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ece1ecdd
Commit
ece1ecdd
authored
Mar 13, 2019
by
Robbert Krebbers
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Get rid of locked value lambdas.
parent
2030d90c
Changes
6
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6 changed files
with
37 additions
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43 deletions
+37
43
HeapLang.md
HeapLang.md
+4
7
theories/heap_lang/lib/assert.v
theories/heap_lang/lib/assert.v
+4
3
theories/heap_lang/lib/par.v
theories/heap_lang/lib/par.v
+1
1
theories/heap_lang/lifting.v
theories/heap_lang/lifting.v
+18
10
theories/heap_lang/notation.v
theories/heap_lang/notation.v
+2
13
theories/heap_lang/proofmode.v
theories/heap_lang/proofmode.v
+8
9
No files found.
HeapLang.md
View file @
ece1ecdd
...
...
@@ 42,10 +42,6 @@ We define a whole lot of shorthands, such as nonrecursive functions (`λ:`),
letbindings, sequential composition, and a more conventional
`match:`
that has
binders in both branches.
Noteworthy is the fact that functions (
`rec:`
,
`λ:`
) in the value scope (
`%V`
)
are
*locked*
. This is to prevent them from being unfolded and reduced too
eagerly.
The widely used
`#`
is a shorthand to turn a basic literal (an integer, a
location, a boolean literal or a unit value) into a value. Since values coerce
to expressions,
`#`
is widely used whenever a Coq value needs to be placed into
...
...
@@ 62,9 +58,10 @@ Tactics to take one or more pure program steps:

`wp_pure`
: Perform one pure reduction step. Pure steps are defined by the
`PureExec`
typeclass and include beta reduction, projections, constructors, as
well as unary and binary arithmetic operators.

`wp_pures`
: Perform as many pure reduction steps as possible.

`wp_pures`
: Perform as many pure reduction steps as possible. This
tactic will
**not**
reduce lambdas/recs that are hidden behind a definition.

`wp_rec`
,
`wp_lam`
: Perform a beta reduction. Unlike
`wp_pure`
, this will
also reduce l
ocked lambdas
.
also reduce l
ambdas that are hidden behind a definition
.

`wp_let`
,
`wp_seq`
: Reduce a letbinding or a sequential composition.

`wp_proj`
: Reduce a projection.

`wp_if_true`
,
`wp_if_false`
,
`wp_if`
: Reduce a conditional expression. The
...
...
@@ 122,7 +119,7 @@ The normal `e1  e2` notation uses expression lambdas, because clearly we want
value lambda). However, the
*specification*
for parallel composition should use
value lambdas, because prior to applying it the term will be reduced as much as
possible to achieve a normal form. To facilitate this, we define a copy of the
`e1  e2`
notation in the value scope that uses
*unlocked*
value lambdas.
`e1  e2`
notation in the value scope that uses value lambdas.
This is not actually a value, but we still but it in the value scope to
differentiate from the other notation that uses expression lambdas. (In the
future, we might decide to add a separate scope for this.) Then, we write the
...
...
theories/heap_lang/lib/assert.v
View file @
ece1ecdd
...
...
@@ 7,11 +7,12 @@ Set Default Proof Using "Type".
Definition
assert
:
val
:
=
λ
:
"v"
,
if
:
"v"
#()
then
#()
else
#
0
#
0
.
(* #0 #0 is unsafe *)
(* just below ;; *)
Notation
"'assert:' e"
:
=
(
assert
(
λ
:
<>,
e
))%
E
(
at
level
99
)
:
expr_scope
.
Notation
"'assert:' e"
:
=
(
assert
(
λ
:
<>,
e
)%
E
)
(
at
level
99
)
:
expr_scope
.
Notation
"'assert:' e"
:
=
(
assert
(
λ
:
<>,
e
)%
V
)
(
at
level
99
)
:
val_scope
.
Lemma
twp_assert
`
{!
heapG
Σ
}
E
(
Φ
:
val
→
iProp
Σ
)
e
:
WP
e
@
E
[{
v
,
⌜
v
=
#
true
⌝
∧
Φ
#()
}]

∗
WP
assert
(
LamV
BAnon
e
)%
V
@
E
[{
Φ
}].
WP
(
assert
:
e
)%
V
@
E
[{
Φ
}].
Proof
.
iIntros
"HΦ"
.
wp_lam
.
wp_apply
(
twp_wand
with
"HΦ"
).
iIntros
(
v
)
"[% ?]"
;
subst
.
by
wp_if
.
...
...
@@ 19,7 +20,7 @@ Qed.
Lemma
wp_assert
`
{!
heapG
Σ
}
E
(
Φ
:
val
→
iProp
Σ
)
e
:
WP
e
@
E
{{
v
,
⌜
v
=
#
true
⌝
∧
▷
Φ
#()
}}

∗
WP
assert
(
LamV
BAnon
e
)%
V
@
E
{{
Φ
}}.
WP
(
assert
:
e
)%
V
@
E
{{
Φ
}}.
Proof
.
iIntros
"HΦ"
.
wp_lam
.
wp_apply
(
wp_wand
with
"HΦ"
).
iIntros
(
v
)
"[% ?]"
;
subst
.
by
wp_if
.
...
...
theories/heap_lang/lib/par.v
View file @
ece1ecdd
...
...
@@ 12,7 +12,7 @@ Definition par : val :=
let
:
"v1"
:
=
join
"handle"
in
(
"v1"
,
"v2"
).
Notation
"e1  e2"
:
=
(
par
(
λ
:
<>,
e1
)%
E
(
λ
:
<>,
e2
)%
E
)
:
expr_scope
.
Notation
"e1  e2"
:
=
(
par
(
LamV
BAnon
e1
%
E
)
(
LamV
BAnon
e2
%
E
)
)
:
val_scope
.
Notation
"e1  e2"
:
=
(
par
(
λ
:
<>,
e1
)%
V
(
λ
:
<>,
e2
)%
V
)
:
val_scope
.
Section
proof
.
Local
Set
Default
Proof
Using
"Type*"
.
...
...
theories/heap_lang/lifting.v
View file @
ece1ecdd
...
...
@@ 91,18 +91,26 @@ Local Ltac solve_pure_exec :=
subst
;
intros
?
;
apply
nsteps_once
,
pure_head_step_pure_step
;
constructor
;
[
solve_exec_safe

solve_exec_puredet
].
(** The behavior of the various [wp_] tactics with regard to lambda differs in
the following way:
 [wp_pures] does *not* reduce lambdas/recs that are hidden behind a definition.
 [wp_rec] and [wp_lam] reduce lambdas/recs that are hidden behind a definition.
To realize this behavior, we define the class [AsRecV v f x erec], which takes a
value [v] as its input, and turns it into a [RecV f x erec] via the instance
[AsRecV_recv : AsRecV (RecV f x e) f x e]. We register this instance via
[Hint Extern] so that it is only used if [v] is syntactically a lambda/rec, and
not if [v] contains a lambda/rec that is hidden behind a definition.
To make sure that [wp_rec] and [wp_lam] do reduce lambdas/recs that are hidden
behind a definition, we activate [AsRecV_recv] by hand in these tactics. *)
Class
AsRecV
(
v
:
val
)
(
f
x
:
binder
)
(
erec
:
expr
)
:
=
as_recv
:
v
=
RecV
f
x
erec
.
Instance
AsRecV_recv
f
x
e
:
AsRecV
(
RecV
f
x
e
)
f
x
e
:
=
eq_refl
.
(* Pure reductions are automatically performed before any wp_ tactics
handling impure operations. Since we do not want these tactics to
unfold locked terms, we do not register this instance explicitely,
but only activate it by hand in the `wp_rec` tactic, where we
*actually* want it to unlock. *)
Lemma
AsRecV_recv_locked
v
f
x
e
:
AsRecV
v
f
x
e
→
AsRecV
(
locked
v
)
f
x
e
.
Proof
.
by
unlock
.
Qed
.
Hint
Mode
AsRecV
!



:
typeclass_instances
.
Definition
AsRecV_recv
f
x
e
:
AsRecV
(
RecV
f
x
e
)
f
x
e
:
=
eq_refl
.
Hint
Extern
0
(
AsRecV
(
RecV
_
_
_
)
_
_
_
)
=>
apply
AsRecV_recv
:
typeclass_instances
.
Instance
pure_recc
f
x
(
erec
:
expr
)
:
PureExec
True
1
(
Rec
f
x
erec
)
(
Val
$
RecV
f
x
erec
).
...
...
theories/heap_lang/notation.v
View file @
ece1ecdd
...
...
@@ 85,7 +85,7 @@ by two spaces in case the whole rec does not fit on a single line. *)
Notation
"'rec:' f x := e"
:
=
(
Rec
f
%
bind
x
%
bind
e
%
E
)
(
at
level
200
,
f
at
level
1
,
x
at
level
1
,
e
at
level
200
,
format
"'[' 'rec:' f x := '/ ' e ']'"
)
:
expr_scope
.
Notation
"'rec:' f x := e"
:
=
(
locked
(
RecV
f
%
bind
x
%
bind
e
%
E
)
)
Notation
"'rec:' f x := e"
:
=
(
RecV
f
%
bind
x
%
bind
e
%
E
)
(
at
level
200
,
f
at
level
1
,
x
at
level
1
,
e
at
level
200
,
format
"'[' 'rec:' f x := '/ ' e ']'"
)
:
val_scope
.
Notation
"'if:' e1 'then' e2 'else' e3"
:
=
(
If
e1
%
E
e2
%
E
e3
%
E
)
...
...
@@ 98,7 +98,7 @@ notations are otherwise not pretty printed back accordingly. *)
Notation
"'rec:' f x y .. z := e"
:
=
(
Rec
f
%
bind
x
%
bind
(
Lam
y
%
bind
..
(
Lam
z
%
bind
e
%
E
)
..))
(
at
level
200
,
f
,
x
,
y
,
z
at
level
1
,
e
at
level
200
,
format
"'[' 'rec:' f x y .. z := '/ ' e ']'"
)
:
expr_scope
.
Notation
"'rec:' f x y .. z := e"
:
=
(
locked
(
RecV
f
%
bind
x
%
bind
(
Lam
y
%
bind
..
(
Lam
z
%
bind
e
%
E
)
..)
))
Notation
"'rec:' f x y .. z := e"
:
=
(
RecV
f
%
bind
x
%
bind
(
Lam
y
%
bind
..
(
Lam
z
%
bind
e
%
E
)
..
))
(
at
level
200
,
f
,
x
,
y
,
z
at
level
1
,
e
at
level
200
,
format
"'[' 'rec:' f x y .. z := '/ ' e ']'"
)
:
val_scope
.
...
...
@@ 111,21 +111,10 @@ Notation "λ: x y .. z , e" := (Lam x%bind (Lam y%bind .. (Lam z%bind e%E) ..))
(
at
level
200
,
x
,
y
,
z
at
level
1
,
e
at
level
200
,
format
"'[' 'λ:' x y .. z , '/ ' e ']'"
)
:
expr_scope
.
(* When parsing lambdas, we want them to be locked (so as to avoid needless
unfolding by tactics and unification). However, unlocked lambdavalues sometimes
appear as part of compound expressions, in which case we want them to be pretty
printed too. We achieve that by using printing only notations for the nonlocked
notation. *)
Notation
"λ: x , e"
:
=
(
LamV
x
%
bind
e
%
E
)
(
at
level
200
,
x
at
level
1
,
e
at
level
200
,
format
"'[' 'λ:' x , '/ ' e ']'"
,
only
printing
)
:
val_scope
.
Notation
"λ: x , e"
:
=
(
locked
(
LamV
x
%
bind
e
%
E
))
(
at
level
200
,
x
at
level
1
,
e
at
level
200
,
format
"'[' 'λ:' x , '/ ' e ']'"
)
:
val_scope
.
Notation
"λ: x y .. z , e"
:
=
(
LamV
x
%
bind
(
Lam
y
%
bind
..
(
Lam
z
%
bind
e
%
E
)
..
))
(
at
level
200
,
x
,
y
,
z
at
level
1
,
e
at
level
200
,
format
"'[' 'λ:' x y .. z , '/ ' e ']'"
,
only
printing
)
:
val_scope
.
Notation
"λ: x y .. z , e"
:
=
(
locked
(
LamV
x
%
bind
(
Lam
y
%
bind
..
(
Lam
z
%
bind
e
%
E
)
..
)))
(
at
level
200
,
x
,
y
,
z
at
level
1
,
e
at
level
200
,
format
"'[' 'λ:' x y .. z , '/ ' e ']'"
)
:
val_scope
.
...
...
theories/heap_lang/proofmode.v
View file @
ece1ecdd
...
...
@@ 105,17 +105,16 @@ Ltac wp_pures :=
repeat
(
wp_pure
_;
[]).
(* The `;[]` makes sure that no sidecondition
magically spawns. *)
(* The handling of betareductions with wp_rec needs special care in
order to allow it to unlock locked `RecV` values: We first put
`AsRecV_recv_locked` in the current environment so that it can be
used as an instance by the typeclass resolution system, then we
perform the reduction, and finally we clear this new hypothesis.
The reason is that we do not want impure wp_ tactics to unfold
locked terms, while we want them to execute arbitrary pure steps. *)
(** Unlike [wp_pures], the tactics [wp_rec] and [wp_lam] should also reduce
lambdas/recs that are hidden behind a definition, i.e. they should use
[AsRecV_recv] as a proper instance instead of a [Hint Extern].
We achieve this by putting [AsRecV_recv] in the current environment so that it
can be used as an instance by the typeclass resolution system. We then perform
the reduction, and finally we clear this new hypothesis. *)
Tactic
Notation
"wp_rec"
:
=
let
H
:
=
fresh
in
assert
(
H
:
=
AsRecV_recv
_locked
)
;
assert
(
H
:
=
AsRecV_recv
)
;
wp_pure
(
App
_
_
)
;
clear
H
.
...
...
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