Commit bd6d237f authored by Sergey Bozhko's avatar Sergey Bozhko

Shorten a needlessly-long proof

parent d52138bf
This diff is collapsed.
......@@ -310,9 +310,9 @@ Module Service.
}
Qed.
(* Last, we prove that overall service of jobs at each time instant is at most 1. *)
(* We prove that the overall service of jobs at each time instant is at most 1. *)
Lemma service_of_jobs_le_1:
forall t1 t2 t P,
forall (t1 t2 t: time) (P: Job -> bool),
\sum_(j <- arrivals_between t1 t2 | P j) service_at sched j t <= 1.
Proof.
intros t1 t2 t P.
......@@ -348,6 +348,23 @@ Module Service.
by rewrite /service_at /scheduled_at SCHED.
}
Qed.
(* We prove that the overall service of jobs within
some time interval [t, t + Δ) is at most Δ. *)
Lemma total_service_of_jobs_le_delta:
forall (t Δ: time) (P: Job -> bool),
\sum_(j <- arrivals_between t (t + Δ) | P j)
service_during sched j t (t + Δ) <= Δ.
Proof.
intros.
have EQ: \sum_(t <= x < t + Δ) 1 = Δ.
{ by rewrite big_const_nat iter_addn mul1n addn0 -{2}[t]addn0 subnDl subn0. }
rewrite -{3}EQ; clear EQ.
rewrite exchange_big //=.
rewrite leq_sum //.
move => t' _.
by apply service_of_jobs_le_1.
Qed.
(* In this section, we introduce a connection between the cumulative
service, cumulative workload, and completion of jobs. *)
......
......@@ -246,14 +246,14 @@ End ExtraLemmas.
Section SumArithmetic.
Lemma sum_seq_diff:
forall (T:eqType) (r: seq T) (F G : T -> nat),
(forall i : T, i \in r -> G i <= F i) ->
\sum_(i <- r) (F i - G i) = \sum_(i <- r) F i - \sum_(i <- r) G i.
forall (T:eqType) (rs: seq T) (F G : T -> nat),
(forall i : T, i \in rs -> G i <= F i) ->
\sum_(i <- rs) (F i - G i) = \sum_(i <- rs) F i - \sum_(i <- rs) G i.
Proof.
intros.
induction r; first by rewrite !big_nil subn0.
induction rs; first by rewrite !big_nil subn0.
rewrite !big_cons subh2.
- apply/eqP; rewrite eqn_add2l; apply/eqP; apply IHr.
- apply/eqP; rewrite eqn_add2l; apply/eqP; apply IHrs.
by intros; apply H; rewrite in_cons; apply/orP; right.
- by apply H; rewrite in_cons; apply/orP; left.
- rewrite big_seq_cond [in X in _ <= X]big_seq_cond.
......@@ -272,6 +272,21 @@ Section SumArithmetic.
move => i; rewrite mem_index_iota; move => /andP [_ LT].
by apply ALL.
Qed.
Lemma sum_pred_diff:
forall (T: eqType) (rs: seq T) (P: T -> bool) (F: T -> nat),
\sum_(r <- rs | P r) F r =
\sum_(r <- rs) F r - \sum_(r <- rs | ~~ P r) F r.
Proof.
clear; intros.
induction rs; first by rewrite !big_nil subn0.
rewrite !big_cons !IHrs; clear IHrs.
case (P a); simpl; last by rewrite subnDl.
rewrite addnBA; first by done.
rewrite big_mkcond leq_sum //.
intros t _.
by case (P t).
Qed.
Lemma telescoping_sum :
forall (T: Type) (F: T->nat) r (x0: T),
......
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