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Paolo G. Giarrusso
examples
Commits
043247ee
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043247ee
authored
Dec 14, 2017
by
Lars Birkedal
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(* In this file we explain how to do the "parallel increment" example (Example
7.5) from the lecture notes in Iris in Coq. *)
(* Contains definitions of the weakest precondition assertion, and its basic
rules. *)
From
iris
.
program_logic
Require
Export
weakestpre
.
(* Definition of invariants and their rules (expressed using the fancy update
modality). *)
From
iris
.
base_logic
.
lib
Require
Export
invariants
.
(* Files related to the interactive proof mode. The first import includes the
general tactics of the proof mode. The second provides some more specialized
tactics particular to the instantiation of Iris to a particular programming
language. *)
From
iris
.
proofmode
Require
Import
tactics
.
From
iris
.
heap_lang
Require
Import
proofmode
.
(* Instantiation of Iris with the particular language. The notation file
contains many shorthand notations for the programming language constructs, and
the lang file contains the actual language syntax. *)
From
iris
.
heap_lang
Require
Import
notation
lang
.
(* We also import the parallel composition construct. This is not a primitive of
the language, but is instead derived. This file contains its definition, and
the proof rule associated with it. *)
From
iris
.
heap_lang
.
lib
Require
Import
par
.
(* We define our terms. The Iris Coq library defines many notations for
programming language constructs, e.g., lambdas, allocation, accessing and so
on. The complete list of notations can be found in
theories/heap_lang/notations.v file in the iriscoq repository.
The # in the notation is used to embed literals, e.g., variables, numbers, as
values of the programmin language. *)
Definition
incr
(
ℓ
:
loc
)
:
expr
:
=
#
ℓ
<
!#
ℓ
+
#
1
.
Section
proof
.
(* In order to do the proof we need to assume certain things about the
instantiation of Iris. The particular, even the heap is handled in an
analogous way as other ghost state. This line states that we assume the
Iris instantiation has sufficient structure to manipulate the heap, e.g.,
it allows us to use the pointsto predicate. *)
Context
`
{!
heapG
Σ
}.
(* Recall that parallel composition construct is defined in terms of fork. To
prove the expected rules for this construct we need some particular ghost
state in the instantiation of Iris, as explained in the lecture notes. The
following line states that we have this ghost state. *)
Context
`
{!
spawnG
Σ
}.
(* The variable Σ has to do with what ghost state is available, and the type
of Iris propositions (written Prop in the lecture notes) depends on this Σ.
But since Σ is the same throughout the development we shall define
shorthand notation which hides it. *)
Notation
iProp
:
=
(
iProp
Σ
).
(* As in the paper proof we will need an invariant to share access to a
location. This invariant will be allocated in this namespace which is a
parameter of the whole development. *)
Context
(
N
:
namespace
).
(* We now start with the details particular to the example. Everything above
was essentially boilerplate which will be there in most verifications. *)
(* The invariant we are going to use is the following. Note that this is the
body of the invariant, not the invariant assertion. Note the scope
delimiter "%I". This is to tell Coq to parse the logical formula as an Iris
assertion, i.e., to interpret the connectives as Iris connectives, and not
perhaps as Coq connectives, or something else. The final piece of syntax is
⌜ ... ⌝. This is used to embed Coq propositions as Iris assertions. That
is, ≥ is the ordinary greaterthanorequal relation on natural numbers,
and we make it an Iris assertion by using ⌜ ... ⌝. Semantically, the
embedding is such that the embedded assertion either holds for all
resources, or none. *)
Definition
incr_inv
(
ℓ
:
loc
)
(
n
:
Z
)
:
iProp
:
=
(
∃
(
m
:
Z
),
⌜
n
≤
m
⌝
∗
ℓ
↦
#
m
)%
I
.
(** The main proofs. *)
(* As in example 7.5 in the notes we will show the following specification.
The specification is parametrized by any location ℓ and natural number n *)
Lemma
parallel_incr_spec
(
ℓ
:
loc
)
(
n
:
Z
)
:
{{{
ℓ
↦
#
n
}}}
(
incr
ℓ
)

(
incr
ℓ
)
;;
!#
ℓ
{{{
m
,
RET
#
m
;
⌜
n
≤
m
⌝
}}}.
Proof
.
(* We first unfold the triple notation. Recall its definition from Section 9
of the lecture notes. *)
iIntros
(
Φ
)
"Hpt HΦ"
.
(* As in the paper proof we now allocate an invariant (in namespace N) and
transfer the points to predicate into it. This is achieved by using the
rule/lemma inv_alloc. But since the allocation of invariants involves the
fancy update modality we use the iMod tactic around the lemma. This
tactic knows about the structural rules of the update modalities, and the
interaction of the update modality and the weakest precondition
assertion, and thus it automatically removes the modality as much as
possible.
The inv_alloc rule has three parameters. The namespace in which to
allocate, the mask which is to be used on the fancy update modality (see
the rule in the notes), and the body of the invariant to be allocated.
Here we leave the mask implicit, since it is determined by the current
goal (the weakest precondition has a mask, which, if not explicitly
stated, is the top mask, containing all the invariant names.
The next ingredient in the line below is the "with" construct. This tells
the iMod tactic which of the assumptions are to be used to satisfy the
assumptions of the inv_alloc rule/lemma. See the ProofMode.md in the
iriscoq repository for the precise syntax of the pattern which follows
the "with" keyword. Here we are using the simplest pattern, we are just
going to use one of our assumptions.
Finally we have the 'as "#Hinv"'. This tells the iMod tactic to name the
conclusion of the inv_alloc rule "HInv", and add it as one of the
persistent assumptions. The # symbol is a pattern used to denote
persistent assumptions. When we wish to move an assertion to the
persistent context the interactive proof mode uses Coq's typeclass search
to try and determine whether the assertion is indeed persistent. In this
case the assertion is an invariant, hence it is. *)
iMod
(
inv_alloc
N
_
(
incr_inv
ℓ
n
)
with
"[Hpt]"
)
as
"#HInv"
.
(* Now we have two subgoals. We need to prove the assumptions of the
inv_alloc rule first. *)

iNext
;
iExists
n
;
iFrame
.
(* This is easy to prove. We use the next introduction rule (the tactic
iNext). And then we have to prove that ℓ ↦ #n ⊢ ∃ m, ℓ ↦ #m ∗ m ≥ n. We
pick n as the witness of the existential, and then we have to prove ℓ ↦
#n ⊢ ℓ ↦ #n ∗ n ≥ n. We do this by first "framing" away the ℓ ↦ #n. The
iFrame tactic is essentially the rule ∗mono.
After these we are left with the goal ⌜n ≥ n⌝. To prove such goals we,
most of the time, wish to exit the interactive proof mode and go to the
ordinary Coq proof mode. This is achieved via the iIntros "!%" tactic.
Here, "!%" is another pattern, as described in ProofMode.md. Hence,
after this we are left with proving n ≥ n, which is solved by the lia
tactic (linear integer arithmetic solver). *)
iIntros
"!%"
.
lia
.

(* Next we use wp_bind rule, as in the paper proof. This rule is
implemented by the wp_bind tactic, which does some bookkeeping so that
the bind rule is as easy to use as on paper. The argument to the tactic
is the expression we wish to focus on. In this case the first part of
the sequencing expression. Note the scope delimiter %E. This is the
scope delimiter for expressions of the language with which Iris is
instantiated. *)
wp_bind
(
incr
ℓ

incr
ℓ
)%
E
.
(* Now we are in a position to use the parallel composition rule, which is
proved in the par library we have imported above. The rule/lemma is
called wp_par. The two arguments are the conclusions of the two
parallel threads. Here they are simply True, as in the paper proof when
we used the htpar rule. The first two subgoals are bookkeeping
subgoals about closedness (absence of free variables) of expression
incr ℓ and incr ℓ. They are easily dispatched by the done tactic. *)
iApply
(
wp_par
(
λ
_
,
⌜
True
⌝
)%
I
(
λ
_
,
⌜
True
⌝
)%
I
)
;
[
done

done

..].
(* We now have three subgoals. The first two are proofs that each thread
does the correct thing, and the final goal is to show that the combined
conclusion of the two threads implies the desired conclusion. This last
part is a peculiarity of the wp_par rule as stated in Coq. It builds in
the rule of consequence, which we would otherwise have to use manually, as
we did when proving on paper. *)
+
(* The proof that the first thread is correct is exactly the same as the
proof on paper. The expression is not atomic, so we cannot open the
invariant immediately. Thus we do as usual, we use the bind rule to
reshape it, then open the invariant and then ... There is a minor
technicality we need to do first. We need to unfold the definition of incr.*)
rewrite
/
incr
.
wp_bind
(!#
ℓ
)%
E
.
(* Now we can open the invariant to read a value stored at location ℓ. *)
(* Opening invariants is done using the iInv tactic. In the most basic
form it takes a namespace as the first argument, and two named
assertions. These are the two parts of the inv_open rule (the two
parts of the conclusion). *)
iInv
N
as
"H"
"Hclose"
.
(* After this we can read the value, since we have the resources. The
tactic wp_load implements the rule for reading a memory location. But
first, we need to eliminate the existential "H" (the incr_inv
assertion) to get a points to predicate. This we do using the
iDestruct tactic. The pattern this time is quite advanced. The
assertion named H is ▷∃ m, ⌜m ≥ n⌝ ∗ ℓ ↦ m. The assertion ∃m, ⌜m ≥ n⌝
∗ ℓ ↦ m is timeless. Hence because the conclusion is the weakest
precondition assertion we can remove the later, which is the purpose
of > in the beginning of the pattern. The rest of the pattern is (m)
and [% Hpt]. The (m) states to name the variable we get after
destructing the existential, m. Finally we have to deal with ⌜m ≥ n⌝
∗ ℓ ↦ m. The pattern [% Hpt] states to move the first part, ⌜m ≥ n⌝
to the pure Coq context (the % character), and it states that the
second part should be named Hpt. *)
iDestruct
"H"
as
(
m
)
">[% Hpt]"
.
(* Now we have a points to assertion in context, so we can use wp_load rule. *)
wp_load
.
(* After reading we must again close the invariant. For this we have the
"Hclose" assertion, which we got when opening the invariant.
We close the invariant by transferring the points to predicate back.
And since closing involves manipulation of fancy updates we use the
iMod tactic. The as "_" pattern states to ignore the conclusion of
"Hclose" assertion. The interesting part of Hclose is the change of
masks, and the conclusion is simply True, which we can ignore. *)
iMod
(
"Hclose"
with
"[Hpt]"
)
as
"_"
.
{
iNext
;
iExists
m
;
iFrame
;
iIntros
"!%"
;
auto
.
}
(* We now have to prove ={T}=> wp ... , but the modality is just in the
way. Thus we use the introduction rule to get rid of it in the goal.
iModIntro is the tactic which introduces modalities such as the update
and the fancy update modality. *)
iModIntro
.
(* Next we do as on paper. We compute the value #m + #1 into #(m+1),
which is simple to do with wp_bind and wp_op tactics. In fact, wp_op
on its own would suffice, since it tries to find a basic operation
and use wp_bind automatically if it does.*)
wp_bind
(
_
+
_
)%
E
;
wp_op
.
(* We then repeat the process of opening the invariant, writing, and
closing the invariant. But to illustrate the flexibility of tactics we
join the call to iInv and iDestruct into one, with a complex pattern. *)
iInv
N
as
(
k
)
">[% Hpt]"
"Hclose"
.
wp_store
.
iMod
(
"Hclose"
with
"[Hpt]"
)
as
"_"
.
{
iNext
;
iExists
(
m
+
1
)
;
iFrame
;
iIntros
"!%"
;
lia
.
}
(* And we are left with proving ={T}=> True, which is trivial. *)
done
.
+
(* The second thread is exactly the same as the first, so the proof is
the same. We could have factored out the proof into a separate lemma,
but instead we here give a shorter version using more advanced
features of the tactics. The reader should compare it with the
previous proof. *)
rewrite
/
incr
.
wp_bind
(!#
ℓ
)%
E
.
iInv
N
as
(
m
)
">[% Hpt]"
"Hclose"
.
wp_load
.
iMod
(
"Hclose"
with
"[Hpt]"
)
as
"_"
.
{
iExists
m
;
iFrame
;
done
.
}
iModIntro
.
wp_op
.
iInv
N
as
(
k
)
">[% Hpt]"
"Hclose"
.
wp_store
.
iMod
(
"Hclose"
with
"[Hpt]"
)
as
"_"
;
last
done
.
{
iExists
(
m
+
1
)
;
iFrame
;
iIntros
"!%"
;
lia
.
}
+
(* And the last goal is before us. We first simplify to get rid of
superflous assumptions and variables. As above, the pattern _ means
ignore this assumption. It is always safe to ignore True as an
assumption. The iIntros tactic can additionally be used to introduce
the forall quantifiers by giving variable names as the first
argument. Here we do not care about the variable names so we give ?
as the pattern, which lets Coq generate a variable name for us. *)
iIntros
(?
?)
"_"
.
(* To be able to use the wp_tactics the goal needs to be of the form ▷
..., thus we first remove the later modality using the later
introduction rule, implemented by the iNext tactic. After that we
have to deal with the application of a function with a dummy argument,
i.e., sequencing. The wp_lam tactic handles it. *)
iNext
.
wp_lam
.
(* The last interesting part of the proof is before us. We do exactly as
we did on paper. We open the invariant, and read the value. And the
invariant will tell us that the value is at least n. We can then apply
the "continuation" HΦ to conclude the proof. *)
iInv
N
as
(
m
)
">[% Hpt]"
"Hclose"
.
wp_load
.
iMod
(
"Hclose"
with
"[Hpt]"
)
as
"_"
.
{
iExists
m
;
iFrame
;
done
.
}
iModIntro
.
(* As stated above, to conclude the proof we apply the "continuation"
HΦ. This illustrates another new feature of the tactics. Observe that
HΦ is a universally quantified formula. To instantiate the variables
we use the $! notation as follows. *)
iApply
(
"HΦ"
$!
m
).
(* And we are left with proving the premise of the continuation, which
is easy, since it is one of our assumptions which we obtained when
opening the invariant. *)
done
.
Qed
.
End
proof
.
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