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Sergey Bozhko
rtproofs
Commits
f1879960
Commit
f1879960
authored
Jul 18, 2019
by
Björn Brandenburg
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add prefix_map transformation and two supporting lemmas
parent
bf4412d0
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restructuring/analysis/transform/prefix.v
restructuring/analysis/transform/prefix.v
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restructuring/analysis/transform/prefix.v
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f1879960
From
mathcomp
Require
Import
ssrnat
ssrbool
fintype
.
From
rt
.
restructuring
.
behavior
Require
Import
schedule
facts
.
all
.
(** This file provides an operation that transforms a finite prefix of
a given schedule by applying a pointwise transformation to each
instant up to a given horizon. *)
Section
SchedulePrefixMap
.
(* For any type of jobs and... *)
Context
{
Job
:
JobType
}.
(* ... any given type of processor states ... *)
Context
{
PState
:
eqType
}.
Context
`
{
ProcessorState
Job
PState
}.
(* ... we define a procedure that applies a given function to every
point in a given finite prefix of the schedule.
The pointwise transformation f is given a schedule and the point
to transform, and must yield a transformed schedule that is used
in subsequent operations. *)
Fixpoint
prefix_map
(
sched
:
schedule
PState
)
(
f
:
schedule
PState
>
instant
>
schedule
PState
)
(
horizon
:
instant
)
:
=
match
horizon
with

0
=>
sched

S
t
=>
let
prefix
:
=
prefix_map
sched
f
t
in
f
prefix
t
end
.
(** We provide a utility lemma that establishes that, if the
pointwise transformation maintains a given property of the
original schedule, then the prefix_map does so as well. *)
Section
PropertyPreservation
.
(* Given any property of schedules... *)
Variable
P
:
schedule
PState
>
Prop
.
(* ...and a pointwise transformation [f],... *)
Variable
f
:
schedule
PState
>
instant
>
schedule
PState
.
(* ...if [f] maintains property [P],... *)
Hypothesis
H_f_maintains_P
:
forall
sched
t
,
P
sched
>
P
(
f
sched
t
).
(* ...then so does a prefix map of [f]. *)
Lemma
prefix_map_property_invariance
:
forall
sched
h
,
P
sched
>
P
(
prefix_map
sched
f
h
).
Proof
.
move
=>
sched
h
P_sched
.
induction
h
;
first
by
rewrite
/
prefix_map
.
rewrite
/
prefix_map
/
prefix_map
.
by
apply
:
H_f_maintains_P
.
Qed
.
End
PropertyPreservation
.
(** Next, we consider the case where the pointwise transformation
establishes a new property stepbystep. *)
Section
PointwiseProperty
.
(* Given any property of schedules [P],... *)
Variable
P
:
schedule
PState
>
Prop
.
(* ...any pointwise property [Q],... *)
Variable
Q
:
schedule
PState
>
instant
>
Prop
.
(* ...and a pointwise transformation [f],... *)
Variable
f
:
schedule
PState
>
instant
>
schedule
PState
.
(* ...if [f] maintains [P]... *)
Hypothesis
H_f_maintains_P
:
forall
sched
t_ref
,
P
sched
>
P
(
f
sched
t_ref
).
(* ...and establishes [Q] (provided that [P] holds)... *)
Hypothesis
H_f_grows_Q
:
forall
sched
t_ref
,
P
sched
>
(
forall
t'
,
t'
<
t_ref
>
Q
sched
t'
)
>
forall
t'
,
t'
<=
t_ref
>
Q
(
f
sched
t_ref
)
t'
.
(* ...then the prefixmap operation will "grow" [Q] up to the horizon. *)
Lemma
prefix_map_pointwise_property
:
forall
sched
horizon
,
P
sched
>
forall
t
,
t
<
horizon
>
Q
(
prefix_map
sched
f
horizon
)
t
.
Proof
.
move
=>
sched
h
P_holds
.
induction
h
as
[
h
Q_holds_before_h
]
;
first
by
rewrite
/
prefix_map
.
rewrite
/
prefix_map
/
prefix_map
=>
t
.
rewrite
ltnS
=>
LE_t_h
.
apply
H_f_grows_Q
=>
//.
by
apply
prefix_map_property_invariance
.
Qed
.
End
PointwiseProperty
.
End
SchedulePrefixMap
.
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