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examples
Commits
9dc89990
Commit
9dc89990
authored
Jan 11, 2019
by
Robbert Krebbers
Browse files
Make use of nested specialization patterns.
parent
380623ef
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theories/logrel_heaplang/ltyping.v
View file @
9dc89990
...
...
@@ -225,23 +225,22 @@ Section types_properties.
(
Γ
⊨
e1
:
A1
→
A2
)
-
∗
(
Γ
⊨
e2
:
A1
)
-
∗
Γ
⊨
e1
e2
:
A2
.
Proof
.
iIntros
"#H1 #H2"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H1"
with
"HΓ"
).
iSpecialize
(
"H2"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"H2"
)
;
iIntros
(
w
)
"HA1"
.
wp_apply
(
wp_wand
with
"H1"
)
;
iIntros
(
v
)
"#HA"
.
by
iApply
"HA"
.
wp_apply
(
wp_wand
with
"(H2 [//])"
)
;
iIntros
(
w
)
"#HA1"
.
wp_apply
(
wp_wand
with
"(H1 [//])"
)
;
iIntros
(
v
)
"#HA"
.
by
iApply
"HA"
.
Qed
.
Lemma
ltyped_fold
Γ
e
(
B
:
ltyC
-
n
>
ltyC
)
:
(
Γ
⊨
e
:
B
(
lty_rec
B
))
-
∗
Γ
⊨
e
:
lty_rec
B
.
Proof
.
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"
H
"
)
;
iIntros
(
w
)
"#HB"
.
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
wp_apply
(
wp_wand
with
"
(H [//])
"
)
;
iIntros
(
w
)
"#HB"
.
by
iEval
(
rewrite
lty_rec_unfold
/
lty_car
/=).
Qed
.
Lemma
ltyped_unfold
Γ
e
(
B
:
ltyC
-
n
>
ltyC
)
:
(
Γ
⊨
e
:
lty_rec
B
)
-
∗
Γ
⊨
rec_unfold
e
:
B
(
lty_rec
B
).
Proof
.
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"
H
"
)
;
iIntros
(
w
)
"HB"
.
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
wp_apply
(
wp_wand
with
"
(H [//])
"
)
;
iIntros
(
w
)
"HB"
.
iEval
(
rewrite
lty_rec_unfold
/
lty_car
/=)
in
"HB"
.
by
wp_lam
.
Qed
.
...
...
@@ -252,85 +251,81 @@ Section types_properties.
Qed
.
Lemma
ltyped_tapp
Γ
e
C
A
:
(
Γ
⊨
e
:
∀
A
,
C
A
)
-
∗
Γ
⊨
e
#()
:
C
A
.
Proof
.
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"
H
"
)
;
iIntros
(
w
)
"#HB"
.
by
iApply
(
"HB"
$!
A
).
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
wp_apply
(
wp_wand
with
"
(H [//])
"
)
;
iIntros
(
w
)
"#HB"
.
by
iApply
(
"HB"
$!
A
).
Qed
.
Lemma
ltyped_pack
Γ
e
C
A
:
(
Γ
⊨
e
:
C
A
)
-
∗
Γ
⊨
e
:
∃
A
,
C
A
.
Proof
.
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"
H
"
)
;
iIntros
(
w
)
"#HB"
.
by
iExists
A
.
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
wp_apply
(
wp_wand
with
"
(H [//])
"
)
;
iIntros
(
w
)
"#HB"
.
by
iExists
A
.
Qed
.
Lemma
ltyped_unpack
Γ
e1
e2
C
B
:
(
Γ
⊨
e1
:
∃
A
,
C
A
)
-
∗
(
∀
A
,
Γ
⊨
e2
:
C
A
→
B
)
-
∗
Γ
⊨
e2
e1
:
B
.
Proof
.
iIntros
"#H1 #H2"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H1"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"H1"
)
;
iIntros
(
v
)
;
iDestruct
1
as
(
A
)
"#HC"
.
iSpecialize
(
"H2"
$!
A
with
"HΓ"
).
wp_apply
(
wp_wand
with
"H2"
)
;
iIntros
(
w
)
"HCB"
.
by
iApply
"HCB"
.
iIntros
"#H1 #H2"
(
vs
)
"!# #HΓ /="
.
wp_apply
(
wp_wand
with
"(H1 [//])"
)
;
iIntros
(
v
)
;
iDestruct
1
as
(
A
)
"#HC"
.
wp_apply
(
wp_wand
with
"(H2 [//])"
)
;
iIntros
(
w
)
"HCB"
.
by
iApply
"HCB"
.
Qed
.
Lemma
ltyped_pair
Γ
e1
e2
A1
A2
:
(
Γ
⊨
e1
:
A1
)
-
∗
(
Γ
⊨
e2
:
A2
)
-
∗
Γ
⊨
(
e1
,
e2
)
:
A1
*
A2
.
Proof
.
iIntros
"#H1 #H2"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H1"
with
"HΓ"
).
iSpecialize
(
"H2"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"H2"
)
;
iIntros
(
w2
)
"#HA2"
.
wp_apply
(
wp_wand
with
"H1"
)
;
iIntros
(
w1
)
"#HA1"
.
wp_apply
(
wp_wand
with
"(H2 [//])"
)
;
iIntros
(
w2
)
"#HA2"
.
wp_apply
(
wp_wand
with
"(H1 [//])"
)
;
iIntros
(
w1
)
"#HA1"
.
wp_pures
.
iExists
w1
,
w2
.
auto
.
Qed
.
Lemma
ltyped_fst
Γ
e
A1
A2
:
(
Γ
⊨
e
:
A1
*
A2
)
-
∗
Γ
⊨
Fst
e
:
A1
.
Proof
.
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"
H
"
)
;
iIntros
(
w
).
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
wp_apply
(
wp_wand
with
"
(H [//])
"
)
;
iIntros
(
w
).
iDestruct
1
as
(
w1
w2
->)
"[??]"
.
by
wp_pures
.
Qed
.
Lemma
ltyped_snd
Γ
e
A1
A2
:
(
Γ
⊨
e
:
A1
*
A2
)
-
∗
Γ
⊨
Snd
e
:
A2
.
Proof
.
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"
H
"
)
;
iIntros
(
w
).
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
wp_apply
(
wp_wand
with
"
(H [//])
"
)
;
iIntros
(
w
).
iDestruct
1
as
(
w1
w2
->)
"[??]"
.
by
wp_pures
.
Qed
.
Lemma
ltyped_injl
Γ
e
A1
A2
:
(
Γ
⊨
e
:
A1
)
-
∗
Γ
⊨
InjL
e
:
A1
+
A2
.
Proof
.
iIntros
"#H"
(
vs
)
"!# HΓ /="
.
iSpecialize
(
"H"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"
H
"
)
;
iIntros
(
w
)
"#HA"
.
iIntros
"#H"
(
vs
)
"!#
#
HΓ /="
.
wp_apply
(
wp_wand
with
"
(H [//])
"
)
;
iIntros
(
w
)
"#HA"
.
wp_pures
.
iLeft
.
iExists
w
.
auto
.
Qed
.
Lemma
ltyped_injr
Γ
e
A1
A2
:
(
Γ
⊨
e
:
A2
)
-
∗
Γ
⊨
InjR
e
:
A1
+
A2
.
Proof
.
iIntros
"#H"
(
vs
)
"!# HΓ /="
.
iSpecialize
(
"H"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"
H
"
)
;
iIntros
(
w
)
"#HA"
.
iIntros
"#H"
(
vs
)
"!#
#
HΓ /="
.
wp_apply
(
wp_wand
with
"
(H [//])
"
)
;
iIntros
(
w
)
"#HA"
.
wp_pures
.
iRight
.
iExists
w
.
auto
.
Qed
.
Lemma
ltyped_case
Γ
e
e1
e2
A1
A2
B
:
(
Γ
⊨
e
:
A1
+
A2
)
-
∗
(
Γ
⊨
e1
:
A1
→
B
)
-
∗
(
Γ
⊨
e2
:
A2
→
B
)
-
∗
Γ
⊨
Case
e
e1
e2
:
B
.
Proof
.
iIntros
"#H #H1 #H2"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H"
with
"HΓ"
).
iSpecialize
(
"H1"
with
"HΓ"
).
iSpecialize
(
"H2"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"H"
)
;
iIntros
(
w
)
"#[HA|HA]"
.
iIntros
"#H #H1 #H2"
(
vs
)
"!# #HΓ /="
.
wp_apply
(
wp_wand
with
"(H [//])"
)
;
iIntros
(
w
)
"#[HA|HA]"
.
-
iDestruct
"HA"
as
(
w1
->)
"HA"
.
wp_pures
.
wp_apply
(
wp_wand
with
"H1"
)
;
iIntros
(
v
)
"#HAB"
.
by
iApply
"HAB"
.
wp_apply
(
wp_wand
with
"
(
H1
[//])
"
)
;
iIntros
(
v
)
"#HAB"
.
by
iApply
"HAB"
.
-
iDestruct
"HA"
as
(
w2
->)
"HA"
.
wp_pures
.
wp_apply
(
wp_wand
with
"H2"
)
;
iIntros
(
v
)
"#HAB"
.
by
iApply
"HAB"
.
wp_apply
(
wp_wand
with
"
(
H2
[//])
"
)
;
iIntros
(
v
)
"#HAB"
.
by
iApply
"HAB"
.
Qed
.
Lemma
ltyped_un_op
Γ
e
op
A
B
:
LTyUnOp
op
A
B
→
(
Γ
⊨
e
:
A
)
-
∗
Γ
⊨
UnOp
op
e
:
B
.
Proof
.
intros
?.
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"
H
"
)
;
iIntros
(
v
)
"#HA"
.
intros
?.
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
wp_apply
(
wp_wand
with
"
(H [//])
"
)
;
iIntros
(
v
)
"#HA"
.
iDestruct
(
lty_un_op
with
"HA"
)
as
(
w
?)
"#HB"
.
by
wp_unop
.
Qed
.
Lemma
ltyped_bin_op
Γ
e1
e2
op
A1
A2
B
:
LTyBinOp
op
A1
A2
B
→
(
Γ
⊨
e1
:
A1
)
-
∗
(
Γ
⊨
e2
:
A2
)
-
∗
Γ
⊨
BinOp
op
e1
e2
:
B
.
Proof
.
intros
.
iIntros
"#H1 #H2"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H1"
with
"HΓ"
).
iSpecialize
(
"H2"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"H2"
)
;
iIntros
(
v2
)
"#HA2"
.
wp_apply
(
wp_wand
with
"H1"
)
;
iIntros
(
v1
)
"#HA1"
.
wp_apply
(
wp_wand
with
"(H2 [//])"
)
;
iIntros
(
v2
)
"#HA2"
.
wp_apply
(
wp_wand
with
"(H1 [//])"
)
;
iIntros
(
v1
)
"#HA1"
.
iDestruct
(
lty_bin_op
with
"HA1 HA2"
)
as
(
w
?)
"#HB"
.
by
wp_binop
.
Qed
.
...
...
@@ -338,21 +333,21 @@ Section types_properties.
(
Γ
⊨
e
:
lty_bool
)
-
∗
(
Γ
⊨
e1
:
B
)
-
∗
(
Γ
⊨
e2
:
B
)
-
∗
Γ
⊨
(
if
:
e
then
e1
else
e2
)
:
B
.
Proof
.
iIntros
"#H #H1 #H2"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H"
with
"HΓ"
).
iIntros
"#H #H1 #H2"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H1"
with
"HΓ"
).
iSpecialize
(
"H2"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"
H
"
)
;
iIntros
(
w
).
iDestruct
1
as
([])
"->"
;
by
wp_if
.
wp_apply
(
wp_wand
with
"
(H [//])
"
)
;
iIntros
(
w
).
iDestruct
1
as
([])
"->"
;
by
wp_if
.
Qed
.
Lemma
ltyped_fork
Γ
e
:
(
Γ
⊨
e
:
())
-
∗
Γ
⊨
Fork
e
:
().
Proof
.
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H"
with
"HΓ"
).
wp_apply
wp_fork
;
last
done
.
by
iApply
(
wp_wand
with
"
H
"
).
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
wp_apply
wp_fork
;
last
done
.
by
iApply
(
wp_wand
with
"
(H [//])
"
).
Qed
.
Lemma
ltyped_alloc
Γ
e
A
:
(
Γ
⊨
e
:
A
)
-
∗
Γ
⊨
ref
e
:
ref
A
.
Proof
.
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H"
with
"HΓ"
).
wp_bind
(
subst_map
_
e
).
iApply
(
wp_wand
with
"
H
"
)
;
iIntros
(
w
)
"HA"
.
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
wp_bind
(
subst_map
_
e
).
iApply
(
wp_wand
with
"
(H [//])
"
)
;
iIntros
(
w
)
"HA"
.
iApply
wp_fupd
.
wp_alloc
l
as
"Hl"
.
iMod
(
inv_alloc
(
tyN
.@
l
)
_
(
∃
v
,
l
↦
v
∗
A
v
)%
I
with
"[Hl HA]"
)
as
"#?"
.
{
iExists
w
.
iFrame
.
}
...
...
@@ -360,8 +355,8 @@ Section types_properties.
Qed
.
Lemma
ltyped_load
Γ
e
A
:
(
Γ
⊨
e
:
ref
A
)
-
∗
Γ
⊨
!
e
:
A
.
Proof
.
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H"
with
"HΓ"
).
wp_bind
(
subst_map
_
e
).
iApply
(
wp_wand
with
"
H
"
)
;
iIntros
(
w
).
iIntros
"#H"
(
vs
)
"!# #HΓ /="
.
wp_bind
(
subst_map
_
e
).
iApply
(
wp_wand
with
"
(H [//])
"
)
;
iIntros
(
w
).
iDestruct
1
as
(
l
->)
"#?"
.
iInv
(
tyN
.@
l
)
as
(
v
)
"[>Hl HA]"
.
wp_load
.
eauto
10
.
Qed
.
...
...
@@ -369,18 +364,16 @@ Section types_properties.
(
Γ
⊨
e1
:
ref
A
)
-
∗
(
Γ
⊨
e2
:
A
)
-
∗
Γ
⊨
(
e1
<-
e2
)
:
().
Proof
.
iIntros
"#H1 #H2"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H1"
with
"HΓ"
).
iSpecialize
(
"H2"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"H2"
)
;
iIntros
(
w2
)
"#HA"
.
wp_apply
(
wp_wand
with
"H1"
)
;
iIntros
(
w1
)
;
iDestruct
1
as
(
l
->)
"#?"
.
wp_apply
(
wp_wand
with
"(H2 [//])"
)
;
iIntros
(
w2
)
"#HA"
.
wp_apply
(
wp_wand
with
"(H1 [//])"
)
;
iIntros
(
w1
)
;
iDestruct
1
as
(
l
->)
"#?"
.
iInv
(
tyN
.@
l
)
as
(
v
)
"[>Hl _]"
.
wp_store
.
eauto
10
.
Qed
.
Lemma
ltyped_faa
Γ
e1
e2
:
(
Γ
⊨
e1
:
ref
lty_int
)
-
∗
(
Γ
⊨
e2
:
lty_int
)
-
∗
Γ
⊨
FAA
e1
e2
:
lty_int
.
Proof
.
iIntros
"#H1 #H2"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H1"
with
"HΓ"
).
iSpecialize
(
"H2"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"H2"
)
;
iIntros
(
w2
)
;
iDestruct
1
as
(
n
)
"->"
.
wp_apply
(
wp_wand
with
"H1"
)
;
iIntros
(
w1
)
;
iDestruct
1
as
(
l
->)
"#?"
.
wp_apply
(
wp_wand
with
"(H2 [//])"
)
;
iIntros
(
w2
)
;
iDestruct
1
as
(
n
)
"->"
.
wp_apply
(
wp_wand
with
"(H1 [//])"
)
;
iIntros
(
w1
)
;
iDestruct
1
as
(
l
->)
"#?"
.
iInv
(
tyN
.@
l
)
as
(
v
)
"[>Hl Hv]"
;
iDestruct
"Hv"
as
(
n'
)
"> ->"
.
wp_faa
.
iModIntro
.
eauto
10
.
Qed
.
...
...
@@ -388,12 +381,11 @@ Section types_properties.
LTyUnboxed
A
→
(
Γ
⊨
e1
:
ref
A
)
-
∗
(
Γ
⊨
e2
:
A
)
-
∗
(
Γ
⊨
e3
:
A
)
-
∗
Γ
⊨
CAS
e1
e2
e3
:
lty_bool
.
Proof
.
intros
.
iIntros
"#H1 #H2 #H3"
(
vs
)
"!# #HΓ /="
.
iSpecialize
(
"H1"
with
"HΓ"
).
iSpecialize
(
"H2"
with
"HΓ"
).
iSpecialize
(
"H3"
with
"HΓ"
).
wp_apply
(
wp_wand
with
"H3"
)
;
iIntros
(
w3
)
"HA3"
.
wp_apply
(
wp_wand
with
"H2"
)
;
iIntros
(
w2
)
"HA2"
.
intros
.
iIntros
"#H1 #H2 #H3"
(
vs
)
"!# #HΓ /="
.
wp_apply
(
wp_wand
with
"(H3 [//])"
)
;
iIntros
(
w3
)
"HA3"
.
wp_apply
(
wp_wand
with
"(H2 [//])"
)
;
iIntros
(
w2
)
"HA2"
.
iDestruct
(
lty_unboxed
with
"HA2"
)
as
%?.
wp_apply
(
wp_wand
with
"H1"
)
;
iIntros
(
w1
)
;
iDestruct
1
as
(
l
->)
"#?"
.
wp_apply
(
wp_wand
with
"
(
H1
[//])
"
)
;
iIntros
(
w1
)
;
iDestruct
1
as
(
l
->)
"#?"
.
iInv
(
tyN
.@
l
)
as
(
v
)
"[>Hl Hv]"
.
destruct
(
decide
(
v
=
w2
))
as
[->|].
-
wp_cas_suc
.
eauto
10
.
...
...
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