coercion.v

Require Export prosa.model.priority.definitions.

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(** ** Automatic FP ➔ JLFP ➔ JLDP Conversion *) (** Since there are natural interpretations of FP and JLFP policies as JLFP and JLDP policies, respectively, we define conversions that express these generalizations. In practice, this means that Coq will be able to automatically satisfy a JLDP assumption if a JLFP or FP policy is in scope. *) (** First, any FP policy can be interpreted as an JLFP policy by comparing jobs according to the priorities of their respective tasks. The priority is lowered to 10 in order to avoid conflict with other instances of JLFP policies which take FP policies as argument.*) #[global] Instance FP_to_JLFP {Job : JobType} {Task : TaskType} {tasks : JobTask Job Task} (FP : FP_policy Task) : JLFP_policy Job | 10 := fun j1 j2 => hep_task (job_task j1) (job_task j2). (** Second, any JLFP policy implies a JLDP policy that simply ignores the time parameter. Analogously, priority is lowered to 10 in order to avoid conflict with other instances of JLDP policies which take JLFP policies as argument.*) #[global] Instance JLFP_to_JLDP {Job : JobType} (JLFP : JLFP_policy Job) : JLDP_policy Job | 10 := fun _ j1 j2 => hep_job j1 j2. (** We add coercions to enable automatic conversion from [JLFP] to [JLDP]... *) Coercion JLFP_to_JLDP : JLFP_policy >-> JLDP_policy. (** ...and from [FP] to [JLFP]. *) #[warning="-uniform-inheritance"] Coercion FP_to_JLFP : FP_policy >-> JLFP_policy. (** We now prove lemmas about conversions between the properties of these priority classes. *) Section PriorityRelationsConversion. (** Consider any type of tasks ... *) Context {Task : TaskType}. (** ... and any type of jobs associated with these tasks. *) Context {Job : JobType}. Context `{JobTask Job Task}. (** To simplify later proofs, we make two trivial remarks on the FP->JLFP->JLDP coercion. *) (** First, when considering a JLFP policy, [hep_job_at] and [hep_job] are by definition equivalent. *) Remark hep_job_at_jlfp : forall `{JLFP_policy Job} j j' t, hep_job_at t j j' = hep_job j j'.

Task: TaskType
Job: JobType
H: JobTask Job Task
forall (H0 : JLFP_policy Job) (j j' : Job) (t : instant), hep_job_at t j j' = hep_job j j'

Proof.

Task: TaskType
Job: JobType
H: JobTask Job Task
forall (H0 : JLFP_policy Job) (j j' : Job) (t : instant), hep_job_at t j j' = hep_job j j'

by move=> ? j j' t; rewrite /hep_job_at/JLFP_to_JLDP. Qed. (** Second, when considering an FP policy, [hep_job_at] and [hep_task] are by definition equivalent. *) Remark hep_job_at_fp : forall `{FP_policy Task} j j' t, hep_job_at t j j' = hep_task (job_task j) (job_task j').

Task: TaskType
Job: JobType
H: JobTask Job Task
forall (H0 : FP_policy Task) (j j' : Job) (t : instant), hep_job_at t j j' = hep_task (job_task j) (job_task j')

Proof.

Task: TaskType
Job: JobType
H: JobTask Job Task
forall (H0 : FP_policy Task) (j j' : Job) (t : instant), hep_job_at t j j' = hep_task (job_task j) (job_task j')

by move=> ? j j' t; rewrite hep_job_at_jlfp/hep_job/FP_to_JLFP. Qed. (** We observe that lifting an [FP_policy] to a [JLFP_policy] trivially maintains reflexivity,... *) Lemma reflexive_priorities_FP_implies_JLFP : forall (fp: FP_policy Task), reflexive_task_priorities fp -> reflexive_job_priorities (FP_to_JLFP fp).

Task: TaskType
Job: JobType
H: JobTask Job Task
forall fp : FP_policy Task, reflexive_task_priorities fp -> reflexive_job_priorities fp

Proof.

Task: TaskType
Job: JobType
H: JobTask Job Task
forall fp : FP_policy Task, reflexive_task_priorities fp -> reflexive_job_priorities fp

by move=> ? refl_fp ?; apply: refl_fp. Qed. (** ...transitivity,... *) Lemma transitive_priorities_FP_implies_JLFP : forall (fp: FP_policy Task), transitive_task_priorities fp -> transitive_job_priorities (FP_to_JLFP fp).

Task: TaskType
Job: JobType
H: JobTask Job Task
forall fp : FP_policy Task, transitive_task_priorities fp -> transitive_job_priorities fp

Proof.

Task: TaskType
Job: JobType
H: JobTask Job Task
forall fp : FP_policy Task, transitive_task_priorities fp -> transitive_job_priorities fp

by move=> ? tran_jlfp ? ? ?; apply: tran_jlfp. Qed. (** ...and totality of priorities. *) Lemma total_priorities_FP_implies_JLFP : forall (fp: FP_policy Task), total_task_priorities fp -> total_job_priorities (FP_to_JLFP fp).

Task: TaskType
Job: JobType
H: JobTask Job Task
forall fp : FP_policy Task, total_task_priorities fp -> total_job_priorities fp

Proof.

Task: TaskType
Job: JobType
H: JobTask Job Task
forall fp : FP_policy Task, total_task_priorities fp -> total_job_priorities fp

by move=> ? tran_fp ? ?; apply: tran_fp. Qed. (** Analogously, lifting a [JLFP_policy] to a [JLDP_policy] also maintains reflexivity,... *) Lemma reflexive_priorities_JLFP_implies_JLDP : forall (jlfp: JLFP_policy Job), reflexive_job_priorities jlfp -> reflexive_priorities (JLFP_to_JLDP jlfp).

Task: TaskType
Job: JobType
H: JobTask Job Task
forall jlfp : JLFP_policy Job, reflexive_job_priorities jlfp -> reflexive_priorities jlfp

Proof.

Task: TaskType
Job: JobType
H: JobTask Job Task
forall jlfp : JLFP_policy Job, reflexive_job_priorities jlfp -> reflexive_priorities jlfp

by move=> ? refl_fp ?; apply: refl_fp. Qed. (** ...transitivity,... *) Lemma transitive_priorities_JLFP_implies_JLDP : forall (jlfp : JLFP_policy Job), transitive_job_priorities jlfp -> transitive_priorities (JLFP_to_JLDP jlfp).

Task: TaskType
Job: JobType
H: JobTask Job Task
forall jlfp : JLFP_policy Job, transitive_job_priorities jlfp -> transitive_priorities jlfp

Proof.

Task: TaskType
Job: JobType
H: JobTask Job Task
forall jlfp : JLFP_policy Job, transitive_job_priorities jlfp -> transitive_priorities jlfp

by move=> ? tran_jlfp ? ? ?; apply: tran_jlfp. Qed. (** ...and totality of priorities. *) Lemma total_priorities_JLFP_implies_JLDP : forall (jlfp : JLFP_policy Job), total_job_priorities jlfp -> total_priorities (JLFP_to_JLDP jlfp).

Task: TaskType
Job: JobType
H: JobTask Job Task
forall jlfp : JLFP_policy Job, total_job_priorities jlfp -> total_priorities jlfp

Proof.

Task: TaskType
Job: JobType
H: JobTask Job Task
forall jlfp : JLFP_policy Job, total_job_priorities jlfp -> total_priorities jlfp

by move=> ? tran_fp ? ?; apply: tran_fp. Qed. End PriorityRelationsConversion. (** We add the above lemmas into the "Hint Database" basic_rt_facts, so Coq will be able to apply them automatically. *) Global Hint Resolve reflexive_priorities_FP_implies_JLFP reflexive_priorities_JLFP_implies_JLDP transitive_priorities_FP_implies_JLFP transitive_priorities_JLFP_implies_JLDP total_priorities_FP_implies_JLFP total_priorities_JLFP_implies_JLDP : basic_rt_facts.