Skip to content
GitLab
Menu
Projects
Groups
Snippets
Loading...
Help
Help
Support
Community forum
Keyboard shortcuts
?
Submit feedback
Contribute to GitLab
Sign in / Register
Toggle navigation
Menu
Open sidebar
Rodolphe Lepigre
Iris
Commits
c6668f89
Commit
c6668f89
authored
Aug 05, 2016
by
Ralf Jung
Browse files
counterexample no longer needs duplicable ghost state
parent
230444d4
Changes
1
Hide whitespace changes
Inline
Sidebyside
Showing
1 changed file
with
91 additions
and
36 deletions
+91
36
program_logic/counter_examples.v
program_logic/counter_examples.v
+91
36
No files found.
program_logic/counter_examples.v
View file @
c6668f89
...
...
@@ 114,8 +114,8 @@ Module inv. Section inv.
Hypothesis
finished_agree
:
forall
n
m
,
finished
n
★
finished
m
⊢
n
=
m
.
Hypothesis
started_
persistent
:
forall
n
,
PersistentP
(
started
n
)
.
Hypothesis
finished_
persistent
:
forall
n
,
PersistentP
(
finished
n
)
.
Hypothesis
started_
dup
:
forall
n
,
started
n
⊢
started
n
★
started
n
.
Hypothesis
finished_
dup
:
forall
n
,
finished
n
⊢
finished
n
★
finished
n
.
(* We have that we cannot view shift from the initial state to false
(because the initial state is actually achievable). *)
...
...
@@ 191,60 +191,110 @@ Module inv. Section inv.
apply
pvs1_mono
.
by
rewrite

HP
(
uPred
.
exist_intro
a
).
Qed
.
(* "Weak box"  a weak form of □ for nonpersistent assertions. *)
Definition
wbox
P
:
iProp
:
=
∃
Q
,
Q
★
□
(
Q
→
P
)
★
□
(
Q
→
Q
★
Q
).
Lemma
wbox_dup
P
:
wbox
P
⊢
wbox
P
★
wbox
P
.
Proof
.
iIntros
"H"
.
iDestruct
"H"
as
(
Q
)
"(HQ & #HP & #Hdup)"
.
iDestruct
(
"Hdup"
with
"HQ"
)
as
"[HQ HQ']"
.
iSplitL
"HQ"
;
iExists
Q
;
iSplit
;
eauto
.
Qed
.
Lemma
wbox_out
P
:
wbox
P
⊢
P
.
Proof
.
iIntros
"H"
.
iDestruct
"H"
as
(
Q
)
"(HQ & #HP & _)"
.
iApply
"HP"
.
done
.
Qed
.
(** Now to the actual counterexample. We start with a weird for of saved propositions. *)
Definition
saved
(
i
:
name
)
(
P
:
iProp
)
:
iProp
:
=
∃
F
:
name
→
iProp
,
P
=
F
i
★
started
i
★
inv
i
(
auth_fresh
∨
∃
j
,
auth_start
j
∨
(
finished
j
★
□
F
j
)).
inv
i
(
auth_fresh
∨
∃
j
,
auth_start
j
∨
(
finished
j
★
wbox
(
F
j
))).
Lemma
saved_dup
i
P
:
saved
i
P
⊢
saved
i
P
★
saved
i
P
.
Proof
.
iIntros
"H"
.
iDestruct
"H"
as
(
F
)
"(#? & Hs & #?)"
.
iDestruct
(
started_dup
with
"Hs"
)
as
"[Hs Hs']"
.
iSplitL
"Hs"
.

iExists
F
.
eauto
.

iExists
F
.
eauto
.
Qed
.
Lemma
saved_alloc
(
P
:
name
→
iProp
)
:
auth_fresh
★
fresh
⊢
pvs1
(
∃
i
,
saved
i
(
P
i
)).
Proof
.
iIntros
"[Haf Hf]"
.
iVs
(
inv_alloc
(
auth_fresh
∨
∃
j
,
auth_start
j
∨
(
finished
j
★
□
P
j
))
with
"[Haf]"
)
as
(
i
)
"#Hi"
.
iIntros
"[Haf Hf]"
.
iVs
(
inv_alloc
(
auth_fresh
∨
∃
j
,
auth_start
j
∨
(
finished
j
★
wbox
(
P
j
))
)
with
"[Haf]"
)
as
(
i
)
"#Hi"
.
{
iLeft
.
done
.
}
iExists
i
.
iApply
inv_open'
.
iSplit
;
first
done
.
iIntros
"[HafHas]"
;
last
first
.
{
iExFalso
.
iDestruct
"Has"
as
(
j
)
"[Has  [Haf _]]"
.

iApply
fresh_not_start
.
iSplitL
"Has"
;
done
.

iApply
fresh_not_finished
.
iSplitL
"Haf"
;
done
.
}
iVs
((
fresh_start
i
)
with
"[Hf Haf]"
)
as
"[Has #Hs]"
;
first
by
iFrame
.
iApply
pvs0_intro
.
iSplitL
.
iVs
((
fresh_start
i
)
with
"[Hf Haf]"
)
as
"[Has Hs]"
;
first
by
iFrame
.
iDestruct
(
started_dup
with
"Hs"
)
as
"[Hs Hs']"
.
iApply
pvs0_intro
.
iSplitR
"Hs'"
.

iRight
.
iExists
i
.
iLeft
.
done
.

iApply
pvs1_intro
.
iExists
P
.
iSplit
;
first
done
.
by
iFrame
"#"
.

iApply
pvs1_intro
.
iExists
P
.
iSplit
;
first
done
.
by
iFrame
.
Qed
.
Lemma
saved_cast
i
P
Q
:
saved
i
P
★
saved
i
Q
★
□
P
⊢
pvs1
(
□
Q
).
saved
i
P
★
saved
i
Q
★
wbox
P
⊢
pvs1
(
wbox
Q
).
Proof
.
iIntros
"(
#
HsP &
#
HsQ &
#
HP)"
.
iDestruct
"HsP"
as
(
FP
)
"(% & HsP & HiP)"
.
iIntros
"(HsP & HsQ & HP)"
.
iDestruct
"HsP"
as
(
FP
)
"(% & HsP &
#
HiP)"
.
iApply
(
inv_open'
i
).
iSplit
;
first
done
.
iIntros
"[HaPHaP]"
.
{
iExFalso
.
iApply
started_not_fresh
.
iSplit
;
done
.
}
{
iExFalso
.
iApply
started_not_fresh
.
iSplit
L
"HaP"
;
done
.
}
(* Can I state a viewshift and immediately run it? *)
iAssert
(
pvs0
(
finished
i
))
with
"[HaP]"
as
"Hf"
.
iAssert
(
pvs0
(
finished
i
))
with
"[HaP
HsP
]"
as
"Hf"
.
{
iDestruct
"HaP"
as
(
j
)
"[Hs  [Hf _]]"
.

iApply
start_finish
.
(* FIXME: iPoseProof as "%" calls the assertion "%" instead of moving to the Coq context. *)
iPoseProof
(
started_start_agree
with
"[#]"
)
as
"H"
;
first
by
iSplit
.
iDestruct
"H"
as
%<.
done
.

iApply
pvs0_intro
.
iPoseProof
(
started_finished_agree
with
"[#]"
)
as
"H"
;
first
by
iSplit
.
iDestruct
"H"
as
%<.
done
.
}
iVs
"Hf"
as
"#Hf"
.
iApply
pvs0_intro
.
iSplitL
.
{
iRight
.
iExists
i
.
iRight
.
subst
.
eauto
.
}

iApply
start_finish
.
iDestruct
(
started_start_agree
with
"[#]"
)
as
"%"
;
first
by
iSplitL
"Hs"
.
subst
j
.
done
.

iApply
pvs0_intro
.
iDestruct
(
started_finished_agree
with
"[#]"
)
as
"%"
;
first
by
iSplitL
"Hf"
.
subst
j
.
done
.
}
iVs
"Hf"
as
"Hf"
.
iApply
pvs0_intro
.
iDestruct
(
finished_dup
with
"Hf"
)
as
"[Hf Hf']"
.
iSplitL
"Hf' HP"
.
{
iRight
.
iExists
i
.
iRight
.
subst
.
iSplitL
"Hf'"
;
done
.
}
iDestruct
"HsQ"
as
(
FQ
)
"(% & HsQ & HiQ)"
.
iApply
(
inv_open'
i
).
iSplit
;
first
iExact
"HiQ"
.
iIntros
"[HaQ  HaQ]"
.
{
iExFalso
.
iApply
started_not_fresh
.
iSplit
;
done
.
}
iDestruct
"HaQ"
as
(
j
)
"[HaS  #[Hf' HQ]]"
.
{
iExFalso
.
iApply
finished_not_start
.
eauto
.
}
iApply
pvs0_intro
.
iSplitL
.
{
iRight
.
iExists
j
.
eauto
.
}
{
iExFalso
.
iApply
started_not_fresh
.
iSplitL
"HaQ"
;
done
.
}
iDestruct
"HaQ"
as
(
j
)
"[HaS  [Hf' HQ]]"
.
{
iExFalso
.
iApply
finished_not_start
.
iSplitL
"HaS"
;
done
.
}
iApply
pvs0_intro
.
iDestruct
(
finished_dup
with
"Hf'"
)
as
"[Hf' Hf'']"
.
iDestruct
(
wbox_dup
with
"HQ"
)
as
"[HQ HQ']"
.
iSplitL
"Hf'' HQ'"
.
{
iRight
.
iExists
j
.
iRight
.
by
iSplitR
"HQ'"
.
}
iPoseProof
(
finished_agree
with
"[#]"
)
as
"H"
.
{
iFrame
"Hf Hf'"
.
done
.
}
iDestruct
"H"
as
%<.
iApply
pvs1_intro
.
subst
Q
.
done
.
Qed
.
(** And now we tie a bad knot. *)
Notation
"¬ P"
:
=
(
□
(
P
→
pvs1
False
))%
I
:
uPred_scope
.
Notation
"¬ P"
:
=
(
wbox
(
P

★
pvs1
False
))%
I
:
uPred_scope
.
Definition
A
i
:
iProp
:
=
∃
P
,
¬
P
★
saved
i
P
.
Instance
:
forall
i
,
PersistentP
(
A
i
)
:
=
_
.
Lemma
A_dup
i
:
A
i
⊢
A
i
★
A
i
.
Proof
.
iIntros
"HA"
.
iDestruct
"HA"
as
(
P
)
"[HNP HsP]"
.
iDestruct
(
wbox_dup
with
"HNP"
)
as
"[HNP HNP']"
.
iDestruct
(
saved_dup
with
"HsP"
)
as
"[HsP HsP']"
.
iSplitL
"HNP HsP"
;
iExists
P
.

by
iSplitL
"HNP"
.

by
iSplitL
"HNP'"
.
Qed
.
Lemma
A_wbox
i
:
A
i
⊢
wbox
(
A
i
).
Proof
.
iIntros
"H"
.
iExists
(
A
i
).
iSplitL
"H"
;
first
done
.
iSplit
;
first
by
iIntros
"!# ?"
.
iIntros
"!# HA"
.
by
iApply
A_dup
.
Qed
.
Lemma
A_alloc
:
auth_fresh
★
fresh
⊢
pvs1
(
∃
i
,
saved
i
(
A
i
)).
...
...
@@ 253,28 +303,33 @@ Module inv. Section inv.
Lemma
alloc_NA
i
:
saved
i
(
A
i
)
⊢
(
¬
A
i
).
Proof
.
iIntros
"#Hi !# #HAi"
.
iPoseProof
"HAi"
as
"HAi'"
.
iIntros
"Hi"
.
iExists
(
saved
i
(
A
i
)).
iSplitL
"Hi"
;
first
done
.
iSplit
;
last
by
(
iIntros
"!# ?"
;
iApply
saved_dup
).
iIntros
"!# Hi HAi"
.
iDestruct
(
A_dup
with
"HAi"
)
as
"[HAi HAi']"
.
iDestruct
"HAi'"
as
(
P
)
"[HNP Hi']"
.
iVs
((
saved_cast
i
)
with
"[]"
)
as
"HP"
.
{
iSplit
;
first
iExact
"Hi"
.
iSplit
;
first
iExact
"Hi'"
.
done
.
}
iDestruct
"HP"
as
"#HP"
.
by
iApply
"HNP"
.
iVs
((
saved_cast
i
)
with
"[Hi Hi' HAi]"
)
as
"HP"
.
{
iSplitL
"Hi"
;
first
done
.
iSplitL
"Hi'"
;
first
done
.
by
iApply
A_wbox
.
}
iPoseProof
(
wbox_out
with
"HNP"
)
as
"HNP"
.
iApply
"HNP"
.
iApply
wbox_out
.
done
.
Qed
.
Lemma
alloc_A
i
:
saved
i
(
A
i
)
⊢
A
i
.
Proof
.
iIntros
"#Hi"
.
iPoseProof
(
alloc_NA
with
"[]"
)
as
"HNA"
;
first
done
.
(* Patterns in iPoseProof don't seem to work; adding a "#" here also does the wrong thing.
Or maybe iPoseProof is the wrong tactic  but then which is the right one? *)
iDestruct
"HNA"
as
"#HNA"
.
iExists
(
A
i
).
iSplit
;
done
.
iIntros
"Hi"
.
iDestruct
(
saved_dup
with
"Hi"
)
as
"[Hi Hi']"
.
iPoseProof
(
alloc_NA
with
"Hi"
)
as
"HNA"
.
iExists
(
A
i
).
iSplitL
"HNA"
;
done
.
Qed
.
Lemma
contradiction
:
False
.
Proof
.
apply
soundness
.
iIntros
"H"
.
iVs
(
A_alloc
with
"H"
)
as
"H"
.
iDestruct
"H"
as
(
i
)
"#H"
.
iPoseProof
(
alloc_NA
with
"H"
)
as
"HN"
.
iApply
"HN"
.
(* FIXME: "iApply alloc_NA" does not work. *)
iVs
(
A_alloc
with
"H"
)
as
"H"
.
iDestruct
"H"
as
(
i
)
"H"
.
iDestruct
(
saved_dup
with
"H"
)
as
"[H H']"
.
iPoseProof
(
alloc_NA
with
"H"
)
as
"HN"
.
iPoseProof
(
wbox_out
with
"HN"
)
as
"HN"
.
iApply
"HN"
.
iApply
alloc_A
.
done
.
Qed
.
...
...
Write
Preview
Markdown
is supported
0%
Try again
or
attach a new file
.
Attach a file
Cancel
You are about to add
0
people
to the discussion. Proceed with caution.
Finish editing this message first!
Cancel
Please
register
or
sign in
to comment