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c6668f89
Commit
c6668f89
authored
Aug 05, 2016
by
Ralf Jung
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counterexample no longer needs duplicable ghost state
parent
230444d4
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program_logic/counter_examples.v
program_logic/counter_examples.v
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program_logic/counter_examples.v
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c6668f89
...
@@ 114,8 +114,8 @@ Module inv. Section inv.
...
@@ 114,8 +114,8 @@ Module inv. Section inv.
Hypothesis
finished_agree
:
Hypothesis
finished_agree
:
forall
n
m
,
finished
n
★
finished
m
⊢
n
=
m
.
forall
n
m
,
finished
n
★
finished
m
⊢
n
=
m
.
Hypothesis
started_
persistent
:
forall
n
,
PersistentP
(
started
n
)
.
Hypothesis
started_
dup
:
forall
n
,
started
n
⊢
started
n
★
started
n
.
Hypothesis
finished_
persistent
:
forall
n
,
PersistentP
(
finished
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
(* We have that we cannot view shift from the initial state to false
(because the initial state is actually achievable). *)
(because the initial state is actually achievable). *)
...
@@ 191,60 +191,110 @@ Module inv. Section inv.
...
@@ 191,60 +191,110 @@ Module inv. Section inv.
apply
pvs1_mono
.
by
rewrite

HP
(
uPred
.
exist_intro
a
).
apply
pvs1_mono
.
by
rewrite

HP
(
uPred
.
exist_intro
a
).
Qed
.
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. *)
(** Now to the actual counterexample. We start with a weird for of saved propositions. *)
Definition
saved
(
i
:
name
)
(
P
:
iProp
)
:
iProp
:
=
Definition
saved
(
i
:
name
)
(
P
:
iProp
)
:
iProp
:
=
∃
F
:
name
→
iProp
,
P
=
F
i
★
started
i
★
∃
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
)
:
Lemma
saved_alloc
(
P
:
name
→
iProp
)
:
auth_fresh
★
fresh
⊢
pvs1
(
∃
i
,
saved
i
(
P
i
)).
auth_fresh
★
fresh
⊢
pvs1
(
∃
i
,
saved
i
(
P
i
)).
Proof
.
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
.
}
{
iLeft
.
done
.
}
iExists
i
.
iApply
inv_open'
.
iSplit
;
first
done
.
iIntros
"[HafHas]"
;
last
first
.
iExists
i
.
iApply
inv_open'
.
iSplit
;
first
done
.
iIntros
"[HafHas]"
;
last
first
.
{
iExFalso
.
iDestruct
"Has"
as
(
j
)
"[Has  [Haf _]]"
.
{
iExFalso
.
iDestruct
"Has"
as
(
j
)
"[Has  [Haf _]]"
.

iApply
fresh_not_start
.
iSplitL
"Has"
;
done
.

iApply
fresh_not_start
.
iSplitL
"Has"
;
done
.

iApply
fresh_not_finished
.
iSplitL
"Haf"
;
done
.
}

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

iRight
.
iExists
i
.
iLeft
.
done
.

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
.
Qed
.
Lemma
saved_cast
i
P
Q
:
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
.
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
.
iApply
(
inv_open'
i
).
iSplit
;
first
done
.
iIntros
"[HaPHaP]"
.
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? *)
(* 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 _]]"
.
{
iDestruct
"HaP"
as
(
j
)
"[Hs  [Hf _]]"
.

iApply
start_finish
.
(* FIXME: iPoseProof as "%" calls the assertion "%" instead of moving to the Coq context. *)

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

iApply
pvs0_intro
.
iPoseProof
(
started_finished_agree
with
"[#]"
)
as
"H"
;
first
by
iSplit
.

iApply
pvs0_intro
.
iDestruct
"H"
as
%<.
done
.
}
iDestruct
(
started_finished_agree
with
"[#]"
)
as
"%"
;
first
by
iSplitL
"Hf"
.
iVs
"Hf"
as
"#Hf"
.
iApply
pvs0_intro
.
iSplitL
.
subst
j
.
done
.
}
{
iRight
.
iExists
i
.
iRight
.
subst
.
eauto
.
}
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)"
.
iDestruct
"HsQ"
as
(
FQ
)
"(% & HsQ & HiQ)"
.
iApply
(
inv_open'
i
).
iSplit
;
first
iExact
"HiQ"
.
iApply
(
inv_open'
i
).
iSplit
;
first
iExact
"HiQ"
.
iIntros
"[HaQ  HaQ]"
.
iIntros
"[HaQ  HaQ]"
.
{
iExFalso
.
iApply
started_not_fresh
.
iSplit
;
done
.
}
{
iExFalso
.
iApply
started_not_fresh
.
iSplitL
"HaQ"
;
done
.
}
iDestruct
"HaQ"
as
(
j
)
"[HaS  #[Hf' HQ]]"
.
iDestruct
"HaQ"
as
(
j
)
"[HaS  [Hf' HQ]]"
.
{
iExFalso
.
iApply
finished_not_start
.
eauto
.
}
{
iExFalso
.
iApply
finished_not_start
.
iSplitL
"HaS"
;
done
.
}
iApply
pvs0_intro
.
iSplitL
.
iApply
pvs0_intro
.
{
iRight
.
iExists
j
.
eauto
.
}
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"
.
iPoseProof
(
finished_agree
with
"[#]"
)
as
"H"
.
{
iFrame
"Hf Hf'"
.
done
.
}
{
iFrame
"Hf Hf'"
.
done
.
}
iDestruct
"H"
as
%<.
iApply
pvs1_intro
.
subst
Q
.
done
.
iDestruct
"H"
as
%<.
iApply
pvs1_intro
.
subst
Q
.
done
.
Qed
.
Qed
.
(** And now we tie a bad knot. *)
(** 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
.
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
:
Lemma
A_alloc
:
auth_fresh
★
fresh
⊢
pvs1
(
∃
i
,
saved
i
(
A
i
)).
auth_fresh
★
fresh
⊢
pvs1
(
∃
i
,
saved
i
(
A
i
)).
...
@@ 253,28 +303,33 @@ Module inv. Section inv.
...
@@ 253,28 +303,33 @@ Module inv. Section inv.
Lemma
alloc_NA
i
:
Lemma
alloc_NA
i
:
saved
i
(
A
i
)
⊢
(
¬
A
i
).
saved
i
(
A
i
)
⊢
(
¬
A
i
).
Proof
.
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']"
.
iDestruct
"HAi'"
as
(
P
)
"[HNP Hi']"
.
iVs
((
saved_cast
i
)
with
"[]"
)
as
"HP"
.
iVs
((
saved_cast
i
)
with
"[Hi Hi' HAi]"
)
as
"HP"
.
{
iSplit
;
first
iExact
"Hi"
.
iSplit
;
first
iExact
"Hi'"
.
done
.
}
{
iSplitL
"Hi"
;
first
done
.
iSplitL
"Hi'"
;
first
done
.
by
iApply
A_wbox
.
}
iDestruct
"HP"
as
"#HP"
.
by
iApply
"HNP"
.
iPoseProof
(
wbox_out
with
"HNP"
)
as
"HNP"
.
iApply
"HNP"
.
iApply
wbox_out
.
done
.
Qed
.
Qed
.
Lemma
alloc_A
i
:
Lemma
alloc_A
i
:
saved
i
(
A
i
)
⊢
A
i
.
saved
i
(
A
i
)
⊢
A
i
.
Proof
.
Proof
.
iIntros
"#Hi"
.
iPoseProof
(
alloc_NA
with
"[]"
)
as
"HNA"
;
first
done
.
iIntros
"Hi"
.
iDestruct
(
saved_dup
with
"Hi"
)
as
"[Hi Hi']"
.
(* Patterns in iPoseProof don't seem to work; adding a "#" here also does the wrong thing.
iPoseProof
(
alloc_NA
with
"Hi"
)
as
"HNA"
.
Or maybe iPoseProof is the wrong tactic  but then which is the right one? *)
iExists
(
A
i
).
iSplitL
"HNA"
;
done
.
iDestruct
"HNA"
as
"#HNA"
.
iExists
(
A
i
).
iSplit
;
done
.
Qed
.
Qed
.
Lemma
contradiction
:
False
.
Lemma
contradiction
:
False
.
Proof
.
Proof
.
apply
soundness
.
iIntros
"H"
.
apply
soundness
.
iIntros
"H"
.
iVs
(
A_alloc
with
"H"
)
as
"H"
.
iDestruct
"H"
as
(
i
)
"#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. *)
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
.
iApply
alloc_A
.
done
.
Qed
.
Qed
.
...
...
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