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Jonas Kastberg
iris
Commits
c48ae35d
Commit
c48ae35d
authored
Jan 17, 2017
by
Robbert Krebbers
Browse files
Add examples of IPM paper to tests/ipm_paper.v.
parent
b5fa9b29
Changes
3
Hide whitespace changes
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_CoqProject
View file @
c48ae35d
...
...
@@ 119,7 +119,7 @@ theories/tests/proofmode.v
theories/tests/barrier_client.v
theories/tests/list_reverse.v
theories/tests/tree_sum.v
theories/tests/
count
er.v
theories/tests/
ipm_pap
er.v
theories/proofmode/strings.v
theories/proofmode/coq_tactics.v
theories/proofmode/environments.v
...
...
theories/tests/counter.v
deleted
100644 → 0
View file @
b5fa9b29
(* This file contains a formalization of the monotone counter, but with an
explicit contruction of the monoid, as we have also done in the proof mode
paper. This should simplify explaining and understanding what is happening.
A version that uses the authoritative monoid and natural number monoid
under max can be found in `heap_lang/lib/counter.v`. *)
From
iris
.
program_logic
Require
Export
weakestpre
.
From
iris
.
heap_lang
Require
Export
lang
.
From
iris
.
program_logic
Require
Export
hoare
.
From
iris
.
proofmode
Require
Import
tactics
.
From
iris
.
heap_lang
Require
Import
proofmode
notation
.
Set
Default
Proof
Using
"Type"
.
Import
uPred
.
Definition
newcounter
:
val
:
=
λ
:
<>,
ref
#
0
.
Definition
incr
:
val
:
=
rec
:
"incr"
"l"
:
=
let
:
"n"
:
=
!
"l"
in
if
:
CAS
"l"
"n"
(#
1
+
"n"
)
then
#()
else
"incr"
"l"
.
Definition
read
:
val
:
=
λ
:
"l"
,
!
"l"
.
(** The CMRA we need. *)
Inductive
M
:
=
Auth
:
nat
→
M

Frag
:
nat
→
M

Bot
.
Section
M
.
Arguments
cmra_op
_
!
_
!
_
/.
Arguments
op
_
_
!
_
!
_
/.
Arguments
core
_
_
!
_
/.
Canonical
Structure
M_C
:
ofeT
:
=
leibnizC
M
.
Instance
M_valid
:
Valid
M
:
=
λ
x
,
x
≠
Bot
.
Instance
M_op
:
Op
M
:
=
λ
x
y
,
match
x
,
y
with

Auth
n
,
Frag
j

Frag
j
,
Auth
n
=>
if
decide
(
j
≤
n
)%
nat
then
Auth
n
else
Bot

Frag
i
,
Frag
j
=>
Frag
(
max
i
j
)

_
,
_
=>
Bot
end
.
Instance
M_pcore
:
PCore
M
:
=
λ
x
,
Some
match
x
with
Auth
j

Frag
j
=>
Frag
j

_
=>
Bot
end
.
Instance
M_empty
:
Empty
M
:
=
Frag
0
.
Definition
M_ra_mixin
:
RAMixin
M
.
Proof
.
apply
ra_total_mixin
;
try
solve_proper

eauto
.

intros
[
n1

i1
]
[
n2

i2
]
[
n3

i3
]
;
repeat
(
simpl
;
case_decide
)
;
f_equal
/=
;
lia
.

intros
[
n1

i1
]
[
n2

i2
]
;
repeat
(
simpl
;
case_decide
)
;
f_equal
/=
;
lia
.

intros
[
n

i
]
;
repeat
(
simpl
;
case_decide
)
;
f_equal
/=
;
lia
.

by
intros
[
n

i
].

intros
[
n1

i1
]
y
[[
n2

i2
]
?]
;
exists
(
core
y
)
;
simplify_eq
/=
;
repeat
(
simpl
;
case_decide
)
;
f_equal
/=
;
lia
.

intros
[
n1

i1
]
[
n2

i2
]
;
simpl
;
by
try
case_decide
.
Qed
.
Canonical
Structure
M_R
:
cmraT
:
=
discreteR
M
M_ra_mixin
.
Definition
M_ucmra_mixin
:
UCMRAMixin
M
.
Proof
.
split
;
try
(
done

apply
_
).
intros
[??]
;
simpl
;
try
case_decide
;
f_equal
/=
;
lia
.
Qed
.
Canonical
Structure
M_UR
:
ucmraT
:
=
discreteUR
M
M_ra_mixin
M_ucmra_mixin
.
Global
Instance
frag_persistent
n
:
Persistent
(
Frag
n
).
Proof
.
by
constructor
.
Qed
.
Lemma
auth_frag_valid
j
n
:
✓
(
Auth
n
⋅
Frag
j
)
→
(
j
≤
n
)%
nat
.
Proof
.
simpl
.
case_decide
.
done
.
by
intros
[].
Qed
.
Lemma
auth_frag_op
(
j
n
:
nat
)
:
(
j
≤
n
)%
nat
→
Auth
n
=
Auth
n
⋅
Frag
j
.
Proof
.
intros
.
by
rewrite
/=
decide_True
.
Qed
.
Lemma
M_update
n
:
Auth
n
~~>
Auth
(
S
n
).
Proof
.
apply
cmra_discrete_update
=>[
m

j
]
/=
?
;
repeat
case_decide
;
done

lia
.
Qed
.
End
M
.
Class
counterG
Σ
:
=
CounterG
{
counter_tokG
:
>
inG
Σ
M_UR
}.
Definition
counter
Σ
:
gFunctors
:
=
#[
GFunctor
M_UR
].
Instance
subG_counter
Σ
{
Σ
}
:
subG
counter
Σ
Σ
→
counterG
Σ
.
Proof
.
solve_inG
.
Qed
.
Section
proof
.
Context
`
{!
heapG
Σ
,
!
counterG
Σ
}.
Implicit
Types
l
:
loc
.
Definition
I
(
γ
:
gname
)
(
l
:
loc
)
:
iProp
Σ
:
=
(
∃
c
:
nat
,
l
↦
#
c
∗
own
γ
(
Auth
c
))%
I
.
Definition
C
(
l
:
loc
)
(
n
:
nat
)
:
iProp
Σ
:
=
(
∃
N
γ
,
inv
N
(
I
γ
l
)
∧
own
γ
(
Frag
n
))%
I
.
(** The main proofs. *)
Global
Instance
C_persistent
l
n
:
PersistentP
(
C
l
n
).
Proof
.
apply
_
.
Qed
.
Lemma
newcounter_spec
:
{{
True
}}
newcounter
#()
{{
v
,
∃
l
,
⌜
v
=
#
l
⌝
∧
C
l
0
}}.
Proof
.
iIntros
"!# _ /="
.
rewrite

wp_fupd
/
newcounter
/=.
wp_seq
.
wp_alloc
l
as
"Hl"
.
iMod
(
own_alloc
(
Auth
0
))
as
(
γ
)
"Hγ"
;
first
done
.
rewrite
(
auth_frag_op
0
0
)
//
;
iDestruct
"Hγ"
as
"[Hγ Hγf]"
.
set
(
N
:
=
nroot
.@
"C"
).
iMod
(
inv_alloc
N
_
(
I
γ
l
)
with
"[Hl Hγ]"
)
as
"#?"
.
{
iIntros
"!>"
.
iExists
0
%
nat
.
by
iFrame
.
}
iModIntro
.
rewrite
/
C
;
eauto
10
.
Qed
.
Lemma
incr_spec
l
n
:
{{
C
l
n
}}
incr
#
l
{{
v
,
⌜
v
=
#()
⌝
∧
C
l
(
S
n
)
}}.
Proof
.
iIntros
"!# Hl /="
.
iL
ö
b
as
"IH"
.
wp_rec
.
iDestruct
"Hl"
as
(
N
γ
)
"[#Hinv Hγf]"
.
wp_bind
(!
_
)%
E
;
iInv
N
as
(
c
)
"[Hl Hγ]"
"Hclose"
.
wp_load
.
iMod
(
"Hclose"
with
"[Hl Hγ]"
)
;
[
iNext
;
iExists
c
;
by
iFrame
].
iModIntro
.
wp_let
.
wp_op
.
wp_bind
(
CAS
_
_
_
).
iInv
N
as
(
c'
)
">[Hl Hγ]"
"Hclose"
.
destruct
(
decide
(
c'
=
c
))
as
[>].

iDestruct
(
own_valid_2
with
"Hγ Hγf"
)
as
%?%
auth_frag_valid
.
iMod
(
own_update_2
with
"Hγ Hγf"
)
as
"Hγ"
.
{
rewrite

auth_frag_op
.
apply
M_update
.
done
.
}
rewrite
(
auth_frag_op
(
S
n
)
(
S
c
))
;
last
omega
.
iDestruct
"Hγ"
as
"[Hγ Hγf]"
.
wp_cas_suc
.
iMod
(
"Hclose"
with
"[Hl Hγ]"
).
{
iNext
.
iExists
(
S
c
).
rewrite
Nat2Z
.
inj_succ
Z
.
add_1_l
.
by
iFrame
.
}
iModIntro
.
wp_if
.
rewrite
{
3
}/
C
;
eauto
10
.

wp_cas_fail
;
first
(
intros
[=]
;
abstract
omega
).
iMod
(
"Hclose"
with
"[Hl Hγ]"
)
;
[
iNext
;
iExists
c'
;
by
iFrame
].
iModIntro
.
wp_if
.
iApply
(
"IH"
with
"[Hγf]"
).
rewrite
{
3
}/
C
;
eauto
10
.
Qed
.
Lemma
read_spec
l
n
:
{{
C
l
n
}}
read
#
l
{{
v
,
∃
m
:
nat
,
⌜
v
=
#
m
∧
n
≤
m
⌝
∧
C
l
m
}}.
Proof
.
iIntros
"!# Hl /="
.
iDestruct
"Hl"
as
(
N
γ
)
"[#Hinv Hγf]"
.
rewrite
/
read
/=.
wp_let
.
iInv
N
as
(
c
)
"[Hl Hγ]"
"Hclose"
.
wp_load
.
iDestruct
(
own_valid
γ
(
Frag
n
⋅
Auth
c
)
with
"[]"
)
as
%
?%
auth_frag_valid
.
{
iApply
own_op
.
by
iFrame
.
}
rewrite
(
auth_frag_op
c
c
)
;
last
lia
;
iDestruct
"Hγ"
as
"[Hγ Hγf']"
.
iMod
(
"Hclose"
with
"[Hl Hγ]"
)
;
[
iNext
;
iExists
c
;
by
iFrame
].
iModIntro
;
rewrite
/
C
;
eauto
10
with
omega
.
Qed
.
End
proof
.
theories/tests/ipm_paper.v
0 → 100644
View file @
c48ae35d
From
iris
.
base_logic
Require
Import
base_logic
.
From
iris
.
proofmode
Require
Import
tactics
.
From
iris
.
program_logic
Require
Export
hoare
.
From
iris
.
heap_lang
Require
Import
proofmode
notation
.
Set
Default
Proof
Using
"Type"
.
(** The proofs from Section 3.1 *)
Section
demo
.
Context
{
M
:
ucmraT
}.
Notation
iProp
:
=
(
uPred
M
).
(* The version in Coq *)
Lemma
and_exist
A
(
P
R
:
Prop
)
(
Ψ
:
A
→
Prop
)
:
P
∧
(
∃
a
,
Ψ
a
)
∧
R
→
∃
a
,
P
∧
Ψ
a
.
Proof
.
intros
[
HP
[
H
Ψ
HR
]].
destruct
H
Ψ
as
[
x
H
Ψ
].
exists
x
.
split
.
assumption
.
assumption
.
Qed
.
(* The version in IPM *)
Lemma
sep_exist
A
(
P
R
:
iProp
)
(
Ψ
:
A
→
iProp
)
:
P
∗
(
∃
a
,
Ψ
a
)
∗
R
⊢
∃
a
,
Ψ
a
∗
P
.
Proof
.
iIntros
"[HP [HΨ HR]]"
.
iDestruct
"HΨ"
as
(
x
)
"HΨ"
.
iExists
x
.
iSplitL
"HΨ"
.
iAssumption
.
iAssumption
.
Qed
.
(* The short version in IPM, as in the paper *)
Lemma
sep_exist_short
A
(
P
R
:
iProp
)
(
Ψ
:
A
→
iProp
)
:
P
∗
(
∃
a
,
Ψ
a
)
∗
R
⊢
∃
a
,
Ψ
a
∗
P
.
Proof
.
iIntros
"[HP [HΨ HR]]"
.
iFrame
"HP"
.
iAssumption
.
Qed
.
(* An even shorter version in IPM, using the frame introduction pattern `$` *)
Lemma
sep_exist_shorter
A
(
P
R
:
iProp
)
(
Ψ
:
A
→
iProp
)
:
P
∗
(
∃
a
,
Ψ
a
)
∗
R
⊢
∃
a
,
Ψ
a
∗
P
.
Proof
.
by
iIntros
"[$ [??]]"
.
Qed
.
End
demo
.
(** The proofs from Section 3.2 *)
(** In the Iris development we often write specifications directly using weakest
preconditions, in sort of `CPS` style, so that they can be applied easier when
proving client code. A version of [list_reverse] in that style can be found in
the file [theories/tests/list_reverse.v]. *)
Section
list_reverse
.
Context
`
{!
heapG
Σ
}.
Notation
iProp
:
=
(
iProp
Σ
).
Implicit
Types
l
:
loc
.
Fixpoint
is_list
(
hd
:
val
)
(
xs
:
list
val
)
:
iProp
:
=
match
xs
with

[]
=>
⌜
hd
=
NONEV
⌝

x
::
xs
=>
∃
l
hd'
,
⌜
hd
=
SOMEV
#
l
⌝
∗
l
↦
(
x
,
hd'
)
∗
is_list
hd'
xs
end
%
I
.
Definition
rev
:
val
:
=
rec
:
"rev"
"hd"
"acc"
:
=
match
:
"hd"
with
NONE
=>
"acc"

SOME
"l"
=>
let
:
"tmp1"
:
=
Fst
!
"l"
in
let
:
"tmp2"
:
=
Snd
!
"l"
in
"l"
<
(
"tmp1"
,
"acc"
)
;;
"rev"
"tmp2"
"hd"
end
.
Lemma
rev_acc_ht
hd
acc
xs
ys
:
{{
is_list
hd
xs
∗
is_list
acc
ys
}}
rev
hd
acc
{{
w
,
is_list
w
(
reverse
xs
++
ys
)
}}.
Proof
.
iIntros
"!# [Hxs Hys]"
.
iL
ö
b
as
"IH"
forall
(
hd
acc
xs
ys
).
wp_rec
;
wp_let
.
destruct
xs
as
[
x
xs
]
;
iSimplifyEq
.

(* nil *)
by
wp_match
.

(* cons *)
iDestruct
"Hxs"
as
(
l
hd'
)
"(% & Hx & Hxs)"
;
iSimplifyEq
.
wp_match
.
wp_load
.
wp_proj
.
wp_let
.
wp_load
.
wp_proj
.
wp_let
.
wp_store
.
rewrite
reverse_cons

assoc
.
iApply
(
"IH"
$!
hd'
(
InjRV
#
l
)
xs
(
x
::
ys
)
with
"Hxs [Hx Hys]"
).
iExists
l
,
acc
;
by
iFrame
.
Qed
.
Lemma
rev_ht
hd
xs
:
{{
is_list
hd
xs
}}
rev
hd
NONE
{{
w
,
is_list
w
(
reverse
xs
)
}}.
Proof
.
iIntros
"!# Hxs"
.
rewrite
(
right_id_L
[]
(++)
(
reverse
xs
)).
iApply
(
rev_acc_ht
hd
NONEV
with
"[Hxs]"
)
;
simpl
;
by
iFrame
.
Qed
.
End
list_reverse
.
(** The proofs from Section 5 *)
(** This part contains a formalization of the monotone counter, but with an
explicit contruction of the monoid, as we have also done in the proof mode
paper. This should simplify explaining and understanding what is happening.
A version that uses the authoritative monoid and natural number monoid
under max can be found in [theories/heap_lang/lib/counter.v]. *)
(** The invariant rule in the paper is in fact derived from mask changing
update modalities (which we did not cover in the paper). Normally we use these
mask changing update modalities directly in our proofs, but in this file we use
the first prove the rule as a lemma, and then use that. *)
Lemma
wp_inv_open
`
{
irisG
Λ
Σ
}
N
E
P
e
Φ
:
nclose
N
⊆
E
→
atomic
e
→
inv
N
P
∗
(
▷
P

∗
WP
e
@
E
∖
↑
N
{{
v
,
▷
P
∗
Φ
v
}})
⊢
WP
e
@
E
{{
Φ
}}.
Proof
.
iIntros
(??)
"[#Hinv Hwp]"
.
iMod
(
inv_open
E
N
P
with
"Hinv"
)
as
"[HP Hclose]"
=>//.
iApply
wp_wand_r
;
iSplitL
"HP Hwp"
;
[
by
iApply
"Hwp"
].
iIntros
(
v
)
"[HP $]"
.
by
iApply
"Hclose"
.
Qed
.
Definition
newcounter
:
val
:
=
λ
:
<>,
ref
#
0
.
Definition
incr
:
val
:
=
rec
:
"incr"
"l"
:
=
let
:
"n"
:
=
!
"l"
in
if
:
CAS
"l"
"n"
(#
1
+
"n"
)
then
#()
else
"incr"
"l"
.
Definition
read
:
val
:
=
λ
:
"l"
,
!
"l"
.
(** The CMRA we need. *)
Inductive
M
:
=
Auth
:
nat
→
M

Frag
:
nat
→
M

Bot
.
Section
M
.
Arguments
cmra_op
_
!
_
!
_
/.
Arguments
op
_
_
!
_
!
_
/.
Arguments
core
_
_
!
_
/.
Canonical
Structure
M_C
:
ofeT
:
=
leibnizC
M
.
Instance
M_valid
:
Valid
M
:
=
λ
x
,
x
≠
Bot
.
Instance
M_op
:
Op
M
:
=
λ
x
y
,
match
x
,
y
with

Auth
n
,
Frag
j

Frag
j
,
Auth
n
=>
if
decide
(
j
≤
n
)%
nat
then
Auth
n
else
Bot

Frag
i
,
Frag
j
=>
Frag
(
max
i
j
)

_
,
_
=>
Bot
end
.
Instance
M_pcore
:
PCore
M
:
=
λ
x
,
Some
match
x
with
Auth
j

Frag
j
=>
Frag
j

_
=>
Bot
end
.
Instance
M_empty
:
Empty
M
:
=
Frag
0
.
Definition
M_ra_mixin
:
RAMixin
M
.
Proof
.
apply
ra_total_mixin
;
try
solve_proper

eauto
.

intros
[
n1

i1
]
[
n2

i2
]
[
n3

i3
]
;
repeat
(
simpl
;
case_decide
)
;
f_equal
/=
;
lia
.

intros
[
n1

i1
]
[
n2

i2
]
;
repeat
(
simpl
;
case_decide
)
;
f_equal
/=
;
lia
.

intros
[
n

i
]
;
repeat
(
simpl
;
case_decide
)
;
f_equal
/=
;
lia
.

by
intros
[
n

i
].

intros
[
n1

i1
]
y
[[
n2

i2
]
?]
;
exists
(
core
y
)
;
simplify_eq
/=
;
repeat
(
simpl
;
case_decide
)
;
f_equal
/=
;
lia
.

intros
[
n1

i1
]
[
n2

i2
]
;
simpl
;
by
try
case_decide
.
Qed
.
Canonical
Structure
M_R
:
cmraT
:
=
discreteR
M
M_ra_mixin
.
Definition
M_ucmra_mixin
:
UCMRAMixin
M
.
Proof
.
split
;
try
(
done

apply
_
).
intros
[??]
;
simpl
;
try
case_decide
;
f_equal
/=
;
lia
.
Qed
.
Canonical
Structure
M_UR
:
ucmraT
:
=
discreteUR
M
M_ra_mixin
M_ucmra_mixin
.
Global
Instance
frag_persistent
n
:
Persistent
(
Frag
n
).
Proof
.
by
constructor
.
Qed
.
Lemma
auth_frag_valid
j
n
:
✓
(
Auth
n
⋅
Frag
j
)
→
(
j
≤
n
)%
nat
.
Proof
.
simpl
.
case_decide
.
done
.
by
intros
[].
Qed
.
Lemma
auth_frag_op
(
j
n
:
nat
)
:
(
j
≤
n
)%
nat
→
Auth
n
=
Auth
n
⋅
Frag
j
.
Proof
.
intros
.
by
rewrite
/=
decide_True
.
Qed
.
Lemma
M_update
n
:
Auth
n
~~>
Auth
(
S
n
).
Proof
.
apply
cmra_discrete_update
=>[
m

j
]
/=
?
;
repeat
case_decide
;
done

lia
.
Qed
.
End
M
.
Class
counterG
Σ
:
=
CounterG
{
counter_tokG
:
>
inG
Σ
M_UR
}.
Definition
counter
Σ
:
gFunctors
:
=
#[
GFunctor
(
constRF
M_UR
)].
Instance
subG_counter
Σ
{
Σ
}
:
subG
counter
Σ
Σ
→
counterG
Σ
.
Proof
.
intros
[?%
subG_inG
_
]%
subG_inv
.
split
;
apply
_
.
Qed
.
Section
counter_proof
.
Context
`
{!
heapG
Σ
,
!
counterG
Σ
}.
Implicit
Types
l
:
loc
.
Definition
I
(
γ
:
gname
)
(
l
:
loc
)
:
iProp
Σ
:
=
(
∃
c
:
nat
,
l
↦
#
c
∗
own
γ
(
Auth
c
))%
I
.
Definition
C
(
l
:
loc
)
(
n
:
nat
)
:
iProp
Σ
:
=
(
∃
N
γ
,
inv
N
(
I
γ
l
)
∧
own
γ
(
Frag
n
))%
I
.
(** The main proofs. *)
Global
Instance
C_persistent
l
n
:
PersistentP
(
C
l
n
).
Proof
.
apply
_
.
Qed
.
Lemma
newcounter_spec
:
{{
True
}}
newcounter
#()
{{
v
,
∃
l
,
⌜
v
=
#
l
⌝
∧
C
l
0
}}.
Proof
.
iIntros
"!# _ /="
.
rewrite

wp_fupd
/
newcounter
/=.
wp_seq
.
wp_alloc
l
as
"Hl"
.
iMod
(
own_alloc
(
Auth
0
))
as
(
γ
)
"Hγ"
;
first
done
.
rewrite
(
auth_frag_op
0
0
)
//
;
iDestruct
"Hγ"
as
"[Hγ Hγf]"
.
set
(
N
:
=
nroot
.@
"counter"
).
iMod
(
inv_alloc
N
_
(
I
γ
l
)
with
"[Hl Hγ]"
)
as
"#?"
.
{
iIntros
"!>"
.
iExists
0
%
nat
.
by
iFrame
.
}
iModIntro
.
rewrite
/
C
;
eauto
10
.
Qed
.
Lemma
incr_spec
l
n
:
{{
C
l
n
}}
incr
#
l
{{
v
,
⌜
v
=
#()
⌝
∧
C
l
(
S
n
)
}}.
Proof
.
iIntros
"!# Hl /="
.
iL
ö
b
as
"IH"
.
wp_rec
.
iDestruct
"Hl"
as
(
N
γ
)
"[#Hinv Hγf]"
.
wp_bind
(!
_
)%
E
.
iApply
wp_inv_open
;
last
iFrame
"Hinv"
;
auto
.
iDestruct
1
as
(
c
)
"[Hl Hγ]"
.
wp_load
.
iSplitL
"Hl Hγ"
;
[
iNext
;
iExists
c
;
by
iFrame
].
wp_let
.
wp_op
.
wp_bind
(
CAS
_
_
_
).
iApply
wp_inv_open
;
last
iFrame
"Hinv"
;
auto
.
iDestruct
1
as
(
c'
)
">[Hl Hγ]"
.
destruct
(
decide
(
c'
=
c
))
as
[>].

iCombine
"Hγ"
"Hγf"
as
"Hγ"
.
iDestruct
(
own_valid
with
"Hγ"
)
as
%?%
auth_frag_valid
;
rewrite

auth_frag_op
//.
iMod
(
own_update
with
"Hγ"
)
as
"Hγ"
;
first
apply
M_update
.
rewrite
(
auth_frag_op
(
S
n
)
(
S
c
))
;
last
lia
;
iDestruct
"Hγ"
as
"[Hγ Hγf]"
.
wp_cas_suc
.
iSplitL
"Hl Hγ"
.
{
iNext
.
iExists
(
S
c
).
rewrite
Nat2Z
.
inj_succ
Z
.
add_1_l
.
by
iFrame
.
}
wp_if
.
rewrite
{
3
}/
C
;
eauto
10
.

wp_cas_fail
;
first
(
intros
[=]
;
abstract
omega
).
iSplitL
"Hl Hγ"
;
[
iNext
;
iExists
c'
;
by
iFrame
].
wp_if
.
iApply
(
"IH"
with
"[Hγf]"
).
rewrite
{
3
}/
C
;
eauto
10
.
Qed
.
Lemma
read_spec
l
n
:
{{
C
l
n
}}
read
#
l
{{
v
,
∃
m
:
nat
,
⌜
v
=
#
m
∧
n
≤
m
⌝
∧
C
l
m
}}.
Proof
.
iIntros
"!# Hl /="
.
iDestruct
"Hl"
as
(
N
γ
)
"[#Hinv Hγf]"
.
rewrite
/
read
/=.
wp_let
.
iApply
wp_inv_open
;
last
iFrame
"Hinv"
;
auto
.
iDestruct
1
as
(
c
)
"[Hl Hγ]"
.
wp_load
.
iDestruct
(
own_valid
γ
(
Frag
n
⋅
Auth
c
)
with
"[]"
)
as
%
?%
auth_frag_valid
.
{
iApply
own_op
.
by
iFrame
.
}
rewrite
(
auth_frag_op
c
c
)
;
last
lia
;
iDestruct
"Hγ"
as
"[Hγ Hγf']"
.
iSplitL
"Hl Hγ"
;
[
iNext
;
iExists
c
;
by
iFrame
].
rewrite
/
C
;
eauto
10
with
omega
.
Qed
.
End
counter_proof
.
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