# Library rt.implementation.uni.susp.dynamic.job

Require Import rt.util.all.
From mathcomp Require Import ssreflect ssrbool ssrnat eqtype seq.

Module ConcreteJob.

Section Defs.

(* Definition of a concrete task. *)
Record concrete_job :=
{
job_id: nat;
job_arrival: nat;
job_cost: nat;
}.

(* To make it compatible with ssreflect, we define a decidable
equality for concrete jobs. *)

Definition job_eqdef (j1 j2: concrete_job) :=
(job_id j1 == job_id j2) &&
(job_arrival j1 == job_arrival j2) &&
(job_cost j1 == job_cost j2) &&

(* Next, we prove that job_eqdef is indeed an equality, ... *)
Lemma eqn_job : Equality.axiom job_eqdef.
Proof.
unfold Equality.axiom; intros x y.
destruct (job_eqdef x y) eqn:EQ.
{
apply ReflectT; unfold job_eqdef in ×.
move: EQ ⇒ /andP [/andP [/andP [/andP [/eqP ID /eqP ARR] /eqP COST] /eqP DL] /eqP TASK].
by destruct x, y; simpl in *; subst.
}
{
apply ReflectF.
unfold job_eqdef, not in *; intro BUG.
apply negbT in EQ; rewrite negb_and in EQ.
destruct x, y.
rewrite negb_and in EQ.
move: EQ ⇒ /orP [EQ | /eqP TASK].
move: EQ ⇒ /orP [EQ | /eqP DL].
rewrite negb_and in EQ.
move: EQ ⇒ /orP [EQ | /eqP COST].
rewrite negb_and in EQ.
move: EQ ⇒ /orP [/eqP ID | /eqP ARR].
by apply ID; inversion BUG.
by apply ARR; inversion BUG.
by apply COST; inversion BUG.
by apply DL; inversion BUG.