Commit 9fa2acb3 authored by Vedant Chavda's avatar Vedant Chavda Committed by Björn Brandenburg

add notion of hyperperiod for periodic tasks

parent fedef925
Require Export prosa.model.task.arrival.periodic.
Require Export prosa.util.lcmseq.
From mathcomp Require Import div.
(** In this file we define the notion of a hyperperiod for periodic tasks. *)
Section Hyperperiod.
(** Consider periodic tasks. *)
Context {Task : TaskType} `{PeriodicModel Task}.
(** Consider any task set [ts]... *)
Variable ts : TaskSet Task.
(** ... and any task [tsk] that belongs to this task set. *)
Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.
(** The hyperperiod of a task set is defined as the least common multiple
(LCM) of the periods of all tasks in the task set. **)
Definition hyperperiod : duration := lcml (map task_period ts).
(** Consequently, a task set's hyperperiod is an integral multiple
of each task's period in the task set. **)
Lemma hyperperiod_int_mult_of_any_task :
exists k, hyperperiod = k * task_period tsk.
Proof.
apply lcm_seq_is_mult_of_all_ints.
apply map_f.
by apply H_tsk_in_ts.
Qed.
End Hyperperiod.
......@@ -48,3 +48,4 @@ runtime
bursty
TODO
mathcomp
hyperperiod
From mathcomp Require Export ssreflect seq div ssrbool ssrnat eqtype ssrfun.
(** A function to calculate the least common multiple
of all integers in a sequence [xs], denoted by [lcml xs] **)
Definition lcml (xs : seq nat) : nat := foldr lcmn 1 xs.
(** Any integer [a] that is contained in the sequence [xs] divides [lcml xs]. **)
Lemma int_divides_lcm_in_seq :
forall (a : nat) (xs : seq nat), a %| lcml (a :: xs).
Proof.
intros.
rewrite /lcml.
induction xs.
- rewrite /foldr.
now apply dvdn_lcml.
- rewrite -cat1s.
rewrite foldr_cat /foldr.
by apply dvdn_lcml.
Qed.
(** Also, [lcml xs1] divides [lcml xs2] if [xs2] contains one extra element as compared to [xs1]. *)
Lemma lcm_seq_divides_lcm_super :
forall (x : nat) (xs : seq nat),
lcml xs %| lcml (x :: xs).
Proof.
intros.
rewrite /lcml.
induction xs; first by auto.
rewrite -cat1s foldr_cat /foldr.
by apply dvdn_lcmr.
Qed.
(** All integers in a sequence [xs] divide [lcml xs]. *)
Lemma lcm_seq_is_mult_of_all_ints :
forall (sq : seq nat) (a : nat), a \in sq -> exists k, lcml sq = k * a.
Proof.
intros xs x IN.
induction xs as [ | z sq IH_DIVIDES]; first by easy.
rewrite in_cons in IN.
move : IN => /orP [/eqP EQ | IN].
+ apply /dvdnP.
rewrite EQ /lcml.
by apply int_divides_lcm_in_seq.
+ move : (IH_DIVIDES IN) => [k EQ].
exists ((foldr lcmn 1 (z :: sq)) %/ (foldr lcmn 1 sq) * k).
rewrite -mulnA -EQ divnK /lcml //.
by apply lcm_seq_divides_lcm_super.
Qed.
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