fully_nonpreemptive.v

From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing]
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[notation-overridden,parsing]
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[notation-overridden,parsing]
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[notation-overridden,parsing]
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[notation-overridden,parsing]
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[notation-overridden,parsing]
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nat_scope. [notation-overridden,parsing]
Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing]

Require Import prosa.model.readiness.basic.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.results.edf.rta.bounded_nps.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.analysis.facts.preemption.task.nonpreemptive.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.analysis.facts.preemption.rtc_threshold.nonpreemptive.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.analysis.facts.readiness.basic.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.model.task.preemption.fully_nonpreemptive.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Import prosa.model.priority.edf.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]

(** * RTA for Fully Non-Preemptive EDF *)
(** In this module we prove the RTA theorem for the fully non-preemptive EDF model. *)

(** ** Setup and Assumptions *)

Section RTAforFullyNonPreemptiveEDFModelwithArrivalCurves.
  
  (** 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.

  (** We assume that jobs and tasks are fully nonpreemptive. *)
  #[local] Existing Instance fully_nonpreemptive_job_model.
  #[local] Existing Instance fully_nonpreemptive_task_model.
  #[local] Existing Instance fully_nonpreemptive_rtc_threshold.

  (** Consider any arrival sequence with consistent, non-duplicate arrivals. *)
  Variable arr_seq : arrival_sequence Job.
  Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
  Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq 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 non-preemptive uniprocessor schedule of this arrival sequence ... *)
  Variable sched : schedule (ideal.processor_state Job).
  Hypothesis H_sched_valid: valid_schedule sched arr_seq.
  Hypothesis H_nonpreemptive_sched : nonpreemptive_schedule sched.

  (** 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 policy defined by the
      job_preemptable function (i.e., jobs have bounded nonpreemptive
      segments). *)
  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.
  
  (** We also define a bound for the priority inversion caused by jobs with lower priority. *)
  Let blocking_bound A :=
    \max_(tsk_o <- ts | (blocking_relevant tsk_o)
                         && (task_deadline tsk_o > task_deadline tsk + A))
     (task_cost tsk_o - ε).
  
  (** Next, we define an upper bound on interfering workload received from jobs 
       of other tasks with higher-than-or-equal priority. *)
  Let bound_on_total_hep_workload A Δ :=
    \sum_(tsk_o <- ts | tsk_o != tsk)
     rbf tsk_o (minn ((A + ε) + task_deadline tsk - task_deadline tsk_o) Δ).
  
  (** 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: nat.
  Hypothesis H_R_is_maximum:
    forall A,
      is_in_search_space A -> 
      exists F,
        A + F >= blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - ε))
                + bound_on_total_hep_workload A (A + F) /\
        R >= F + (task_cost tsk - ε).

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

  Let response_time_bounded_by := task_response_time_bound arr_seq sched.

  Theorem uniprocessor_response_time_bound_fully_nonpreemptive_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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= 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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
response_time_bounded_by tsk R
    case: (posnP (task_cost tsk)) => [ZERO|POS].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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
response_time_bounded_by tsk R intros j ARR TSK.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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
job_response_time_bound sched j R
      have ZEROj: job_cost j = 0.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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
job_cost j = 0
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
ZEROj: job_cost j = 0
job_response_time_bound sched j 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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
job_cost j = 0 move: (H_valid_job_cost j ARR) => NEQ.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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
NEQ: valid_job_cost j
job_cost j = 0
        rewrite /valid_job_cost in NEQ.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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
NEQ: job_cost j <= task_cost (job_task j)
job_cost j = 0
        move: TSK => /eqP -> in NEQ.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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
NEQ: job_cost j <= task_cost tsk
job_cost j = 0
        rewrite ZERO in NEQ.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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
NEQ: job_cost j <= 0
job_cost j = 0
        by apply/eqP; rewrite -leqn0.
      }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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
ZEROj: job_cost j = 0
job_response_time_bound sched j R
      by rewrite /job_response_time_bound /completed_by ZEROj.
    }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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
response_time_bounded_by tsk R
    try ( eapply uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments with (L0 := L) ) ||
    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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
consistent_arrival_times arr_seq
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
arrival_sequence_uniq arr_seq
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
valid_schedule sched arr_seq
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
work_conserving arr_seq sched
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
respects_JLFP_policy_at_preemption_point arr_seq sched
  (EDF Job)
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
all_jobs_from_taskset arr_seq ?ts
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
arrivals_have_valid_job_costs arr_seq
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
valid_taskset_arrival_curve ?ts max_arrivals
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
taskset_respects_max_arrivals arr_seq ?ts
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
tsk \in ?ts
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
valid_preemption_model arr_seq sched
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
valid_task_run_to_completion_threshold arr_seq tsk
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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
0 < 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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
L = total_request_bound_function ?ts 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_arrival_times_are_consistent: consistent_arrival_times
  arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq 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_nonpreemptive_sched: nonpreemptive_schedule sched
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 A : nat =>
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(task_deadline tsk + A <
 task_deadline tsk_o))
   (task_cost tsk_o - ε): nat -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε +
         task_deadline
           tsk -
         task_deadline
           tsk_o) Δ): nat -> nat -> 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: nat
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : nat,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - ε)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
forall A : duration,
bounded_nps.is_in_search_space ?ts tsk L A ->
exists F : duration,
  bounded_nps.blocking_bound ?ts tsk A +
  (task_request_bound_function tsk (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  \sum_(tsk_o <- 
  ?ts | tsk_o != tsk)
     task_request_bound_function tsk_o
       (minn
          (A + ε + task_deadline tsk -
           task_deadline tsk_o) 
          (A + F)) <= 
  A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
    all: rt_eauto.
  Qed.
  
End RTAforFullyNonPreemptiveEDFModelwithArrivalCurves.