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92f0cde6
Commit
92f0cde6
authored
Aug 05, 2016
by
Ralf Jung
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complete proof that invariants without laters are inconsistent
parent
6e9785bf
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program_logic/counter_examples.v
program_logic/counter_examples.v
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program_logic/counter_examples.v
View file @
92f0cde6
...
...
@@ 3,7 +3,7 @@ From iris.proofmode Require Import tactics.
(** This proves that we need the ▷ in a "Saved Proposition" construction with
namedependend allocation. *)
Section
savedprop
.
Module
savedprop
.
Section
savedprop
.
Context
(
M
:
ucmraT
).
Notation
iProp
:
=
(
uPred
M
).
Notation
"¬ P"
:
=
(
□
(
P
→
False
))%
I
:
uPred_scope
.
...
...
@@ 57,14 +57,13 @@ Section savedprop.
apply
(@
uPred
.
adequacy
M
False
1
)
;
simpl
.
rewrite

uPred
.
later_intro
.
apply
rvs_false
.
Qed
.
End
savedprop
.
End
savedprop
.
End
savedprop
.
(** This proves that we need the ▷ when opening invariants. *)
(** We fork in [uPred M] for any M, but the proof would work in any BI. *)
Section
inv
.
Module
inv
.
Section
inv
.
Context
(
M
:
ucmraT
).
Notation
iProp
:
=
(
uPred
M
).
Notation
"¬ P"
:
=
(
□
(
P
→
False
))%
I
:
uPred_scope
.
Implicit
Types
P
:
iProp
.
(** Assumptions *)
...
...
@@ 161,7 +160,6 @@ Section inv.
Global
Instance
elim_pvs0_pvs0
P
Q
:
ElimVs
(
pvs0
P
)
P
(
pvs0
Q
)
(
pvs0
Q
).
Proof
.
rename
Q0
into
Q
.
rewrite
/
ElimVs
.
etrans
;
last
eapply
pvs0_pvs0
.
rewrite
pvs0_frame_r
.
apply
pvs0_mono
.
by
rewrite
uPred
.
wand_elim_r
.
Qed
.
...
...
@@ 169,7 +167,6 @@ Section inv.
Global
Instance
elim_pvs1_pvs1
P
Q
:
ElimVs
(
pvs1
P
)
P
(
pvs1
Q
)
(
pvs1
Q
).
Proof
.
rename
Q0
into
Q
.
rewrite
/
ElimVs
.
etrans
;
last
eapply
pvs1_pvs1
.
rewrite
pvs1_frame_r
.
apply
pvs1_mono
.
by
rewrite
uPred
.
wand_elim_r
.
Qed
.
...
...
@@ 177,7 +174,6 @@ Section inv.
Global
Instance
elim_pvs0_pvs1
P
Q
:
ElimVs
(
pvs0
P
)
P
(
pvs1
Q
)
(
pvs1
Q
).
Proof
.
rename
Q0
into
Q
.
rewrite
/
ElimVs
.
rewrite
pvs0_pvs1
.
apply
elim_pvs1_pvs1
.
Qed
.
...
...
@@ 195,7 +191,7 @@ Section inv.
apply
pvs1_mono
.
by
rewrite

HP
(
uPred
.
exist_intro
a
).
Qed
.
(** Now to the actual counterexample. *)
(** 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
)).
...
...
@@ 245,39 +241,41 @@ Section inv.
iDestruct
"H"
as
%<.
iApply
pvs1_intro
.
subst
Q
.
done
.
Qed
.
(*
Now, define:
N(P) := box(P ==> False)
A[i] := Exists P. N(P) * i > P
Notice that A[i] => box(A[i]).
OK, now we are going to prove that True ==> False.
(** And now we tie a bad knot. *)
Notation
"¬ P"
:
=
(
□
(
P
→
pvs1
False
))%
I
:
uPred_scope
.
Definition
A
i
:
iProp
:
=
∃
P
,
¬
P
★
saved
i
P
.
Instance
:
forall
i
,
PersistentP
(
A
i
)
:
=
_
.
First we allocate some k s.t. k > A[k], which we know we can do
because of the axiom for >.
Lemma
A_alloc
:
auth_fresh
★
fresh
⊢
pvs1
(
∃
i
,
saved
i
(
A
i
)).
Proof
.
by
apply
saved_alloc
.
Qed
.
Claim 2: N(A[k]).
Proof:
 Suppose A[k]. So, box(A[k]). So, A[k] * A[k].
 So there is some P s.t. A[k] * N(P) * k > P.
 Since k > A(k), by Claim 1 we can view shift to P * N(P).
 Hence, we can view shift to False.
QED.
Notice that in Iris proper all we could get on the third line of the
above proof is later(P) * N(P), which would not be enough to derive
the claim.
Lemma
alloc_NA
i
:
saved
i
(
A
i
)
⊢
(
¬
A
i
).
Proof
.
iIntros
"#Hi !# #HAi"
.
iPoseProof
"HAi"
as
"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"
.
Qed
.
Claim 3: A[k].
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
.
Qed
.
Proof:
 By Claim 2, we have N(A(k)) * k > A[k].
 Thus, picking P := A[k], we have Exists P. N(P) * k > P, i.e. we have A[k].
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. *)
iApply
alloc_A
.
done
.
Qed
.
Claim 2 and Claim 3 together view shift to False.
*)
End
inv
.
End
inv
.
End
inv
.
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