢ P ∈ B$$, then
|
||||
$$Γ \vdash N [ x := P ] ∈ B$$.
|
||||
|
||||
- If $$t$$ is a variable there are two cases to consider,
|
||||
depending on whether $$t$$ is $$x$$ or some other variable.
|
||||
- If $$N$$ is a variable there are two cases to consider,
|
||||
depending on whether $$N$$ is $$x$$ or some other variable.
|
||||
|
||||
- If $$t = x$$, then from the fact that $$\Gamma, x:U \vdash x :
|
||||
T$$ we conclude that $$U = T$$. We must show that $$[x:=v]x =
|
||||
v$$ has type $$T$$ under $$\Gamma$$, given the assumption that
|
||||
$$v$$ has type $$U = T$$ under the empty context. This
|
||||
- If $$N = x$$, then from the fact that $$Γ , x ↦ A ⊢ N ∈ B$$
|
||||
we conclude that $$A = B$$. We must show that $$x [ x := P] =
|
||||
P$$ has type $$A$$ under $$Γ$$, given the assumption that
|
||||
$$P$$ has type $$A$$ under the empty context. This
|
||||
follows from context invariance: if a closed term has type
|
||||
$$T$$ in the empty context, it has that type in any context.
|
||||
$$A$$ in the empty context, it has that type in any context.
|
||||
|
||||
- If $$t$$ is some variable $$y$$ that is not equal to $$x$$, then
|
||||
we need only note that $$y$$ has the same type under $$\Gamma,
|
||||
x:U$$ as under $$\Gamma$$.
|
||||
- If $$N$$ is some variable $$y$$ that is not equal to $$x$$, then
|
||||
we need only note that $$y$$ has the same type under $$Γ , x ↦ A$$
|
||||
as under $$Γ$$.
|
||||
|
||||
- If $$t$$ is an abstraction $$\lambda y:t_{11}. t_{12}$$, then the IH tells us,
|
||||
for all $$\Gamma'$$ and $$T'$$, that if $$\Gamma',x:U \vdash t_{12}:T'$$
|
||||
and $$\vdash v:U$$, then $$\Gamma' \vdash [x:=v]t_{12}:T'$$.
|
||||
- If $$N$$ is an abstraction $$λᵀ y ∈ A′ ⇒ N′$$, then the IH tells us,
|
||||
for all $$Γ′$$́ and $$A′$$, that if $$Γ′ , x ↦ A ⊢ N′ ∈ B′$$
|
||||
and $$∅ ⊢ P ∈ A$$, then $$Γ′ ⊢ N′ [ x := P ] ∈ B′$$.
|
||||
|
||||
The substitution in the conclusion behaves differently
|
||||
depending on whether $$x$$ and $$y$$ are the same variable.
|
||||
|
||||
First, suppose $$x = y$$. Then, by the definition of
|
||||
substitution, $$[x:=v]t = t$$, so we just need to show $$\Gamma \vdash
|
||||
t : T$$. But we know $$\Gamma,x:U \vdash t : T$$, and, since $$y$$
|
||||
does not appear free in $$\lambda y:t_{11}. t_{12}$$, the context invariance
|
||||
lemma yields $$\Gamma \vdash t : T$$.
|
||||
First, suppose $$x ≡ y$$. Then, by the definition of
|
||||
substitution, $$N [ x := P] = N$$, so we just need to show $$Γ ⊢ N ∈ B$$.
|
||||
But we know $$Γ , x ↦ A ⊢ N ∈ B$$ and, since $$x ≡ y$$
|
||||
does not appear free in $$λᵀ y ∈ A′ ⇒ N′$$, the context invariance
|
||||
lemma yields $$Γ ⊢ N ∈ B$$.
|
||||
|
||||
Second, suppose $$x \neq y$$. We know $$\Gamma,x:U,y:t_{11} \vdash
|
||||
t_{12}:t_{12}$$ by inversion of the typing relation, from which
|
||||
$$\Gamma,y:t_{11},x:U \vdash t_{12}:t_{12}$$ follows by the context invariance
|
||||
lemma, so the IH applies, giving us $$\Gamma,y:t_{11} \vdash
|
||||
[x:=v]t_{12}:t_{12}$$. By $$abs$$, $$\Gamma \vdash \lambda y:t_{11}.
|
||||
[x:=v]t_{12}:t_{11}\to t_{12}$$, and by the definition of substitution (noting
|
||||
that $$x \neq y$$), $$\Gamma \vdash \lambda y:t_{11}. [x:=v]t_{12}:t_{11}\to
|
||||
t_{12}$$ as required.
|
||||
Second, suppose $$x ≢ y$$. We know $$Γ , x ↦ A , y ↦ A′ ⊢ N' ∈ B′$$
|
||||
by inversion of the typing relation, from which
|
||||
$$Γ , y ↦ A′ , x ↦ A ⊢ N′ ∈ B′$$ follows by update permute,
|
||||
so the IH applies, giving us $$Γ , y ↦ A′ ⊢ N′ [ x := P ] ∈ B′$$
|
||||
By $$⇒-I$$, we have $$Γ ⊢ λᵀ y ∈ A′ ⇒ (N′ [ x := P ]) ∈ A′ ⇒ B′$$
|
||||
and the definition of substitution (noting $$x ≢ y$$) gives
|
||||
$$Γ ⊢ (λᵀ y ∈ A′ ⇒ N′) [ x := P ] ∈ A′ ⇒ B′$$ as required.
|
||||
|
||||
- If $$t$$ is an application $$t_1 t_2$$, the result follows
|
||||
- If $$N$$ is an application $$L ·ᵀ M$$, the result follows
|
||||
straightforwardly from the definition of substitution and the
|
||||
induction hypotheses.
|
||||
|
||||
|
|
@ -492,7 +489,7 @@ $$\Gamma$$, if $$\Gamma,x:U \vdash t : T$$ and $$\vdash v : U$$, then $$\Gamma
|
|||
One more technical note: This proof is a rare case where an
|
||||
induction on terms, rather than typing derivations, yields a
|
||||
simpler argument. The reason for this is that the assumption
|
||||
$$update Gamma x U \vdash t : T$$ is not completely generic, in the
|
||||
$$Γ , x ↦ A ⊢ N ∈ B$$ is not completely generic, in the
|
||||
sense that one of the "slots" in the typing relation---namely the
|
||||
context---is not just a variable, and this means that Agda's
|
||||
native induction tactic does not give us the induction hypothesis
|
||||
|
|
@ -501,6 +498,11 @@ generalization is a little tricky. The term $$t$$, on the other
|
|||
hand, _is_ completely generic.
|
||||
|
||||
\begin{code}
|
||||
[:=]-preserves-⊢ {Γ} {x} v∶A (var y y∈Γ) with x ≟ y
|
||||
... | yes x=y = {!!}
|
||||
... | no x≠y = {!!}
|
||||
|
||||
|
||||
{-
|
||||
[:=]-preserves-⊢ {Γ} {x} v∶A (var y y∈Γ) with x ≟ y
|
||||
... | yes x=y = {!!}
|
||||
|
|
|
|||
Loading…
Reference in a new issue