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Jeremy Siek 2020-07-17 10:08:38 -04:00
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24
src/plfa/part2/Subtyping.lagda.md
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@ -616,7 +616,7 @@ terms. Thus, the `subst` function is split into two parts: a raw
`subst` function that operators on terms and a `subst-pres` lemma that
proves that substitution preserves types. We define `subst` in this
section and postpone `subst-pres` to the
[Preservation](#subtyping-pres) section. Likewise for `rename`.
[Preservation](#subtyping-preservation) section. Likewise for `rename`.
We begin by defining the `ext` function on renamings.
```
@ -877,13 +877,14 @@ typed (C-suc c) = ⊢suc (typed c)
typed (C-rcd ⊢Ms dks As<:Bs) = ⊢<: (⊢rcd ⊢Ms dks) As<:Bs
```
## Progress
## Progress <a name="subtyping-progress"></a>
The Progress theorem states that a well-typed term may either take a
reduction step or it is already a value. The proof of Progress is like
the one in the [Properties](./Properties.lagda.md); it proceeds by
induction on the typing derivation and most of the cases remain the
same. Below we discuss the new cases for records and subsumption.
the one in the [Properties]({{ site.baseurl }}/Properties/); it
proceeds by induction on the typing derivation and most of the cases
remain the same. Below we discuss the new cases for records and
subsumption.
```
data Progress (M : Term) : Set where
@ -924,19 +925,18 @@ progress (⊢case ⊢L ⊢M ⊢N) with progress ⊢L
... | done VL with canonical ⊢L VL
... | C-zero = step β-zero
... | C-suc CL = step (β-suc (value CL))
progress (⊢μ ⊢M) = step β-μ
progress (⊢μ ⊢M) = step β-μ
progress (⊢# {n}{Γ}{A}{M}{l}{ls}{As}{i}{d} ⊢M ls[i]=l As[i]=A)
with progress ⊢M
... | step M—→M′ = step (ξ-# M—→M′)
... | step M—→M′ = step (ξ-# M—→M′)
... | done VM
with canonical ⊢M VM
... | C-rcd {ks = ms}{As = Bs} ⊢Ms _ (<:-rcd ls⊆ms _)
with lookup-⊆ {i = i} ls⊆ms
... | ⟨ k , ls[i]=ms[k] ⟩ = step (β-# {j = k} (trans (sym ls[i]=ms[k]) ls[i]=l))
progress (⊢rcd x d) = done V-rcd
progress (⊢<: {A = A}{B} ⊢M A<:B) = progress ⊢M
... | ⟨ k , ls[i]=ms[k] ⟩ = step (β-# {j = k}(trans (sym ls[i]=ms[k]) ls[i]=l))
progress (⊢rcd x d) = done V-rcd
progress (⊢<: {A = A}{B} ⊢M A<:B) = progress ⊢M
```
* Case `⊢#`: We have `Γ ⊢ M ⦂ ｛ ls ⦂ As ｝`, `lookup ls i ≡ l`, and `lookup As i ≡ A`.
By the induction hypothesis, either `M —→ M′` or `M` is a value. In the later case we
conclude that `M # l —→ M′ # l` by rule `ξ-#`. On the other hand, if `M` is a value,
@ -950,7 +950,7 @@ progress (⊢<: {A = A}{B} ⊢M A<:B) = progress ⊢M
* Case `⊢<:`: we invoke the induction hypothesis on sub-derivation of `Γ ⊢ M ⦂ A`.
## Preservation <a name="subtyping-pres"></a>
## Preservation <a name="subtyping-preservation"></a>
In this section we prove that when a well-typed term takes a reduction
step, the result is another well-typed term with the same type.