Library prosa.analysis.facts.behavior.service
(* ----------------------------------[ coqtop ]---------------------------------
Welcome to Coq 8.11.2 (June 2020)
----------------------------------------------------------------------------- *)
Require Export prosa.util.all.
Require Export prosa.behavior.all.
Require Export prosa.model.processor.platform_properties.
Require Export prosa.analysis.definitions.schedule_prefix.
Service
Consider any job type and any processor state.
For any given schedule...
...and any given job...
...we establish a number of useful rewriting rules that decompose
the service received during an interval into smaller intervals.
As a trivial base case, no job receives any service during an empty
interval.
Lemma service_during_geq:
∀ t1 t2,
t1 ≥ t2 → service_during sched j t1 t2 = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 631)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 : nat, t2 <= t1 -> service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 t1t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 634)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
t1t2 : t2 <= t1
============================
service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
rewrite /service_during big_geq //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t1 t2,
t1 ≥ t2 → service_during sched j t1 t2 = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 631)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 : nat, t2 <= t1 -> service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 t1t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 634)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
t1t2 : t2 <= t1
============================
service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
rewrite /service_during big_geq //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Equally trivially, no job has received service prior to time zero.
Corollary service0:
service sched j 0 = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 637)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
service sched j 0 = 0
----------------------------------------------------------------------------- *)
Proof.
rewrite /service service_during_geq //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
service sched j 0 = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 637)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
service sched j 0 = 0
----------------------------------------------------------------------------- *)
Proof.
rewrite /service service_during_geq //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Trivially, an interval consisting of one time unit is equivalent to
[service_at].
Lemma service_during_instant:
∀ t,
service_during sched j t t.+1 = service_at sched j t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 647)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t : instant,
service_during sched j t (succn t) = service_at sched j t
----------------------------------------------------------------------------- *)
Proof.
move ⇒ t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 648)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
============================
service_during sched j t (succn t) = service_at sched j t
----------------------------------------------------------------------------- *)
by rewrite /service_during big_nat_recr ?big_geq //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t,
service_during sched j t t.+1 = service_at sched j t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 647)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t : instant,
service_during sched j t (succn t) = service_at sched j t
----------------------------------------------------------------------------- *)
Proof.
move ⇒ t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 648)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
============================
service_during sched j t (succn t) = service_at sched j t
----------------------------------------------------------------------------- *)
by rewrite /service_during big_nat_recr ?big_geq //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Next, we observe that we can look at the service received during an
interval [t1, t3) as the sum of the service during [t1, t2) and [t2, t3)
for any t2 \in [t1, t3]. (The "_cat" suffix denotes the concatenation of
the two intervals.)
Lemma service_during_cat:
∀ t1 t2 t3,
t1 ≤ t2 ≤ t3 →
(service_during sched j t1 t2) + (service_during sched j t2 t3)
= service_during sched j t1 t3.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 662)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 t3 : nat,
t1 <= t2 <= t3 ->
service_during sched j t1 t2 + service_during sched j t2 t3 =
service_during sched j t1 t3
----------------------------------------------------------------------------- *)
Proof.
move ⇒ t1 t2 t3 /andP [t1t2 t2t3].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 705)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2, t3 : nat
t1t2 : t1 <= t2
t2t3 : t2 <= t3
============================
service_during sched j t1 t2 + service_during sched j t2 t3 =
service_during sched j t1 t3
----------------------------------------------------------------------------- *)
by rewrite /service_during -big_cat_nat /=.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t1 t2 t3,
t1 ≤ t2 ≤ t3 →
(service_during sched j t1 t2) + (service_during sched j t2 t3)
= service_during sched j t1 t3.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 662)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 t3 : nat,
t1 <= t2 <= t3 ->
service_during sched j t1 t2 + service_during sched j t2 t3 =
service_during sched j t1 t3
----------------------------------------------------------------------------- *)
Proof.
move ⇒ t1 t2 t3 /andP [t1t2 t2t3].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 705)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2, t3 : nat
t1t2 : t1 <= t2
t2t3 : t2 <= t3
============================
service_during sched j t1 t2 + service_during sched j t2 t3 =
service_during sched j t1 t3
----------------------------------------------------------------------------- *)
by rewrite /service_during -big_cat_nat /=.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Since [service] is just a special case of [service_during], the same holds
for [service].
Lemma service_cat:
∀ t1 t2,
t1 ≤ t2 →
(service sched j t1) + (service_during sched j t1 t2)
= service sched j t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 676)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 : nat,
t1 <= t2 ->
service sched j t1 + service_during sched j t1 t2 = service sched j t2
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 t1t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 679)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
t1t2 : t1 <= t2
============================
service sched j t1 + service_during sched j t1 t2 = service sched j t2
----------------------------------------------------------------------------- *)
rewrite /service service_during_cat //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t1 t2,
t1 ≤ t2 →
(service sched j t1) + (service_during sched j t1 t2)
= service sched j t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 676)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 : nat,
t1 <= t2 ->
service sched j t1 + service_during sched j t1 t2 = service sched j t2
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 t1t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 679)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
t1t2 : t1 <= t2
============================
service sched j t1 + service_during sched j t1 t2 = service sched j t2
----------------------------------------------------------------------------- *)
rewrite /service service_during_cat //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
As a special case, we observe that the service during an interval can be
decomposed into the first instant and the rest of the interval.
Lemma service_during_first_plus_later:
∀ t1 t2,
t1 < t2 →
(service_at sched j t1) + (service_during sched j t1.+1 t2)
= service_during sched j t1 t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 690)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 : nat,
t1 < t2 ->
service_at sched j t1 + service_during sched j (succn t1) t2 =
service_during sched j t1 t2
----------------------------------------------------------------------------- *)
Proof.
move ⇒ t1 t2 t1t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 693)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
t1t2 : t1 < t2
============================
service_at sched j t1 + service_during sched j (succn t1) t2 =
service_during sched j t1 t2
----------------------------------------------------------------------------- *)
have TIMES: t1 ≤ t1.+1 ≤ t2 by rewrite /(_ && _) ifT //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 713)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
t1t2 : t1 < t2
TIMES : t1 <= succn t1 <= t2
============================
service_at sched j t1 + service_during sched j (succn t1) t2 =
service_during sched j t1 t2
----------------------------------------------------------------------------- *)
have SPLIT := service_during_cat t1 t1.+1 t2 TIMES.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 718)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
t1t2 : t1 < t2
TIMES : t1 <= succn t1 <= t2
SPLIT : service_during sched j t1 (succn t1) +
service_during sched j (succn t1) t2 = service_during sched j t1 t2
============================
service_at sched j t1 + service_during sched j (succn t1) t2 =
service_during sched j t1 t2
----------------------------------------------------------------------------- *)
by rewrite -service_during_instant //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t1 t2,
t1 < t2 →
(service_at sched j t1) + (service_during sched j t1.+1 t2)
= service_during sched j t1 t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 690)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 : nat,
t1 < t2 ->
service_at sched j t1 + service_during sched j (succn t1) t2 =
service_during sched j t1 t2
----------------------------------------------------------------------------- *)
Proof.
move ⇒ t1 t2 t1t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 693)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
t1t2 : t1 < t2
============================
service_at sched j t1 + service_during sched j (succn t1) t2 =
service_during sched j t1 t2
----------------------------------------------------------------------------- *)
have TIMES: t1 ≤ t1.+1 ≤ t2 by rewrite /(_ && _) ifT //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 713)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
t1t2 : t1 < t2
TIMES : t1 <= succn t1 <= t2
============================
service_at sched j t1 + service_during sched j (succn t1) t2 =
service_during sched j t1 t2
----------------------------------------------------------------------------- *)
have SPLIT := service_during_cat t1 t1.+1 t2 TIMES.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 718)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
t1t2 : t1 < t2
TIMES : t1 <= succn t1 <= t2
SPLIT : service_during sched j t1 (succn t1) +
service_during sched j (succn t1) t2 = service_during sched j t1 t2
============================
service_at sched j t1 + service_during sched j (succn t1) t2 =
service_during sched j t1 t2
----------------------------------------------------------------------------- *)
by rewrite -service_during_instant //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Symmetrically, we have the same for the end of the interval.
Lemma service_during_last_plus_before:
∀ t1 t2,
t1 ≤ t2 →
(service_during sched j t1 t2) + (service_at sched j t2)
= service_during sched j t1 t2.+1.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 704)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 : nat,
t1 <= t2 ->
service_during sched j t1 t2 + service_at sched j t2 =
service_during sched j t1 (succn t2)
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 t1t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 707)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
t1t2 : t1 <= t2
============================
service_during sched j t1 t2 + service_at sched j t2 =
service_during sched j t1 (succn t2)
----------------------------------------------------------------------------- *)
rewrite -(service_during_cat t1 t2 t2.+1); last by rewrite /(_ && _) ifT //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 710)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
t1t2 : t1 <= t2
============================
service_during sched j t1 t2 + service_at sched j t2 =
service_during sched j t1 t2 + service_during sched j t2 (succn t2)
----------------------------------------------------------------------------- *)
rewrite service_during_instant //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t1 t2,
t1 ≤ t2 →
(service_during sched j t1 t2) + (service_at sched j t2)
= service_during sched j t1 t2.+1.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 704)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 : nat,
t1 <= t2 ->
service_during sched j t1 t2 + service_at sched j t2 =
service_during sched j t1 (succn t2)
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 t1t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 707)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
t1t2 : t1 <= t2
============================
service_during sched j t1 t2 + service_at sched j t2 =
service_during sched j t1 (succn t2)
----------------------------------------------------------------------------- *)
rewrite -(service_during_cat t1 t2 t2.+1); last by rewrite /(_ && _) ifT //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 710)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
t1t2 : t1 <= t2
============================
service_during sched j t1 t2 + service_at sched j t2 =
service_during sched j t1 t2 + service_during sched j t2 (succn t2)
----------------------------------------------------------------------------- *)
rewrite service_during_instant //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
And hence also for [service].
Corollary service_last_plus_before:
∀ t,
(service sched j t) + (service_at sched j t)
= service sched j t.+1.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 717)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t : instant,
service sched j t + service_at sched j t = service sched j (succn t)
----------------------------------------------------------------------------- *)
Proof.
move⇒ t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 718)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
============================
service sched j t + service_at sched j t = service sched j (succn t)
----------------------------------------------------------------------------- *)
rewrite /service.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 725)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
============================
service_during sched j 0 t + service_at sched j t =
service_during sched j 0 (succn t)
----------------------------------------------------------------------------- *)
rewrite -service_during_last_plus_before //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t,
(service sched j t) + (service_at sched j t)
= service sched j t.+1.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 717)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t : instant,
service sched j t + service_at sched j t = service sched j (succn t)
----------------------------------------------------------------------------- *)
Proof.
move⇒ t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 718)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
============================
service sched j t + service_at sched j t = service sched j (succn t)
----------------------------------------------------------------------------- *)
rewrite /service.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 725)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
============================
service_during sched j 0 t + service_at sched j t =
service_during sched j 0 (succn t)
----------------------------------------------------------------------------- *)
rewrite -service_during_last_plus_before //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Finally, we deconstruct the service received during an interval [t1, t3)
into the service at a midpoint t2 and the service in the intervals before
and after.
Lemma service_split_at_point:
∀ t1 t2 t3,
t1 ≤ t2 < t3 →
(service_during sched j t1 t2) + (service_at sched j t2) + (service_during sched j t2.+1 t3)
= service_during sched j t1 t3.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 735)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 t3 : nat,
t1 <= t2 < t3 ->
service_during sched j t1 t2 + service_at sched j t2 +
service_during sched j (succn t2) t3 = service_during sched j t1 t3
----------------------------------------------------------------------------- *)
Proof.
move ⇒ t1 t2 t3 /andP [t1t2 t2t3].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 778)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2, t3 : nat
t1t2 : t1 <= t2
t2t3 : t2 < t3
============================
service_during sched j t1 t2 + service_at sched j t2 +
service_during sched j (succn t2) t3 = service_during sched j t1 t3
----------------------------------------------------------------------------- *)
rewrite -addnA service_during_first_plus_later// service_during_cat// /(_ && _) ifT//.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 858)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2, t3 : nat
t1t2 : t1 <= t2
t2t3 : t2 < t3
============================
t2 <= t3
----------------------------------------------------------------------------- *)
by exact: ltnW.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End Composition.
∀ t1 t2 t3,
t1 ≤ t2 < t3 →
(service_during sched j t1 t2) + (service_at sched j t2) + (service_during sched j t2.+1 t3)
= service_during sched j t1 t3.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 735)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 t3 : nat,
t1 <= t2 < t3 ->
service_during sched j t1 t2 + service_at sched j t2 +
service_during sched j (succn t2) t3 = service_during sched j t1 t3
----------------------------------------------------------------------------- *)
Proof.
move ⇒ t1 t2 t3 /andP [t1t2 t2t3].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 778)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2, t3 : nat
t1t2 : t1 <= t2
t2t3 : t2 < t3
============================
service_during sched j t1 t2 + service_at sched j t2 +
service_during sched j (succn t2) t3 = service_during sched j t1 t3
----------------------------------------------------------------------------- *)
rewrite -addnA service_during_first_plus_later// service_during_cat// /(_ && _) ifT//.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 858)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2, t3 : nat
t1t2 : t1 <= t2
t2t3 : t2 < t3
============================
t2 <= t3
----------------------------------------------------------------------------- *)
by exact: ltnW.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End Composition.
As a common special case, we establish facts about schedules in which a
job receives either 1 or 0 service units at all times.
Consider any job type and any processor state.
Let's consider a unit-service model...
...and a given schedule.
Let [j] be any job that is to be scheduled.
First, we prove that the instantaneous service cannot be greater than 1, ...
Lemma service_at_most_one:
∀ t, service_at sched j t ≤ 1.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 630)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
============================
forall t : instant, service_at sched j t <= 1
----------------------------------------------------------------------------- *)
Proof.
by move⇒ t; rewrite /service_at.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t, service_at sched j t ≤ 1.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 630)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
============================
forall t : instant, service_at sched j t <= 1
----------------------------------------------------------------------------- *)
Proof.
by move⇒ t; rewrite /service_at.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
...which implies that the cumulative service received by job [j] in any
interval of length delta is at most delta.
Lemma cumulative_service_le_delta:
∀ t delta,
service_during sched j t (t + delta) ≤ delta.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 636)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
============================
forall (t : instant) (delta : nat),
service_during sched j t (t + delta) <= delta
----------------------------------------------------------------------------- *)
Proof.
unfold service_during; intros t delta.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 640)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : instant
delta : nat
============================
\sum_(t <= t0 < t + delta) service_at sched j t0 <= delta
----------------------------------------------------------------------------- *)
apply leq_trans with (n := \sum_(t ≤ t0 < t + delta) 1);
last by rewrite sum_of_ones.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 645)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : instant
delta : nat
============================
\sum_(t <= t0 < t + delta) service_at sched j t0 <=
\sum_(t <= t0 < t + delta) 1
----------------------------------------------------------------------------- *)
by apply: leq_sum ⇒ t' _; apply: service_at_most_one.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Section ServiceIsAStepFunction.
∀ t delta,
service_during sched j t (t + delta) ≤ delta.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 636)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
============================
forall (t : instant) (delta : nat),
service_during sched j t (t + delta) <= delta
----------------------------------------------------------------------------- *)
Proof.
unfold service_during; intros t delta.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 640)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : instant
delta : nat
============================
\sum_(t <= t0 < t + delta) service_at sched j t0 <= delta
----------------------------------------------------------------------------- *)
apply leq_trans with (n := \sum_(t ≤ t0 < t + delta) 1);
last by rewrite sum_of_ones.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 645)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : instant
delta : nat
============================
\sum_(t <= t0 < t + delta) service_at sched j t0 <=
\sum_(t <= t0 < t + delta) 1
----------------------------------------------------------------------------- *)
by apply: leq_sum ⇒ t' _; apply: service_at_most_one.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Section ServiceIsAStepFunction.
We show that the service received by any job [j] is a step function.
Lemma service_is_a_step_function:
is_step_function (service sched j).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 640)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
============================
is_step_function (service sched j)
----------------------------------------------------------------------------- *)
Proof.
rewrite /is_step_function ⇒ t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 642)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : nat
============================
service sched j (t + 1) <= service sched j t + 1
----------------------------------------------------------------------------- *)
rewrite addn1 -service_last_plus_before leq_add2l.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 661)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : nat
============================
service_at sched j t <= 1
----------------------------------------------------------------------------- *)
apply service_at_most_one.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
is_step_function (service sched j).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 640)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
============================
is_step_function (service sched j)
----------------------------------------------------------------------------- *)
Proof.
rewrite /is_step_function ⇒ t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 642)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : nat
============================
service sched j (t + 1) <= service sched j t + 1
----------------------------------------------------------------------------- *)
rewrite addn1 -service_last_plus_before leq_add2l.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 661)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : nat
============================
service_at sched j t <= 1
----------------------------------------------------------------------------- *)
apply service_at_most_one.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Next, consider any time [t]...
...and let [s0] be any value less than the service received
by job [j] by time [t].
Then, we show that there exists an earlier time [t0] where job [j] had [s0]
units of service.
Corollary exists_intermediate_service:
∃ t0,
t0 < t ∧
service sched j t0 = s0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 651)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : instant
s0 : duration
H_less_than_s : s0 < service sched j t
============================
exists t0 : nat, t0 < t /\ service sched j t0 = s0
----------------------------------------------------------------------------- *)
Proof.
feed (exists_intermediate_point (service sched j));
[by apply service_is_a_step_function | intros EX].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 661)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : instant
s0 : duration
H_less_than_s : s0 < service sched j t
EX : forall x1 x2 : nat,
x1 <= x2 ->
forall y : nat,
service sched j x1 <= y < service sched j x2 ->
exists x_mid : nat, x1 <= x_mid < x2 /\ service sched j x_mid = y
============================
exists t0 : nat, t0 < t /\ service sched j t0 = s0
----------------------------------------------------------------------------- *)
feed (EX 0 t); first by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 667)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : instant
s0 : duration
H_less_than_s : s0 < service sched j t
EX : forall y : nat,
service sched j 0 <= y < service sched j t ->
exists x_mid : nat, 0 <= x_mid < t /\ service sched j x_mid = y
============================
exists t0 : nat, t0 < t /\ service sched j t0 = s0
----------------------------------------------------------------------------- *)
feed (EX s0);
first by rewrite /service /service_during big_geq //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 673)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : instant
s0 : duration
H_less_than_s : s0 < service sched j t
EX : exists x_mid : nat, 0 <= x_mid < t /\ service sched j x_mid = s0
============================
exists t0 : nat, t0 < t /\ service sched j t0 = s0
----------------------------------------------------------------------------- *)
by move: EX ⇒ /= [x_mid EX]; ∃ x_mid.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End ServiceIsAStepFunction.
End UnitService.
∃ t0,
t0 < t ∧
service sched j t0 = s0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 651)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : instant
s0 : duration
H_less_than_s : s0 < service sched j t
============================
exists t0 : nat, t0 < t /\ service sched j t0 = s0
----------------------------------------------------------------------------- *)
Proof.
feed (exists_intermediate_point (service sched j));
[by apply service_is_a_step_function | intros EX].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 661)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : instant
s0 : duration
H_less_than_s : s0 < service sched j t
EX : forall x1 x2 : nat,
x1 <= x2 ->
forall y : nat,
service sched j x1 <= y < service sched j x2 ->
exists x_mid : nat, x1 <= x_mid < x2 /\ service sched j x_mid = y
============================
exists t0 : nat, t0 < t /\ service sched j t0 = s0
----------------------------------------------------------------------------- *)
feed (EX 0 t); first by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 667)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : instant
s0 : duration
H_less_than_s : s0 < service sched j t
EX : forall y : nat,
service sched j 0 <= y < service sched j t ->
exists x_mid : nat, 0 <= x_mid < t /\ service sched j x_mid = y
============================
exists t0 : nat, t0 < t /\ service sched j t0 = s0
----------------------------------------------------------------------------- *)
feed (EX s0);
first by rewrite /service /service_during big_geq //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 673)
Job : JobType
PState : Type
H : ProcessorState Job PState
H_unit_service : unit_service_proc_model PState
sched : schedule PState
j : Job
t : instant
s0 : duration
H_less_than_s : s0 < service sched j t
EX : exists x_mid : nat, 0 <= x_mid < t /\ service sched j x_mid = s0
============================
exists t0 : nat, t0 < t /\ service sched j t0 = s0
----------------------------------------------------------------------------- *)
by move: EX ⇒ /= [x_mid EX]; ∃ x_mid.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End ServiceIsAStepFunction.
End UnitService.
We establish a basic fact about the monotonicity of service.
Consider any job type and any processor model.
Consider any given schedule...
...and a given job that is to be scheduled.
We observe that the amount of service received is monotonic by definition.
Lemma service_monotonic:
∀ t1 t2,
t1 ≤ t2 →
service sched j t1 ≤ service sched j t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 632)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 : nat, t1 <= t2 -> service sched j t1 <= service sched j t2
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 let1t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 635)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
let1t2 : t1 <= t2
============================
service sched j t1 <= service sched j t2
----------------------------------------------------------------------------- *)
by rewrite -(service_cat sched j t1 t2 let1t2); apply: leq_addr.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End Monotonicity.
∀ t1 t2,
t1 ≤ t2 →
service sched j t1 ≤ service sched j t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 632)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 : nat, t1 <= t2 -> service sched j t1 <= service sched j t2
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 let1t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 635)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : nat
let1t2 : t1 <= t2
============================
service sched j t1 <= service sched j t2
----------------------------------------------------------------------------- *)
by rewrite -(service_cat sched j t1 t2 let1t2); apply: leq_addr.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End Monotonicity.
Consider any job type and any processor model.
Section RelationToScheduled.
Context {Job: JobType}.
Context {PState: Type}.
Context `{ProcessorState Job PState}.
Context {Job: JobType}.
Context {PState: Type}.
Context `{ProcessorState Job PState}.
Consider any given schedule...
...and a given job that is to be scheduled.
We observe that a job that isn't scheduled cannot possibly receive service.
Lemma not_scheduled_implies_no_service:
∀ t,
~~ scheduled_at sched j t → service_at sched j t = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 633)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t : instant, ~~ scheduled_at sched j t -> service_at sched j t = 0
----------------------------------------------------------------------------- *)
Proof.
rewrite /service_at /scheduled_at.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 647)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t : instant,
~~ scheduled_in j (sched t) -> service_in j (sched t) = 0
----------------------------------------------------------------------------- *)
move⇒ t NOT_SCHED.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 649)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
NOT_SCHED : ~~ scheduled_in j (sched t)
============================
service_in j (sched t) = 0
----------------------------------------------------------------------------- *)
rewrite service_implies_scheduled //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t,
~~ scheduled_at sched j t → service_at sched j t = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 633)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t : instant, ~~ scheduled_at sched j t -> service_at sched j t = 0
----------------------------------------------------------------------------- *)
Proof.
rewrite /service_at /scheduled_at.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 647)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t : instant,
~~ scheduled_in j (sched t) -> service_in j (sched t) = 0
----------------------------------------------------------------------------- *)
move⇒ t NOT_SCHED.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 649)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
NOT_SCHED : ~~ scheduled_in j (sched t)
============================
service_in j (sched t) = 0
----------------------------------------------------------------------------- *)
rewrite service_implies_scheduled //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Conversely, if a job receives service, then it must be scheduled.
Lemma service_at_implies_scheduled_at:
∀ t,
service_at sched j t > 0 → scheduled_at sched j t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 641)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t : instant, 0 < service_at sched j t -> scheduled_at sched j t
----------------------------------------------------------------------------- *)
Proof.
move⇒ t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 642)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
============================
0 < service_at sched j t -> scheduled_at sched j t
----------------------------------------------------------------------------- *)
destruct (scheduled_at sched j t) eqn:SCHEDULED; first trivial.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 658)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
SCHEDULED : scheduled_at sched j t = false
============================
0 < service_at sched j t -> false
----------------------------------------------------------------------------- *)
rewrite not_scheduled_implies_no_service // negbT //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t,
service_at sched j t > 0 → scheduled_at sched j t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 641)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t : instant, 0 < service_at sched j t -> scheduled_at sched j t
----------------------------------------------------------------------------- *)
Proof.
move⇒ t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 642)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
============================
0 < service_at sched j t -> scheduled_at sched j t
----------------------------------------------------------------------------- *)
destruct (scheduled_at sched j t) eqn:SCHEDULED; first trivial.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 658)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
SCHEDULED : scheduled_at sched j t = false
============================
0 < service_at sched j t -> false
----------------------------------------------------------------------------- *)
rewrite not_scheduled_implies_no_service // negbT //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Thus, if the cumulative amount of service changes, then it must be
scheduled, too.
Lemma service_delta_implies_scheduled:
∀ t,
service sched j t < service sched j t.+1 → scheduled_at sched j t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 652)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t : instant,
service sched j t < service sched j (succn t) -> scheduled_at sched j t
----------------------------------------------------------------------------- *)
Proof.
move ⇒ t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 653)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
============================
service sched j t < service sched j (succn t) -> scheduled_at sched j t
----------------------------------------------------------------------------- *)
rewrite -service_last_plus_before -{1}(addn0 (service sched j t)) ltn_add2l.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 677)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
============================
0 < service_at sched j t -> scheduled_at sched j t
----------------------------------------------------------------------------- *)
apply: service_at_implies_scheduled_at.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t,
service sched j t < service sched j t.+1 → scheduled_at sched j t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 652)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t : instant,
service sched j t < service sched j (succn t) -> scheduled_at sched j t
----------------------------------------------------------------------------- *)
Proof.
move ⇒ t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 653)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
============================
service sched j t < service sched j (succn t) -> scheduled_at sched j t
----------------------------------------------------------------------------- *)
rewrite -service_last_plus_before -{1}(addn0 (service sched j t)) ltn_add2l.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 677)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t : instant
============================
0 < service_at sched j t -> scheduled_at sched j t
----------------------------------------------------------------------------- *)
apply: service_at_implies_scheduled_at.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
We observe that a job receives cumulative service during some interval iff
it receives services at some specific time in the interval.
Lemma service_during_service_at:
∀ t1 t2,
service_during sched j t1 t2 > 0
↔
∃ t,
t1 ≤ t < t2 ∧
service_at sched j t > 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 663)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 : instant,
0 < service_during sched j t1 t2 <->
(exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t)
----------------------------------------------------------------------------- *)
Proof.
split.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 667)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
============================
0 < service_during sched j t1 t2 ->
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
subgoal 2 (ID 668) is:
(exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t) ->
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 667)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
============================
0 < service_during sched j t1 t2 ->
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
move⇒ NONZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 669)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
NONZERO : 0 < service_during sched j t1 t2
============================
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
case (boolP([∃ t: 'I_t2,
(t ≥ t1) && (service_at sched j t > 0)])) ⇒ [EX | ALL].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 682)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
NONZERO : 0 < service_during sched j t1 t2
EX : ~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2)
(, (t1 <= t) && (0 < service_at sched j t)) t)
============================
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
subgoal 2 (ID 683) is:
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
- move: EX ⇒ /existsP [x /andP [GE SERV]].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 763)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
NONZERO : 0 < service_during sched j t1 t2
x : ordinal_finType t2
GE : t1 <= x
SERV : 0 < service_at sched j x
============================
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
subgoal 2 (ID 683) is:
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
∃ x; split ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 767)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
NONZERO : 0 < service_during sched j t1 t2
x : ordinal_finType t2
GE : t1 <= x
SERV : 0 < service_at sched j x
============================
t1 <= x < t2
subgoal 2 (ID 683) is:
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
apply /andP; by split.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 683)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
NONZERO : 0 < service_during sched j t1 t2
ALL : ~~
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2)
(, (t1 <= t) && (0 < service_at sched j t)) t)
============================
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
- rewrite negb_exists in ALL; move: ALL ⇒ /forallP ALL.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 869)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
NONZERO : 0 < service_during sched j t1 t2
ALL : forall x : ordinal_finType t2,
~~ ((t1 <= x) && (0 < service_at sched j x))
============================
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
rewrite /service_during big_nat_cond in NONZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 901)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
ALL : forall x : ordinal_finType t2,
~~ ((t1 <= x) && (0 < service_at sched j x))
NONZERO : 0 <
\sum_(t1 <= i < t2 | (t1 <= i < t2) && true) service_at sched j i
============================
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
rewrite big1 ?ltn0 // in NONZERO ⇒ i.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 988)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
ALL : forall x : ordinal_finType t2,
~~ ((t1 <= x) && (0 < service_at sched j x))
i : nat
============================
(t1 <= i < t2) && true -> service_at sched j i = 0
----------------------------------------------------------------------------- *)
rewrite andbT; move ⇒ /andP [GT LT].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1033)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
ALL : forall x : ordinal_finType t2,
~~ ((t1 <= x) && (0 < service_at sched j x))
i : nat
GT : t1 <= i
LT : i < t2
============================
service_at sched j i = 0
----------------------------------------------------------------------------- *)
specialize (ALL (Ordinal LT)); simpl in ALL.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1038)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
i : nat
LT : i < t2
ALL : ~~ ((t1 <= i) && (0 < service_at sched j i))
GT : t1 <= i
============================
service_at sched j i = 0
----------------------------------------------------------------------------- *)
rewrite GT andTb -eqn0Ngt in ALL.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1066)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
i : nat
LT : i < t2
GT : t1 <= i
ALL : service_at sched j i == 0
============================
service_at sched j i = 0
----------------------------------------------------------------------------- *)
apply /eqP ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 668)
subgoal 1 (ID 668) is:
(exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t) ->
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 668)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
============================
(exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t) ->
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 668)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
============================
(exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t) ->
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
move⇒ [t [TT SERVICE]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1142)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : 0 < service_at sched j t
============================
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
case (boolP (0 < service_during sched j t1 t2)) ⇒ // NZ.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1176)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : 0 < service_at sched j t
NZ : ~~ (0 < service_during sched j t1 t2)
============================
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
exfalso.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1177)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : 0 < service_at sched j t
NZ : ~~ (0 < service_during sched j t1 t2)
============================
False
----------------------------------------------------------------------------- *)
rewrite -eqn0Ngt in NZ.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1198)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : 0 < service_at sched j t
NZ : service_during sched j t1 t2 == 0
============================
False
----------------------------------------------------------------------------- *)
move/eqP: NZ.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1227)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : 0 < service_at sched j t
============================
service_during sched j t1 t2 = 0 -> False
----------------------------------------------------------------------------- *)
rewrite big_nat_eq0 ⇒ IS_ZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1253)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : 0 < service_at sched j t
IS_ZERO : forall i : nat, t1 <= i < t2 -> service_at sched j i = 0
============================
False
----------------------------------------------------------------------------- *)
have NO_SERVICE := IS_ZERO t TT.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1258)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : 0 < service_at sched j t
IS_ZERO : forall i : nat, t1 <= i < t2 -> service_at sched j i = 0
NO_SERVICE : service_at sched j t = 0
============================
False
----------------------------------------------------------------------------- *)
apply lt0n_neq0 in SERVICE.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1259)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : service_at sched j t != 0
IS_ZERO : forall i : nat, t1 <= i < t2 -> service_at sched j i = 0
NO_SERVICE : service_at sched j t = 0
============================
False
----------------------------------------------------------------------------- *)
by move/neqP in SERVICE; contradiction.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t1 t2,
service_during sched j t1 t2 > 0
↔
∃ t,
t1 ≤ t < t2 ∧
service_at sched j t > 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 663)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 : instant,
0 < service_during sched j t1 t2 <->
(exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t)
----------------------------------------------------------------------------- *)
Proof.
split.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 667)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
============================
0 < service_during sched j t1 t2 ->
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
subgoal 2 (ID 668) is:
(exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t) ->
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 667)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
============================
0 < service_during sched j t1 t2 ->
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
move⇒ NONZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 669)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
NONZERO : 0 < service_during sched j t1 t2
============================
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
case (boolP([∃ t: 'I_t2,
(t ≥ t1) && (service_at sched j t > 0)])) ⇒ [EX | ALL].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 682)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
NONZERO : 0 < service_during sched j t1 t2
EX : ~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2)
(, (t1 <= t) && (0 < service_at sched j t)) t)
============================
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
subgoal 2 (ID 683) is:
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
- move: EX ⇒ /existsP [x /andP [GE SERV]].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 763)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
NONZERO : 0 < service_during sched j t1 t2
x : ordinal_finType t2
GE : t1 <= x
SERV : 0 < service_at sched j x
============================
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
subgoal 2 (ID 683) is:
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
∃ x; split ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 767)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
NONZERO : 0 < service_during sched j t1 t2
x : ordinal_finType t2
GE : t1 <= x
SERV : 0 < service_at sched j x
============================
t1 <= x < t2
subgoal 2 (ID 683) is:
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
apply /andP; by split.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 683)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
NONZERO : 0 < service_during sched j t1 t2
ALL : ~~
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2)
(, (t1 <= t) && (0 < service_at sched j t)) t)
============================
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
- rewrite negb_exists in ALL; move: ALL ⇒ /forallP ALL.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 869)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
NONZERO : 0 < service_during sched j t1 t2
ALL : forall x : ordinal_finType t2,
~~ ((t1 <= x) && (0 < service_at sched j x))
============================
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
rewrite /service_during big_nat_cond in NONZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 901)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
ALL : forall x : ordinal_finType t2,
~~ ((t1 <= x) && (0 < service_at sched j x))
NONZERO : 0 <
\sum_(t1 <= i < t2 | (t1 <= i < t2) && true) service_at sched j i
============================
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
rewrite big1 ?ltn0 // in NONZERO ⇒ i.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 988)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
ALL : forall x : ordinal_finType t2,
~~ ((t1 <= x) && (0 < service_at sched j x))
i : nat
============================
(t1 <= i < t2) && true -> service_at sched j i = 0
----------------------------------------------------------------------------- *)
rewrite andbT; move ⇒ /andP [GT LT].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1033)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
ALL : forall x : ordinal_finType t2,
~~ ((t1 <= x) && (0 < service_at sched j x))
i : nat
GT : t1 <= i
LT : i < t2
============================
service_at sched j i = 0
----------------------------------------------------------------------------- *)
specialize (ALL (Ordinal LT)); simpl in ALL.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1038)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
i : nat
LT : i < t2
ALL : ~~ ((t1 <= i) && (0 < service_at sched j i))
GT : t1 <= i
============================
service_at sched j i = 0
----------------------------------------------------------------------------- *)
rewrite GT andTb -eqn0Ngt in ALL.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1066)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
i : nat
LT : i < t2
GT : t1 <= i
ALL : service_at sched j i == 0
============================
service_at sched j i = 0
----------------------------------------------------------------------------- *)
apply /eqP ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 668)
subgoal 1 (ID 668) is:
(exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t) ->
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 668)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
============================
(exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t) ->
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 668)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
============================
(exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t) ->
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
move⇒ [t [TT SERVICE]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1142)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : 0 < service_at sched j t
============================
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
case (boolP (0 < service_during sched j t1 t2)) ⇒ // NZ.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1176)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : 0 < service_at sched j t
NZ : ~~ (0 < service_during sched j t1 t2)
============================
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
exfalso.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1177)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : 0 < service_at sched j t
NZ : ~~ (0 < service_during sched j t1 t2)
============================
False
----------------------------------------------------------------------------- *)
rewrite -eqn0Ngt in NZ.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1198)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : 0 < service_at sched j t
NZ : service_during sched j t1 t2 == 0
============================
False
----------------------------------------------------------------------------- *)
move/eqP: NZ.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1227)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : 0 < service_at sched j t
============================
service_during sched j t1 t2 = 0 -> False
----------------------------------------------------------------------------- *)
rewrite big_nat_eq0 ⇒ IS_ZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1253)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : 0 < service_at sched j t
IS_ZERO : forall i : nat, t1 <= i < t2 -> service_at sched j i = 0
============================
False
----------------------------------------------------------------------------- *)
have NO_SERVICE := IS_ZERO t TT.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1258)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : 0 < service_at sched j t
IS_ZERO : forall i : nat, t1 <= i < t2 -> service_at sched j i = 0
NO_SERVICE : service_at sched j t = 0
============================
False
----------------------------------------------------------------------------- *)
apply lt0n_neq0 in SERVICE.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1259)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t : nat
TT : t1 <= t < t2
SERVICE : service_at sched j t != 0
IS_ZERO : forall i : nat, t1 <= i < t2 -> service_at sched j i = 0
NO_SERVICE : service_at sched j t = 0
============================
False
----------------------------------------------------------------------------- *)
by move/neqP in SERVICE; contradiction.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Thus, any job that receives some service during an interval must be
scheduled at some point during the interval...
Corollary cumulative_service_implies_scheduled:
∀ t1 t2,
service_during sched j t1 t2 > 0 →
∃ t,
t1 ≤ t < t2 ∧
scheduled_at sched j t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 674)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 : instant,
0 < service_during sched j t1 t2 ->
exists t : nat, t1 <= t < t2 /\ scheduled_at sched j t
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 676)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
============================
0 < service_during sched j t1 t2 ->
exists t : nat, t1 <= t < t2 /\ scheduled_at sched j t
----------------------------------------------------------------------------- *)
rewrite service_during_service_at.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 701)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
============================
(exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t) ->
exists t : nat, t1 <= t < t2 /\ scheduled_at sched j t
----------------------------------------------------------------------------- *)
move⇒ [t' [TIMES SERVICED]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 722)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t' : nat
TIMES : t1 <= t' < t2
SERVICED : 0 < service_at sched j t'
============================
exists t : nat, t1 <= t < t2 /\ scheduled_at sched j t
----------------------------------------------------------------------------- *)
∃ t'; split ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 727)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t' : nat
TIMES : t1 <= t' < t2
SERVICED : 0 < service_at sched j t'
============================
scheduled_at sched j t'
----------------------------------------------------------------------------- *)
by apply: service_at_implies_scheduled_at.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t1 t2,
service_during sched j t1 t2 > 0 →
∃ t,
t1 ≤ t < t2 ∧
scheduled_at sched j t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 674)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t1 t2 : instant,
0 < service_during sched j t1 t2 ->
exists t : nat, t1 <= t < t2 /\ scheduled_at sched j t
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 676)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
============================
0 < service_during sched j t1 t2 ->
exists t : nat, t1 <= t < t2 /\ scheduled_at sched j t
----------------------------------------------------------------------------- *)
rewrite service_during_service_at.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 701)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
============================
(exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t) ->
exists t : nat, t1 <= t < t2 /\ scheduled_at sched j t
----------------------------------------------------------------------------- *)
move⇒ [t' [TIMES SERVICED]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 722)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t' : nat
TIMES : t1 <= t' < t2
SERVICED : 0 < service_at sched j t'
============================
exists t : nat, t1 <= t < t2 /\ scheduled_at sched j t
----------------------------------------------------------------------------- *)
∃ t'; split ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 727)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
t' : nat
TIMES : t1 <= t' < t2
SERVICED : 0 < service_at sched j t'
============================
scheduled_at sched j t'
----------------------------------------------------------------------------- *)
by apply: service_at_implies_scheduled_at.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
...which implies that any job with positive cumulative service must have
been scheduled at some point.
Corollary positive_service_implies_scheduled_before:
∀ t,
service sched j t > 0 → ∃ t', (t' < t ∧ scheduled_at sched j t').
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 684)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t : instant,
0 < service sched j t -> exists t' : nat, t' < t /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
Proof.
move⇒ t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 685)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t2 : instant
============================
0 < service sched j t2 ->
exists t' : nat, t' < t2 /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
rewrite /service ⇒ NONZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 693)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t2 : instant
NONZERO : 0 < service_during sched j 0 t2
============================
exists t' : nat, t' < t2 /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
have EX_SCHED := cumulative_service_implies_scheduled 0 t2 NONZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 698)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t2 : instant
NONZERO : 0 < service_during sched j 0 t2
EX_SCHED : exists t : nat, 0 <= t < t2 /\ scheduled_at sched j t
============================
exists t' : nat, t' < t2 /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
by move: EX_SCHED ⇒ [t [TIMES SCHED_AT]]; ∃ t; split.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t,
service sched j t > 0 → ∃ t', (t' < t ∧ scheduled_at sched j t').
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 684)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
============================
forall t : instant,
0 < service sched j t -> exists t' : nat, t' < t /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
Proof.
move⇒ t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 685)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t2 : instant
============================
0 < service sched j t2 ->
exists t' : nat, t' < t2 /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
rewrite /service ⇒ NONZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 693)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t2 : instant
NONZERO : 0 < service_during sched j 0 t2
============================
exists t' : nat, t' < t2 /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
have EX_SCHED := cumulative_service_implies_scheduled 0 t2 NONZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 698)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t2 : instant
NONZERO : 0 < service_during sched j 0 t2
EX_SCHED : exists t : nat, 0 <= t < t2 /\ scheduled_at sched j t
============================
exists t' : nat, t' < t2 /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
by move: EX_SCHED ⇒ [t [TIMES SCHED_AT]]; ∃ t; split.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
If we can assume that a scheduled job always receives service,
we can further prove the converse.
Assume [j] always receives some positive service.
In other words, not being scheduled is equivalent to receiving zero
service.
Lemma no_service_not_scheduled:
∀ t,
~~ scheduled_at sched j t ↔ service_at sched j t = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 696)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
forall t : instant, ~~ scheduled_at sched j t <-> service_at sched j t = 0
----------------------------------------------------------------------------- *)
Proof.
move⇒ t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 697)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t : instant
============================
~~ scheduled_at sched j t <-> service_at sched j t = 0
----------------------------------------------------------------------------- *)
rewrite /scheduled_at /service_at.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 711)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t : instant
============================
~~ scheduled_in j (sched t) <-> service_in j (sched t) = 0
----------------------------------------------------------------------------- *)
split ⇒ [NOT_SCHED | NO_SERVICE].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 715)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t : instant
NOT_SCHED : ~~ scheduled_in j (sched t)
============================
service_in j (sched t) = 0
subgoal 2 (ID 716) is:
~~ scheduled_in j (sched t)
----------------------------------------------------------------------------- *)
- by rewrite service_implies_scheduled //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 716)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t : instant
NO_SERVICE : service_in j (sched t) = 0
============================
~~ scheduled_in j (sched t)
----------------------------------------------------------------------------- *)
- apply (contra (H_scheduled_implies_serviced j (sched t))).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 731)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t : instant
NO_SERVICE : service_in j (sched t) = 0
============================
~~ (0 < service_in j (sched t))
----------------------------------------------------------------------------- *)
by rewrite -eqn0Ngt; apply /eqP ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t,
~~ scheduled_at sched j t ↔ service_at sched j t = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 696)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
forall t : instant, ~~ scheduled_at sched j t <-> service_at sched j t = 0
----------------------------------------------------------------------------- *)
Proof.
move⇒ t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 697)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t : instant
============================
~~ scheduled_at sched j t <-> service_at sched j t = 0
----------------------------------------------------------------------------- *)
rewrite /scheduled_at /service_at.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 711)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t : instant
============================
~~ scheduled_in j (sched t) <-> service_in j (sched t) = 0
----------------------------------------------------------------------------- *)
split ⇒ [NOT_SCHED | NO_SERVICE].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 715)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t : instant
NOT_SCHED : ~~ scheduled_in j (sched t)
============================
service_in j (sched t) = 0
subgoal 2 (ID 716) is:
~~ scheduled_in j (sched t)
----------------------------------------------------------------------------- *)
- by rewrite service_implies_scheduled //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 716)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t : instant
NO_SERVICE : service_in j (sched t) = 0
============================
~~ scheduled_in j (sched t)
----------------------------------------------------------------------------- *)
- apply (contra (H_scheduled_implies_serviced j (sched t))).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 731)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t : instant
NO_SERVICE : service_in j (sched t) = 0
============================
~~ (0 < service_in j (sched t))
----------------------------------------------------------------------------- *)
by rewrite -eqn0Ngt; apply /eqP ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Then, if a job does not receive any service during an interval, it
is not scheduled.
Lemma no_service_during_implies_not_scheduled:
∀ t1 t2,
service_during sched j t1 t2 = 0 →
∀ t,
t1 ≤ t < t2 → ~~ scheduled_at sched j t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 708)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
forall t1 t2 : instant,
service_during sched j t1 t2 = 0 ->
forall t : nat, t1 <= t < t2 -> ~~ scheduled_at sched j t
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 ZERO_SUM t /andP [GT_t1 LT_t2].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 752)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2 : instant
ZERO_SUM : service_during sched j t1 t2 = 0
t : nat
GT_t1 : t1 <= t
LT_t2 : t < t2
============================
~~ scheduled_at sched j t
----------------------------------------------------------------------------- *)
rewrite no_service_not_scheduled.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 762)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2 : instant
ZERO_SUM : service_during sched j t1 t2 = 0
t : nat
GT_t1 : t1 <= t
LT_t2 : t < t2
============================
service_at sched j t = 0
----------------------------------------------------------------------------- *)
move: ZERO_SUM.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 764)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2 : instant
t : nat
GT_t1 : t1 <= t
LT_t2 : t < t2
============================
service_during sched j t1 t2 = 0 -> service_at sched j t = 0
----------------------------------------------------------------------------- *)
rewrite /service_during big_nat_eq0 ⇒ IS_ZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 799)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2 : instant
t : nat
GT_t1 : t1 <= t
LT_t2 : t < t2
IS_ZERO : forall i : nat, t1 <= i < t2 -> service_at sched j i = 0
============================
service_at sched j t = 0
----------------------------------------------------------------------------- *)
by apply (IS_ZERO t); apply /andP; split ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t1 t2,
service_during sched j t1 t2 = 0 →
∀ t,
t1 ≤ t < t2 → ~~ scheduled_at sched j t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 708)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
forall t1 t2 : instant,
service_during sched j t1 t2 = 0 ->
forall t : nat, t1 <= t < t2 -> ~~ scheduled_at sched j t
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 ZERO_SUM t /andP [GT_t1 LT_t2].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 752)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2 : instant
ZERO_SUM : service_during sched j t1 t2 = 0
t : nat
GT_t1 : t1 <= t
LT_t2 : t < t2
============================
~~ scheduled_at sched j t
----------------------------------------------------------------------------- *)
rewrite no_service_not_scheduled.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 762)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2 : instant
ZERO_SUM : service_during sched j t1 t2 = 0
t : nat
GT_t1 : t1 <= t
LT_t2 : t < t2
============================
service_at sched j t = 0
----------------------------------------------------------------------------- *)
move: ZERO_SUM.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 764)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2 : instant
t : nat
GT_t1 : t1 <= t
LT_t2 : t < t2
============================
service_during sched j t1 t2 = 0 -> service_at sched j t = 0
----------------------------------------------------------------------------- *)
rewrite /service_during big_nat_eq0 ⇒ IS_ZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 799)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2 : instant
t : nat
GT_t1 : t1 <= t
LT_t2 : t < t2
IS_ZERO : forall i : nat, t1 <= i < t2 -> service_at sched j i = 0
============================
service_at sched j t = 0
----------------------------------------------------------------------------- *)
by apply (IS_ZERO t); apply /andP; split ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Conversely, if a job is not scheduled during an interval, then
it does not receive any service in that interval
Lemma not_scheduled_during_implies_zero_service:
∀ t1 t2,
(∀ t, t1 ≤ t < t2 → ~~ scheduled_at sched j t) →
service_during sched j t1 t2 = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 720)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
forall t1 t2 : nat,
(forall t : nat, t1 <= t < t2 -> ~~ scheduled_at sched j t) ->
service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
Proof.
intros t1 t2 NSCHED.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 723)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2 : nat
NSCHED : forall t : nat, t1 <= t < t2 -> ~~ scheduled_at sched j t
============================
service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
apply big_nat_eq0; move⇒ t NEQ.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 729)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2 : nat
NSCHED : forall t : nat, t1 <= t < t2 -> ~~ scheduled_at sched j t
t : nat
NEQ : t1 <= t < t2
============================
service_at sched j t = 0
----------------------------------------------------------------------------- *)
by apply no_service_not_scheduled, NSCHED.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t1 t2,
(∀ t, t1 ≤ t < t2 → ~~ scheduled_at sched j t) →
service_during sched j t1 t2 = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 720)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
forall t1 t2 : nat,
(forall t : nat, t1 <= t < t2 -> ~~ scheduled_at sched j t) ->
service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
Proof.
intros t1 t2 NSCHED.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 723)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2 : nat
NSCHED : forall t : nat, t1 <= t < t2 -> ~~ scheduled_at sched j t
============================
service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
apply big_nat_eq0; move⇒ t NEQ.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 729)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2 : nat
NSCHED : forall t : nat, t1 <= t < t2 -> ~~ scheduled_at sched j t
t : nat
NEQ : t1 <= t < t2
============================
service_at sched j t = 0
----------------------------------------------------------------------------- *)
by apply no_service_not_scheduled, NSCHED.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
If a job is scheduled at some point in an interval, it receives
positive cumulative service during the interval...
Lemma scheduled_implies_cumulative_service:
∀ t1 t2,
(∃ t,
t1 ≤ t < t2 ∧
scheduled_at sched j t) →
service_during sched j t1 t2 > 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 731)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
forall t1 t2 : nat,
(exists t : nat, t1 <= t < t2 /\ scheduled_at sched j t) ->
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 [t' [TIMES SCHED]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 754)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2, t' : nat
TIMES : t1 <= t' < t2
SCHED : scheduled_at sched j t'
============================
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
rewrite service_during_service_at.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 765)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2, t' : nat
TIMES : t1 <= t' < t2
SCHED : scheduled_at sched j t'
============================
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
∃ t'; split ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 770)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2, t' : nat
TIMES : t1 <= t' < t2
SCHED : scheduled_at sched j t'
============================
0 < service_at sched j t'
----------------------------------------------------------------------------- *)
move: SCHED.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 794)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2, t' : nat
TIMES : t1 <= t' < t2
============================
scheduled_at sched j t' -> 0 < service_at sched j t'
----------------------------------------------------------------------------- *)
rewrite /scheduled_at /service_at.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 808)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2, t' : nat
TIMES : t1 <= t' < t2
============================
scheduled_in j (sched t') -> 0 < service_in j (sched t')
----------------------------------------------------------------------------- *)
by apply (H_scheduled_implies_serviced j (sched t')).
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t1 t2,
(∃ t,
t1 ≤ t < t2 ∧
scheduled_at sched j t) →
service_during sched j t1 t2 > 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 731)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
forall t1 t2 : nat,
(exists t : nat, t1 <= t < t2 /\ scheduled_at sched j t) ->
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 [t' [TIMES SCHED]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 754)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2, t' : nat
TIMES : t1 <= t' < t2
SCHED : scheduled_at sched j t'
============================
0 < service_during sched j t1 t2
----------------------------------------------------------------------------- *)
rewrite service_during_service_at.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 765)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2, t' : nat
TIMES : t1 <= t' < t2
SCHED : scheduled_at sched j t'
============================
exists t : nat, t1 <= t < t2 /\ 0 < service_at sched j t
----------------------------------------------------------------------------- *)
∃ t'; split ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 770)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2, t' : nat
TIMES : t1 <= t' < t2
SCHED : scheduled_at sched j t'
============================
0 < service_at sched j t'
----------------------------------------------------------------------------- *)
move: SCHED.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 794)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2, t' : nat
TIMES : t1 <= t' < t2
============================
scheduled_at sched j t' -> 0 < service_at sched j t'
----------------------------------------------------------------------------- *)
rewrite /scheduled_at /service_at.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 808)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t1, t2, t' : nat
TIMES : t1 <= t' < t2
============================
scheduled_in j (sched t') -> 0 < service_in j (sched t')
----------------------------------------------------------------------------- *)
by apply (H_scheduled_implies_serviced j (sched t')).
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
...which again applies to total service, too.
Corollary scheduled_implies_nonzero_service:
∀ t,
(∃ t',
t' < t ∧
scheduled_at sched j t') →
service sched j t > 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 741)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
forall t : nat,
(exists t' : nat, t' < t /\ scheduled_at sched j t') ->
0 < service sched j t
----------------------------------------------------------------------------- *)
Proof.
move⇒ t [t' [TT SCHED]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 763)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t, t' : nat
TT : t' < t
SCHED : scheduled_at sched j t'
============================
0 < service sched j t
----------------------------------------------------------------------------- *)
rewrite /service.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 770)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t, t' : nat
TT : t' < t
SCHED : scheduled_at sched j t'
============================
0 < service_during sched j 0 t
----------------------------------------------------------------------------- *)
apply scheduled_implies_cumulative_service.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 771)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t, t' : nat
TT : t' < t
SCHED : scheduled_at sched j t'
============================
exists t0 : nat, 0 <= t0 < t /\ scheduled_at sched j t0
----------------------------------------------------------------------------- *)
∃ t'; split ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End GuaranteedService.
∀ t,
(∃ t',
t' < t ∧
scheduled_at sched j t') →
service sched j t > 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 741)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
forall t : nat,
(exists t' : nat, t' < t /\ scheduled_at sched j t') ->
0 < service sched j t
----------------------------------------------------------------------------- *)
Proof.
move⇒ t [t' [TT SCHED]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 763)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t, t' : nat
TT : t' < t
SCHED : scheduled_at sched j t'
============================
0 < service sched j t
----------------------------------------------------------------------------- *)
rewrite /service.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 770)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t, t' : nat
TT : t' < t
SCHED : scheduled_at sched j t'
============================
0 < service_during sched j 0 t
----------------------------------------------------------------------------- *)
apply scheduled_implies_cumulative_service.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 771)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H_scheduled_implies_serviced : ideal_progress_proc_model PState
t, t' : nat
TT : t' < t
SCHED : scheduled_at sched j t'
============================
exists t0 : nat, 0 <= t0 < t /\ scheduled_at sched j t0
----------------------------------------------------------------------------- *)
∃ t'; split ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End GuaranteedService.
Furthermore, if we know that jobs are not released early, then we can
narrow the interval during which they must have been scheduled.
Assume that jobs must arrive to execute.
We prove that any job with positive cumulative service at time [t] must
have been scheduled some time since its arrival and before time [t].
Lemma positive_service_implies_scheduled_since_arrival:
∀ t,
service sched j t > 0 →
∃ t', (job_arrival j ≤ t' < t ∧ scheduled_at sched j t').
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 700)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
============================
forall t : instant,
0 < service sched j t ->
exists t' : nat, job_arrival j <= t' < t /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
Proof.
move⇒ t SERVICE.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 702)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : instant
SERVICE : 0 < service sched j t
============================
exists t' : nat, job_arrival j <= t' < t /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
have EX_SCHED := positive_service_implies_scheduled_before t SERVICE.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 707)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : instant
SERVICE : 0 < service sched j t
EX_SCHED : exists t' : nat, t' < t /\ scheduled_at sched j t'
============================
exists t' : nat, job_arrival j <= t' < t /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
inversion EX_SCHED as [t'' [TIMES SCHED_AT]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 720)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : instant
SERVICE : 0 < service sched j t
EX_SCHED : exists t' : nat, t' < t /\ scheduled_at sched j t'
t'' : nat
TIMES : t'' < t
SCHED_AT : scheduled_at sched j t''
============================
exists t' : nat, job_arrival j <= t' < t /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
∃ t''; split; last by assumption.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 724)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : instant
SERVICE : 0 < service sched j t
EX_SCHED : exists t' : nat, t' < t /\ scheduled_at sched j t'
t'' : nat
TIMES : t'' < t
SCHED_AT : scheduled_at sched j t''
============================
job_arrival j <= t'' < t
----------------------------------------------------------------------------- *)
rewrite /(_ && _) ifT //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 742)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : instant
SERVICE : 0 < service sched j t
EX_SCHED : exists t' : nat, t' < t /\ scheduled_at sched j t'
t'' : nat
TIMES : t'' < t
SCHED_AT : scheduled_at sched j t''
============================
job_arrival j <= t''
----------------------------------------------------------------------------- *)
move: H_jobs_must_arrive.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 765)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : instant
SERVICE : 0 < service sched j t
EX_SCHED : exists t' : nat, t' < t /\ scheduled_at sched j t'
t'' : nat
TIMES : t'' < t
SCHED_AT : scheduled_at sched j t''
============================
jobs_must_arrive_to_execute sched -> job_arrival j <= t''
----------------------------------------------------------------------------- *)
rewrite /jobs_must_arrive_to_execute /has_arrived ⇒ ARR.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 778)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : instant
SERVICE : 0 < service sched j t
EX_SCHED : exists t' : nat, t' < t /\ scheduled_at sched j t'
t'' : nat
TIMES : t'' < t
SCHED_AT : scheduled_at sched j t''
ARR : forall (j : Job) (t : instant),
scheduled_at sched j t -> job_arrival j <= t
============================
job_arrival j <= t''
----------------------------------------------------------------------------- *)
by apply: ARR; exact.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Lemma not_scheduled_before_arrival:
∀ t, t < job_arrival j → ~~ scheduled_at sched j t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 707)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
============================
forall t : nat, t < job_arrival j -> ~~ scheduled_at sched j t
----------------------------------------------------------------------------- *)
Proof.
move⇒ t EARLY.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 709)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : nat
EARLY : t < job_arrival j
============================
~~ scheduled_at sched j t
----------------------------------------------------------------------------- *)
apply: (contra (H_jobs_must_arrive j t)).
(* ----------------------------------[ coqtop ]---------------------------------
1 focused subgoal
(shelved: 1) (ID 720)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : nat
EARLY : t < job_arrival j
============================
~~ has_arrived j t
----------------------------------------------------------------------------- *)
rewrite /has_arrived -ltnNge //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t,
service sched j t > 0 →
∃ t', (job_arrival j ≤ t' < t ∧ scheduled_at sched j t').
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 700)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
============================
forall t : instant,
0 < service sched j t ->
exists t' : nat, job_arrival j <= t' < t /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
Proof.
move⇒ t SERVICE.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 702)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : instant
SERVICE : 0 < service sched j t
============================
exists t' : nat, job_arrival j <= t' < t /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
have EX_SCHED := positive_service_implies_scheduled_before t SERVICE.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 707)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : instant
SERVICE : 0 < service sched j t
EX_SCHED : exists t' : nat, t' < t /\ scheduled_at sched j t'
============================
exists t' : nat, job_arrival j <= t' < t /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
inversion EX_SCHED as [t'' [TIMES SCHED_AT]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 720)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : instant
SERVICE : 0 < service sched j t
EX_SCHED : exists t' : nat, t' < t /\ scheduled_at sched j t'
t'' : nat
TIMES : t'' < t
SCHED_AT : scheduled_at sched j t''
============================
exists t' : nat, job_arrival j <= t' < t /\ scheduled_at sched j t'
----------------------------------------------------------------------------- *)
∃ t''; split; last by assumption.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 724)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : instant
SERVICE : 0 < service sched j t
EX_SCHED : exists t' : nat, t' < t /\ scheduled_at sched j t'
t'' : nat
TIMES : t'' < t
SCHED_AT : scheduled_at sched j t''
============================
job_arrival j <= t'' < t
----------------------------------------------------------------------------- *)
rewrite /(_ && _) ifT //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 742)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : instant
SERVICE : 0 < service sched j t
EX_SCHED : exists t' : nat, t' < t /\ scheduled_at sched j t'
t'' : nat
TIMES : t'' < t
SCHED_AT : scheduled_at sched j t''
============================
job_arrival j <= t''
----------------------------------------------------------------------------- *)
move: H_jobs_must_arrive.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 765)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : instant
SERVICE : 0 < service sched j t
EX_SCHED : exists t' : nat, t' < t /\ scheduled_at sched j t'
t'' : nat
TIMES : t'' < t
SCHED_AT : scheduled_at sched j t''
============================
jobs_must_arrive_to_execute sched -> job_arrival j <= t''
----------------------------------------------------------------------------- *)
rewrite /jobs_must_arrive_to_execute /has_arrived ⇒ ARR.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 778)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : instant
SERVICE : 0 < service sched j t
EX_SCHED : exists t' : nat, t' < t /\ scheduled_at sched j t'
t'' : nat
TIMES : t'' < t
SCHED_AT : scheduled_at sched j t''
ARR : forall (j : Job) (t : instant),
scheduled_at sched j t -> job_arrival j <= t
============================
job_arrival j <= t''
----------------------------------------------------------------------------- *)
by apply: ARR; exact.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Lemma not_scheduled_before_arrival:
∀ t, t < job_arrival j → ~~ scheduled_at sched j t.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 707)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
============================
forall t : nat, t < job_arrival j -> ~~ scheduled_at sched j t
----------------------------------------------------------------------------- *)
Proof.
move⇒ t EARLY.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 709)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : nat
EARLY : t < job_arrival j
============================
~~ scheduled_at sched j t
----------------------------------------------------------------------------- *)
apply: (contra (H_jobs_must_arrive j t)).
(* ----------------------------------[ coqtop ]---------------------------------
1 focused subgoal
(shelved: 1) (ID 720)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : nat
EARLY : t < job_arrival j
============================
~~ has_arrived j t
----------------------------------------------------------------------------- *)
rewrite /has_arrived -ltnNge //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
We show that job [j] does not receive service at any time [t] prior to its
arrival.
Lemma service_before_job_arrival_zero:
∀ t,
t < job_arrival j →
service_at sched j t = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 716)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
============================
forall t : nat, t < job_arrival j -> service_at sched j t = 0
----------------------------------------------------------------------------- *)
Proof.
move⇒ t NOT_ARR.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 718)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : nat
NOT_ARR : t < job_arrival j
============================
service_at sched j t = 0
----------------------------------------------------------------------------- *)
rewrite not_scheduled_implies_no_service // not_scheduled_before_arrival //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t,
t < job_arrival j →
service_at sched j t = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 716)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
============================
forall t : nat, t < job_arrival j -> service_at sched j t = 0
----------------------------------------------------------------------------- *)
Proof.
move⇒ t NOT_ARR.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 718)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : nat
NOT_ARR : t < job_arrival j
============================
service_at sched j t = 0
----------------------------------------------------------------------------- *)
rewrite not_scheduled_implies_no_service // not_scheduled_before_arrival //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Note that the same property applies to the cumulative service.
Lemma cumulative_service_before_job_arrival_zero :
∀ t1 t2 : instant,
t2 ≤ job_arrival j →
service_during sched j t1 t2 = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 724)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
============================
forall t1 t2 : instant,
t2 <= job_arrival j -> service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 EARLY.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 727)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : instant
EARLY : t2 <= job_arrival j
============================
service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
rewrite /service_during.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 734)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : instant
EARLY : t2 <= job_arrival j
============================
\sum_(t1 <= t < t2) service_at sched j t = 0
----------------------------------------------------------------------------- *)
have ZERO_SUM: \sum_(t1 ≤ t < t2) service_at sched j t = \sum_(t1 ≤ t < t2) 0;
last by rewrite ZERO_SUM sum0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 748)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : instant
EARLY : t2 <= job_arrival j
============================
\sum_(t1 <= t < t2) service_at sched j t = \sum_(t1 <= t < t2) 0
----------------------------------------------------------------------------- *)
rewrite big_nat_cond [in RHS]big_nat_cond.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 782)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : instant
EARLY : t2 <= job_arrival j
============================
\sum_(t1 <= i < t2 | (t1 <= i < t2) && true) service_at sched j i =
\sum_(t1 <= i < t2 | (t1 <= i < t2) && true) 0
----------------------------------------------------------------------------- *)
apply: eq_bigr ⇒ i /andP [TIMES _].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 875)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : instant
EARLY : t2 <= job_arrival j
i : nat
TIMES : t1 <= i < t2
============================
service_at sched j i = 0
----------------------------------------------------------------------------- *)
move: TIMES ⇒ /andP [le_t1_i lt_i_t2].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 917)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : instant
EARLY : t2 <= job_arrival j
i : nat
le_t1_i : t1 <= i
lt_i_t2 : i < t2
============================
service_at sched j i = 0
----------------------------------------------------------------------------- *)
apply (service_before_job_arrival_zero i).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 918)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : instant
EARLY : t2 <= job_arrival j
i : nat
le_t1_i : t1 <= i
lt_i_t2 : i < t2
============================
i < job_arrival j
----------------------------------------------------------------------------- *)
by apply leq_trans with (n := t2); auto.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t1 t2 : instant,
t2 ≤ job_arrival j →
service_during sched j t1 t2 = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 724)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
============================
forall t1 t2 : instant,
t2 <= job_arrival j -> service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 EARLY.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 727)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : instant
EARLY : t2 <= job_arrival j
============================
service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
rewrite /service_during.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 734)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : instant
EARLY : t2 <= job_arrival j
============================
\sum_(t1 <= t < t2) service_at sched j t = 0
----------------------------------------------------------------------------- *)
have ZERO_SUM: \sum_(t1 ≤ t < t2) service_at sched j t = \sum_(t1 ≤ t < t2) 0;
last by rewrite ZERO_SUM sum0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 748)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : instant
EARLY : t2 <= job_arrival j
============================
\sum_(t1 <= t < t2) service_at sched j t = \sum_(t1 <= t < t2) 0
----------------------------------------------------------------------------- *)
rewrite big_nat_cond [in RHS]big_nat_cond.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 782)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : instant
EARLY : t2 <= job_arrival j
============================
\sum_(t1 <= i < t2 | (t1 <= i < t2) && true) service_at sched j i =
\sum_(t1 <= i < t2 | (t1 <= i < t2) && true) 0
----------------------------------------------------------------------------- *)
apply: eq_bigr ⇒ i /andP [TIMES _].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 875)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : instant
EARLY : t2 <= job_arrival j
i : nat
TIMES : t1 <= i < t2
============================
service_at sched j i = 0
----------------------------------------------------------------------------- *)
move: TIMES ⇒ /andP [le_t1_i lt_i_t2].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 917)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : instant
EARLY : t2 <= job_arrival j
i : nat
le_t1_i : t1 <= i
lt_i_t2 : i < t2
============================
service_at sched j i = 0
----------------------------------------------------------------------------- *)
apply (service_before_job_arrival_zero i).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 918)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : instant
EARLY : t2 <= job_arrival j
i : nat
le_t1_i : t1 <= i
lt_i_t2 : i < t2
============================
i < job_arrival j
----------------------------------------------------------------------------- *)
by apply leq_trans with (n := t2); auto.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Hence, one can ignore the service received by a job before its arrival
time...
Lemma ignore_service_before_arrival:
∀ t1 t2,
t1 ≤ job_arrival j →
t2 ≥ job_arrival j →
service_during sched j t1 t2 = service_during sched j (job_arrival j) t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 741)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
============================
forall t1 t2 : nat,
t1 <= job_arrival j ->
job_arrival j <= t2 ->
service_during sched j t1 t2 = service_during sched j (job_arrival j) t2
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 le_t1 le_t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 745)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : nat
le_t1 : t1 <= job_arrival j
le_t2 : job_arrival j <= t2
============================
service_during sched j t1 t2 = service_during sched j (job_arrival j) t2
----------------------------------------------------------------------------- *)
rewrite -(service_during_cat sched j t1 (job_arrival j) t2).
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 758)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : nat
le_t1 : t1 <= job_arrival j
le_t2 : job_arrival j <= t2
============================
service_during sched j t1 (job_arrival j) +
service_during sched j (job_arrival j) t2 =
service_during sched j (job_arrival j) t2
subgoal 2 (ID 759) is:
t1 <= job_arrival j <= t2
----------------------------------------------------------------------------- *)
rewrite cumulative_service_before_job_arrival_zero //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 759)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : nat
le_t1 : t1 <= job_arrival j
le_t2 : job_arrival j <= t2
============================
t1 <= job_arrival j <= t2
----------------------------------------------------------------------------- *)
by apply/andP; split; exact.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t1 t2,
t1 ≤ job_arrival j →
t2 ≥ job_arrival j →
service_during sched j t1 t2 = service_during sched j (job_arrival j) t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 741)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
============================
forall t1 t2 : nat,
t1 <= job_arrival j ->
job_arrival j <= t2 ->
service_during sched j t1 t2 = service_during sched j (job_arrival j) t2
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 le_t1 le_t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 745)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : nat
le_t1 : t1 <= job_arrival j
le_t2 : job_arrival j <= t2
============================
service_during sched j t1 t2 = service_during sched j (job_arrival j) t2
----------------------------------------------------------------------------- *)
rewrite -(service_during_cat sched j t1 (job_arrival j) t2).
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 758)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : nat
le_t1 : t1 <= job_arrival j
le_t2 : job_arrival j <= t2
============================
service_during sched j t1 (job_arrival j) +
service_during sched j (job_arrival j) t2 =
service_during sched j (job_arrival j) t2
subgoal 2 (ID 759) is:
t1 <= job_arrival j <= t2
----------------------------------------------------------------------------- *)
rewrite cumulative_service_before_job_arrival_zero //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 759)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t1, t2 : nat
le_t1 : t1 <= job_arrival j
le_t2 : job_arrival j <= t2
============================
t1 <= job_arrival j <= t2
----------------------------------------------------------------------------- *)
by apply/andP; split; exact.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
... which we can also state in terms of cumulative service.
Corollary no_service_before_arrival:
∀ t,
t ≤ job_arrival j → service sched j t = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 750)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
============================
forall t : nat, t <= job_arrival j -> service sched j t = 0
----------------------------------------------------------------------------- *)
Proof.
move⇒ t EARLY.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 752)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : nat
EARLY : t <= job_arrival j
============================
service sched j t = 0
----------------------------------------------------------------------------- *)
rewrite /service cumulative_service_before_job_arrival_zero //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End AfterArrival.
∀ t,
t ≤ job_arrival j → service sched j t = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 750)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
============================
forall t : nat, t <= job_arrival j -> service sched j t = 0
----------------------------------------------------------------------------- *)
Proof.
move⇒ t EARLY.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 752)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
H0 : JobArrival Job
H_jobs_must_arrive : jobs_must_arrive_to_execute sched
t : nat
EARLY : t <= job_arrival j
============================
service sched j t = 0
----------------------------------------------------------------------------- *)
rewrite /service cumulative_service_before_job_arrival_zero //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End AfterArrival.
In this section, we prove some lemmas about time instants with same
service.
Consider any time instants [t1] and [t2]...
...where [t1] is no later than [t2]...
...and where job [j] has received the same amount of service.
First, we observe that this means that the job receives no service
during [t1, t2)...
Lemma constant_service_implies_no_service_during:
service_during sched j t1 t2 = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 698)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
============================
service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
Proof.
move: H_same_service.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 699)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
============================
service sched j t1 = service sched j t2 -> service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
rewrite -(service_cat sched j t1 t2) // -[service sched j t1 in LHS]addn0 ⇒ /eqP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 806)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
============================
service sched j t1 + 0 == service sched j t1 + service_during sched j t1 t2 ->
service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
by rewrite eqn_add2l ⇒ /eqP //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
service_during sched j t1 t2 = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 698)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
============================
service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
Proof.
move: H_same_service.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 699)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
============================
service sched j t1 = service sched j t2 -> service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
rewrite -(service_cat sched j t1 t2) // -[service sched j t1 in LHS]addn0 ⇒ /eqP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 806)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
============================
service sched j t1 + 0 == service sched j t1 + service_during sched j t1 t2 ->
service_during sched j t1 t2 = 0
----------------------------------------------------------------------------- *)
by rewrite eqn_add2l ⇒ /eqP //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
...which of course implies that it does not receive service at any
point, either.
Lemma constant_service_implies_not_scheduled:
∀ t,
t1 ≤ t < t2 → service_at sched j t = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 705)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
============================
forall t : nat, t1 <= t < t2 -> service_at sched j t = 0
----------------------------------------------------------------------------- *)
Proof.
move⇒ t /andP [GE_t1 LT_t2].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 746)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : nat
GE_t1 : t1 <= t
LT_t2 : t < t2
============================
service_at sched j t = 0
----------------------------------------------------------------------------- *)
move: constant_service_implies_no_service_during.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 747)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : nat
GE_t1 : t1 <= t
LT_t2 : t < t2
============================
service_during sched j t1 t2 = 0 -> service_at sched j t = 0
----------------------------------------------------------------------------- *)
rewrite /service_during big_nat_eq0 ⇒ IS_ZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 782)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : nat
GE_t1 : t1 <= t
LT_t2 : t < t2
IS_ZERO : forall i : nat, t1 <= i < t2 -> service_at sched j i = 0
============================
service_at sched j t = 0
----------------------------------------------------------------------------- *)
apply IS_ZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 783)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : nat
GE_t1 : t1 <= t
LT_t2 : t < t2
IS_ZERO : forall i : nat, t1 <= i < t2 -> service_at sched j i = 0
============================
t1 <= t < t2
----------------------------------------------------------------------------- *)
apply /andP; split ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t,
t1 ≤ t < t2 → service_at sched j t = 0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 705)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
============================
forall t : nat, t1 <= t < t2 -> service_at sched j t = 0
----------------------------------------------------------------------------- *)
Proof.
move⇒ t /andP [GE_t1 LT_t2].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 746)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : nat
GE_t1 : t1 <= t
LT_t2 : t < t2
============================
service_at sched j t = 0
----------------------------------------------------------------------------- *)
move: constant_service_implies_no_service_during.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 747)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : nat
GE_t1 : t1 <= t
LT_t2 : t < t2
============================
service_during sched j t1 t2 = 0 -> service_at sched j t = 0
----------------------------------------------------------------------------- *)
rewrite /service_during big_nat_eq0 ⇒ IS_ZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 782)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : nat
GE_t1 : t1 <= t
LT_t2 : t < t2
IS_ZERO : forall i : nat, t1 <= i < t2 -> service_at sched j i = 0
============================
service_at sched j t = 0
----------------------------------------------------------------------------- *)
apply IS_ZERO.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 783)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : nat
GE_t1 : t1 <= t
LT_t2 : t < t2
IS_ZERO : forall i : nat, t1 <= i < t2 -> service_at sched j i = 0
============================
t1 <= t < t2
----------------------------------------------------------------------------- *)
apply /andP; split ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
We show that job [j] receives service at some point [t < t1]
iff [j] receives service at some point [t' < t2].
Lemma same_service_implies_serviced_at_earlier_times:
[∃ t: 'I_t1, service_at sched j t > 0] =
[∃ t': 'I_t2, service_at sched j t' > 0].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 722)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t) t) =
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, 0 < service_at sched j t') t')
----------------------------------------------------------------------------- *)
Proof.
apply /idP/idP ⇒ /existsP [t SERVICED].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 833)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t1
SERVICED : 0 < service_at sched j t
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, 0 < service_at sched j t') t')
subgoal 2 (ID 834) is:
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t0 : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t0) t0)
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 833)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t1
SERVICED : 0 < service_at sched j t
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, 0 < service_at sched j t') t')
----------------------------------------------------------------------------- *)
have LE: t < t2
by apply: (leq_trans _ H_t1_le_t2) ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 848)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t1
SERVICED : 0 < service_at sched j t
LE : t < t2
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, 0 < service_at sched j t') t')
----------------------------------------------------------------------------- *)
by apply /existsP; ∃ (Ordinal LE).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 834)
subgoal 1 (ID 834) is:
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t0 : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t0) t0)
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 834)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t2
SERVICED : 0 < service_at sched j t
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t0 : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t0) t0)
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 834)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t2
SERVICED : 0 < service_at sched j t
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t0 : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t0) t0)
----------------------------------------------------------------------------- *)
case (boolP (t < t1)) ⇒ LE.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 882)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t2
SERVICED : 0 < service_at sched j t
LE : t < t1
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t0 : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t0) t0)
subgoal 2 (ID 883) is:
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t0 : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t0) t0)
----------------------------------------------------------------------------- *)
- by apply /existsP; ∃ (Ordinal LE).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 883)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t2
SERVICED : 0 < service_at sched j t
LE : ~~ (t < t1)
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t0 : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t0) t0)
----------------------------------------------------------------------------- *)
- exfalso.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 912)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t2
SERVICED : 0 < service_at sched j t
LE : ~~ (t < t1)
============================
False
----------------------------------------------------------------------------- *)
have TIMES: t1 ≤ t < t2
by apply /andP; split ⇒ //; rewrite leqNgt //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 968)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t2
SERVICED : 0 < service_at sched j t
LE : ~~ (t < t1)
TIMES : t1 <= t < t2
============================
False
----------------------------------------------------------------------------- *)
have NO_SERVICE := constant_service_implies_not_scheduled t TIMES.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 973)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t2
SERVICED : 0 < service_at sched j t
LE : ~~ (t < t1)
TIMES : t1 <= t < t2
NO_SERVICE : service_at sched j t = 0
============================
False
----------------------------------------------------------------------------- *)
by rewrite NO_SERVICE in SERVICED.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
[∃ t: 'I_t1, service_at sched j t > 0] =
[∃ t': 'I_t2, service_at sched j t' > 0].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 722)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t) t) =
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, 0 < service_at sched j t') t')
----------------------------------------------------------------------------- *)
Proof.
apply /idP/idP ⇒ /existsP [t SERVICED].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 833)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t1
SERVICED : 0 < service_at sched j t
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, 0 < service_at sched j t') t')
subgoal 2 (ID 834) is:
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t0 : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t0) t0)
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 833)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t1
SERVICED : 0 < service_at sched j t
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, 0 < service_at sched j t') t')
----------------------------------------------------------------------------- *)
have LE: t < t2
by apply: (leq_trans _ H_t1_le_t2) ⇒ //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 848)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t1
SERVICED : 0 < service_at sched j t
LE : t < t2
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, 0 < service_at sched j t') t')
----------------------------------------------------------------------------- *)
by apply /existsP; ∃ (Ordinal LE).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 834)
subgoal 1 (ID 834) is:
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t0 : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t0) t0)
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 834)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t2
SERVICED : 0 < service_at sched j t
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t0 : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t0) t0)
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 834)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t2
SERVICED : 0 < service_at sched j t
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t0 : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t0) t0)
----------------------------------------------------------------------------- *)
case (boolP (t < t1)) ⇒ LE.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 882)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t2
SERVICED : 0 < service_at sched j t
LE : t < t1
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t0 : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t0) t0)
subgoal 2 (ID 883) is:
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t0 : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t0) t0)
----------------------------------------------------------------------------- *)
- by apply /existsP; ∃ (Ordinal LE).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 883)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t2
SERVICED : 0 < service_at sched j t
LE : ~~ (t < t1)
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t0 : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j t0) t0)
----------------------------------------------------------------------------- *)
- exfalso.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 912)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t2
SERVICED : 0 < service_at sched j t
LE : ~~ (t < t1)
============================
False
----------------------------------------------------------------------------- *)
have TIMES: t1 ≤ t < t2
by apply /andP; split ⇒ //; rewrite leqNgt //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 968)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t2
SERVICED : 0 < service_at sched j t
LE : ~~ (t < t1)
TIMES : t1 <= t < t2
============================
False
----------------------------------------------------------------------------- *)
have NO_SERVICE := constant_service_implies_not_scheduled t TIMES.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 973)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
t : ordinal_finType t2
SERVICED : 0 < service_at sched j t
LE : ~~ (t < t1)
TIMES : t1 <= t < t2
NO_SERVICE : service_at sched j t = 0
============================
False
----------------------------------------------------------------------------- *)
by rewrite NO_SERVICE in SERVICED.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Then, under the assumption that scheduled jobs receives service,
we can translate this into a claim about scheduled_at.
Assume [j] always receives some positive service.
We show that job [j] is scheduled at some point [t < t1] iff [j] is scheduled
at some point [t' < t2].
Lemma same_service_implies_scheduled_at_earlier_times:
[∃ t: 'I_t1, scheduled_at sched j t] =
[∃ t': 'I_t2, scheduled_at sched j t'].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 741)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, scheduled_at sched j t) t) =
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, scheduled_at sched j t') t')
----------------------------------------------------------------------------- *)
Proof.
have CONV: ∀ B, [∃ b: 'I_B, scheduled_at sched j b]
= [∃ b: 'I_B, service_at sched j b > 0].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 759)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
forall B : nat,
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B) (, scheduled_at sched j b) b) =
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B) (, 0 < service_at sched j b) b)
subgoal 2 (ID 761) is:
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, scheduled_at sched j t) t) =
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, scheduled_at sched j t') t')
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 759)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
forall B : nat,
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B) (, scheduled_at sched j b) b) =
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B) (, 0 < service_at sched j b) b)
----------------------------------------------------------------------------- *)
move⇒ B.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 762)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
B : nat
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B) (, scheduled_at sched j b) b) =
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B) (, 0 < service_at sched j b) b)
----------------------------------------------------------------------------- *)
apply/idP/idP ⇒ /existsP [b P]; apply /existsP; ∃ b.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 924)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
B : nat
b : ordinal_finType B
P : scheduled_at sched j b
============================
0 < service_at sched j b
subgoal 2 (ID 926) is:
scheduled_at sched j b
----------------------------------------------------------------------------- *)
- by move: P; rewrite /scheduled_at /service_at;
apply (H_scheduled_implies_serviced j (sched b)).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 926)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
B : nat
b : ordinal_finType B
P : 0 < service_at sched j b
============================
scheduled_at sched j b
----------------------------------------------------------------------------- *)
- by apply service_at_implies_scheduled_at.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 761)
subgoal 1 (ID 761) is:
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, scheduled_at sched j t) t) =
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, scheduled_at sched j t') t')
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 761)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
CONV : forall B : nat,
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B)
(, scheduled_at sched j b) b) =
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B)
(, 0 < service_at sched j b) b)
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, scheduled_at sched j t) t) =
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, scheduled_at sched j t') t')
----------------------------------------------------------------------------- *)
rewrite !CONV.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 949)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
CONV : forall B : nat,
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B)
(, scheduled_at sched j b) b) =
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B)
(, 0 < service_at sched j b) b)
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun b : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j b) b) =
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun b : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, 0 < service_at sched j b) b)
----------------------------------------------------------------------------- *)
apply same_service_implies_serviced_at_earlier_times.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End TimesWithSameService.
End RelationToScheduled.
Section ServiceInTwoSchedules.
[∃ t: 'I_t1, scheduled_at sched j t] =
[∃ t': 'I_t2, scheduled_at sched j t'].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 741)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, scheduled_at sched j t) t) =
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, scheduled_at sched j t') t')
----------------------------------------------------------------------------- *)
Proof.
have CONV: ∀ B, [∃ b: 'I_B, scheduled_at sched j b]
= [∃ b: 'I_B, service_at sched j b > 0].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 759)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
forall B : nat,
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B) (, scheduled_at sched j b) b) =
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B) (, 0 < service_at sched j b) b)
subgoal 2 (ID 761) is:
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, scheduled_at sched j t) t) =
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, scheduled_at sched j t') t')
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 759)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
============================
forall B : nat,
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B) (, scheduled_at sched j b) b) =
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B) (, 0 < service_at sched j b) b)
----------------------------------------------------------------------------- *)
move⇒ B.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 762)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
B : nat
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B) (, scheduled_at sched j b) b) =
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B) (, 0 < service_at sched j b) b)
----------------------------------------------------------------------------- *)
apply/idP/idP ⇒ /existsP [b P]; apply /existsP; ∃ b.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 924)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
B : nat
b : ordinal_finType B
P : scheduled_at sched j b
============================
0 < service_at sched j b
subgoal 2 (ID 926) is:
scheduled_at sched j b
----------------------------------------------------------------------------- *)
- by move: P; rewrite /scheduled_at /service_at;
apply (H_scheduled_implies_serviced j (sched b)).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 926)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
B : nat
b : ordinal_finType B
P : 0 < service_at sched j b
============================
scheduled_at sched j b
----------------------------------------------------------------------------- *)
- by apply service_at_implies_scheduled_at.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 761)
subgoal 1 (ID 761) is:
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, scheduled_at sched j t) t) =
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, scheduled_at sched j t') t')
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 761)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
CONV : forall B : nat,
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B)
(, scheduled_at sched j b) b) =
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B)
(, 0 < service_at sched j b) b)
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun t : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, scheduled_at sched j t) t) =
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun t' : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, scheduled_at sched j t') t')
----------------------------------------------------------------------------- *)
rewrite !CONV.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 949)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched : schedule PState
j : Job
t1, t2 : instant
H_t1_le_t2 : t1 <= t2
H_same_service : service sched j t1 = service sched j t2
H_scheduled_implies_serviced : ideal_progress_proc_model PState
CONV : forall B : nat,
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B)
(, scheduled_at sched j b) b) =
~~
FiniteQuant.quant0b (T:=ordinal_finType B)
(fun b : 'I_B =>
FiniteQuant.ex (T:=ordinal_finType B)
(, 0 < service_at sched j b) b)
============================
~~
FiniteQuant.quant0b (T:=ordinal_finType t1)
(fun b : 'I_t1 =>
FiniteQuant.ex (T:=ordinal_finType t1) (, 0 < service_at sched j b) b) =
~~
FiniteQuant.quant0b (T:=ordinal_finType t2)
(fun b : 'I_t2 =>
FiniteQuant.ex (T:=ordinal_finType t2) (, 0 < service_at sched j b) b)
----------------------------------------------------------------------------- *)
apply same_service_implies_serviced_at_earlier_times.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End TimesWithSameService.
End RelationToScheduled.
Section ServiceInTwoSchedules.
Consider any job type and any processor model.
Consider any two given schedules...
Given an interval in which the schedules provide the same service
to a job at each instant, we can prove that the cumulative service
received during the interval has to be the same.
Consider two time instants...
...and a given job that is to be scheduled.
Assume that, in any instant between [t1] and [t2] the service
provided to [j] from the two schedules is the same.
Hypothesis H_sched1_sched2_same_service_at:
∀ t, t1 ≤ t < t2 →
service_at sched1 j t = service_at sched2 j t.
∀ t, t1 ≤ t < t2 →
service_at sched1 j t = service_at sched2 j t.
It follows that the service provided during [t1] and [t2]
is also the same.
Lemma same_service_during:
service_during sched1 j t1 t2 = service_during sched2 j t1 t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 641)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched1, sched2 : schedule PState
t1, t2 : instant
j : Job
H_sched1_sched2_same_service_at : forall t : nat,
t1 <= t < t2 ->
service_at sched1 j t =
service_at sched2 j t
============================
service_during sched1 j t1 t2 = service_during sched2 j t1 t2
----------------------------------------------------------------------------- *)
Proof.
rewrite /service_during.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 648)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched1, sched2 : schedule PState
t1, t2 : instant
j : Job
H_sched1_sched2_same_service_at : forall t : nat,
t1 <= t < t2 ->
service_at sched1 j t =
service_at sched2 j t
============================
\sum_(t1 <= t < t2) service_at sched1 j t =
\sum_(t1 <= t < t2) service_at sched2 j t
----------------------------------------------------------------------------- *)
apply eq_big_nat.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 649)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched1, sched2 : schedule PState
t1, t2 : instant
j : Job
H_sched1_sched2_same_service_at : forall t : nat,
t1 <= t < t2 ->
service_at sched1 j t =
service_at sched2 j t
============================
forall i : nat,
t1 <= i < t2 -> service_at sched1 j i = service_at sched2 j i
----------------------------------------------------------------------------- *)
by apply H_sched1_sched2_same_service_at.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End ServiceDuringEquivalentInterval.
service_during sched1 j t1 t2 = service_during sched2 j t1 t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 641)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched1, sched2 : schedule PState
t1, t2 : instant
j : Job
H_sched1_sched2_same_service_at : forall t : nat,
t1 <= t < t2 ->
service_at sched1 j t =
service_at sched2 j t
============================
service_during sched1 j t1 t2 = service_during sched2 j t1 t2
----------------------------------------------------------------------------- *)
Proof.
rewrite /service_during.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 648)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched1, sched2 : schedule PState
t1, t2 : instant
j : Job
H_sched1_sched2_same_service_at : forall t : nat,
t1 <= t < t2 ->
service_at sched1 j t =
service_at sched2 j t
============================
\sum_(t1 <= t < t2) service_at sched1 j t =
\sum_(t1 <= t < t2) service_at sched2 j t
----------------------------------------------------------------------------- *)
apply eq_big_nat.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 649)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched1, sched2 : schedule PState
t1, t2 : instant
j : Job
H_sched1_sched2_same_service_at : forall t : nat,
t1 <= t < t2 ->
service_at sched1 j t =
service_at sched2 j t
============================
forall i : nat,
t1 <= i < t2 -> service_at sched1 j i = service_at sched2 j i
----------------------------------------------------------------------------- *)
by apply H_sched1_sched2_same_service_at.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End ServiceDuringEquivalentInterval.
We can leverage the previous lemma to conclude that two schedules
that match in a given interval will also have the same cumulative
service across the interval.
Corollary equal_prefix_implies_same_service_during:
∀ t1 t2,
(∀ t, t1 ≤ t < t2 → sched1 t = sched2 t) →
∀ j, service_during sched1 j t1 t2 = service_during sched2 j t1 t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 640)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched1, sched2 : schedule PState
============================
forall t1 t2 : nat,
(forall t : nat, t1 <= t < t2 -> sched1 t = sched2 t) ->
forall j : Job,
service_during sched1 j t1 t2 = service_during sched2 j t1 t2
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 SCHED_EQ j.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 644)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched1, sched2 : schedule PState
t1, t2 : nat
SCHED_EQ : forall t : nat, t1 <= t < t2 -> sched1 t = sched2 t
j : Job
============================
service_during sched1 j t1 t2 = service_during sched2 j t1 t2
----------------------------------------------------------------------------- *)
apply same_service_during ⇒ t' RANGE.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 647)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched1, sched2 : schedule PState
t1, t2 : nat
SCHED_EQ : forall t : nat, t1 <= t < t2 -> sched1 t = sched2 t
j : Job
t' : nat
RANGE : t1 <= t' < t2
============================
service_at sched1 j t' = service_at sched2 j t'
----------------------------------------------------------------------------- *)
by rewrite /service_at SCHED_EQ.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ t1 t2,
(∀ t, t1 ≤ t < t2 → sched1 t = sched2 t) →
∀ j, service_during sched1 j t1 t2 = service_during sched2 j t1 t2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 640)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched1, sched2 : schedule PState
============================
forall t1 t2 : nat,
(forall t : nat, t1 <= t < t2 -> sched1 t = sched2 t) ->
forall j : Job,
service_during sched1 j t1 t2 = service_during sched2 j t1 t2
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 SCHED_EQ j.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 644)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched1, sched2 : schedule PState
t1, t2 : nat
SCHED_EQ : forall t : nat, t1 <= t < t2 -> sched1 t = sched2 t
j : Job
============================
service_during sched1 j t1 t2 = service_during sched2 j t1 t2
----------------------------------------------------------------------------- *)
apply same_service_during ⇒ t' RANGE.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 647)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched1, sched2 : schedule PState
t1, t2 : nat
SCHED_EQ : forall t : nat, t1 <= t < t2 -> sched1 t = sched2 t
j : Job
t' : nat
RANGE : t1 <= t' < t2
============================
service_at sched1 j t' = service_at sched2 j t'
----------------------------------------------------------------------------- *)
by rewrite /service_at SCHED_EQ.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
For convenience, we restate the corollary also at the level of
[service] for identical prefixes.
Corollary identical_prefix_service:
∀ h,
identical_prefix sched1 sched2 h →
∀ j, service sched1 j h = service sched2 j h.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 654)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched1, sched2 : schedule PState
============================
forall h : instant,
identical_prefix sched1 sched2 h ->
forall j : Job, service sched1 j h = service sched2 j h
----------------------------------------------------------------------------- *)
Proof.
move⇒ h IDENT j; by apply equal_prefix_implies_same_service_during.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End ServiceInTwoSchedules.
∀ h,
identical_prefix sched1 sched2 h →
∀ j, service sched1 j h = service sched2 j h.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 654)
Job : JobType
PState : Type
H : ProcessorState Job PState
sched1, sched2 : schedule PState
============================
forall h : instant,
identical_prefix sched1 sched2 h ->
forall j : Job, service sched1 j h = service sched2 j h
----------------------------------------------------------------------------- *)
Proof.
move⇒ h IDENT j; by apply equal_prefix_implies_same_service_during.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End ServiceInTwoSchedules.