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Simon Friis Vindum
Iris
Commits
594a1dd0
Commit
594a1dd0
authored
Jun 05, 2014
by
Filip Sieczkowski
Browse files
Proved the Hoare triple rules, except the one about timeless props.
parent
a01925c4
Changes
1
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594a1dd0
...
...
@@ 659,7 +659,7 @@ Qed.
(
HStep
:
prim_step
(
ei
,
σ
)
(
ei'
,
σ
'
)),
exists
w''
r'
s'
,
w'
⊑
w''
/\
WP
(
K
[[
ei'
]])
φ
w''
k
r'
/\
erasure
σ
'
m
(
Some
r'
·
rf
)
s'
w''
@
k
)
/\
(
forall
e'
K
(
HDec
:
e
=
K
[[
e'
]]),
(
forall
e'
K
(
HDec
:
e
=
K
[[
fork
e'
]]),
exists
w''
rfk
rret
s'
,
w'
⊑
w''
/\
WP
(
K
[[
fork_ret
]])
φ
w''
k
rret
/\
WP
e'
(
umconst
⊤
)
w''
k
rfk
...
...
@@ 861,7 +861,24 @@ Qed.
Lemma
htRet
e
(
HV
:
is_value
e
)
m
:
valid
(
ht
m
⊤
e
(
eqV
(
exist
_
e
HV
))).
Proof
.
Admitted
.
intros
w'
nn
rr
w
_
n
r'
_
_
_;
clear
w'
nn
rr
.
unfold
wp
;
rewrite
fixp_eq
;
fold
(
wp
m
).
intros
w'
;
intros
;
split
;
[
split
]
;
intros
.

exists
w'
r'
s
;
split
;
[
reflexivity

split
;
[
assumption
]
].
simpl
;
reflexivity
.

assert
(
HT
:
=
values_stuck
_
HV
).
eapply
HT
in
HStep
;
[
contradiction

eassumption
].

subst
e
;
assert
(
HT
:
=
fill_value
_
_
HV
)
;
subst
K
.
revert
HV
;
rewrite
fill_empty
;
intros
.
contradiction
(
fork_not_value
_
HV
).
Qed
.
Implicit
Type
(
C
:
Props
).
Lemma
wpO
m
e
φ
w
r
:
wp
m
e
φ
w
O
r
.
Proof
.
unfold
wp
;
rewrite
fixp_eq
;
fold
(
wp
m
)
;
intros
w'
;
intros
;
now
inversion
HLt
.
Qed
.
(** Bind **)
Program
Definition
plugV
m
φ
φ
'
K
:
=
...
...
@@ 879,53 +896,262 @@ Qed.
rewrite
EQv
;
reflexivity
.
Qed
.
Lemma
htBind
P
φ
φ
'
K
e
m
:
ht
m
P
e
φ
∧
all
(
plugV
m
φ
φ
'
K
)
⊑
ht
m
P
(
K
[[
e
]])
φ
'
.
Lemma
unit_min
r
:
pcm_unit
_
⊑
r
.
Proof
.
Admitted
.
exists
r
;
now
erewrite
comm
,
pcm_op_unit
by
apply
_
.
Qed
.
Lemma
htBind_alt
P
Q
φ
φ
'
K
e
m
(
He
:
Q
⊑
ht
m
P
e
φ
)
(
HK
:
forall
v
,
Q
⊑
ht
m
(
φ
v
)
(
K
[[
`
v
]])
φ
'
)
:
Q
⊑
ht
m
P
(
K
[[
e
]])
φ
'
.
Admitted
.
Definition
wf_nat_ind
:
=
well_founded_induction
Wf_nat
.
lt_wf
.
Implicit
Type
(
C
:
Props
).
Lemma
htBind
P
φ
φ
'
K
e
m
:
ht
m
P
e
φ
∧
all
(
plugV
m
φ
φ
'
K
)
⊑
ht
m
P
(
K
[[
e
]])
φ
'
.
Proof
.
intros
wz
nz
rz
[
He
HK
]
w
HSw
n
r
HLe
_
HP
.
specialize
(
He
_
HSw
_
_
HLe
(
unit_min
_
)
HP
).
rewrite
HSw
,
<
HLe
in
HK
;
clear
wz
nz
HSw
HLe
HP
.
revert
e
w
r
He
HK
;
induction
n
using
wf_nat_ind
;
intros
;
rename
H
into
IH
.
unfold
wp
;
rewrite
fixp_eq
;
fold
(
wp
m
).
unfold
wp
in
He
;
rewrite
fixp_eq
in
He
;
fold
(
wp
m
)
in
He
.
destruct
(
is_value_dec
e
)
as
[
HVal

HNVal
]
;
[
clear
IH
].

intros
w'
;
intros
;
edestruct
He
as
[
HV
_
]
;
try
eassumption
;
[].
clear
He
HE
;
specialize
(
HV
HVal
)
;
destruct
HV
as
[
w''
[
r'
[
s'
[
HSw'
[
H
φ
HE
]
]
]
]
].
(* Fold the goal back into a wp *)
setoid_rewrite
HSw'
.
assert
(
HT
:
wp
m
(
K
[[
e
]])
φ
'
w''
(
S
k
)
r'
)
;
[
unfold
wp
in
HT
;
rewrite
fixp_eq
in
HT
;
fold
(
wp
m
)
in
HT
;
eapply
HT
;
[
reflexivity

unfold
lt
;
reflexivity

eassumption
]
].
clear
HE
;
specialize
(
HK
(
exist
_
e
HVal
)).
do
30
red
in
HK
;
unfold
proj1_sig
in
HK
.
apply
HK
;
[
etransitivity
;
eassumption

apply
HLt

apply
unit_min

assumption
].

intros
w'
;
intros
;
edestruct
He
as
[
_
[
HS
HF
]
]
;
try
eassumption
;
[].
split
;
[
intros
HVal
;
contradiction
HNVal
;
assert
(
HT
:
=
fill_value
_
_
HVal
)
;
subst
K
;
rewrite
fill_empty
in
HVal
;
assumption

split
;
intros
].
+
clear
He
HF
HE
;
edestruct
step_by_value
as
[
K'
EQK
]
;
try
eassumption
;
[].
subst
K0
;
rewrite
<
fill_comp
in
HDec
;
apply
fill_inj2
in
HDec
.
edestruct
HS
as
[
w''
[
r'
[
s'
[
HSw'
[
He
HE
]
]
]
]
]
;
try
eassumption
;
[].
subst
e
;
clear
HStep
HS
.
do
3
eexists
;
split
;
[
eassumption

split
;
[
eassumption
]
].
rewrite
<
fill_comp
;
apply
IH
;
try
assumption
;
[].
unfold
lt
in
HLt
;
rewrite
<
HSw'
,
<
HSw
,
Le
.
le_n_Sn
,
HLt
;
apply
HK
.
+
clear
He
HS
HE
;
edestruct
fork_by_value
as
[
K'
EQK
]
;
try
eassumption
;
[].
subst
K0
;
rewrite
<
fill_comp
in
HDec
;
apply
fill_inj2
in
HDec
.
edestruct
HF
as
[
w''
[
rfk
[
rret
[
s'
[
HSw'
[
HWR
[
HWF
HE
]
]
]
]
]
]
]
;
try
eassumption
;
[]
;
subst
e
;
clear
HF
.
do
4
eexists
;
split
;
[
eassumption

split
;
[
split
;
eassumption
]
].
rewrite
<
fill_comp
;
apply
IH
;
try
assumption
;
[].
unfold
lt
in
HLt
;
rewrite
<
HSw'
,
<
HSw
,
Le
.
le_n_Sn
,
HLt
;
apply
HK
.
Qed
.
(** Consequence **)
Lemma
htCons
C
P
P'
φ
φ
'
m
e
(
HPre
:
C
⊑
vs
m
m
P
P'
)
(
HT
:
C
⊑
ht
m
P'
e
φ
'
)
(
HPost
:
forall
v
,
C
⊑
vs
m
m
(
φ
'
v
)
(
φ
v
))
:
C
⊑
ht
m
P
e
φ
.
Admitted
.
Program
Definition
vsLift
m1
m2
φ
φ
'
:
=
n
[(
fun
v
=>
vs
m1
m2
(
φ
v
)
(
φ
'
v
))].
Next
Obligation
.
intros
v1
v2
EQv
;
unfold
vs
.
rewrite
EQv
;
reflexivity
.
Qed
.
Next
Obligation
.
intros
v1
v2
EQv
;
unfold
vs
.
rewrite
EQv
;
reflexivity
.
Qed
.
Lemma
htCons
P
P'
φ
φ
'
m
e
:
vs
m
m
P
P'
∧
ht
m
P'
e
φ
'
∧
all
(
vsLift
m
m
φ
'
φ
)
⊑
ht
m
P
e
φ
.
Proof
.
intros
wz
nz
rz
[
[
HP
He
]
H
φ
]
w
HSw
n
r
HLe
_
Hp
.
specialize
(
HP
_
HSw
_
_
HLe
(
unit_min
_
)
Hp
).
unfold
wp
;
rewrite
fixp_eq
;
fold
(
wp
m
).
rewrite
<
HLe
,
HSw
in
He
,
H
φ
;
clear
wz
nz
HSw
HLe
Hp
.
intros
w'
;
intros
;
edestruct
HP
with
(
mf
:
=
mask_emp
)
as
[
w''
[
r'
[
s'
[
HSw'
[
Hp'
HE'
]
]
]
]
]
;
try
eassumption
;
[
intros
j
;
tauto

eapply
erasure_equiv
,
HE
;
try
reflexivity
;
unfold
mask_emp
,
const
;
intros
j
;
tauto
].
clear
HP
HE
;
rewrite
HSw
in
He
;
specialize
(
He
w''
HSw'
_
r'
HLt
(
unit_min
_
)
Hp'
).
setoid_rewrite
HSw'
.
assert
(
HT
:
wp
m
e
φ
w''
(
S
k
)
r'
)
;
[
unfold
wp
in
HT
;
rewrite
fixp_eq
in
HT
;
fold
(
wp
m
)
in
HT
;
eapply
HT
;
[
reflexivity

unfold
lt
;
reflexivity
]
;
eapply
erasure_equiv
,
HE'
;
try
reflexivity
;
unfold
mask_emp
,
const
;
intros
j
;
tauto
].
unfold
lt
in
HLt
;
rewrite
HSw
,
HSw'
,
<
HLt
in
H
φ
;
clear

He
H
φ
.
(* Second phase of the proof: got rid of the preconditions,
now induction takes care of the evaluation. *)
rename
r'
into
r
;
rename
w''
into
w
.
revert
w
r
e
He
H
φ
;
generalize
(
S
k
)
as
n
;
clear
k
;
induction
n
using
wf_nat_ind
.
rename
H
into
IH
;
intros
;
unfold
wp
;
rewrite
fixp_eq
;
fold
(
wp
m
).
unfold
wp
in
He
;
rewrite
fixp_eq
in
He
;
fold
(
wp
m
).
intros
w'
;
intros
;
edestruct
He
as
[
HV
[
HS
HF
]
]
;
try
eassumption
;
[].
split
;
[
intros
HVal
;
clear
HS
HF
IH

split
;
intros
;
[
clear
HV
HF

clear
HV
HS
]
].

clear
He
HE
;
specialize
(
HV
HVal
)
;
destruct
HV
as
[
w''
[
r'
[
s'
[
HSw'
[
H
φ
'
HE
]
]
]
]
].
eapply
H
φ
in
H
φ
'
;
[
etransitivity
;
eassumption

apply
HLt

apply
unit_min
].
clear
w
n
HSw
H
φ
HLt
;
edestruct
H
φ
'
with
(
mf
:
=
mask_emp
)
as
[
w
[
r''
[
s''
[
HSw
[
H
φ
HE'
]
]
]
]
]
;
[
reflexivity

apply
le_n

intros
j
;
tauto

eapply
erasure_equiv
,
HE
;
try
reflexivity
;
unfold
mask_emp
,
const
;
intros
j
;
tauto
].
exists
w
r''
s''
;
split
;
[
etransitivity
;
eassumption

split
;
[
assumption
]
].
eapply
erasure_equiv
,
HE'
;
try
reflexivity
.
unfold
mask_emp
,
const
;
intros
j
;
tauto
.

clear
HE
He
;
edestruct
HS
as
[
w''
[
r'
[
s'
[
HSw'
[
He
HE
]
]
]
]
]
;
try
eassumption
;
clear
HS
;
fold
(
wp
m
)
in
He
.
do
3
eexists
;
split
;
[
eassumption

split
;
[
eassumption
]
].
apply
IH
;
try
assumption
;
[].
unfold
lt
in
HLt
;
rewrite
Le
.
le_n_Sn
,
HLt
,
<
HSw'
,
<
HSw
;
apply
H
φ
.

clear
HE
He
;
fold
(
wp
m
)
in
HF
;
edestruct
HF
as
[
w''
[
rfk
[
rret
[
s'
[
HSw'
[
HWF
[
HWR
HE
]
]
]
]
]
]
]
;
[
eassumption
].
clear
HF
;
do
4
eexists
;
split
;
[
eassumption

split
;
[
split
;
eassumption
]
].
apply
IH
;
try
assumption
;
[].
unfold
lt
in
HLt
;
rewrite
Le
.
le_n_Sn
,
HLt
,
<
HSw'
,
<
HSw
;
apply
H
φ
.
Qed
.
Lemma
htACons
C
P
P'
φ
φ
'
m
m'
e
Lemma
htACons
m
m'
e
P
P'
φ
φ
'
(
HAt
:
atomic
e
)
(
HSub
:
m'
⊆
m
)
(
HPre
:
C
⊑
vs
m
m'
P
P'
)
(
HT
:
C
⊑
ht
m'
P'
e
φ
'
)
(
HPost
:
forall
v
,
C
⊑
vs
m'
m
(
φ
'
v
)
(
φ
v
))
:
C
⊑
ht
m
P
e
φ
.
Admitted
.
(
HSub
:
m'
⊆
m
)
:
vs
m
m'
P
P'
∧
ht
m'
P'
e
φ
'
∧
all
(
vsLift
m'
m
φ
'
φ
)
⊑
ht
m
P
e
φ
.
Proof
.
intros
wz
nz
rz
[
[
HP
He
]
H
φ
]
w
HSw
n
r
HLe
_
Hp
.
specialize
(
HP
_
HSw
_
_
HLe
(
unit_min
_
)
Hp
).
unfold
wp
;
rewrite
fixp_eq
;
fold
(
wp
m
).
split
;
[
intros
HV
;
contradiction
(
atomic_not_value
e
)
].
split
;
[
intros
;
subst
;
contradiction
(
fork_not_atomic
K
e'
)
].
intros
;
rewrite
<
HLe
,
HSw
in
He
,
H
φ
;
clear
wz
nz
HSw
HLe
Hp
.
edestruct
HP
with
(
mf
:
=
mask_emp
)
as
[
w''
[
r'
[
s'
[
HSw'
[
Hp'
HE'
]
]
]
]
]
;
[
eassumption

eassumption

intros
j
;
tauto

eapply
erasure_equiv
,
HE
;
try
reflexivity
;
unfold
mask_emp
,
const
;
intros
j
;
tauto
].
clear
HP
HE
;
rewrite
HSw0
in
He
;
specialize
(
He
w''
HSw'
_
r'
HLt
(
unit_min
_
)
Hp'
).
unfold
lt
in
HLt
;
rewrite
HSw0
,
<
HLt
in
H
φ
;
clear
w
n
HSw0
HLt
Hp'
.
unfold
wp
in
He
;
rewrite
fixp_eq
in
He
;
fold
(
wp
m'
)
in
He
.
edestruct
He
as
[
_
[
HS
_
]
]
;
[
reflexivity

unfold
lt
;
reflexivity

eapply
erasure_equiv
,
HE'
;
try
reflexivity
;
unfold
mask_emp
,
const
;
intros
j
;
tauto
].
edestruct
HS
as
[
w
[
r''
[
s''
[
HSw
[
He'
HE
]
]
]
]
]
;
try
eassumption
;
clear
He
HS
HE'
.
destruct
k
as
[
k
]
;
[
exists
w'
r'
s'
;
split
;
[
reflexivity

split
;
[
apply
wpO

exact
I
]
]
].
edestruct
atomic_step
as
[
EQk
HVal
]
;
try
eassumption
;
subst
K
.
rewrite
fill_empty
in
*
;
subst
ei
.
setoid_rewrite
HSw'
;
setoid_rewrite
HSw
.
rewrite
HSw'
,
HSw
in
H
φ
;
clear

HE
He'
H
φ
HSub
HVal
.
unfold
wp
in
He'
;
rewrite
fixp_eq
in
He'
;
fold
(
wp
m'
)
in
He'
.
edestruct
He'
as
[
HV
_
]
;
[
reflexivity

apply
le_n

eassumption
].
clear
HE
He'
;
specialize
(
HV
HVal
)
;
destruct
HV
as
[
w'
[
r
[
s
[
HSw
[
H
φ
'
HE
]
]
]
]
].
eapply
H
φ
in
H
φ
'
;
[
assumption

apply
Le
.
le_n_Sn

apply
unit_min
].
clear
H
φ
;
edestruct
H
φ
'
with
(
mf
:
=
mask_emp
)
as
[
w''
[
r'
[
s'
[
HSw'
[
H
φ
HE'
]
]
]
]
]
;
[
reflexivity

apply
le_n

intros
j
;
tauto

eapply
erasure_equiv
,
HE
;
try
reflexivity
;
unfold
mask_emp
,
const
;
intros
j
;
tauto
].
clear
H
φ
'
HE
;
exists
w''
r'
s'
;
split
;
[
etransitivity
;
eassumption

split
;
[
eapply
erasure_equiv
,
HE'
;
try
reflexivity
;
unfold
mask_emp
,
const
;
intros
j
;
tauto
]
].
clear

H
φ
;
unfold
wp
;
rewrite
fixp_eq
;
fold
(
wp
m
).
intros
w
;
intros
;
split
;
[
intros
HVal'

split
;
intros
;
exfalso
].

do
3
eexists
;
split
;
[
reflexivity

split
;
[
eassumption
]
].
unfold
lt
in
HLt
;
rewrite
HLt
,
<
HSw
.
eapply
φ
,
H
φ
;
[
apply
le_n
]
;
simpl
;
reflexivity
.

eapply
values_stuck
;
eassumption
.

clear

HDec
HVal
;
subst
;
assert
(
HT
:
=
fill_value
_
_
HVal
)
;
subst
.
rewrite
fill_empty
in
HVal
;
now
apply
fork_not_value
in
HVal
.
Qed
.
(** Framing **)
Lemma
htFrame
m
P
R
e
φ
:
ht
m
P
e
φ
⊑
ht
m
(
P
*
R
)
e
(
lift_bin
sc
φ
(
umconst
R
)).
Admitted
.
Proof
.
intros
w
n
rz
He
w'
HSw
n'
r
HLe
_
[
r1
[
r2
[
EQr
[
HP
HLR
]
]
]
].
specialize
(
He
_
HSw
_
_
HLe
(
unit_min
_
)
HP
).
clear
P
w
n
rz
HSw
HLe
HP
;
rename
w'
into
w
;
rename
n'
into
n
.
revert
e
w
r1
r
EQr
HLR
He
;
induction
n
using
wf_nat_ind
;
intros
;
rename
H
into
IH
.
unfold
wp
;
rewrite
fixp_eq
;
fold
(
wp
m
)
;
intros
w'
;
intros
.
unfold
wp
in
He
;
rewrite
fixp_eq
in
He
;
fold
(
wp
m
)
in
He
.
rewrite
<
EQr
,
<
assoc
in
HE
;
edestruct
He
as
[
HV
[
HS
HF
]
]
;
try
eassumption
;
[].
clear
He
;
split
;
[
intros
HVal
;
clear
HS
HF
IH
HE

split
;
intros
;
[
clear
HV
HF
HE

clear
HV
HS
HE
]
].

specialize
(
HV
HVal
)
;
destruct
HV
as
[
w''
[
r1'
[
s'
[
HSw'
[
H
φ
HE
]
]
]
]
].
rewrite
assoc
in
HE
;
destruct
(
Some
r1'
·
Some
r2
)
as
[
r'
]
eqn
:
EQr'
;
[
eapply
erasure_not_empty
in
HE
;
[
contradiction

now
erewrite
!
pcm_op_zero
by
apply
_
]
].
do
3
eexists
;
split
;
[
eassumption

split
;
[
eassumption
]
].
exists
r1'
r2
;
split
;
[
now
rewrite
EQr'

split
;
[
assumption
]
].
unfold
lt
in
HLt
;
rewrite
HLt
,
<
HSw'
,
<
HSw
;
apply
HLR
.

edestruct
HS
as
[
w''
[
r1'
[
s'
[
HSw'
[
He
HE
]
]
]
]
]
;
try
eassumption
;
[]
;
clear
HS
.
destruct
k
as
[
k
]
;
[
exists
w'
r1'
s'
;
split
;
[
reflexivity

split
;
[
apply
wpO

exact
I
]
]
].
rewrite
assoc
in
HE
;
destruct
(
Some
r1'
·
Some
r2
)
as
[
r'
]
eqn
:
EQr'
;
[
eapply
erasure_not_empty
in
HE
;
[
contradiction

now
erewrite
!
pcm_op_zero
by
apply
_
]
].
do
3
eexists
;
split
;
[
eassumption

split
;
[
eassumption
]
].
eapply
IH
;
try
eassumption
;
[
rewrite
<
EQr'
;
reflexivity
].
unfold
lt
in
HLt
;
rewrite
Le
.
le_n_Sn
,
HLt
,
<
HSw'
,
<
HSw
;
apply
HLR
.

specialize
(
HF
_
_
HDec
)
;
destruct
HF
as
[
w''
[
rfk
[
rret
[
s'
[
HSw'
[
HWF
[
HWR
HE
]
]
]
]
]
]
].
destruct
k
as
[
k
]
;
[
exists
w'
rfk
rret
s'
;
split
;
[
reflexivity

split
;
[
apply
wpO

split
;
[
apply
wpO

exact
I
]
]
]
].
rewrite
assoc
,
<
(
assoc
(
Some
rfk
))
in
HE
;
destruct
(
Some
rret
·
Some
r2
)
as
[
rret'
]
eqn
:
EQrret
;
[
eapply
erasure_not_empty
in
HE
;
[
contradiction

now
erewrite
(
comm
_
0
),
!
pcm_op_zero
by
apply
_
]
].
do
4
eexists
;
split
;
[
eassumption

split
;
[
split
;
eassumption
]
].
eapply
IH
;
try
eassumption
;
[
rewrite
<
EQrret
;
reflexivity
].
unfold
lt
in
HLt
;
rewrite
Le
.
le_n_Sn
,
HLt
,
<
HSw'
,
<
HSw
;
apply
HLR
.
Qed
.
Lemma
htAFrame
m
P
R
e
φ
(
HAt
:
atomic
e
)
:
ht
m
P
e
φ
⊑
ht
m
(
P
*
▹
R
)
e
(
lift_bin
sc
φ
(
umconst
R
)).
Admitted
.
Proof
.
intros
w
n
rz
He
w'
HSw
n'
r
HLe
_
[
r1
[
r2
[
EQr
[
HP
HLR
]
]
]
].
specialize
(
He
_
HSw
_
_
HLe
(
unit_min
_
)
HP
).
clear
rz
n
HLe
;
unfold
wp
;
rewrite
fixp_eq
;
fold
(
wp
m
).
clear
w
HSw
;
rename
n'
into
n
;
rename
w'
into
w
;
intros
w'
;
intros
.
split
;
[
intros
;
exfalso

split
;
intros
;
[
exfalso
]
].

contradiction
(
atomic_not_value
e
).

destruct
k
as
[
k
]
;
[
exists
w'
r
s
;
split
;
[
reflexivity

split
;
[
apply
wpO

exact
I
]
]
].
unfold
wp
in
He
;
rewrite
fixp_eq
in
He
;
fold
(
wp
m
)
in
He
.
rewrite
<
EQr
,
<
assoc
in
HE
.
edestruct
He
as
[
_
[
HeS
_
]
]
;
try
eassumption
;
[].
edestruct
HeS
as
[
w''
[
r1'
[
s'
[
HSw'
[
He'
HE'
]
]
]
]
]
;
try
eassumption
;
[].
clear
HE
He
HeS
;
rewrite
assoc
in
HE'
.
destruct
(
Some
r1'
·
Some
r2
)
as
[
r'
]
eqn
:
EQr'
;
[
eapply
erasure_not_empty
in
HE'
;
[
contradiction

now
erewrite
!
pcm_op_zero
by
apply
_
]
].
do
3
eexists
;
split
;
[
eassumption

split
;
[
eassumption
]
].
edestruct
atomic_step
as
[
EQK
HVal
]
;
try
eassumption
;
[]
;
subst
K
.
unfold
lt
in
HLt
;
rewrite
<
HLt
,
HSw
,
HSw'
in
HLR
;
simpl
in
HLR
.
clear

He'
HVal
EQr'
HLR
;
rename
w''
into
w
.
unfold
wp
;
rewrite
fixp_eq
;
fold
(
wp
m
)
;
intros
w'
;
intros
.
split
;
[
intros
HVal'

split
;
intros
;
exfalso
].
+
unfold
wp
in
He'
;
rewrite
fixp_eq
in
He'
;
fold
(
wp
m
)
in
He'
.
rewrite
<
EQr'
,
<
assoc
in
HE
;
edestruct
He'
as
[
HV
_
]
;
try
eassumption
;
[].
revert
HVal'
;
rewrite
fill_empty
in
*
;
intros
;
specialize
(
HV
HVal'
).
destruct
HV
as
[
w''
[
r1''
[
s''
[
HSw'
[
H
φ
HE'
]
]
]
]
].
rewrite
assoc
in
HE'
;
destruct
(
Some
r1''
·
Some
r2
)
as
[
r''
]
eqn
:
EQr''
;
[
eapply
erasure_not_empty
in
HE'
;
[
contradiction

now
erewrite
!
pcm_op_zero
by
apply
_
]
].
do
3
eexists
;
split
;
[
eassumption

split
;
[
eassumption
]
].
exists
r1''
r2
;
split
;
[
now
rewrite
EQr''

split
;
[
assumption
]
].
unfold
lt
in
HLt
;
rewrite
<
HLt
,
HSw
,
HSw'
in
HLR
;
apply
HLR
.
+
rewrite
fill_empty
in
HDec
;
eapply
values_stuck
;
eassumption
.
+
rewrite
fill_empty
in
HDec
;
subst
;
clear

HVal
.
assert
(
HT
:
=
fill_value
_
_
HVal
)
;
subst
K
;
rewrite
fill_empty
in
HVal
.
contradiction
(
fork_not_value
e'
).

subst
;
clear

HAt
;
eapply
fork_not_atomic
;
eassumption
.
Qed
.
(** Fork **)
Lemma
htFork
m
P
R
e
φ
:
ht
m
P
e
φ
⊑
ht
m
(
P
*
▹
R
)
(
fork
e
)
(
lift_bin
sc
(
eqV
(
exist
_
fork_ret
fork_ret_is_value
))
(
umconst
R
)).
Admitted
.
Lemma
htFork
m
P
R
e
:
ht
m
P
e
(
umconst
⊤
)
⊑
ht
m
(
P
*
▹
R
)
(
fork
e
)
(
lift_bin
sc
(
eqV
(
exist
_
fork_ret
fork_ret_is_value
))
(
umconst
R
)).
Proof
.
intros
w
n
rz
He
w'
HSw
n'
r
HLe
_
[
r1
[
r2
[
EQr
[
HP
HLR
]
]
]
].
specialize
(
He
_
HSw
_
_
HLe
(
unit_min
_
)
HP
).
clear
rz
n
HLe
;
unfold
wp
;
rewrite
fixp_eq
;
fold
(
wp
m
).
clear
w
HSw
;
rename
n'
into
n
;
rename
w'
into
w
;
intros
w'
;
intros
.
split
;
[
intros
;
contradiction
(
fork_not_value
e
)

split
;
intros
;
[
exfalso
]
].

assert
(
HT
:
=
fill_fork
_
_
_
HDec
)
;
subst
K
;
rewrite
fill_empty
in
HDec
;
subst
.
eapply
fork_stuck
with
(
K
:
=
ε
)
;
[
eassumption
]
;
reflexivity
.

assert
(
HT
:
=
fill_fork
_
_
_
HDec
)
;
subst
K
;
rewrite
fill_empty
in
HDec
.
apply
fork_inj
in
HDec
;
subst
e'
;
rewrite
<
EQr
in
HE
.
unfold
lt
in
HLt
;
rewrite
<
HLt
,
<
Le
.
le_n_Sn
,
HSw
in
He
.
rewrite
<
Le
.
le_n_Sn
in
HE
.
do
4
eexists
;
split
;
[
reflexivity

split
;
[
split
;
eassumption
]
].
rewrite
fill_empty
;
unfold
wp
;
rewrite
fixp_eq
;
fold
(
wp
m
).
rewrite
<
HLt
,
HSw
in
HLR
;
simpl
in
HLR
.
clear

HLR
;
intros
w''
;
intros
;
split
;
[
intros

split
;
intros
;
exfalso
].
+
do
3
eexists
;
split
;
[
reflexivity

split
;
[
eassumption
]
].
exists
(
pcm_unit
_
)
r2
;
split
;
[
now
erewrite
pcm_op_unit
by
apply
_
].
split
;
[
unfold
lt
in
HLt
;
rewrite
<
HLt
,
HSw
in
HLR
;
apply
HLR
].
simpl
;
reflexivity
.
+
eapply
values_stuck
;
eassumption

exact
fork_ret_is_value
.
+
assert
(
HV
:
=
fork_ret_is_value
)
;
rewrite
HDec
in
HV
;
clear
HDec
.
assert
(
HT
:
=
fill_value
_
_
HV
)
;
subst
K
;
rewrite
fill_empty
in
HV
.
eapply
fork_not_value
;
eassumption
.
Qed
.
(** Not stated: the Shift (timeless) rule *)
End
HoareTripleProperties
.
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