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 "_ + _" 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]
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]
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.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]
Require Import prosa.model.schedule.work_conserving.
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.task.preemption.parameters.
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.model.rbf.
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.model.arrival_curves.
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.model.sequential.
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.busy_interval.ideal.priority_inversion_bounded.
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_pi.
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 EDF  with Bounded Non-Preemptive Segments *)

(** In this section we instantiate the Abstract RTA for EDF-schedulers
    with Bounded Priority Inversion to EDF-schedulers for ideal
    uni-processor model of real-time tasks with arbitrary
    arrival models _and_ bounded non-preemptive segments. *)

(** Recall that Abstract RTA for EDF-schedulers with Bounded Priority
    Inversion does not specify the cause of priority inversion. In
    this section, we prove that the priority inversion caused by
    execution of non-preemptive segments is bounded. Thus the Abstract
    RTA for EDF-schedulers is applicable to this instantiation. *)
Section RTAforEDFwithBoundedNonpreemptiveSegmentsWithArrivalCurves.

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

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

  (** For clarity, let's denote the relative deadline of a task as [D]. *)
  Let D tsk := task_deadline tsk.

  (** Consider the EDF policy that indicates a higher-or-equal priority relation.
     Note that we do not relate the EDF policy with the scheduler. However, we
     define functions for Interference and Interfering Workload that actively use
     the concept of priorities. *)
  Let EDF := EDF Job.

  (** 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.

  (** Next, consider any valid ideal uni-processor schedule of this arrival sequence ... *)
  Variable sched : schedule (ideal.processor_state Job).
  Hypothesis H_sched_valid : valid_schedule sched arr_seq.

  (** In addition, we assume the existence of a function mapping jobs
      to their preemption points ... *)
  Context `{JobPreemptable Job}.

  (** ... and assume that it defines a valid preemption model with
      bounded non-preemptive segments. *)
  Hypothesis H_valid_model_with_bounded_nonpreemptive_segments:
    valid_model_with_bounded_nonpreemptive_segments arr_seq 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 scheduling policy at every preemption point. *)
  Hypothesis H_respects_policy : respects_JLFP_policy_at_preemption_point arr_seq sched EDF.

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

  (** ... assume that all jobs come from the 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.

  (** Consider a valid preemption model... *)
  Hypothesis H_valid_preemption_model:
    valid_preemption_model arr_seq sched.

  (** ...and a valid task run-to-completion threshold function. That
     is, [task_rtct tsk] is (1) no bigger than [tsk]'s cost, (2) for
     any job of task [tsk] [job_rtct] is bounded by [task_rtct]. *)
  Hypothesis H_valid_run_to_completion_threshold:
    valid_task_run_to_completion_threshold arr_seq tsk.

  (** We introduce as an 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.

  (** 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) + D tsk - D tsk_o) Δ).

  (** Let's define some local names for clarity. *)
  Let max_length_of_priority_inversion :=
    max_length_of_priority_inversion arr_seq.
  Let response_time_bounded_by := task_response_time_bound arr_seq sched.

  (** For a job with the relative arrival offset [A] within its busy window, we
      define the following blocking bound. Only other tasks that potentially
      release non-zero-cost jobs are relevant, so we define a predicate to
      exclude pathological cases. *)
  Definition blocking_relevant (tsk_o : Task) :=
    (max_arrivals tsk_o ε > 0) && (task_cost tsk_o > 0).
  Definition blocking_bound (A : duration)  :=
    \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk_o > D tsk + A))
     (task_max_nonpreemptive_segment tsk_o - ε).

  (** ** Search Space *)

  (** If priority inversion is caused exclusively by non-preemptive sections,
      then we do not need to consider the priority-inversion bound in the search
      space. Hence we define the following search space, which refines the more
      general [bounded_pi.is_in_search_space] for our specific setting. *)
  Definition is_in_search_space (L A : duration) :=
    (A < L) && (task_rbf_changes_at tsk A
                || bound_on_total_hep_workload_changes_at ts tsk A).

  (** For the following proof, we exploit the fact that the blocking bound is
      monotonically decreasing in [A], which we note here. *)
  Fact blocking_bound_decreasing :
    forall A1 A2,
      A1 <= A2 ->
      blocking_bound A1 >= blocking_bound A2.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
forall A1 A2 : nat,
A1 <= A2 -> blocking_bound A2 <= blocking_bound A1
  Proof.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
forall A1 A2 : nat,
A1 <= A2 -> blocking_bound A2 <= blocking_bound A1
    move=> A1 A2 LEQ.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A1, A2: nat
LEQ: A1 <= A2
blocking_bound A2 <= blocking_bound A1
    rewrite /blocking_bound.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A1, A2: nat
LEQ: A1 <= A2
\max_(tsk_o <- ts | blocking_relevant tsk_o &&
                    (D tsk + A2 < D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε) <=
\max_(tsk_o <- ts | blocking_relevant tsk_o &&
                    (D tsk + A1 < D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε)
    apply: bigmax_subset => tsk_o IN /andP[/andP[OTHER LT] ARR].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A1, A2: nat
LEQ: A1 <= A2
tsk_o: Task
IN: tsk_o \in ts
OTHER: 0 < max_arrivals tsk_o ε
LT: 0 < task_cost tsk_o
ARR: D tsk + A2 < D tsk_o
blocking_relevant tsk_o && (D tsk + A1 < D tsk_o)
    by repeat (apply /andP; split) => //; lia.
  Qed.

  (** To use the refined search space with the abstract theorem, we must show
      that it still includes all relevant points. To this end, we first observe
      that a step in the blocking bound implies the existence of a task that
      could release a job with an absolute deadline equal to the absolute
      deadline of the job under analysis. *)
  Lemma task_with_equal_deadline_exists :
    forall {A},
      priority_inversion_changes_at blocking_bound A ->
      exists tsk_o, (tsk_o \in ts)
                 && (blocking_relevant tsk_o)
                 && (tsk_o != tsk)
                 && (D tsk_o == D tsk + A).
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
forall A : duration,
priority_inversion_changes_at blocking_bound A ->
exists tsk_o : Task,
  (tsk_o \in ts) && blocking_relevant tsk_o &&
  (tsk_o != tsk) && (D tsk_o == D tsk + A)
  Proof.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
forall A : duration,
priority_inversion_changes_at blocking_bound A ->
exists tsk_o : Task,
  (tsk_o \in ts) && blocking_relevant tsk_o &&
  (tsk_o != tsk) && (D tsk_o == D tsk + A)
    move=> A.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A: duration
priority_inversion_changes_at blocking_bound A ->
exists tsk_o : Task,
  (tsk_o \in ts) && blocking_relevant tsk_o &&
  (tsk_o != tsk) && (D tsk_o == D tsk + A) rewrite /priority_inversion_changes_at => NEQ.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A: duration
NEQ: blocking_bound (A - ε) != blocking_bound A
exists tsk_o : Task,
  (tsk_o \in ts) && blocking_relevant tsk_o &&
  (tsk_o != tsk) && (D tsk_o == D tsk + A)
    have LEQ: blocking_bound A <= blocking_bound (A - ε) by apply: blocking_bound_decreasing; lia.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A: duration
NEQ: blocking_bound (A - ε) != blocking_bound A
LEQ: blocking_bound A <= blocking_bound (A - ε)
exists tsk_o : Task,
  (tsk_o \in ts) && blocking_relevant tsk_o &&
  (tsk_o != tsk) && (D tsk_o == D tsk + A)
    have LT: blocking_bound A < blocking_bound (A - ε) by lia.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A: duration
NEQ: blocking_bound (A - ε) != blocking_bound A
LEQ: blocking_bound A <= blocking_bound (A - ε)
LT: blocking_bound A < blocking_bound (A - ε)
exists tsk_o : Task,
  (tsk_o \in ts) && blocking_relevant tsk_o &&
  (tsk_o != tsk) && (D tsk_o == D tsk + A)
    move: LT; rewrite /blocking_bound => LT {LEQ} {NEQ}.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A: duration
LT: \max_(tsk_o <- ts | 
blocking_relevant tsk_o && 
(D tsk + A < D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε) <
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(D tsk + (A - ε) < D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε)
exists tsk_o : Task,
  (tsk_o \in ts) && blocking_relevant tsk_o &&
  (tsk_o != tsk) && (D tsk_o == D tsk + A)
    move: (bigmax_witness_diff LT) => [tsk_o [IN [NOT HOLDS]]].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A: duration
LT: \max_(tsk_o <- ts | 
blocking_relevant tsk_o && 
(D tsk + A < D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε) <
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(D tsk + (A - ε) < D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε)
tsk_o: Task
IN: tsk_o \in ts
NOT: ~~
(blocking_relevant tsk_o &&
 (D tsk + A < D tsk_o))
HOLDS: blocking_relevant tsk_o &&
(D tsk + (A - ε) < D tsk_o)
exists tsk_o : Task,
  (tsk_o \in ts) && blocking_relevant tsk_o &&
  (tsk_o != tsk) && (D tsk_o == D tsk + A)
    move: HOLDS => /andP[REL LTeps].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A: duration
LT: \max_(tsk_o <- ts | 
blocking_relevant tsk_o && 
(D tsk + A < D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε) <
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(D tsk + (A - ε) < D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε)
tsk_o: Task
IN: tsk_o \in ts
NOT: ~~
(blocking_relevant tsk_o &&
 (D tsk + A < D tsk_o))
REL: blocking_relevant tsk_o
LTeps: D tsk + (A - ε) < D tsk_o
exists tsk_o : Task,
  (tsk_o \in ts) && blocking_relevant tsk_o &&
  (tsk_o != tsk) && (D tsk_o == D tsk + A)
    exists tsk_o; repeat (apply /andP; split) => //;
      first by apply /eqP => EQ; move: LTeps; rewrite EQ; lia.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A: duration
LT: \max_(tsk_o <- ts | 
blocking_relevant tsk_o && 
(D tsk + A < D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε) <
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(D tsk + (A - ε) < D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε)
tsk_o: Task
IN: tsk_o \in ts
NOT: ~~
(blocking_relevant tsk_o &&
 (D tsk + A < D tsk_o))
REL: blocking_relevant tsk_o
LTeps: D tsk + (A - ε) < D tsk_o
D tsk_o == D tsk + A
    move: NOT; rewrite negb_and => /orP[/negP // |].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A: duration
LT: \max_(tsk_o <- ts | 
blocking_relevant tsk_o && 
(D tsk + A < D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε) <
\max_(tsk_o <- ts | 
blocking_relevant tsk_o &&
(D tsk + (A - ε) < D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε)
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
LTeps: D tsk + (A - ε) < D tsk_o
~~ (D tsk + A < D tsk_o) -> D tsk_o == D tsk + A
    by move: LTeps; rewrite /ε => LTeps; lia.
  Qed.

  (** With the above setup in place, we can show that the search space defined
      above by [is_in_search_space] covers the the more abstract search space
      defined by [bounded_pi.is_in_search_space]. *)
  Lemma search_space_inclusion :
     forall {A L},
       bounded_pi.is_in_search_space ts tsk blocking_bound L A ->
       is_in_search_space L A.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
forall A L : duration,
bounded_pi.is_in_search_space ts tsk blocking_bound L
  A -> is_in_search_space L A
   Proof.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
forall A L : duration,
bounded_pi.is_in_search_space ts tsk blocking_bound L
  A -> is_in_search_space L A
     move=> A L /andP[BOUND STEP].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
|| task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts
     tsk A
is_in_search_space L A
     apply /andP; split => //; apply /orP.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
|| task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts
     tsk A
task_rbf_changes_at tsk A \/
bound_on_total_hep_workload_changes_at ts tsk A
     move: STEP => /orP[/orP[STEP|RBF] | IBF]; [right| by left| by right].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
bound_on_total_hep_workload_changes_at ts tsk A
     move: (task_with_equal_deadline_exists STEP) => [tsk_o /andP[/andP[/andP[IN REL] OTHER] EQ]].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
OTHER: tsk_o != tsk
EQ: D tsk_o == D tsk + A
bound_on_total_hep_workload_changes_at ts tsk A
     rewrite /bound_on_total_hep_workload_changes_at.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
OTHER: tsk_o != tsk
EQ: D tsk_o == D tsk + A
has
  (fun tsko : Task =>
   (tsk != tsko) &&
   (task_request_bound_function tsko
      (A + task_deadline tsk - task_deadline tsko)
    != task_request_bound_function tsko
         (A + ε + task_deadline tsk -
          task_deadline tsko))) ts
     apply /hasP; exists tsk_o => //.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
OTHER: tsk_o != tsk
EQ: D tsk_o == D tsk + A
(tsk != tsk_o) &&
(task_request_bound_function tsk_o
   (A + task_deadline tsk - task_deadline tsk_o)
 != task_request_bound_function tsk_o
      (A + ε + task_deadline tsk - task_deadline tsk_o))
     apply /andP; split; first by rewrite eq_sym.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
OTHER: tsk_o != tsk
EQ: D tsk_o == D tsk + A
task_request_bound_function tsk_o
  (A + task_deadline tsk - task_deadline tsk_o)
!= task_request_bound_function tsk_o
     (A + ε + task_deadline tsk - task_deadline tsk_o)
     move: EQ.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
OTHER: tsk_o != tsk
D tsk_o == D tsk + A ->
task_request_bound_function tsk_o
  (A + task_deadline tsk - task_deadline tsk_o)
!= task_request_bound_function tsk_o
     (A + ε + task_deadline tsk - task_deadline tsk_o) rewrite /D => /eqP EQ.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
OTHER: tsk_o != tsk
EQ: task_deadline tsk_o = task_deadline tsk + A
task_request_bound_function tsk_o
  (A + task_deadline tsk - task_deadline tsk_o)
!= task_request_bound_function tsk_o
     (A + ε + task_deadline tsk - task_deadline tsk_o)
     rewrite /task_request_bound_function EQ.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
OTHER: tsk_o != tsk
EQ: task_deadline tsk_o = task_deadline tsk + A
task_cost tsk_o *
max_arrivals tsk_o
  (A + task_deadline tsk - (task_deadline tsk + A))
!= task_cost tsk_o *
   max_arrivals tsk_o
     (A + ε + task_deadline tsk -
      (task_deadline tsk + A))
     move: REL; rewrite /blocking_relevant => /andP [ARRIVES COST].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
OTHER: tsk_o != tsk
EQ: task_deadline tsk_o = task_deadline tsk + A
ARRIVES: 0 < max_arrivals tsk_o ε
COST: 0 < task_cost tsk_o
task_cost tsk_o *
max_arrivals tsk_o
  (A + task_deadline tsk - (task_deadline tsk + A))
!= task_cost tsk_o *
   max_arrivals tsk_o
     (A + ε + task_deadline tsk -
      (task_deadline tsk + A))
     rewrite eqn_pmul2l //.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
OTHER: tsk_o != tsk
EQ: task_deadline tsk_o = task_deadline tsk + A
ARRIVES: 0 < max_arrivals tsk_o ε
COST: 0 < task_cost tsk_o
max_arrivals tsk_o
  (A + task_deadline tsk - (task_deadline tsk + A))
!= max_arrivals tsk_o
     (A + ε + task_deadline tsk -
      (task_deadline tsk + A))
     have -> : A + task_deadline tsk - (task_deadline tsk + A)
              = 0 by lia.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
OTHER: tsk_o != tsk
EQ: task_deadline tsk_o = task_deadline tsk + A
ARRIVES: 0 < max_arrivals tsk_o ε
COST: 0 < task_cost tsk_o
max_arrivals tsk_o 0
!= max_arrivals tsk_o
     (A + ε + task_deadline tsk -
      (task_deadline tsk + A))
     have -> : A + ε + task_deadline tsk - (task_deadline tsk + A)
              = ε by lia.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
OTHER: tsk_o != tsk
EQ: task_deadline tsk_o = task_deadline tsk + A
ARRIVES: 0 < max_arrivals tsk_o ε
COST: 0 < task_cost tsk_o
max_arrivals tsk_o 0 != max_arrivals tsk_o ε
     by move: (H_valid_arrival_curve tsk_o IN) => [-> _]; lia.
   Qed.


  (** ** Priority inversion is bounded *)
  (** In this section, we prove that a priority inversion for task [tsk] is bounded by
      the maximum length of non-preemptive segments among the tasks with lower priority. *)
  Section PriorityInversionIsBounded.

    (** First, we observe that the maximum non-preemptive segment length of any
        task that releases a job with an earlier absolute deadline (w.r.t. a
        given job [j]) and non-zero execution cost upper-bounds the maximum
        possible length of priority inversion (of said job [j]).  *)
    Lemma priority_inversion_is_bounded_by_max_np_segment :
      forall {j t1},
        max_length_of_priority_inversion j t1
        <= \max_(j_lp <- arrivals_between arr_seq 0 t1 | (~~ EDF j_lp j)
                                                         && (job_cost j_lp > 0))
           (task_max_nonpreemptive_segment (job_task j_lp) - ε).
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
forall (j : Job) (t1 : instant),
max_length_of_priority_inversion j t1 <=
\max_(j_lp <- arrivals_between arr_seq 0 t1 | ~~
                                              EDF j_lp
                                                j &&
                                              (0 <
                                               job_cost
                                                 j_lp))
   (task_max_nonpreemptive_segment (job_task j_lp) - ε)
    Proof.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
forall (j : Job) (t1 : instant),
max_length_of_priority_inversion j t1 <=
\max_(j_lp <- arrivals_between arr_seq 0 t1 | ~~
                                              EDF j_lp
                                                j &&
                                              (0 <
                                               job_cost
                                                 j_lp))
   (task_max_nonpreemptive_segment (job_task j_lp) - ε)
      move=> j t1.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1: instant
max_length_of_priority_inversion j t1 <=
\max_(j_lp <- arrivals_between arr_seq 0 t1 | ~~
                                              EDF j_lp
                                                j &&
                                              (0 <
                                               job_cost
                                                 j_lp))
   (task_max_nonpreemptive_segment (job_task j_lp) - ε)
      rewrite /max_length_of_priority_inversion /max_length_of_priority_inversion.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1: instant
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1 <=
\max_(j_lp <- arrivals_between arr_seq 0 t1 | ~~
                                              EDF j_lp
                                                j &&
                                              (0 <
                                               job_cost
                                                 j_lp))
   (task_max_nonpreemptive_segment (job_task j_lp) - ε)
      apply: leq_big_max => j' JINB NOTHEP.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1: instant
j': Job
JINB: j' \in arrivals_before arr_seq t1
NOTHEP: ~~ hep_job j' j && (0 < job_cost j')
job_max_nonpreemptive_segment j' - ε <=
task_max_nonpreemptive_segment (job_task j') - ε
      rewrite leq_sub2r //.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1: instant
j': Job
JINB: j' \in arrivals_before arr_seq t1
NOTHEP: ~~ hep_job j' j && (0 < job_cost j')
job_max_nonpreemptive_segment j' <=
task_max_nonpreemptive_segment (job_task j')
      apply in_arrivals_implies_arrived in JINB.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1: instant
j': Job
JINB: arrives_in arr_seq j'
NOTHEP: ~~ hep_job j' j && (0 < job_cost j')
job_max_nonpreemptive_segment j' <=
task_max_nonpreemptive_segment (job_task j')
      by apply H_valid_model_with_bounded_nonpreemptive_segments.
    Qed.

    (** Second, we prove that the maximum length of a priority inversion of a
        given job [j] is indeed bounded by defined the blocking bound. *)
    Lemma priority_inversion_is_bounded_by_blocking:
      forall j t1 t2,
        arrives_in arr_seq j ->
        job_of_task tsk j ->
        busy_interval_prefix  arr_seq sched j t1 t2 ->
        max_length_of_priority_inversion j t1 <= blocking_bound (job_arrival j - t1).
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
forall (j : Job) (t1 t2 : instant),
arrives_in arr_seq j ->
job_of_task tsk j ->
busy_interval_prefix arr_seq sched j t1 t2 ->
max_length_of_priority_inversion j t1 <=
blocking_bound (job_arrival j - t1)
    Proof.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
forall (j : Job) (t1 t2 : instant),
arrives_in arr_seq j ->
job_of_task tsk j ->
busy_interval_prefix arr_seq sched j t1 t2 ->
max_length_of_priority_inversion j t1 <=
blocking_bound (job_arrival j - t1)
      intros j t1 t2 ARR TSK BUSY; unfold max_length_of_priority_inversion, blocking_bound.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval_prefix arr_seq sched j t1 t2
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1 <=
\max_(tsk_o <- ts | blocking_relevant tsk_o &&
                    (D tsk + (job_arrival j - t1) <
                     D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε)
      destruct BUSY as [TT [QT [_ LE]]]; move: LE => /andP [GE LT].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1 <=
\max_(tsk_o <- ts | blocking_relevant tsk_o &&
                    (D tsk + (job_arrival j - t1) <
                     D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε)
      apply: leq_trans; first by apply: priority_inversion_is_bounded_by_max_np_segment.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
\max_(j_lp <- arrivals_between arr_seq 0 t1 | ~~
                                              EDF j_lp
                                                j &&
                                              (0 <
                                               job_cost
                                                 j_lp))
   (task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_o <- ts | blocking_relevant tsk_o &&
                    (D tsk + (job_arrival j - t1) <
                     D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε)
      apply /bigmax_leq_seqP => j' JINB NOTHEP.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
JINB: j' \in arrivals_between arr_seq 0 t1
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
task_max_nonpreemptive_segment (job_task j') - ε <=
\max_(tsk_o <- ts | blocking_relevant tsk_o &&
                    (D tsk + (job_arrival j - t1) <
                     D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε)
      have ARR': arrives_in arr_seq j'
        by apply: in_arrivals_implies_arrived; exact: JINB.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
JINB: j' \in arrivals_between arr_seq 0 t1
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
task_max_nonpreemptive_segment (job_task j') - ε <=
\max_(tsk_o <- ts | blocking_relevant tsk_o &&
                    (D tsk + (job_arrival j - t1) <
                     D tsk_o))
   (task_max_nonpreemptive_segment tsk_o - ε)
      apply leq_bigmax_cond_seq with (x := (job_task j')) (F := fun tsk => task_max_nonpreemptive_segment tsk - 1);
        first by apply H_all_jobs_from_taskset.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
JINB: j' \in arrivals_between arr_seq 0 t1
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
blocking_relevant (job_task j') &&
(D tsk + (job_arrival j - t1) < D (job_task j'))
      eapply in_arrivals_implies_arrived_between in JINB; last by rt_eauto.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
JINB: arrived_between j' 0 t1
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
blocking_relevant (job_task j') &&
(D tsk + (job_arrival j - t1) < D (job_task j'))
      move: JINB; move => /andP [_ TJ'].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
blocking_relevant (job_task j') &&
(D tsk + (job_arrival j - t1) < D (job_task j'))
      repeat (apply/andP; split); last first.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
D tsk + (job_arrival j - t1) < D (job_task j')
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
0 < task_cost (job_task j')
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
0 < max_arrivals (job_task j') ε
      {
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
D tsk + (job_arrival j - t1) < D (job_task j') rewrite /EDF -ltnNge in NOTHEP.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
NOTHEP: (job_deadline j < job_deadline j') &&
(0 < job_cost j')
D tsk + (job_arrival j - t1) < D (job_task j')
        move: TSK => /eqP <-.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
NOTHEP: (job_deadline j < job_deadline j') &&
(0 < job_cost j')
D (job_task j) + (job_arrival j - t1) <
D (job_task j')
        have ARRLE: job_arrival j' < job_arrival j by apply leq_trans with t1.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
NOTHEP: (job_deadline j < job_deadline j') &&
(0 < job_cost j')
ARRLE: job_arrival j' < job_arrival j
D (job_task j) + (job_arrival j - t1) <
D (job_task j')
        move: NOTHEP; rewrite /job_deadline /absolute_deadline.job_deadline_from_task_deadline /D.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
ARRLE: job_arrival j' < job_arrival j
(job_arrival j + task_deadline (job_task j) <
 job_arrival j' + task_deadline (job_task j')) &&
(0 < job_cost j') ->
task_deadline (job_task j) + (job_arrival j - t1) <
task_deadline (job_task j')
        by lia. }
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
0 < task_cost (job_task j')
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
0 < max_arrivals (job_task j') ε
      {
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
0 < task_cost (job_task j') move: NOTHEP => /andP [_ NZ].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
NZ: 0 < job_cost j'
0 < task_cost (job_task j')
        move: (H_valid_job_cost j' ARR'); rewrite /valid_job_cost.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
NZ: 0 < job_cost j'
job_cost j' <= task_cost (job_task j') ->
0 < task_cost (job_task j')
        by lia. }
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
0 < max_arrivals (job_task j') ε
      {
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
0 < max_arrivals (job_task j') ε apply: non_pathological_max_arrivals; last first.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
arrives_in ?Goal ?Goal1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
job_of_task (job_task j') ?Goal1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
respects_max_arrivals 
  ?Goal (job_task j') 
  (max_arrivals (job_task j'))
          -
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
arrives_in ?Goal ?Goal1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
job_of_task (job_task j') ?Goal1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
respects_max_arrivals 
  ?Goal (job_task j') 
  (max_arrivals (job_task j')) exact: ARR'.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
job_of_task (job_task j') j'
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
respects_max_arrivals arr_seq 
  (job_task j') 
  (max_arrivals (job_task j'))
          -
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
job_of_task (job_task j') j'
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
respects_max_arrivals arr_seq 
  (job_task j') 
  (max_arrivals (job_task j')) by rewrite /job_of_task.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
respects_max_arrivals arr_seq (job_task j')
  (max_arrivals (job_task j'))
          -
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
respects_max_arrivals arr_seq (job_task j')
  (max_arrivals (job_task j')) by apply H_is_arrival_curve, H_all_jobs_from_taskset, ARR'. }
    Qed.

    (** Using the lemma above, we prove that the priority inversion of the task is bounded by
       the maximum length of a nonpreemptive section of lower-priority tasks. *)
    Lemma priority_inversion_is_bounded:
      priority_inversion_is_bounded_by arr_seq sched tsk blocking_bound.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
priority_inversion_is_bounded_by arr_seq sched tsk
  blocking_bound
    Proof.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
priority_inversion_is_bounded_by arr_seq sched tsk
  blocking_bound
      move => j ARR TSK POS t1 t2 PREF; move: (PREF) => [_ [_ [_ /andP [T _]]]].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
cumulative_priority_inversion arr_seq sched j t1 t2 <=
blocking_bound (job_arrival j - t1)
      move: H_sched_valid => [COARR MBR].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
cumulative_priority_inversion arr_seq sched j t1 t2 <=
blocking_bound (job_arrival j - t1)
      destruct (leqP (t2 - t1) (blocking_bound (job_arrival j - t1))) as [NEQ|NEQ].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: t2 - t1 <= blocking_bound (job_arrival j - t1)
cumulative_priority_inversion arr_seq sched j t1 t2 <=
blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
cumulative_priority_inversion arr_seq sched j t1 t2 <=
blocking_bound (job_arrival j - t1)
      {
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: t2 - t1 <= blocking_bound (job_arrival j - t1)
cumulative_priority_inversion arr_seq sched j t1 t2 <=
blocking_bound (job_arrival j - t1) apply leq_trans with (t2 - t1); last by done.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: t2 - t1 <= blocking_bound (job_arrival j - t1)
cumulative_priority_inversion arr_seq sched j t1 t2 <=
t2 - t1
        rewrite /cumulative_priority_inversion.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: t2 - t1 <= blocking_bound (job_arrival j - t1)
\sum_(t1 <= t < t2)
   priority_inversion_dec arr_seq sched j t <= t2 - t1
        rewrite -[X in _ <= X]addn0 -[t2 - t1]mul1n -iter_addn -big_const_nat.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: t2 - t1 <= blocking_bound (job_arrival j - t1)
\sum_(t1 <= t < t2)
   priority_inversion_dec arr_seq sched j t <=
\sum_(t1 <= i < t2) 1
        by rewrite leq_sum //; intros t _; destruct (priority_inversion_dec).
      }
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
cumulative_priority_inversion arr_seq sched j t1 t2 <=
blocking_bound (job_arrival j - t1)
      edestruct @preemption_time_exists as [ppt [PPT NEQ2]]; rt_eauto.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
NEQ2: t1 <= ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 t2 <=
blocking_bound (job_arrival j - t1)
      move: NEQ2 => /andP [GE LE].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 t2 <=
blocking_bound (job_arrival j - t1)
      apply leq_trans with (cumulative_priority_inversion arr_seq sched j t1 ppt);
        last apply leq_trans with (ppt - t1).
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 t2 <=
cumulative_priority_inversion arr_seq sched j t1 ppt
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 ppt <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
      -
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 t2 <=
cumulative_priority_inversion arr_seq sched j t1 ppt
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 ppt <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1) rewrite /cumulative_priority_inversion.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
\sum_(t1 <= t < t2)
   priority_inversion_dec arr_seq sched j t <=
\sum_(t1 <= t < ppt)
   priority_inversion_dec arr_seq sched j t
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 ppt <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
        rewrite (@big_cat_nat _ _ _ ppt) //=; last first.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt <= t2
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
\sum_(t1 <= i < ppt)
   priority_inversion_dec arr_seq sched j i +
\sum_(ppt <= i < t2)
   priority_inversion_dec arr_seq sched j i <=
\sum_(t1 <= t < ppt)
   priority_inversion_dec arr_seq sched j t
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 ppt <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
        {
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt <= t2 rewrite ltn_subRL in NEQ.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
NEQ: t1 + blocking_bound (job_arrival j - t1) < t2
ppt <= t2
          apply leq_trans with (t1 + blocking_bound (job_arrival j - t1)); last by apply ltnW.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
NEQ: t1 + blocking_bound (job_arrival j - t1) < t2
ppt <= t1 + blocking_bound (job_arrival j - t1)
          apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
NEQ: t1 + blocking_bound (job_arrival j - t1) < t2
t1 + max_length_of_priority_inversion j t1 <=
t1 + blocking_bound (job_arrival j - t1)
          by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2; apply/eqP. }
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
\sum_(t1 <= i < ppt)
   priority_inversion_dec arr_seq sched j i +
\sum_(ppt <= i < t2)
   priority_inversion_dec arr_seq sched j i <=
\sum_(t1 <= t < ppt)
   priority_inversion_dec arr_seq sched j t
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 ppt <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
        rewrite -[X in _ <= X]addn0 leq_add2l leqn0.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
\sum_(ppt <= i < t2)
   priority_inversion_dec arr_seq sched j i == 0
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 ppt <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
        rewrite big_nat_cond big1 //; move => t /andP [/andP [GEt LTt] _ ].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
priority_inversion_dec arr_seq sched j t = 0
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 ppt <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
        edestruct @not_quiet_implies_exists_scheduled_hp_job
          with (K := ppt - t1) (t := t) as [j_hp [ARRB [HP SCHEDHP]]]; rt_eauto.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
exists pr_t : instant,
  preemption_time sched pr_t /\
  t1 <= pr_t <= t1 + (ppt - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
t1 + (ppt - t1) <= t < t2
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
j_hp: Job
ARRB: arrived_between j_hp t1 t.+1
HP: hep_job j_hp j
SCHEDHP: scheduled_at sched j_hp t
priority_inversion_dec arr_seq sched j t = 0
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 ppt <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
        {
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
exists pr_t : instant,
  preemption_time sched pr_t /\
  t1 <= pr_t <= t1 + (ppt - t1) by exists ppt; split; [done | rewrite subnKC //; apply/andP; split]. }
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
t1 + (ppt - t1) <= t < t2
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
j_hp: Job
ARRB: arrived_between j_hp t1 t.+1
HP: hep_job j_hp j
SCHEDHP: scheduled_at sched j_hp t
priority_inversion_dec arr_seq sched j t = 0
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 ppt <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
        {
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
t1 + (ppt - t1) <= t < t2 by rewrite subnKC //; apply/andP; split. }
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
j_hp: Job
ARRB: arrived_between j_hp t1 t.+1
HP: hep_job j_hp j
SCHEDHP: scheduled_at sched j_hp t
priority_inversion_dec arr_seq sched j t = 0
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 ppt <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
        apply/eqP; rewrite eqb0; apply/negP; move => /priority_inversion_P INV.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
j_hp: Job
ARRB: arrived_between j_hp t1 t.+1
HP: hep_job j_hp j
SCHEDHP: scheduled_at sched j_hp t
INV: jobs_come_from_arrival_sequence sched arr_seq ->
jobs_must_arrive_to_execute sched ->
consistent_arrival_times arr_seq ->
priority_inversion sched j t
False
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 ppt <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
        feed_n 3 INV; rt_eauto; last move: INV => [_ [j_lp /andP[SCHED PRIO]]].
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
j_hp: Job
ARRB: arrived_between j_hp t1 t.+1
HP: hep_job j_hp j
SCHEDHP: scheduled_at sched j_hp t
j_lp: Job
SCHED: scheduled_at sched j_lp t
PRIO: ~~ hep_job j_lp j
False
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 ppt <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
        enough (EQ : j_lp = j_hp); first by subst; rewrite HP in PRIO.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
j_hp: Job
ARRB: arrived_between j_hp t1 t.+1
HP: hep_job j_hp j
SCHEDHP: scheduled_at sched j_hp t
j_lp: Job
SCHED: scheduled_at sched j_lp t
PRIO: ~~ hep_job j_lp j
j_lp = j_hp
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 ppt <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
        by eapply ideal_proc_model_is_a_uniprocessor_model; rt_eauto.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 ppt <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
      -
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
cumulative_priority_inversion arr_seq sched j t1 ppt <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1) rewrite /cumulative_priority_inversion.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
\sum_(t1 <= t < ppt)
   priority_inversion_dec arr_seq sched j t <=
ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
        rewrite -[X in _ <= X]addn0 -[ppt - t1]mul1n -iter_addn -big_const_nat.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
\sum_(t1 <= t < ppt)
   priority_inversion_dec arr_seq sched j t <=
\sum_(t1 <= i < ppt) 1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
        by rewrite leq_sum //; intros t _; destruct (priority_inversion_dec).
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
      -
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1) rewrite leq_subLR.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
ppt <= t1 + blocking_bound (job_arrival j - t1)
        apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
COARR: jobs_come_from_arrival_sequence sched arr_seq
MBR: jobs_must_be_ready_to_execute sched
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <=
t1 +
priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq j t1
t1 + max_length_of_priority_inversion j t1 <=
t1 + blocking_bound (job_arrival j - t1)
        by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2; apply/eqP.
    Qed.

  End PriorityInversionIsBounded.

  (** ** Response-Time Bound *)
  (** In this section, we prove that the maximum among the solutions of the response-time
      bound recurrence is a response-time bound for [tsk]. *)
  Section ResponseTimeBound.

    (** 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.

    (** 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 L A ->
        exists (F : duration),
          A + F >= blocking_bound A
                  + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk))
                  + bound_on_total_hep_workload  A (A + F) /\
          R >= F + (task_cost tsk - task_rtct tsk).

    (** Then, using the results for the general RTA for EDF-schedulers, we establish a
         response-time bound for the more concrete model of bounded nonpreemptive segments.
         Note that in case of the general RTA for EDF-schedulers, we just _assume_ that
         the priority inversion is bounded. In this module we provide the preemption model
         with bounded nonpreemptive segments and _prove_ that the priority inversion is
         bounded. *)
    Theorem uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments:
      response_time_bounded_by tsk R.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space L A ->
exists F : duration,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
response_time_bounded_by tsk R
    Proof.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space L A ->
exists F : duration,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
response_time_bounded_by tsk R
      eapply uniprocessor_response_time_bound_edf; rt_eauto.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space L A ->
exists F : duration,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
priority_inversion_is_bounded_by arr_seq sched tsk
  ?priority_inversion_bound
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space L A ->
exists F : duration,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
forall A : duration,
bounded_pi.is_in_search_space ts tsk
  ?priority_inversion_bound L A ->
exists F : duration,
  ?priority_inversion_bound 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
      -
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space L A ->
exists F : duration,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
priority_inversion_is_bounded_by arr_seq sched tsk
  ?priority_inversion_bound
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space L A ->
exists F : duration,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
forall A : duration,
bounded_pi.is_in_search_space ts tsk
  ?priority_inversion_bound L A ->
exists F : duration,
  ?priority_inversion_bound 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 by  apply priority_inversion_is_bounded.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space L A ->
exists F : duration,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
forall A : duration,
bounded_pi.is_in_search_space ts tsk blocking_bound L
  A ->
exists F : duration,
  blocking_bound 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
      -
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space L A ->
exists F : duration,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
forall A : duration,
bounded_pi.is_in_search_space ts tsk blocking_bound L
  A ->
exists F : duration,
  blocking_bound 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 move=> A BPI_SP.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched EDF
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
H5: 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
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat =>
\sum_(tsk_o <- ts | 
tsk_o != tsk)
   rbf tsk_o
     (minn
        (A + ε + D tsk -
        D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion_bounded.max_length_of_priority_inversion
  arr_seq: Job ->
instant -> nat
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space L A ->
exists F : duration,
  blocking_bound A +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  bound_on_total_hep_workload A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
A: duration
BPI_SP: bounded_pi.is_in_search_space ts tsk
  blocking_bound L A
exists F : duration,
  blocking_bound 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
        by apply H_R_is_maximum, search_space_inclusion.
    Qed.

  End ResponseTimeBound.

End RTAforEDFwithBoundedNonpreemptiveSegmentsWithArrivalCurves.