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Arthur Azevedo de Amorim
Tutorial POPL20
Commits
8aa143bb
Commit
8aa143bb
authored
Jan 17, 2020
by
Amin Timany
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coqdoc for safety.v
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theories/safety.v
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theories/safety.v
View file @
8aa143bb
From
iris
.
heap_lang
Require
Export
adequacy
.
From
tutorial_popl20
Require
Export
fundamental
.
(** * Safety of semantic types and type safety based on that
We prove that any _closed_ expression that is semantically typed
is safe, i.e., it does not crash. Based on this theorem we then
prove _type safety_, i.e., any _closed_ syntactically welltyped
program does not get stuck. Type safety is a simple consequence of
the fundamental theorem of logical relations together with the
safety for semantic typing.
*)
(** The following lemma states that given a closed program [e], heap
[σ], and _Coq_ predicate [φ : val → Prop], if there is a semantic
type [A] such that [A] implies [φ], and [e] is semantically typed
at type [A], then we have [adequate NotStuck e σ (λ v σ, φ
v)]. The proposition [adequate NotStuck e σ (λ v σ, φ v)] means
that [e], starting in heap [σ] does not get stuck, and if [e]
reduces to a value [v], we have [φ v]. *)
Lemma
sem_gen_type_safety
`
{
heapPreG
Σ
}
e
σ
φ
:
(
∀
`
{
heapG
Σ
},
∃
A
:
sem_ty
Σ
,
(
∀
v
,
A
v

∗
⌜φ
v
⌝
)
∧
(
∅
⊨
e
:
A
))
→
adequate
NotStuck
e
σ
(
λ
v
σ
,
φ
v
).
...
...
@@ 13,6 +31,9 @@ Proof.
by
iIntros
;
iApply
HA
.
Qed
.
(** This lemma states that semantically typed closed programs do not
get stuck. It is a simple consequence of the lemma
[sem_gen_type_safety] above. *)
Lemma
sem_type_safety
`
{
heapPreG
Σ
}
e
σ
es
σ
'
e'
:
(
∀
`
{
heapG
Σ
},
∃
A
,
∅
⊨
e
:
A
)
→
rtc
erased_step
([
e
],
σ
)
(
es
,
σ
'
)
→
e'
∈
es
→
...
...
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