Require Import prosa.model.readiness.basic.
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Require Import prosa.results.edf.rta.bounded_nps.
Require Export prosa.analysis.facts.preemption.task.preemptive.
Require Export prosa.analysis.facts.preemption.rtc_threshold.preemptive.
Require Export prosa.analysis.facts.readiness.basic.
Require Import prosa.model.task.preemption.fully_preemptive.
Require Import prosa.model.priority.edf.

(** * RTA for Fully Preemptive EDF *)
(** In this section we prove the RTA theorem for the fully preemptive EDF model *)

(** ** Setup and Assumptions *)

Section RTAforFullyPreemptiveEDFModelwithArrivalCurves.

  (** We assume that jobs and tasks are fully preemptive. *)
  #[local] Existing Instance fully_preemptive_job_model.
  #[local] Existing Instance fully_preemptive_task_model.
  #[local] Existing Instance fully_preemptive_rtc_threshold.

  (** Consider any type of tasks ... *)
  Context {Task : TaskType}.
  Context `{TaskCost Task}.
  Context `{TaskDeadline Task}.

  (**  ... and any type of jobs associated with these tasks. *)
  Context {Job : JobType}.
  Context `{JobTask Job Task}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

  (** We assume the classic (i.e., Liu & Layland) model of readiness
      without jitter or self-suspensions, wherein pending jobs are
      always ready. *)
  #[local] Existing Instance basic_ready_instance.

  (** Consider any arrival sequence with consistent, non-duplicate arrivals. *)
  Variable arr_seq : arrival_sequence Job.
  Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.

  (** Consider an arbitrary task set ts, ... *)
  Variable ts : list Task.

  (** ... assume that all jobs come from this task set, ... *)
  Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.

  (** ... and the cost of a job cannot be larger than the task cost. *)
  Hypothesis H_valid_job_cost:
    arrivals_have_valid_job_costs arr_seq.

  (** Let max_arrivals be a family of valid arrival curves, i.e., for
     any task [tsk] in ts [max_arrival tsk] is (1) an arrival bound of
     [tsk], and (2) it is a monotonic function that equals 0 for the
     empty interval delta = 0. *)
  Context `{MaxArrivals Task}.
  Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.
  Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.

  (** Let [tsk] be any task in ts that is to be analyzed. *)
  Variable tsk : Task.
  Hypothesis H_tsk_in_ts : tsk \in ts.

  (** Next, consider any valid ideal uniprocessor schedule of the arrival sequence ... *)
  Variable sched : schedule (ideal.processor_state Job).
  Hypothesis H_sched_valid: valid_schedule sched arr_seq.
  Hypothesis H_jobs_come_from_arrival_sequence:
    jobs_come_from_arrival_sequence sched arr_seq.

  (** Next, we assume that the schedule is a work-conserving schedule... *)
  Hypothesis H_work_conserving : work_conserving arr_seq sched.

  (** ... and the schedule respects the scheduling policy. *)
  Hypothesis H_respects_policy : respects_JLFP_policy_at_preemption_point arr_seq sched (EDF Job).

  (** ** Total Workload and Length of Busy Interval *)

  (** We introduce the abbreviation [rbf] for the task request bound function,
      which is defined as [task_cost(T) × max_arrivals(T,Δ)] for a task T. *)
  Let rbf := task_request_bound_function.

  (** Next, we introduce [task_rbf] as an abbreviation
      for the task request bound function of task [tsk]. *)
  Let task_rbf := rbf tsk.

  (** Using the sum of individual request bound functions, we define the request bound
      function of all tasks (total request bound function). *)
  Let total_rbf := total_request_bound_function ts.

  (** If jobs are fully preemptive, lower priority jobs do not cause priority inversion.
      Hence, the blocking bound is always 0 for any [A]. *)
  Let blocking_bound (A : duration) := 0.

  (** Let L be any positive fixed point of the busy interval recurrence. *)
  Variable L : duration.
  Hypothesis H_L_positive : L > 0.
  Hypothesis H_fixed_point : L = total_rbf L.

  (** ** Response-Time Bound *)

  (** To reduce the time complexity of the analysis, recall the notion of search space. *)
  Let is_in_search_space := bounded_nps.is_in_search_space ts tsk L.

  (** Consider any value [R], and assume that for any given arrival
      offset [A] in the search space, there is a solution of the
      response-time bound recurrence which is bounded by [R]. *)
  Variable R : duration.
  Hypothesis H_R_is_maximum:
    forall (A : duration),
      is_in_search_space A ->
      exists (F : duration),
        A + F >= task_rbf (A + ε) + bound_on_athep_workload ts tsk A (A + F) /\
        R >= F.

  (** Now, we can leverage the results for the abstract model with
      bounded non-preemptive segments to establish a response-time
      bound for the more concrete model of fully preemptive
      scheduling. *)

  Let response_time_bounded_by := task_response_time_bound arr_seq sched.

  Theorem uniprocessor_response_time_bound_fully_preemptive_edf:
    response_time_bounded_by tsk R.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
Job: JobType
H1: JobTask Job Task
H2: JobArrival Job
H3: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H4: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (EDF Job)
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
blocking_bound:= fun=> 0: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
is_in_search_space:= bounded_nps.is_in_search_space ts
  tsk L: duration -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  task_rbf (A + 1) +
  bound_on_athep_workload ts tsk A
    (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
response_time_bounded_by tsk R
  Proof.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
Job: JobType
H1: JobTask Job Task
H2: JobArrival Job
H3: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H4: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (EDF Job)
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
blocking_bound:= fun=> 0: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
is_in_search_space:= bounded_nps.is_in_search_space ts
  tsk L: duration -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  task_rbf (A + 1) +
  bound_on_athep_workload ts tsk A
    (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
response_time_bounded_by tsk R
    eapply uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments with (L:=L) => //.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
Job: JobType
H1: JobTask Job Task
H2: JobArrival Job
H3: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H4: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (EDF Job)
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
blocking_bound:= fun=> 0: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
is_in_search_space:= bounded_nps.is_in_search_space ts
  tsk L: duration -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  task_rbf (A + 1) +
  bound_on_athep_workload ts tsk A
    (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
forall A : duration,
bounded_nps.is_in_search_space ts tsk L A ->
exists F : duration,
  edf.blocking_bound ts tsk A +
  (task_request_bound_function tsk (A + 1) -
   (task_cost tsk - task_rtct tsk)) +
  bound_on_athep_workload ts tsk A (A + F) <= A + F /\
  F + (task_cost tsk - task_rtct tsk) <= R
    move => A /andP [LT CHANGE].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
Job: JobType
H1: JobTask Job Task
H2: JobArrival Job
H3: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H4: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (EDF Job)
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
blocking_bound:= fun=> 0: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
is_in_search_space:= bounded_nps.is_in_search_space ts
  tsk L: duration -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  task_rbf (A + 1) +
  bound_on_athep_workload ts tsk A
    (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A: duration
LT: A < L
CHANGE: task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts
     tsk A
exists F : duration,
  edf.blocking_bound ts tsk A +
  (task_request_bound_function tsk (A + 1) -
   (task_cost tsk - task_rtct tsk)) +
  bound_on_athep_workload ts tsk A (A + F) <= A + F /\
  F + (task_cost tsk - task_rtct tsk) <= R
    have BLOCK: forall A', edf.blocking_bound ts tsk A' = blocking_bound A'.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
Job: JobType
H1: JobTask Job Task
H2: JobArrival Job
H3: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H4: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (EDF Job)
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
blocking_bound:= fun=> 0: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
is_in_search_space:= bounded_nps.is_in_search_space ts
  tsk L: duration -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  task_rbf (A + 1) +
  bound_on_athep_workload ts tsk A
    (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A: duration
LT: A < L
CHANGE: task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts
     tsk A
forall A' : duration,
edf.blocking_bound ts tsk A' = blocking_bound A'
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
Job: JobType
H1: JobTask Job Task
H2: JobArrival Job
H3: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H4: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (EDF Job)
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
blocking_bound:= fun=> 0: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
is_in_search_space:= bounded_nps.is_in_search_space ts
  tsk L: duration -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  task_rbf (A + 1) +
  bound_on_athep_workload ts tsk A
    (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A: duration
LT: A < L
CHANGE: task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts
     tsk A
BLOCK: forall A' : duration,
edf.blocking_bound ts tsk A' =
blocking_bound A'
exists F : duration,
  edf.blocking_bound ts tsk A +
  (task_request_bound_function tsk (A + 1) -
   (task_cost tsk - task_rtct tsk)) +
  bound_on_athep_workload ts tsk A (A + F) <= 
  A + F /\ 
  F + (task_cost tsk - task_rtct tsk) <= R
    {
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
Job: JobType
H1: JobTask Job Task
H2: JobArrival Job
H3: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H4: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (EDF Job)
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
blocking_bound:= fun=> 0: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
is_in_search_space:= bounded_nps.is_in_search_space ts
  tsk L: duration -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  task_rbf (A + 1) +
  bound_on_athep_workload ts tsk A
    (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A: duration
LT: A < L
CHANGE: task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts
     tsk A
forall A' : duration,
edf.blocking_bound ts tsk A' = blocking_bound A' by move=> A'; rewrite /edf.blocking_bound /parameters.task_max_nonpreemptive_segment
         /fully_preemptive_task_model subnn big1_eq. }
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
Job: JobType
H1: JobTask Job Task
H2: JobArrival Job
H3: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H4: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (EDF Job)
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
blocking_bound:= fun=> 0: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
is_in_search_space:= bounded_nps.is_in_search_space ts
  tsk L: duration -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  task_rbf (A + 1) +
  bound_on_athep_workload ts tsk A
    (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A: duration
LT: A < L
CHANGE: task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts
     tsk A
BLOCK: forall A' : duration,
edf.blocking_bound ts tsk A' =
blocking_bound A'
exists F : duration,
  edf.blocking_bound ts tsk A +
  (task_request_bound_function tsk (A + 1) -
   (task_cost tsk - task_rtct tsk)) +
  bound_on_athep_workload ts tsk A (A + F) <= A + F /\
  F + (task_cost tsk - task_rtct tsk) <= R
    specialize (H_R_is_maximum A); feed H_R_is_maximum; first by apply/andP; split; done.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
Job: JobType
H1: JobTask Job Task
H2: JobArrival Job
H3: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H4: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (EDF Job)
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
blocking_bound:= fun=> 0: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
is_in_search_space:= bounded_nps.is_in_search_space ts
  tsk L: duration -> bool
R, A: duration
H_R_is_maximum: exists F : duration,
  task_rbf (A + 1) +
  bound_on_athep_workload ts tsk A
    (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LT: A < L
CHANGE: task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts
     tsk A
BLOCK: forall A' : duration,
edf.blocking_bound ts tsk A' =
blocking_bound A'
exists F : duration,
  edf.blocking_bound ts tsk A +
  (task_request_bound_function tsk (A + 1) -
   (task_cost tsk - task_rtct tsk)) +
  bound_on_athep_workload ts tsk A (A + F) <= A + F /\
  F + (task_cost tsk - task_rtct tsk) <= R
    move: H_R_is_maximum => [F [FIX BOUND]].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
Job: JobType
H1: JobTask Job Task
H2: JobArrival Job
H3: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H4: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (EDF Job)
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
blocking_bound:= fun=> 0: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
is_in_search_space:= bounded_nps.is_in_search_space ts
  tsk L: duration -> bool
R, A: duration
H_R_is_maximum: exists F : duration,
  task_rbf (A + 1) +
  bound_on_athep_workload ts tsk A
    (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LT: A < L
CHANGE: task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts
     tsk A
BLOCK: forall A' : duration,
edf.blocking_bound ts tsk A' =
blocking_bound A'
F: duration
FIX: task_rbf (A + 1) +
bound_on_athep_workload ts tsk A (A + F) <=
A + F
BOUND: F <= R
exists F : duration,
  edf.blocking_bound ts tsk A +
  (task_request_bound_function tsk (A + 1) -
   (task_cost tsk - task_rtct tsk)) +
  bound_on_athep_workload ts tsk A (A + F) <= A + F /\
  F + (task_cost tsk - task_rtct tsk) <= R
    exists F; split.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
Job: JobType
H1: JobTask Job Task
H2: JobArrival Job
H3: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H4: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (EDF Job)
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
blocking_bound:= fun=> 0: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
is_in_search_space:= bounded_nps.is_in_search_space ts
  tsk L: duration -> bool
R, A: duration
H_R_is_maximum: exists F : duration,
  task_rbf (A + 1) +
  bound_on_athep_workload ts tsk A
    (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LT: A < L
CHANGE: task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts
     tsk A
BLOCK: forall A' : duration,
edf.blocking_bound ts tsk A' =
blocking_bound A'
F: duration
FIX: task_rbf (A + 1) +
bound_on_athep_workload ts tsk A (A + F) <=
A + F
BOUND: F <= R
edf.blocking_bound ts tsk A +
(task_request_bound_function tsk (A + 1) -
 (task_cost tsk - task_rtct tsk)) +
bound_on_athep_workload ts tsk A (A + F) <= A + F
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
Job: JobType
H1: JobTask Job Task
H2: JobArrival Job
H3: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H4: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (EDF Job)
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
blocking_bound:= fun=> 0: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
is_in_search_space:= bounded_nps.is_in_search_space ts
  tsk L: duration -> bool
R, A: duration
H_R_is_maximum: exists F : duration,
  task_rbf (A + 1) +
  bound_on_athep_workload ts tsk A
    (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LT: A < L
CHANGE: task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts
     tsk A
BLOCK: forall A' : duration,
edf.blocking_bound ts tsk A' =
blocking_bound A'
F: duration
FIX: task_rbf (A + 1) +
bound_on_athep_workload ts tsk A (A + F) <=
A + F
BOUND: F <= R
F + (task_cost tsk - task_rtct tsk) <= R
    +
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
Job: JobType
H1: JobTask Job Task
H2: JobArrival Job
H3: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H4: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (EDF Job)
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
blocking_bound:= fun=> 0: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
is_in_search_space:= bounded_nps.is_in_search_space ts
  tsk L: duration -> bool
R, A: duration
H_R_is_maximum: exists F : duration,
  task_rbf (A + 1) +
  bound_on_athep_workload ts tsk A
    (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LT: A < L
CHANGE: task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts
     tsk A
BLOCK: forall A' : duration,
edf.blocking_bound ts tsk A' =
blocking_bound A'
F: duration
FIX: task_rbf (A + 1) +
bound_on_athep_workload ts tsk A (A + F) <=
A + F
BOUND: F <= R
edf.blocking_bound ts tsk A +
(task_request_bound_function tsk (A + 1) -
 (task_cost tsk - task_rtct tsk)) +
bound_on_athep_workload ts tsk A (A + F) <= A + F by rewrite BLOCK add0n subnn subn0.
    +
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
Job: JobType
H1: JobTask Job Task
H2: JobArrival Job
H3: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H4: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (EDF Job)
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
blocking_bound:= fun=> 0: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
is_in_search_space:= bounded_nps.is_in_search_space ts
  tsk L: duration -> bool
R, A: duration
H_R_is_maximum: exists F : duration,
  task_rbf (A + 1) +
  bound_on_athep_workload ts tsk A
    (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LT: A < L
CHANGE: task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts
     tsk A
BLOCK: forall A' : duration,
edf.blocking_bound ts tsk A' =
blocking_bound A'
F: duration
FIX: task_rbf (A + 1) +
bound_on_athep_workload ts tsk A (A + F) <=
A + F
BOUND: F <= R
F + (task_cost tsk - task_rtct tsk) <= R by rewrite subnn addn0.
  Qed.

End RTAforFullyPreemptiveEDFModelwithArrivalCurves.