Require Import prosa.util.int.
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New coercion path [GRing.subring_closed_semi;
                   GRing.semiring_closedD] : GRing.subring_closed >-> GRing.addr_closed is ambiguous with existing 
[GRing.subring_closedB; GRing.zmod_closedD] : GRing.subring_closed >-> GRing.addr_closed.
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[ambiguous-paths,coercions,default]
Notation "_ - _" was already used in scope
distn_scope. [notation-overridden,parsing,default]
Require Export prosa.analysis.abstract.ideal.cumulative_bounds.
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Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing,default]
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nat_scope. [notation-overridden,parsing,default]
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nat_scope. [notation-overridden,parsing,default]
Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing,default]
Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing,default]
Require Export prosa.analysis.facts.priority.elf.
Require Export prosa.analysis.facts.interference.
Require Export prosa.analysis.facts.busy_interval.carry_in.
Require Export prosa.analysis.facts.readiness.basic.
Require Export prosa.analysis.facts.priority.jlfp_with_fp.
Require Export prosa.analysis.facts.model.rbf.
Require Export prosa.analysis.facts.busy_interval.pi.
Require Export prosa.analysis.facts.busy_interval.pi_cond.
Require Export prosa.analysis.abstract.ideal.abstract_seq_rta.
Require Export prosa.analysis.facts.model.task_cost.

(** * Response-Time Analysis for the ELF Scheduling Policy *)

(** In the following, we derive a response-time analysis for ELF schedulers,
    assuming a workload of sporadic real-time tasks characterized by arbitrary
    arrival curves executing upon an ideal uniprocessor. To this end, we
    instantiate the _abstract Response-Time Analysis_ (aRTA) as provided in the
    [prosa.analysis.abstract] module. *)

(** Note that this analysis is currently specific to workloads where each task has bounded
    nonpreemptive segments. This specificity will be further explained by the assumptions
    which depend on it. The analysis catering to the more general model with bounded priority
    inversion remains future work. *)

Section AbstractRTAforELFwithArrivalCurves.

  (** ** A. Defining the System Model *)

  (** Before any formal claims can be stated, an initial setup is needed to
      define the system model under consideration. To this end, we next
      introduce and define the following notions using Prosa's standard
      definitions and behavioral semantics:
      - tasks, jobs, and their parameters,
      - the sequence of job arrivals,
      - worst-case execution time (WCET) and the absence of self-suspensions,
      - the set of tasks under analysis,
      - the task under analysis, and, finally,
      - an arbitrary schedule of the task set. *)

  (** *** Tasks and Jobs  *)

  (** Consider any type of tasks, each characterized by a WCET [task_cost],
      a run-to-completion threshold [task_rtct], a maximum non-preemptive segment
      length [task_max_nonpreemptive_segment], an arrival curve [max_arrivals], and
      a relative priority point [task_priority_point] ... *)
  Context  {Task : TaskType} `{TaskCost Task} `{TaskRunToCompletionThreshold Task}
    `{TaskMaxNonpreemptiveSegment Task} `{MaxArrivals Task} `{PriorityPoint Task}.

  (** ... and any type of jobs associated with these tasks, where each job has
      an arrival time [job_arrival], a cost [job_cost], and an arbitrary
      preemption model indicated by [job_preemptable]. *)
  Context  {Job : JobType} `{JobTask Job Task} {Arrival : JobArrival Job}
    {Cost : JobCost Job} `{JobPreemptable Job}.

  (** *** The Job Arrival Sequence *)

  (** Consider any valid arrival sequence [arr_seq]. *)
  Variable arr_seq : arrival_sequence Job.
  Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.

  (** *** Absence of Self-Suspensions and WCET Compliance *)

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

  (** We further require that a job's cost cannot exceed its task's stated
      WCET. *)
  Hypothesis H_valid_job_cost : arrivals_have_valid_job_costs arr_seq.

  (** *** The Task Set *)

  (** We consider an arbitrary task set [ts]... *)
  Variable ts : list Task.
  Hypothesis H_task_set : uniq ts.

  (** ... and assume that all jobs stem from tasks in this task set. *)
  Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.

  (** Furthermore, we assume that [max_arrivals] is a family of valid arrival
      curves that constrains the arrival sequence [arr_seq], i.e., for any task
      [tsk] in [ts], [max_arrival tsk] is (1) an arrival bound of [tsk], and ... *)
  Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.

  (** ... (2) a monotonic function that equals [0] for the empty interval [delta = 0]. *)
  Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.

  (** *** The Task Under Analysis *)

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

  (** We assume that [tsk] is described by a valid task _run-to-completion
      threshold_. That is, there exists a task parameter [task_rtct] such
      that [task_rtct tsk] is
      - (1) no larger than [tsk]'s WCET, and
      - (2) for any job of task [tsk], the job's run-to-completion threshold
            [job_rtct] is bounded by [task_rtct tsk]. *)
  Hypothesis H_valid_run_to_completion_threshold :
    valid_task_run_to_completion_threshold arr_seq tsk.

  (** *** The Schedule *)

  (** Finally, we consider any arbitrary, valid ideal uni-processor schedule of
      the given arrival sequence [arr_seq]. *)
  Variable sched : schedule (ideal.processor_state Job).
  Hypothesis H_sched_valid : valid_schedule sched arr_seq.

  (** Now, we assume that the fixed-priority policy [FP] that parameterizes the
      ELF policy is... *)
  Variable FP : FP_policy Task.

  (** ... reflexive, transitive, and total. *)
  Hypothesis H_reflexive_priorities : reflexive_task_priorities FP.
  Hypothesis H_transitive_priorities : transitive_task_priorities FP.
  Hypothesis H_total_priorities : total_task_priorities FP.

  (** We further assume that the schedule complies with the preemption model ... *)
  Hypothesis H_valid_preemption_model : valid_preemption_model arr_seq sched.

  (** ... and finally, that it respects the [ELF] scheduling policy. *)
  Hypothesis H_respects_policy : respects_JLFP_policy_at_preemption_point arr_seq sched (ELF FP).

  (** ** B. Interference and Interfering Workload *)

  (** With the system model in place, the next step is to encode the scheduling
      policy and preemption model such that aRTA becomes applicable. To this
      end, we encode the semantics of the scheduling policy and preemption model
      by instantiating two functions, called [interference] and
      [interfering_workload].

      We can simply reuse the existing general definitions of _interference_ and
      _interfering workload_ that apply to job-level fixed-priority (JLFP)
      policy (as provided in the module [analysis.abstract.ideal.iw_instantiation]).*)
  #[local] Instance ideal_elf_interference : Interference Job :=
      ideal_jlfp_interference arr_seq sched.

  #[local] Instance ideal_elf_interfering_workload : InterferingWorkload Job :=
      ideal_jlfp_interfering_workload arr_seq sched.

  (** ** C. Classic and Abstract Work Conservation *)

  (** We assume that the schedule is work-conserving in the classic sense. *)
  Hypothesis H_work_conserving : work_conserving.work_conserving arr_seq sched.

  (** This allows us to apply [instantiated_i_and_w_are_coherent_with_schedule]
      to conclude that abstract work-conservation also holds. *)

  (** ** D. The Priority Inversion Bound and its Validity *)

    (** In this file, we break the priority inversion experienced by any job into two
      categories :
      - (1) priority inversion caused by jobs belonging to tasks with lower priority than [tsk]
      - (2) priority inversion caused by jobs belonging to tasks with equal priority as [tsk]
    Note that, by definition of the ELF policy, no job from a task with higher priority than [tsk]
    can cause priority inversion. *)

  (** We define a predicate to identify jobs from lower-priority tasks,
      or tasks for which [tsk] has higher priority. *)
  Let is_lower_priority j' := hp_task tsk (job_task j').

  (** We assume there exists a bound on the maximum length of priority inversion from these jobs
      that is incurred by any job of task [tsk]. *)
  Variable priority_inversion_lp_tasks_bound : duration.
  Hypothesis H_priority_inversion_from_lp_tasks_is_bounded :
    priority_inversion_cond_is_bounded_by arr_seq sched tsk
      is_lower_priority (constant priority_inversion_lp_tasks_bound).

  (** Similarly, we define a predicate to select jobs whose tasks have equal priority as [tsk]... *)
  Let is_equal_priority j' := ep_task tsk (job_task j').

  (** ... and assume that there exists a bound on the maximum length of priority inversion
      caused by them to any job of [tsk]. *)
  Variable priority_inversion_ep_tasks_bound : duration -> duration.
  Hypothesis H_priority_inversion_from_ep_tasks_is_bounded :
    priority_inversion_cond_is_bounded_by arr_seq sched tsk
      is_equal_priority priority_inversion_ep_tasks_bound.

  (** We then define the priority inversion bound as a maximum of the bounds on the two categories. *)
  (** Note that this definition is only applicable when all tasks have bounded nonpreemptive segments. *)
  Definition priority_inversion_bound (A: duration) := maxn priority_inversion_lp_tasks_bound
                                                            (priority_inversion_ep_tasks_bound A).

  (** Now, we define the following predicate to identify tasks that can release jobs. *)
  Let blocking_relevant (tsk_o : Task) :=
    (max_arrivals tsk_o ε > 0) && (task_cost tsk_o > 0).

  (** We also define a predicate to identify equal priority tasks that cannot cause
      priority inversion for a job [j], given that [j]'s busy interval starts that instant [t1]. *)
  Let is_ep_causing_intf (j: Job) (t1 : instant) (tsk_other:Task) :=
    ((job_arrival j)%:R - t1%:R + task_priority_point tsk - task_priority_point tsk_other >=0)%R .

  (** Using the above, we define the condition that an equal-priority task must satisfy
      for any of its jobs to cause blocking (or priority inversion) to a job [j] in [j]'s
      busy interval starting at [t1]. *)
  Let ep_task_blocking_relevant tsk_other j t1 :=
    ep_task tsk_other tsk && ~~ is_ep_causing_intf j t1 tsk_other &&
      blocking_relevant tsk_other.

  (** Finally, we assume that the [priority_inversion_ep_tasks_bound] is bounded by the
      maximum [task_cost] of tasks which satisfy the above condition. Note that this
      assumption is valid only for the model where tasks have bounded nonpreemptive
      segments. *)
  Hypothesis H_priority_inversion_from_ep_tasks_concrete_bound :
    forall j t1,
      job_task j = tsk ->
      priority_inversion_ep_tasks_bound (job_arrival j - t1)
        <= \max_(i <- ts | ep_task_blocking_relevant i j t1) task_cost i.

  (** Having defined bounds on two separate categories of priority inversion, we now show that
      the defined [priority_inversion_bound] upper-bounds the priority inversion faced by any job
      belonging to [tsk], regardless of its cause. *)
  Lemma priority_inversion_is_bounded :
    priority_inversion_is_bounded_by arr_seq sched tsk priority_inversion_bound.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
priority_inversion_is_bounded_by arr_seq sched tsk
  priority_inversion_bound
  Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
priority_inversion_is_bounded_by arr_seq sched tsk
  priority_inversion_bound
    move=> j ARR TSK POS t1 t2 BUSY_PREF.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
BUSY_PREF: classical.busy_interval_prefix arr_seq
  sched j t1 t2
cumulative_priority_inversion arr_seq sched j t1 t2 <=
priority_inversion_bound (job_arrival j - t1)
    have [-> //|/andP[_ /hasP[jlp jlp_at_t1 nHEPj]]]:
      cumulative_priority_inversion arr_seq sched j t1 t2 = 0 \/ priority_inversion arr_seq sched j t1
      by apply: busy_interval_pi_cases.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
BUSY_PREF: classical.busy_interval_prefix arr_seq
  sched j t1 t2
jlp: Job
jlp_at_t1: jlp \in scheduled_jobs_at arr_seq sched t1
nHEPj: ~~ hep_job jlp j
cumulative_priority_inversion arr_seq sched j t1 t2 <=
priority_inversion_bound (job_arrival j - t1)
    rewrite scheduled_jobs_at_iff in jlp_at_t1 =>//.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
BUSY_PREF: classical.busy_interval_prefix arr_seq
  sched j t1 t2
jlp: Job
nHEPj: ~~ hep_job jlp j
jlp_at_t1: scheduled_at sched jlp t1
cumulative_priority_inversion arr_seq sched j t1 t2 <=
priority_inversion_bound (job_arrival j - t1)
    have [lp | ep]: ~~ hep_task (job_task jlp) (job_task j)
                    \/ ep_task (job_task jlp) (job_task j)
      by apply/orP; rewrite -nhp_ep_nhep_task =>//; move: nHEPj; apply /contra/hp_task_implies_hep_job.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
BUSY_PREF: classical.busy_interval_prefix arr_seq
  sched j t1 t2
jlp: Job
nHEPj: ~~ hep_job jlp j
jlp_at_t1: scheduled_at sched jlp t1
lp: ~~ hep_task (job_task jlp) (job_task j)
cumulative_priority_inversion arr_seq sched j t1 t2 <=
priority_inversion_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
BUSY_PREF: classical.busy_interval_prefix arr_seq
  sched j t1 t2
jlp: Job
nHEPj: ~~ hep_job jlp j
jlp_at_t1: scheduled_at sched jlp t1
ep: ep_task (job_task jlp) (job_task j)
cumulative_priority_inversion arr_seq sched j t1 t2 <=
priority_inversion_bound (job_arrival j - t1)
    {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
BUSY_PREF: classical.busy_interval_prefix arr_seq
  sched j t1 t2
jlp: Job
nHEPj: ~~ hep_job jlp j
jlp_at_t1: scheduled_at sched jlp t1
lp: ~~ hep_task (job_task jlp) (job_task j)
cumulative_priority_inversion arr_seq sched j t1 t2 <=
priority_inversion_bound (job_arrival j - t1) have -> : cumulative_priority_inversion arr_seq sched j t1 t2
                = cumulative_priority_inversion_cond arr_seq sched j is_lower_priority t1 t2
        by apply: cum_task_pi_eq => //; rewrite /is_lower_priority -not_hep_hp_task //; by move: (TSK) => /eqP <-.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
BUSY_PREF: classical.busy_interval_prefix arr_seq
  sched j t1 t2
jlp: Job
nHEPj: ~~ hep_job jlp j
jlp_at_t1: scheduled_at sched jlp t1
lp: ~~ hep_task (job_task jlp) (job_task j)
cumulative_priority_inversion_cond arr_seq sched j
  is_lower_priority t1 t2 <=
priority_inversion_bound (job_arrival j - t1)
      rewrite leq_max; apply/orP; left.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
BUSY_PREF: classical.busy_interval_prefix arr_seq
  sched j t1 t2
jlp: Job
nHEPj: ~~ hep_job jlp j
jlp_at_t1: scheduled_at sched jlp t1
lp: ~~ hep_task (job_task jlp) (job_task j)
cumulative_priority_inversion_cond arr_seq sched j
  is_lower_priority t1 t2 <=
priority_inversion_lp_tasks_bound
      by apply: H_priority_inversion_from_lp_tasks_is_bounded. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
BUSY_PREF: classical.busy_interval_prefix arr_seq
  sched j t1 t2
jlp: Job
nHEPj: ~~ hep_job jlp j
jlp_at_t1: scheduled_at sched jlp t1
ep: ep_task (job_task jlp) (job_task j)
cumulative_priority_inversion arr_seq sched j t1 t2 <=
priority_inversion_bound (job_arrival j - t1)
    {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
BUSY_PREF: classical.busy_interval_prefix arr_seq
  sched j t1 t2
jlp: Job
nHEPj: ~~ hep_job jlp j
jlp_at_t1: scheduled_at sched jlp t1
ep: ep_task (job_task jlp) (job_task j)
cumulative_priority_inversion arr_seq sched j t1 t2 <=
priority_inversion_bound (job_arrival j - t1) have -> : cumulative_priority_inversion arr_seq sched j t1 t2
                = cumulative_priority_inversion_cond arr_seq sched j is_equal_priority t1 t2
        by apply: cum_task_pi_eq => //; rewrite /is_equal_priority ep_task_sym; move: (TSK) => /eqP <-.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
BUSY_PREF: classical.busy_interval_prefix arr_seq
  sched j t1 t2
jlp: Job
nHEPj: ~~ hep_job jlp j
jlp_at_t1: scheduled_at sched jlp t1
ep: ep_task (job_task jlp) (job_task j)
cumulative_priority_inversion_cond arr_seq sched j
  is_equal_priority t1 t2 <=
priority_inversion_bound (job_arrival j - t1)
      rewrite leq_max; apply/orP; right.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
BUSY_PREF: classical.busy_interval_prefix arr_seq
  sched j t1 t2
jlp: Job
nHEPj: ~~ hep_job jlp j
jlp_at_t1: scheduled_at sched jlp t1
ep: ep_task (job_task jlp) (job_task j)
cumulative_priority_inversion_cond arr_seq sched j
  is_equal_priority t1 t2 <=
priority_inversion_ep_tasks_bound (job_arrival j - t1)
      by apply: H_priority_inversion_from_ep_tasks_is_bounded. }
  Qed.

  (** ** E. Maximum Busy-Window Length *)

  (** The next step is to establish a bound on the maximum busy-window length,
      which aRTA requires to be given. *)

  (** Using the sum of individual request bound functions, we define
      the request bound function of all tasks with higher-or-equal
      priority [FP] (with respect to [tsk]). *)
  Let total_hep_rbf := total_hep_request_bound_function_FP ts tsk.

  (** To this end, we assume that we are given a positive value [L] ...*)
  Variable L : duration.
  Hypothesis H_L_positive : L > 0.

  (** ... that is a fixed point of the following equation. *)
  Hypothesis H_fixed_point : L = priority_inversion_lp_tasks_bound + total_hep_rbf L.

  (** Given this definition of [L], we prove that [L] bounds the length of the busy window. *)
  Lemma instantiated_busy_intervals_are_bounded :
    busy_intervals_are_bounded_by arr_seq sched tsk L.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
busy_intervals_are_bounded_by arr_seq sched tsk L
  Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
busy_intervals_are_bounded_by arr_seq sched tsk L
    move => j ARR TSK POS; move: (TSK) => /eqP TSK'.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
exists t1 t2 : nat,
  t1 <= job_arrival j < t2 /\
  t2 <= t1 + L /\ busy_interval sched j t1 t2
    edestruct (exists_busy_interval) with (delta := L) (JLFP := ELF (FP)) as [t1 [t2 [T1 [T2 BI]]]] => //.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
priority_inversion_of_job_is_bounded_by arr_seq sched
  j ?priority_inversion_bound
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
forall t : nat,
?priority_inversion_bound (job_arrival j - t) +
workload_of_higher_or_equal_priority_jobs j
  (arrivals_between arr_seq t (t + L)) <= L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t1, t2: nat
T1: t1 <= job_arrival j < t2
T2: t2 <= t1 + L
BI: classical.busy_interval arr_seq sched j t1 t2
exists t1 t2 : nat,
  t1 <= job_arrival j < t2 /\
  t2 <= t1 + L /\ busy_interval sched j t1 t2
    {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
priority_inversion_of_job_is_bounded_by arr_seq sched
  j ?priority_inversion_bound by apply: priority_inversion_is_bounded. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
forall t : nat,
priority_inversion_bound (job_arrival j - t) +
workload_of_higher_or_equal_priority_jobs j
  (arrivals_between arr_seq t (t + L)) <= L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t1, t2: nat
T1: t1 <= job_arrival j < t2
T2: t2 <= t1 + L
BI: classical.busy_interval arr_seq sched j t1 t2
exists t1 t2 : nat,
  t1 <= job_arrival j < t2 /\
  t2 <= t1 + L /\ busy_interval sched j t1 t2
    {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
forall t : nat,
priority_inversion_bound (job_arrival j - t) +
workload_of_higher_or_equal_priority_jobs j
  (arrivals_between arr_seq t (t + L)) <= L rewrite {2}H_fixed_point => t.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
priority_inversion_bound (job_arrival j - t) +
workload_of_higher_or_equal_priority_jobs j
  (arrivals_between arr_seq t (t + L)) <=
priority_inversion_lp_tasks_bound + total_hep_rbf L
      apply leq_trans with (priority_inversion_lp_tasks_bound
                            + priority_inversion_ep_tasks_bound (job_arrival j - t)
                            + workload_of_higher_or_equal_priority_jobs j (arrivals_between arr_seq t (t + L)));
        first by rewrite leq_add2r max_leq_add.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
priority_inversion_lp_tasks_bound +
priority_inversion_ep_tasks_bound (job_arrival j - t) +
workload_of_higher_or_equal_priority_jobs j
  (arrivals_between arr_seq t (t + L)) <=
priority_inversion_lp_tasks_bound + total_hep_rbf L
      rewrite -addnA leq_add2l /total_hep_rbf.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
priority_inversion_ep_tasks_bound (job_arrival j - t) +
workload_of_higher_or_equal_priority_jobs j
  (arrivals_between arr_seq t (t + L)) <=
total_hep_request_bound_function_FP ts tsk L
      rewrite hep_rbf_taskwise_partitioning /total_ep_request_bound_function_FP (bigID (is_ep_causing_intf j t)) /=.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
priority_inversion_ep_tasks_bound (job_arrival j - t) +
workload_of_higher_or_equal_priority_jobs j
  (arrivals_between arr_seq t (t + L)) <=
total_hp_request_bound_function_FP ts tsk L +
(\sum_(i <- ts | ep_task i tsk &&
                 is_ep_causing_intf j t i)
    task_request_bound_function i L +
 \sum_(i <- ts | ep_task i tsk &&
                 ~~ is_ep_causing_intf j t i)
    task_request_bound_function i L)
      rewrite hep_workload_partitioning_taskwise =>//; rewrite -[leqRHS]addnACl addnA.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
priority_inversion_ep_tasks_bound (job_arrival j - t) +
workload_of_jobs (hep_from_hp_task j)
  (arrivals_between arr_seq t (t + L)) +
workload_of_jobs (hep_from_ep_task j)
  (arrivals_between arr_seq t (t + L)) <=
\sum_(i <- ts | ep_task i tsk &&
                ~~ is_ep_causing_intf j t i)
   task_request_bound_function i L +
total_hp_request_bound_function_FP ts tsk L +
\sum_(i <- ts | ep_task i tsk &&
                is_ep_causing_intf j t i)
   task_request_bound_function i L
      apply: leq_add; first apply: leq_add.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
priority_inversion_ep_tasks_bound (job_arrival j - t) <=
\sum_(i <- ts | ep_task i tsk &&
                ~~ is_ep_causing_intf j t i)
   task_request_bound_function i L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
workload_of_jobs 
  (hep_from_hp_task j)
  (arrivals_between arr_seq t (t + L)) <=
total_hp_request_bound_function_FP ts tsk L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
workload_of_jobs 
  (hep_from_ep_task j)
  (arrivals_between arr_seq t (t + L)) <=
\sum_(i <- ts | 
ep_task i tsk && 
is_ep_causing_intf j t i)
   task_request_bound_function i L
      {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
priority_inversion_ep_tasks_bound (job_arrival j - t) <=
\sum_(i <- ts | ep_task i tsk &&
                ~~ is_ep_causing_intf j t i)
   task_request_bound_function i L apply: (leq_trans (H_priority_inversion_from_ep_tasks_concrete_bound _ _ _)) =>//.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
\max_(i <- ts | ep_task_blocking_relevant i j t)
   task_cost i <=
\sum_(i <- ts | ep_task i tsk &&
                ~~ is_ep_causing_intf j t i)
   task_request_bound_function i L
        apply/(leq_trans (bigmax_leq_sum _ _ _ _)).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
\sum_(i <- ts | ep_task_blocking_relevant i j t)
   task_cost i <=
\sum_(i <- ts | ep_task i tsk &&
                ~~ is_ep_causing_intf j t i)
   task_request_bound_function i L
        apply leq_trans with
          (\sum_(i <- ts | ep_task_blocking_relevant i j t) task_request_bound_function i L); last first.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
\sum_(i <- ts | ep_task_blocking_relevant i j t)
   task_request_bound_function i L <=
\sum_(i <- ts | ep_task i tsk &&
                ~~ is_ep_causing_intf j t i)
   task_request_bound_function i L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
\sum_(i <- ts | 
ep_task_blocking_relevant i j t) 
task_cost i <=
\sum_(i <- ts | 
ep_task_blocking_relevant i j t)
   task_request_bound_function i L
        {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
\sum_(i <- ts | ep_task_blocking_relevant i j t)
   task_request_bound_function i L <=
\sum_(i <- ts | ep_task i tsk &&
                ~~ is_ep_causing_intf j t i)
   task_request_bound_function i L apply: sub_le_big => //=; first by move => ? ?; apply leq_addr.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
forall i : Task,
ep_task_blocking_relevant i j t ->
ep_task i tsk && ~~ is_ep_causing_intf j t i
          by move => tsk' /andP[]. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
\sum_(i <- ts | ep_task_blocking_relevant i j t)
   task_cost i <=
\sum_(i <- ts | ep_task_blocking_relevant i j t)
   task_request_bound_function i L
        rewrite  big_seq_cond [leqRHS]big_seq_cond.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
\sum_(i <- ts | (i \in ts) &&
                ep_task_blocking_relevant i j t)
   task_cost i <=
\sum_(i <- ts | (i \in ts) &&
                ep_task_blocking_relevant i j t)
   task_request_bound_function i L
        apply: leq_sum =>tsk_other /andP[? /andP[_ /andP[MA _]]].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
tsk_other: Task
_a_: tsk_other \in ts
MA: 0 < max_arrivals tsk_other 1
task_cost tsk_other <=
task_request_bound_function tsk_other L
        apply leq_trans with (task_request_bound_function tsk_other ε).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
tsk_other: Task
_a_: tsk_other \in ts
MA: 0 < max_arrivals tsk_other 1
task_cost tsk_other <=
task_request_bound_function tsk_other 1
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
tsk_other: Task
_a_: tsk_other \in ts
MA: 0 < max_arrivals tsk_other 1
task_request_bound_function tsk_other 1 <=
task_request_bound_function tsk_other L
        -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
tsk_other: Task
_a_: tsk_other \in ts
MA: 0 < max_arrivals tsk_other 1
task_cost tsk_other <=
task_request_bound_function tsk_other 1 by apply: leq_pmulr.
        -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
tsk_other: Task
_a_: tsk_other \in ts
MA: 0 < max_arrivals tsk_other 1
task_request_bound_function tsk_other 1 <=
task_request_bound_function tsk_other L by apply/leq_mul/(H_valid_arrival_curve _ _).2. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
workload_of_jobs (hep_from_hp_task j)
  (arrivals_between arr_seq t (t + L)) <=
total_hp_request_bound_function_FP ts tsk L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
workload_of_jobs 
  (hep_from_ep_task j)
  (arrivals_between arr_seq t (t + L)) <=
\sum_(i <- ts | 
ep_task i tsk && 
is_ep_causing_intf j t i)
   task_request_bound_function i L
      {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
workload_of_jobs (hep_from_hp_task j)
  (arrivals_between arr_seq t (t + L)) <=
total_hp_request_bound_function_FP ts tsk L rewrite hep_hp_workload_hp =>//.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
workload_of_jobs (from_hp_task j)
  (arrivals_between arr_seq t (t + L)) <=
total_hp_request_bound_function_FP ts tsk L rewrite /from_hp_task TSK'; exact: sum_of_jobs_le_sum_rbf. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
workload_of_jobs (hep_from_ep_task j)
  (arrivals_between arr_seq t (t + L)) <=
\sum_(i <- ts | ep_task i tsk &&
                is_ep_causing_intf j t i)
   task_request_bound_function i L
      {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
workload_of_jobs (hep_from_ep_task j)
  (arrivals_between arr_seq t (t + L)) <=
\sum_(i <- ts | ep_task i tsk &&
                is_ep_causing_intf j t i)
   task_request_bound_function i L apply: leq_trans; last by apply: sum_of_jobs_le_sum_rbf.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
workload_of_jobs (hep_from_ep_task j)
  (arrivals_between arr_seq t (t + L)) <=
\sum_(j' <- arrivals_between arr_seq t (t + L) | (fun
                                                 tsk' : Task
                                                 =>
                                                 ep_task
                                                 tsk'
                                                 tsk &&
                                                 is_ep_causing_intf
                                                 j t
                                                 tsk')
                                                 (job_task
                                                 j'))
   job_cost j'
        rewrite /workload_of_jobs big_seq_cond [leqRHS]big_seq_cond.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
\sum_(i <- arrivals_between arr_seq t (t + L) | (i
                                                  \in 
                                                 arrivals_between
                                                 arr_seq
                                                 t
                                                 (t +
                                                 L)) &&
                                                hep_from_ep_task
                                                 j i)
   job_cost i <=
\sum_(i <- arrivals_between arr_seq t (t + L) | [&&
                                                 i
                                                  \in 
                                                 arrivals_between
                                                 arr_seq
                                                 t
                                                 (t +
                                                 L),
                                                 ep_task
                                                 (job_task
                                                 i)
                                                 tsk
                                                  & 
                                                is_ep_causing_intf
                                                 j t
                                                 (job_task
                                                 i)])
   job_cost i
        apply: sub_le_big => //; first by move => ? ?; apply: leq_addr.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
forall i : Job,
(i \in arrivals_between arr_seq t (t + L)) &&
hep_from_ep_task j i ->
[&& i \in arrivals_between arr_seq t (t + L),
    ep_task (job_task i) tsk
  & is_ep_causing_intf j t (job_task i)]
        move=> j0; case eq: (_ \in _) =>//=.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
j0: Job
eq: (j0 \in arrivals_between arr_seq t (t + L)) =
true
hep_from_ep_task j j0 ->
ep_task (job_task j0) tsk &&
is_ep_causing_intf j t (job_task j0)
        move=> /andP[HEPj EP]; rewrite -TSK'; apply/andP; split =>//.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
j0: Job
eq: (j0 \in arrivals_between arr_seq t (t + L)) =
true
HEPj: hep_job j0 j
EP: ep_task (job_task j0) (job_task j)
is_ep_causing_intf j t (job_task j0)
        move: HEPj.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
j0: Job
eq: (j0 \in arrivals_between arr_seq t (t + L)) =
true
EP: ep_task (job_task j0) (job_task j)
hep_job j0 j -> is_ep_causing_intf j t (job_task j0)
        have -> : hep_job j0 j = (@hep_job _ (GEL Job Task) j0 j) by apply: hep_job_elf_gel.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
j0: Job
eq: (j0 \in arrivals_between arr_seq t (t + L)) =
true
EP: ep_task (job_task j0) (job_task j)
hep_job j0 j -> is_ep_causing_intf j t (job_task j0)
        rewrite /is_ep_causing_intf lerBrDl addrAC lerBrDl addr0 -TSK'.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
j0: Job
eq: (j0 \in arrivals_between arr_seq t (t + L)) =
true
EP: ep_task (job_task j0) (job_task j)
hep_job j0 j ->
(t%:R + task_priority_point (job_task j0) <=
 (job_arrival j)%:R + task_priority_point (job_task j))%R
        apply: le_trans; rewrite lerD2r ler_nat.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t: nat
j0: Job
eq: (j0 \in arrivals_between arr_seq t (t + L)) =
true
EP: ep_task (job_task j0) (job_task j)
t <= job_arrival j0
        apply: job_arrival_between_ge=>//. }}
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t1, t2: nat
T1: t1 <= job_arrival j < t2
T2: t2 <= t1 + L
BI: classical.busy_interval arr_seq sched j t1 t2
exists t1 t2 : nat,
  t1 <= job_arrival j < t2 /\
  t2 <= t1 + L /\ busy_interval sched j t1 t2
    {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t1, t2: nat
T1: t1 <= job_arrival j < t2
T2: t2 <= t1 + L
BI: classical.busy_interval arr_seq sched j t1 t2
exists t1 t2 : nat,
  t1 <= job_arrival j < t2 /\
  t2 <= t1 + L /\ busy_interval sched j t1 t2 exists t1, t2; split=> [//|]; split=> [//|].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
TSK': job_task j = tsk
t1, t2: nat
T1: t1 <= job_arrival j < t2
T2: t2 <= t1 + L
BI: classical.busy_interval arr_seq sched j t1 t2
busy_interval sched j t1 t2
      eapply instantiated_busy_interval_equivalent_busy_interval => //. }
  Qed.

  (** ** F. The Interference-Bound Function *)

  (** Next, we define the interference [task_IBF] and prove that [task_IBF]
      bounds the interference incurred by any job of [tsk]. Note that
      [task_IBF] only takes the interference from jobs of other tasks into
      account i.e., self-interference is not included. *)

  (** We first consider the interference incurred due to strictly
      higher-priority tasks i.e., those which have strictly higher-priority
      according to the FP policy. *)
  Definition total_hp_rbf := total_hp_request_bound_function_FP ts tsk.

  (** We define the following parameterized end point of the interval during
      which interfering jobs of equal-priority tasks must arrive. The implicit
      beginning of the interval is the start of the busy window, i.e., at time
      [t1]. *)
  Definition ep_task_intf_interval (tsk_o : Task) (A : instant) :=
    ((A + ε)%:R + task_priority_point tsk - task_priority_point tsk_o)%R.

  (** Using this interval end point, we define the bound on the total
      equal-priority ([EP]) workload produced during the interval [Δ] as the sum
      of the RBFs of all tasks (with equal priority as [tsk]) in the task set
      [ts] (excluding [tsk]) over the minimum of [Δ] and
      [ep_task_intf_interval]. *)
  Definition bound_on_total_ep_workload (A Δ : duration) :=
    \sum_(tsk_o <- ts | ep_task tsk tsk_o && (tsk_o != tsk))
      task_request_bound_function tsk_o (minn `|Num.max 0%R (ep_task_intf_interval tsk_o A)| Δ).

  (** Finally, [task_IBF] for an interval [Δ] is defined as the sum of
      [priority_inversion_bound], [bound_on_total_ep_workload], and
      [total_hp_rbf] on [Δ]. *)
  Definition task_IBF (A Δ : duration) :=
    priority_inversion_bound A + bound_on_total_ep_workload A Δ + total_hp_rbf Δ.

  (** In this section, we prove the soundness of [task_IBF].*)
  Section BoundingIBF.

    (** Consider any job [j] of task [tsk] that has a positive job cost and is
        in the arrival sequence. *)
    Variable j : Job.
    Hypothesis H_job_of_task : job_of_task tsk j.
    Hypothesis H_job_cost_positive : job_cost_positive j.
    Hypothesis H_j_in_arr_seq : arrives_in arr_seq j.

    (** Assume the busy interval of [j] (in the abstract sense) is given by <<[t1,t2)>>. *)
    Variable t1 t2 : instant.
    Hypothesis H_busy_window : definitions.busy_interval sched j t1 t2.

    (** Consider any arbitrary length [Δ] interval <<[t1, t1+ Δ)>> within the
        busy window. *)
    Variable Δ : duration.
    Hypothesis H_Δ_in_busy : t1 + Δ <= t2.

    (** We define the service needed by jobs belongings to other equal-priority
        tasks, that have higher-or-equal priority than [j]... *)
    Definition service_of_hp_jobs_from_other_ep_tasks (j : Job) (t1 t2 : instant) :=
      service_of_jobs sched (fun jhp => other_ep_task_hep_job jhp j)
        (arrivals_between arr_seq t1 t2) t1 t2.

    (** ...and show that it is equivalent to the cumulative interference
        incurred by [j] due to these jobs. *)
    Lemma cumulative_intf_ep_task_service_equiv :
      cumulative_interference_from_hep_jobs_from_other_ep_tasks arr_seq sched j t1 (t1 + Δ)
      = service_of_hp_jobs_from_other_ep_tasks  j t1 (t1 + Δ).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
cumulative_interference_from_hep_jobs_from_other_ep_tasks
  arr_seq sched j t1 (t1 + Δ) =
service_of_hp_jobs_from_other_ep_tasks j t1 (t1 + Δ)
    Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
cumulative_interference_from_hep_jobs_from_other_ep_tasks
  arr_seq sched j t1 (t1 + Δ) =
service_of_hp_jobs_from_other_ep_tasks j t1 (t1 + Δ)
      rewrite /cumulative_interference_from_hep_jobs_from_other_ep_tasks
        /service_of_hp_jobs_from_other_ep_tasks
        /hep_job_from_other_ep_task_interference.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
\sum_(t1 <= t < t1 + Δ)
   has (other_ep_task_hep_job^~ j)
     (served_jobs_at arr_seq sched t) =
service_of_jobs sched (other_ep_task_hep_job^~ j)
  (arrivals_between arr_seq t1 (t1 + Δ)) t1 (t1 + Δ)
      erewrite cumulative_pred_served_eq_service => //.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
classical.quiet_time arr_seq sched ?j t1
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
forall j' : Job,
other_ep_task_hep_job j' j -> hep_job j' ?j
      -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
classical.quiet_time arr_seq sched ?j t1 apply instantiated_quiet_time_equivalent_quiet_time => //.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
quiet_time sched j t1
        by apply H_busy_window.
      -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
forall j' : Job,
other_ep_task_hep_job j' j -> hep_job j' j by move => j' /andP[/andP[HEP _] _].
    Qed.

    (** Similarly, we define the service required by jobs belonging to other
        strictly higher-priority tasks, that have higher-or-equal priority than
        [j], ... *)
    Definition service_of_hp_jobs_from_other_hp_tasks (j : Job) (t1 t2 : instant) :=
      service_of_jobs sched (fun jhp => hp_task_hep_job jhp j)
        (arrivals_between arr_seq t1 t2) t1 t2.

    (** ... and show that it is equivalent to the cumulative interference
        incurred by [j] due to these jobs. *)
    Lemma cumulative_intf_hp_task_service_equiv :
      cumulative_interference_from_hep_jobs_from_hp_tasks arr_seq sched j t1 (t1 + Δ)
      = service_of_hp_jobs_from_other_hp_tasks j t1 (t1 + Δ).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
cumulative_interference_from_hep_jobs_from_hp_tasks
  arr_seq sched j t1 (t1 + Δ) =
service_of_hp_jobs_from_other_hp_tasks j t1 (t1 + Δ)
    Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
cumulative_interference_from_hep_jobs_from_hp_tasks
  arr_seq sched j t1 (t1 + Δ) =
service_of_hp_jobs_from_other_hp_tasks j t1 (t1 + Δ)
      rewrite /cumulative_interference_from_hep_jobs_from_hp_tasks
              /service_of_hp_jobs_from_other_hp_tasks
              /hep_job_from_hp_task_interference.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
\sum_(t1 <= t < t1 + Δ)
   has (hp_task_hep_job^~ j)
     (served_jobs_at arr_seq sched t) =
service_of_jobs sched (hp_task_hep_job^~ j)
  (arrivals_between arr_seq t1 (t1 + Δ)) t1 (t1 + Δ)
      erewrite cumulative_pred_served_eq_service => //.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
classical.quiet_time arr_seq sched ?j t1
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
forall j' : Job, hp_task_hep_job j' j -> hep_job j' ?j
      -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
classical.quiet_time arr_seq sched ?j t1 apply instantiated_quiet_time_equivalent_quiet_time => //.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
quiet_time sched j t1
        by apply H_busy_window.
      -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
forall j' : Job, hp_task_hep_job j' j -> hep_job j' j by move => j' /andP[HEP HP].
    Qed.

    (** Assume the arrival time of [j] relative to the busy window start is
        given by [A]. *)
    Let A := job_arrival j - t1.

    (** First, consider the case where [ep_task_intf_interval ≤ Δ]. *)
    Section ShortenRange.

      (** Consider any equal-priority task [tsk_o] distinct from [tsk].  Assume
          that [tsk_o] is in [ts]. *)
      Variable tsk_o : Task.
      Hypothesis H_tsko_in_ts : tsk_o \in ts.
      Hypothesis H_neq : tsk_o != tsk.
      Hypothesis H_task_priority : ep_task tsk tsk_o.

      (** We define a predicate to identify higher-or-equal-priority jobs coming
          from the task [tsk]. *)
      Let hep_jobs_from (tsk : Task) :=
        fun (jo : Job) =>
          hep_job jo j
          && ep_task (job_task jo) (job_task j)
          && (job_task jo == tsk).

      (** If [Δ] is greater than [ep_task_intf_interval] for [tsk_o] and [A], ... *)
      Hypothesis H_Δ_ge : (ep_task_intf_interval tsk_o A <= Δ%:R)%R.

      (** ... then the workload of jobs satisfying the predicate [hp_jobs_from]
              in the interval <<[t1,t1 + Δ)>> is bounded by the workload in the
              interval <<[t1, t1 + ep_task_intf_interval [tsk_o] [A])>>. *)
      Lemma total_workload_shorten_range :
        workload_of_jobs [eta hep_jobs_from tsk_o]
          (arrivals_between arr_seq t1 (t1 + Δ))
        <= workload_of_jobs [eta hep_jobs_from tsk_o]
            (arrivals_between arr_seq t1
               `|Num.max 0%R (t1%:R + ep_task_intf_interval tsk_o A)%R|).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
tsk_o: Task
H_tsko_in_ts: tsk_o \in ts
H_neq: tsk_o != tsk
H_task_priority: ep_task tsk tsk_o
hep_jobs_from:= fun (tsk : Task) (jo : Job) =>
hep_job jo j &&
ep_task (job_task jo) (job_task j) &&
(job_task jo == tsk): Task -> Job -> bool
H_Δ_ge: (ep_task_intf_interval tsk_o A <= Δ%:R)%R
workload_of_jobs [eta hep_jobs_from tsk_o]
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
workload_of_jobs [eta hep_jobs_from tsk_o]
  (arrivals_between arr_seq t1
     `|Num.max 0%R
         (t1%:R + ep_task_intf_interval tsk_o A)%R|)
      Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
tsk_o: Task
H_tsko_in_ts: tsk_o \in ts
H_neq: tsk_o != tsk
H_task_priority: ep_task tsk tsk_o
hep_jobs_from:= fun (tsk : Task) (jo : Job) =>
hep_job jo j &&
ep_task (job_task jo) (job_task j) &&
(job_task jo == tsk): Task -> Job -> bool
H_Δ_ge: (ep_task_intf_interval tsk_o A <= Δ%:R)%R
workload_of_jobs [eta hep_jobs_from tsk_o]
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
workload_of_jobs [eta hep_jobs_from tsk_o]
  (arrivals_between arr_seq t1
     `|Num.max 0%R
         (t1%:R + ep_task_intf_interval tsk_o A)%R|)
        have BOUNDED: `|Num.max 0%R (t1%:R + (ep_task_intf_interval tsk_o A))%R| <= t1 + Δ
          by clear - H_Δ_ge; lia.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
tsk_o: Task
H_tsko_in_ts: tsk_o \in ts
H_neq: tsk_o != tsk
H_task_priority: ep_task tsk tsk_o
hep_jobs_from:= fun (tsk : Task) (jo : Job) =>
hep_job jo j &&
ep_task (job_task jo) (job_task j) &&
(job_task jo == tsk): Task -> Job -> bool
H_Δ_ge: (ep_task_intf_interval tsk_o A <= Δ%:R)%R
BOUNDED: `|Num.max 0%R
    (t1%:R + ep_task_intf_interval tsk_o A)%R| <=
t1 + Δ
workload_of_jobs [eta hep_jobs_from tsk_o]
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
workload_of_jobs [eta hep_jobs_from tsk_o]
  (arrivals_between arr_seq t1
     `|Num.max 0%R
         (t1%:R + ep_task_intf_interval tsk_o A)%R|)
        rewrite /hep_jobs_from /hep_job /ELF.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
tsk_o: Task
H_tsko_in_ts: tsk_o \in ts
H_neq: tsk_o != tsk
H_task_priority: ep_task tsk tsk_o
hep_jobs_from:= fun (tsk : Task) (jo : Job) =>
hep_job jo j &&
ep_task (job_task jo) (job_task j) &&
(job_task jo == tsk): Task -> Job -> bool
H_Δ_ge: (ep_task_intf_interval tsk_o A <= Δ%:R)%R
BOUNDED: `|Num.max 0%R
    (t1%:R + ep_task_intf_interval tsk_o A)%R| <=
t1 + Δ
workload_of_jobs
  (fun x : Job =>
   (hp_task (job_task x) (job_task j)
    || hep_task (job_task x) (job_task j) &&
       hep_job x j) &&
   ep_task (job_task x) (job_task j) &&
   (job_task x == tsk_o))
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
workload_of_jobs
  (fun x : Job =>
   (hp_task (job_task x) (job_task j)
    || hep_task (job_task x) (job_task j) &&
       hep_job x j) &&
   ep_task (job_task x) (job_task j) &&
   (job_task x == tsk_o))
  (arrivals_between arr_seq t1
     `|Num.max 0%R
         (t1%:R + ep_task_intf_interval tsk_o A)%R|)
        rewrite (workload_of_jobs_nil_tail _ _ BOUNDED) // => j' IN' ARR'.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
tsk_o: Task
H_tsko_in_ts: tsk_o \in ts
H_neq: tsk_o != tsk
H_task_priority: ep_task tsk tsk_o
hep_jobs_from:= fun (tsk : Task) (jo : Job) =>
hep_job jo j &&
ep_task (job_task jo) (job_task j) &&
(job_task jo == tsk): Task -> Job -> bool
H_Δ_ge: (ep_task_intf_interval tsk_o A <= Δ%:R)%R
BOUNDED: `|Num.max 0%R
    (t1%:R + ep_task_intf_interval tsk_o A)%R| <=
t1 + Δ
j': Job
IN': j' \in arrivals_between arr_seq t1 (t1 + Δ)
ARR': `|Num.max 0%R
    (t1%:R + ep_task_intf_interval tsk_o A)%R| <=
job_arrival j'
~~
((hp_task (job_task j') (job_task j)
  || hep_task (job_task j') (job_task j) &&
     hep_job j' j) &&
 ep_task (job_task j') (job_task j) &&
 (job_task j' == tsk_o))
        apply/contraT => /negPn /andP [/andP [/orP[/andP [_ /negP+]|/andP [_ HEP]] /andP [_ _]] /eqP TSKo] //.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
tsk_o: Task
H_tsko_in_ts: tsk_o \in ts
H_neq: tsk_o != tsk
H_task_priority: ep_task tsk tsk_o
hep_jobs_from:= fun (tsk : Task) (jo : Job) =>
hep_job jo j &&
ep_task (job_task jo) (job_task j) &&
(job_task jo == tsk): Task -> Job -> bool
H_Δ_ge: (ep_task_intf_interval tsk_o A <= Δ%:R)%R
BOUNDED: `|Num.max 0%R
    (t1%:R + ep_task_intf_interval tsk_o A)%R| <=
t1 + Δ
j': Job
IN': j' \in arrivals_between arr_seq t1 (t1 + Δ)
ARR': `|Num.max 0%R
    (t1%:R + ep_task_intf_interval tsk_o A)%R| <=
job_arrival j'
HEP: hep_job j' j
TSKo: job_task j' = tsk_o
false
        move: ARR'; rewrite /ep_task_intf_interval  -TSKo.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
tsk_o: Task
H_tsko_in_ts: tsk_o \in ts
H_neq: tsk_o != tsk
H_task_priority: ep_task tsk tsk_o
hep_jobs_from:= fun (tsk : Task) (jo : Job) =>
hep_job jo j &&
ep_task (job_task jo) (job_task j) &&
(job_task jo == tsk): Task -> Job -> bool
H_Δ_ge: (ep_task_intf_interval tsk_o A <= Δ%:R)%R
BOUNDED: `|Num.max 0%R
    (t1%:R + ep_task_intf_interval tsk_o A)%R| <=
t1 + Δ
j': Job
IN': j' \in arrivals_between arr_seq t1 (t1 + Δ)
HEP: hep_job j' j
TSKo: job_task j' = tsk_o
`|Num.max 0%R
    (t1%:R +
     ((A + 1)%:R + task_priority_point tsk -
      task_priority_point (job_task j')))%R| <=
job_arrival j' -> false
        move: H_job_of_task => /eqP <-.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
tsk_o: Task
H_tsko_in_ts: tsk_o \in ts
H_neq: tsk_o != tsk
H_task_priority: ep_task tsk tsk_o
hep_jobs_from:= fun (tsk : Task) (jo : Job) =>
hep_job jo j &&
ep_task (job_task jo) (job_task j) &&
(job_task jo == tsk): Task -> Job -> bool
H_Δ_ge: (ep_task_intf_interval tsk_o A <= Δ%:R)%R
BOUNDED: `|Num.max 0%R
    (t1%:R + ep_task_intf_interval tsk_o A)%R| <=
t1 + Δ
j': Job
IN': j' \in arrivals_between arr_seq t1 (t1 + Δ)
HEP: hep_job j' j
TSKo: job_task j' = tsk_o
`|Num.max 0%R
    (t1%:R +
     ((A + 1)%:R + task_priority_point (job_task j) -
      task_priority_point (job_task j')))%R| <=
job_arrival j' -> false
        move: HEP; rewrite /hep_job /GEL /job_priority_point.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
tsk_o: Task
H_tsko_in_ts: tsk_o \in ts
H_neq: tsk_o != tsk
H_task_priority: ep_task tsk tsk_o
hep_jobs_from:= fun (tsk : Task) (jo : Job) =>
hep_job jo j &&
ep_task (job_task jo) (job_task j) &&
(job_task jo == tsk): Task -> Job -> bool
H_Δ_ge: (ep_task_intf_interval tsk_o A <= Δ%:R)%R
BOUNDED: `|Num.max 0%R
    (t1%:R + ep_task_intf_interval tsk_o A)%R| <=
t1 + Δ
j': Job
IN': j' \in arrivals_between arr_seq t1 (t1 + Δ)
TSKo: job_task j' = tsk_o
((job_arrival j')%:R +
 task_priority_point (job_task j') <=
 (job_arrival j)%:R + task_priority_point (job_task j))%R ->
`|Num.max 0%R
    (t1%:R +
     ((A + 1)%:R + task_priority_point (job_task j) -
      task_priority_point (job_task j')))%R| <=
job_arrival j' -> false
        by clear; lia.
      Qed.

    End ShortenRange.

    (** Then, we establish that the cumulative interference incurred by [j] due
        to all higher-or-equal-priority jobs from equal-priority tasks is
        bounded by the [bound_on_ep_workload], ... *)
    Lemma bound_on_ep_workload :
      cumulative_interference_from_hep_jobs_from_other_ep_tasks arr_seq sched j t1 (t1 + Δ)
        <= bound_on_total_ep_workload (job_arrival j - t1) Δ.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
cumulative_interference_from_hep_jobs_from_other_ep_tasks
  arr_seq sched j t1 (t1 + Δ) <=
bound_on_total_ep_workload (job_arrival j - t1) Δ
    Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
cumulative_interference_from_hep_jobs_from_other_ep_tasks
  arr_seq sched j t1 (t1 + Δ) <=
bound_on_total_ep_workload (job_arrival j - t1) Δ
      move: H_job_of_task => /eqP TSK.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
cumulative_interference_from_hep_jobs_from_other_ep_tasks
  arr_seq sched j t1 (t1 + Δ) <=
bound_on_total_ep_workload (job_arrival j - t1) Δ
      rewrite cumulative_intf_ep_task_service_equiv /service_of_hp_jobs_from_other_ep_tasks
        /service_of_jobs.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + Δ) | 
other_ep_task_hep_job j0 j)
   service_during sched j0 t1 (t1 + Δ) <=
bound_on_total_ep_workload (job_arrival j - t1) Δ
      apply: leq_trans; first by apply: service_of_jobs_le_workload.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
workload_of_jobs (other_ep_task_hep_job^~ j)
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
bound_on_total_ep_workload (job_arrival j - t1) Δ
      rewrite /bound_on_total_ep_workload /ep_task_hep_job
        /other_ep_task_hep_job /ep_task_hep_job.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
workload_of_jobs
  (fun j0 : Job =>
   hep_job j0 j && ep_task (job_task j0) (job_task j) &&
   (job_task j0 != job_task j))
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
\sum_(tsk_o <- ts | ep_task tsk tsk_o &&
                    (tsk_o != tsk))
   task_request_bound_function tsk_o
     (minn
        `|Num.max 0%R
            (ep_task_intf_interval tsk_o
               (job_arrival j - t1))| Δ)
      rewrite (hep_workload_from_other_ep_partitioned_by_tasks _ _ ts _  _ tsk) //.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
\sum_(tsk_o <- ts | other_ep_task tsk tsk_o)
   workload_of_jobs
     (fun j0 : Job =>
      hep_job_of_ep_other_task j j0 &&
      (job_task j0 == tsk_o))
     (arrivals_between arr_seq t1 (t1 + Δ)) <=
\sum_(tsk_o <- ts | ep_task tsk tsk_o &&
                    (tsk_o != tsk))
   task_request_bound_function tsk_o
     (minn
        `|Num.max 0%R
            (ep_task_intf_interval tsk_o
               (job_arrival j - t1))| Δ)
      apply: leq_sum_seq => tsk_o IN /andP[EP OTHER].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
workload_of_jobs
  (fun j0 : Job =>
   hep_job_of_ep_other_task j j0 &&
   (job_task j0 == tsk_o))
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
task_request_bound_function tsk_o
  (minn
     `|Num.max 0%R
         (ep_task_intf_interval tsk_o
            (job_arrival j - t1))| Δ)
      apply: leq_trans.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
workload_of_jobs
  (fun j0 : Job =>
   hep_job_of_ep_other_task j j0 &&
   (job_task j0 == tsk_o))
  (arrivals_between arr_seq t1 (t1 + Δ)) <= ?Goal
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
?Goal <=
task_request_bound_function tsk_o
  (minn
     `|Num.max 0%R
        (ep_task_intf_interval tsk_o
        (job_arrival j - t1))| Δ)
      {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
workload_of_jobs
  (fun j0 : Job =>
   hep_job_of_ep_other_task j j0 &&
   (job_task j0 == tsk_o))
  (arrivals_between arr_seq t1 (t1 + Δ)) <= ?Goal apply: (workload_of_jobs_weaken _
                  (fun j0 : Job => hep_job j0 j
                                && ep_task (job_task j0) (job_task j)
                                && (job_task j0 == tsk_o)))
             => i /andP[/andP[? ?] ?].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
i: Job
_a_: hep_job i j && ep_task (job_task i) (job_task j)
_b_: job_task i != job_task j
_b1_: job_task i == tsk_o
hep_job i j && ep_task (job_task i) (job_task j) &&
(job_task i == tsk_o)
        by apply/andP; split. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
workload_of_jobs
  (fun j0 : Job =>
   hep_job j0 j && ep_task (job_task j0) (job_task j) &&
   (job_task j0 == tsk_o))
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
task_request_bound_function tsk_o
  (minn
     `|Num.max 0%R
         (ep_task_intf_interval tsk_o
            (job_arrival j - t1))| Δ)
      {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
workload_of_jobs
  (fun j0 : Job =>
   hep_job j0 j && ep_task (job_task j0) (job_task j) &&
   (job_task j0 == tsk_o))
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
task_request_bound_function tsk_o
  (minn
     `|Num.max 0%R
         (ep_task_intf_interval tsk_o
            (job_arrival j - t1))| Δ) have [LEQ|GT] := leqP Δ `|Num.max 0%R (ep_task_intf_interval tsk_o A)%R|; [| apply ltnW in GT].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
LEQ: Δ <=
`|Num.max 0%R (ep_task_intf_interval tsk_o A)|
workload_of_jobs
  (fun j0 : Job =>
   hep_job j0 j && ep_task (job_task j0) (job_task j) &&
   (job_task j0 == tsk_o))
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
task_request_bound_function tsk_o Δ
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
workload_of_jobs
  (fun j0 : Job =>
   hep_job j0 j && ep_task (job_task j0) (job_task j) &&
   (job_task j0 == tsk_o))
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
task_request_bound_function tsk_o
  `|Num.max 0%R (ep_task_intf_interval tsk_o A)|
        {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
LEQ: Δ <=
`|Num.max 0%R (ep_task_intf_interval tsk_o A)|
workload_of_jobs
  (fun j0 : Job =>
   hep_job j0 j && ep_task (job_task j0) (job_task j) &&
   (job_task j0 == tsk_o))
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
task_request_bound_function tsk_o Δ apply: leq_trans; last by eapply task_workload_le_task_rbf.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
LEQ: Δ <=
`|Num.max 0%R (ep_task_intf_interval tsk_o A)|
workload_of_jobs
  (fun j0 : Job =>
   hep_job j0 j && ep_task (job_task j0) (job_task j) &&
   (job_task j0 == tsk_o))
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
workload_of_jobs (job_of_task tsk_o)
  (arrivals_between arr_seq ?t (?t + Δ))
          apply: (workload_of_jobs_weaken _ (fun j0 => (job_task j0 == tsk_o))).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
LEQ: Δ <=
`|Num.max 0%R (ep_task_intf_interval tsk_o A)|
forall j0 : Job,
hep_job j0 j && ep_task (job_task j0) (job_task j) &&
(job_task j0 == tsk_o) -> job_task j0 == tsk_o
          by move => j'/ andP[_ EXACT]. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
workload_of_jobs
  (fun j0 : Job =>
   hep_job j0 j && ep_task (job_task j0) (job_task j) &&
   (job_task j0 == tsk_o))
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
task_request_bound_function tsk_o
  `|Num.max 0%R (ep_task_intf_interval tsk_o A)|
        {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
workload_of_jobs
  (fun j0 : Job =>
   hep_job j0 j && ep_task (job_task j0) (job_task j) &&
   (job_task j0 == tsk_o))
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
task_request_bound_function tsk_o
  `|Num.max 0%R (ep_task_intf_interval tsk_o A)| apply: leq_trans;
            first by apply: total_workload_shorten_range => //; clear - GT H_Δ_in_busy; lia.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
workload_of_jobs
  [eta (fun (tsk : Task) (jo : Job) =>
        hep_job jo j &&
        ep_task (job_task jo) (job_task j) &&
        (job_task jo == tsk)) tsk_o]
  (arrivals_between arr_seq t1
     `|Num.max 0%R
         (t1%:R + ep_task_intf_interval tsk_o A)%R|) <=
task_request_bound_function tsk_o
  `|Num.max 0%R (ep_task_intf_interval tsk_o A)|
          rewrite (workload_of_jobs_equiv_pred _ _ (fun jo : Job => hep_job jo j && (job_task jo == tsk_o))).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
workload_of_jobs
  (fun jo : Job =>
   hep_job jo j && (job_task jo == tsk_o))
  (arrivals_between arr_seq t1
     `|Num.max 0%R
         (t1%:R + ep_task_intf_interval tsk_o A)%R|) <=
task_request_bound_function tsk_o
  `|Num.max 0%R (ep_task_intf_interval tsk_o A)|
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
{in arrivals_between arr_seq t1
      `|Num.max 0%R
        (t1%:R + ep_task_intf_interval tsk_o A)%R|,
  (fun x : Job =>
   hep_job x j && ep_task (job_task x) (job_task j) &&
   (job_task x == tsk_o)) =1
  (fun jo : Job =>
   hep_job jo j && (job_task jo == tsk_o))}
          {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
workload_of_jobs
  (fun jo : Job =>
   hep_job jo j && (job_task jo == tsk_o))
  (arrivals_between arr_seq t1
     `|Num.max 0%R
         (t1%:R + ep_task_intf_interval tsk_o A)%R|) <=
task_request_bound_function tsk_o
  `|Num.max 0%R (ep_task_intf_interval tsk_o A)| case EQ: (0 <= ep_task_intf_interval tsk_o A)%R;
              last by rewrite arrivals_between_geq; [rewrite workload_of_jobs0|clear - EQ; lia].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
EQ: (0 <= ep_task_intf_interval tsk_o A)%R = true
workload_of_jobs
  (fun jo : Job =>
   hep_job jo j && (job_task jo == tsk_o))
  (arrivals_between arr_seq t1
     `|Num.max 0%R
         (t1%:R + ep_task_intf_interval tsk_o A)%R|) <=
task_request_bound_function tsk_o
  `|Num.max 0%R (ep_task_intf_interval tsk_o A)|
            have -> : `|Num.max 0%R (t1%:R + ep_task_intf_interval tsk_o A)%R|
                     = t1 + `|Num.max 0%R (ep_task_intf_interval tsk_o A)|
              by clear -EQ; lia.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
EQ: (0 <= ep_task_intf_interval tsk_o A)%R = true
workload_of_jobs
  (fun jo : Job =>
   hep_job jo j && (job_task jo == tsk_o))
  (arrivals_between arr_seq t1
     (t1 +
      `|Num.max 0%R (ep_task_intf_interval tsk_o A)|)) <=
task_request_bound_function tsk_o
  `|Num.max 0%R (ep_task_intf_interval tsk_o A)|
            by apply: (workload_le_rbf' arr_seq tsk_o _ _  t1 _ (fun jo => hep_job jo j)). }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
{in arrivals_between arr_seq t1
      `|Num.max 0%R
          (t1%:R + ep_task_intf_interval tsk_o A)%R|,
  (fun x : Job =>
   hep_job x j && ep_task (job_task x) (job_task j) &&
   (job_task x == tsk_o)) =1
  (fun jo : Job =>
   hep_job jo j && (job_task jo == tsk_o))}
          {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
{in arrivals_between arr_seq t1
      `|Num.max 0%R
          (t1%:R + ep_task_intf_interval tsk_o A)%R|,
  (fun x : Job =>
   hep_job x j && ep_task (job_task x) (job_task j) &&
   (job_task x == tsk_o)) =1
  (fun jo : Job =>
   hep_job jo j && (job_task jo == tsk_o))} move => j' IN1.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
j': Job
IN1: j'
  \in arrivals_between arr_seq t1
        `|
        Num.max 0%R
        (t1%:R + ep_task_intf_interval tsk_o A)%R|
hep_job j' j && ep_task (job_task j') (job_task j) &&
(job_task j' == tsk_o) =
hep_job j' j && (job_task j' == tsk_o)
            have [TSK'|_] := (eqVneq  (job_task j') tsk_o).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
j': Job
IN1: j'
  \in arrivals_between arr_seq t1
        `|
        Num.max 0%R
        (t1%:R + ep_task_intf_interval tsk_o A)%R|
TSK': job_task j' = tsk_o
hep_job j' j && ep_task (job_task j') (job_task j) &&
true = hep_job j' j && true
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
j': Job
IN1: j'
  \in arrivals_between arr_seq t1
        `|
        Num.max 0%R
        (t1%:R + ep_task_intf_interval tsk_o A)%R|
hep_job j' j && ep_task (job_task j') (job_task j) &&
false = hep_job j' j && false
            -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
j': Job
IN1: j'
  \in arrivals_between arr_seq t1
        `|
        Num.max 0%R
        (t1%:R + ep_task_intf_interval tsk_o A)%R|
TSK': job_task j' = tsk_o
hep_job j' j && ep_task (job_task j') (job_task j) &&
true = hep_job j' j && true by rewrite !andbT TSK TSK' ep_task_sym EP andbT.
            -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
TSK: job_task j = tsk
tsk_o: Task
IN: tsk_o \in ts
EP: ep_task tsk tsk_o
OTHER: tsk_o != tsk
GT: `|Num.max 0%R (ep_task_intf_interval tsk_o A)| <=
Δ
j': Job
IN1: j'
  \in arrivals_between arr_seq t1
        `|
        Num.max 0%R
        (t1%:R + ep_task_intf_interval tsk_o A)%R|
hep_job j' j && ep_task (job_task j') (job_task j) &&
false = hep_job j' j && false by rewrite !andbF. }}}
    Qed.

    (** ... and that the cumulative interference incurred by [j] due to all
        higher-or-equal priority jobs from higher-priority tasks is bounded by
        the [total_hp_rbf]]. *)
    Lemma bound_on_hp_workload :
      cumulative_interference_from_hep_jobs_from_hp_tasks arr_seq sched j t1 (t1 + Δ)
        <= total_hp_rbf Δ.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
cumulative_interference_from_hep_jobs_from_hp_tasks
  arr_seq sched j t1 (t1 + Δ) <= total_hp_rbf Δ
    Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
cumulative_interference_from_hep_jobs_from_hp_tasks
  arr_seq sched j t1 (t1 + Δ) <= total_hp_rbf Δ
      rewrite cumulative_intf_hp_task_service_equiv /total_hp_rbf.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
service_of_hp_jobs_from_other_hp_tasks j t1 (t1 + Δ) <=
total_hp_request_bound_function_FP ts tsk Δ
      apply: leq_trans;
        first by apply: service_of_jobs_le_workload.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
workload_of_jobs (hp_task_hep_job^~ j)
  (arrivals_between arr_seq t1 (t1 + Δ)) <=
total_hp_request_bound_function_FP ts tsk Δ
      rewrite /workload_of_jobs /total_hp_request_bound_function_FP.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + Δ) | 
hp_task_hep_job j0 j) job_cost j0 <=
\sum_(tsk_other <- ts | hp_task tsk_other tsk)
   task_request_bound_function tsk_other Δ
      rewrite [X in X <= _](eq_big (fun j0 => hp_task (job_task j0) tsk) job_cost) //;
        first by apply: sum_of_jobs_le_sum_rbf; eauto.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
hp_task_hep_job^~ j =1
(fun j0 : Job => hp_task (job_task j0) tsk)
      rewrite /hp_task_hep_job  => j'.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
j': Job
hep_job j' j && hp_task (job_task j') (job_task j) =
hp_task (job_task j') tsk
      rewrite andb_idl => [|?].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
j': Job
hp_task (job_task j') (job_task j) =
hp_task (job_task j') tsk
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
j': Job
_Hyp_: hp_task (job_task j') (job_task j)
hep_job j' j
      -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
j': Job
hp_task (job_task j') (job_task j) =
hp_task (job_task j') tsk by move: H_job_of_task => /eqP ->.
      -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
j: Job
H_job_of_task: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
H_j_in_arr_seq: arrives_in arr_seq j
t1, t2: instant
H_busy_window: busy_interval sched j t1 t2
Δ: duration
H_Δ_in_busy: t1 + Δ <= t2
A:= job_arrival j - t1: nat
j': Job
_Hyp_: hp_task (job_task j') (job_task j)
hep_job j' j by apply/orP; left.
    Qed.

  End BoundingIBF.

  (** Finally, we use the above two lemmas to prove that [task_IBF] bounds the
      interference incurred by [tsk]. *)
  Lemma instantiated_task_interference_is_bounded :
    task_interference_is_bounded_by arr_seq sched tsk task_IBF.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
task_interference_is_bounded_by arr_seq sched tsk
  task_IBF
  Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
task_interference_is_bounded_by arr_seq sched tsk
  task_IBF
    move => t1 t2 Δ j ARR TSK BUSY LT NCOMPL A OFF.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
NCOMPL: ~~ completed_by sched j (t1 + Δ)
A: nat
OFF: relative_arrival_time_of_job_is_A sched j A
cumul_cond_interference (nonself arr_seq sched) j t1
  (t1 + Δ) <= task_IBF A Δ
    move: (OFF _ _ BUSY) => EQA; subst A.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
NCOMPL: ~~ completed_by sched j (t1 + Δ)
OFF: relative_arrival_time_of_job_is_A sched j
  (job_arrival j - t1)
cumul_cond_interference (nonself arr_seq sched) j t1
  (t1 + Δ) <= task_IBF (job_arrival j - t1) Δ
    have [ZERO|POS] := posnP (job_cost j).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
NCOMPL: ~~ completed_by sched j (t1 + Δ)
OFF: relative_arrival_time_of_job_is_A sched j
  (job_arrival j - t1)
ZERO: job_cost j = 0
cumul_cond_interference (nonself arr_seq sched) j t1
  (t1 + Δ) <= task_IBF (job_arrival j - t1) Δ
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
NCOMPL: ~~ completed_by sched j (t1 + Δ)
OFF: relative_arrival_time_of_job_is_A sched j
  (job_arrival j - t1)
POS: 0 < job_cost j
cumul_cond_interference 
  (nonself arr_seq sched) j t1 
  (t1 + Δ) <= 
task_IBF (job_arrival j - t1) Δ
    -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
NCOMPL: ~~ completed_by sched j (t1 + Δ)
OFF: relative_arrival_time_of_job_is_A sched j
  (job_arrival j - t1)
ZERO: job_cost j = 0
cumul_cond_interference (nonself arr_seq sched) j t1
  (t1 + Δ) <= task_IBF (job_arrival j - t1) Δ exfalso; move: NCOMPL => /negP COMPL; apply: COMPL.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
OFF: relative_arrival_time_of_job_is_A sched j
  (job_arrival j - t1)
ZERO: job_cost j = 0
completed_by sched j (t1 + Δ)
      by rewrite /completed_by /completed_by ZERO.
    -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
NCOMPL: ~~ completed_by sched j (t1 + Δ)
OFF: relative_arrival_time_of_job_is_A sched j
  (job_arrival j - t1)
POS: 0 < job_cost j
cumul_cond_interference (nonself arr_seq sched) j t1
  (t1 + Δ) <= task_IBF (job_arrival j - t1) Δ rewrite -/(cumul_task_interference _ _ _ _ _).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
NCOMPL: ~~ completed_by sched j (t1 + Δ)
OFF: relative_arrival_time_of_job_is_A sched j
  (job_arrival j - t1)
POS: 0 < job_cost j
cumul_task_interference arr_seq sched j t1 (t1 + Δ) <=
task_IBF (job_arrival j - t1) Δ
      rewrite (leqRW (cumulative_task_interference_split _ _ _ _ _ _ _ _ _ _ _ _ _ )) //=.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
NCOMPL: ~~ completed_by sched j (t1 + Δ)
OFF: relative_arrival_time_of_job_is_A sched j
  (job_arrival j - t1)
POS: 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
  (t1 + Δ) +
cumulative_another_task_hep_job_interference arr_seq
  sched j t1 (t1 + Δ) <=
task_IBF (job_arrival j - t1) Δ
      rewrite /task_IBF -addnA.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
NCOMPL: ~~ completed_by sched j (t1 + Δ)
OFF: relative_arrival_time_of_job_is_A sched j
  (job_arrival j - t1)
POS: 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
  (t1 + Δ) +
cumulative_another_task_hep_job_interference arr_seq
  sched j t1 (t1 + Δ) <=
priority_inversion_bound (job_arrival j - t1) +
(bound_on_total_ep_workload (job_arrival j - t1) Δ +
 total_hp_rbf Δ)
      apply: leq_add;
        first by apply: cumulative_priority_inversion_is_bounded priority_inversion_is_bounded =>//.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
NCOMPL: ~~ completed_by sched j (t1 + Δ)
OFF: relative_arrival_time_of_job_is_A sched j
  (job_arrival j - t1)
POS: 0 < job_cost j
cumulative_another_task_hep_job_interference arr_seq
  sched j t1 (t1 + Δ) <=
bound_on_total_ep_workload (job_arrival j - t1) Δ +
total_hp_rbf Δ
      rewrite cumulative_hep_interference_split_tasks_new // addnC.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
NCOMPL: ~~ completed_by sched j (t1 + Δ)
OFF: relative_arrival_time_of_job_is_A sched j
  (job_arrival j - t1)
POS: 0 < job_cost j
cumulative_interference_from_hep_jobs_from_other_ep_tasks
  arr_seq sched j t1 (t1 + Δ) +
cumulative_interference_from_hep_jobs_from_hp_tasks
  arr_seq sched j t1 (t1 + Δ) <=
bound_on_total_ep_workload (job_arrival j - t1) Δ +
total_hp_rbf Δ
      apply: leq_add.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
NCOMPL: ~~ completed_by sched j (t1 + Δ)
OFF: relative_arrival_time_of_job_is_A sched j
  (job_arrival j - t1)
POS: 0 < job_cost j
cumulative_interference_from_hep_jobs_from_other_ep_tasks
  arr_seq sched j t1 (t1 + Δ) <=
bound_on_total_ep_workload (job_arrival j - t1) Δ
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
NCOMPL: ~~ completed_by sched j (t1 + Δ)
OFF: relative_arrival_time_of_job_is_A sched j
  (job_arrival j - t1)
POS: 0 < job_cost j
cumulative_interference_from_hep_jobs_from_hp_tasks
  arr_seq sched j t1 
  (t1 + Δ) <= 
total_hp_rbf Δ
      +
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
NCOMPL: ~~ completed_by sched j (t1 + Δ)
OFF: relative_arrival_time_of_job_is_A sched j
  (job_arrival j - t1)
POS: 0 < job_cost j
cumulative_interference_from_hep_jobs_from_other_ep_tasks
  arr_seq sched j t1 (t1 + Δ) <=
bound_on_total_ep_workload (job_arrival j - t1) Δ by apply: bound_on_ep_workload.
      +
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
t1, t2: instant
Δ: nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval sched j t1 t2
LT: t1 + Δ < t2
NCOMPL: ~~ completed_by sched j (t1 + Δ)
OFF: relative_arrival_time_of_job_is_A sched j
  (job_arrival j - t1)
POS: 0 < job_cost j
cumulative_interference_from_hep_jobs_from_hp_tasks
  arr_seq sched j t1 (t1 + Δ) <= total_hp_rbf Δ by apply: bound_on_hp_workload.
  Qed.

  (** ** G. Defining the Search Space *)

  (** In this section, we define the concrete search space for [ELF]. Then, we
      prove that, if a given [A] is in the abstract search space, then it is
      also included in the concrete search space. *)

  (** For [tsk], the total interference bound is defined as the sum of the interference due to
      - (1) jobs belonging to the same task (self interference) and
      - (2) jobs belonging to other tasks [task_IBF]. *)
  Let total_interference_bound (A Δ : duration) :=
    task_request_bound_function tsk (A + ε) - task_cost tsk + task_IBF A Δ.

  (** In the case of ELF, of the four terms that constitute the total
      interference bound, only the [priority_inversion_bound], task RBF and the
      bound on total equal-priority workload are dependent on the offset [A]. *)

  (** Therefore, in order to define the concrete search space, we define
      predicates that capture when these values change for successive
      values of the offset [A]. *)
  Definition task_rbf_changes_at (A : duration) :=
    task_request_bound_function tsk  A != task_request_bound_function tsk (A + ε).

  Definition bound_on_total_ep_workload_changes_at A :=
    has (fun tsk_o => ep_task tsk tsk_o
                   && (tsk_o != tsk)
                   && (ep_task_intf_interval tsk_o (A - ε) != ep_task_intf_interval tsk_o A))
      ts.

  Definition priority_inversion_changes_at (A : duration) :=
    priority_inversion_bound (A - ε) != priority_inversion_bound A.

  (** Finally, we define the concrete search space as the set containing all
      points less than [L] at which any of the bounds on priority inversion,
      task RBF, or total equal-priority workload changes. *)
  Definition is_in_search_space (A : duration) :=
    (A < L) && (priority_inversion_changes_at A
                || task_rbf_changes_at A
                || bound_on_total_ep_workload_changes_at A).

  (** In this section, we prove that, for any job [j] of task [tsk], if [A] is
      in the abstract search space, then it is also in the concrete search space. *)
  Section ConcreteSearchSpace.

    (** To rule out pathological cases with the concrete search space,
        we assume that the task cost is positive and the arrival curve
        is non-pathological. *)
    Hypothesis H_task_cost_pos : 0 < task_cost tsk.
    Hypothesis H_arrival_curve_pos : 0 < max_arrivals tsk ε.

    (** Any point [A] in the abstract search space... *)
    Variable A : duration.
    Hypothesis H_A_is_in_abstract_search_space :
      search_space.is_in_search_space L total_interference_bound A.

    (** ... is also in the concrete search space. *)
    Lemma A_is_in_concrete_search_space :
      is_in_search_space A.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
is_in_search_space A
    Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
is_in_search_space A
      move: H_A_is_in_abstract_search_space  => [-> | [/andP [POSA LTL] [x [LTx INSP2]]]]; apply/andP; split => //.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
priority_inversion_changes_at 0
|| task_rbf_changes_at 0
|| bound_on_total_ep_workload_changes_at 0
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound (A - 1) x <>
total_interference_bound A x
priority_inversion_changes_at A
|| task_rbf_changes_at A
|| bound_on_total_ep_workload_changes_at A
      {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
priority_inversion_changes_at 0
|| task_rbf_changes_at 0
|| bound_on_total_ep_workload_changes_at 0 apply/orP; left; apply/orP; right.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
task_rbf_changes_at 0
        rewrite /task_rbf_changes_at task_rbf_0_zero // eq_sym -lt0n add0n.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
0 < task_request_bound_function tsk 1
        by apply task_rbf_epsilon_gt_0 => //.
      }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound (A - 1) x <>
total_interference_bound A x
priority_inversion_changes_at A
|| task_rbf_changes_at A
|| bound_on_total_ep_workload_changes_at A
      apply: contraT; rewrite !negb_or => /andP [/andP [/negPn/eqP PI /negPn/eqP RBF]  WL].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound (A - 1) x <>
total_interference_bound A x
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
WL: ~~ bound_on_total_ep_workload_changes_at A
false
      exfalso; apply: INSP2.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
WL: ~~ bound_on_total_ep_workload_changes_at A
total_interference_bound (A - 1) x =
total_interference_bound A x
      rewrite /total_interference_bound subnK // RBF.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
WL: ~~ bound_on_total_ep_workload_changes_at A
task_request_bound_function tsk (A + 1) -
task_cost tsk + task_IBF (A - 1) x =
task_request_bound_function tsk (A + 1) -
task_cost tsk + task_IBF A x
      apply/eqP; rewrite eqn_add2l /task_IBF PI !addnACl eqn_add2r.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
WL: ~~ bound_on_total_ep_workload_changes_at A
bound_on_total_ep_workload (A - 1) x ==
bound_on_total_ep_workload A x
      rewrite /bound_on_total_ep_workload.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
WL: ~~ bound_on_total_ep_workload_changes_at A
\sum_(tsk_o <- ts | ep_task tsk tsk_o &&
                    (tsk_o != tsk))
   task_request_bound_function tsk_o
     (minn
        `|Num.max 0%R
            (ep_task_intf_interval tsk_o (A - 1))| x) ==
\sum_(tsk_o <- ts | ep_task tsk tsk_o &&
                    (tsk_o != tsk))
   task_request_bound_function tsk_o
     (minn
        `|Num.max 0%R (ep_task_intf_interval tsk_o A)|
        x)
      apply/eqP; rewrite big_seq_cond [RHS]big_seq_cond.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
WL: ~~ bound_on_total_ep_workload_changes_at A
\sum_(i <- ts | [&& i \in ts, ep_task tsk i
                  & i != tsk])
   task_request_bound_function i
     (minn
        `|Num.max 0%R
            (ep_task_intf_interval i (A - 1))| x) =
\sum_(i <- ts | [&& i \in ts, ep_task tsk i
                  & i != tsk])
   task_request_bound_function i
     (minn `|Num.max 0%R (ep_task_intf_interval i A)|
        x)
      apply: eq_big => // tsk_i /andP [TS OTHER].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
WL: ~~ bound_on_total_ep_workload_changes_at A
tsk_i: Task
TS: tsk_i \in ts
OTHER: ep_task tsk tsk_i && (tsk_i != tsk)
task_request_bound_function tsk_i
  (minn
     `|Num.max 0%R
         (ep_task_intf_interval tsk_i (A - 1))| x) =
task_request_bound_function tsk_i
  (minn `|Num.max 0%R (ep_task_intf_interval tsk_i A)|
     x)
      move: WL; rewrite /bound_on_total_ep_workload_changes_at => /hasPn WL.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
tsk_i: Task
TS: tsk_i \in ts
OTHER: ep_task tsk tsk_i && (tsk_i != tsk)
WL: {in ts,
  forall x : Task,
  ~~
  (ep_task tsk x && (x != tsk) &&
   (ep_task_intf_interval x (A - 1)
    != ep_task_intf_interval x A))}
task_request_bound_function tsk_i
  (minn
     `|Num.max 0%R
         (ep_task_intf_interval tsk_i (A - 1))| x) =
task_request_bound_function tsk_i
  (minn `|Num.max 0%R (ep_task_intf_interval tsk_i A)|
     x)
      move: {WL} (WL tsk_i TS) =>  /nandP [/negbTE EQ|/negPn/eqP WL].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
tsk_i: Task
TS: tsk_i \in ts
OTHER: ep_task tsk tsk_i && (tsk_i != tsk)
EQ: ep_task tsk tsk_i && (tsk_i != tsk) = false
task_request_bound_function tsk_i
  (minn
     `|Num.max 0%R
         (ep_task_intf_interval tsk_i (A - 1))| x) =
task_request_bound_function tsk_i
  (minn `|Num.max 0%R (ep_task_intf_interval tsk_i A)|
     x)
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
tsk_i: Task
TS: tsk_i \in ts
OTHER: ep_task tsk tsk_i && (tsk_i != tsk)
WL: ep_task_intf_interval tsk_i (A - 1) =
ep_task_intf_interval tsk_i A
task_request_bound_function tsk_i
  (minn
     `|Num.max 0%R
        (ep_task_intf_interval tsk_i (A - 1))| x) =
task_request_bound_function tsk_i
  (minn `|
     Num.max 0%R (ep_task_intf_interval tsk_i A)| x)
      {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
tsk_i: Task
TS: tsk_i \in ts
OTHER: ep_task tsk tsk_i && (tsk_i != tsk)
EQ: ep_task tsk tsk_i && (tsk_i != tsk) = false
task_request_bound_function tsk_i
  (minn
     `|Num.max 0%R
         (ep_task_intf_interval tsk_i (A - 1))| x) =
task_request_bound_function tsk_i
  (minn `|Num.max 0%R (ep_task_intf_interval tsk_i A)|
     x) by move: OTHER; rewrite EQ. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
tsk_i: Task
TS: tsk_i \in ts
OTHER: ep_task tsk tsk_i && (tsk_i != tsk)
WL: ep_task_intf_interval tsk_i (A - 1) =
ep_task_intf_interval tsk_i A
task_request_bound_function tsk_i
  (minn
     `|Num.max 0%R
         (ep_task_intf_interval tsk_i (A - 1))| x) =
task_request_bound_function tsk_i
  (minn `|Num.max 0%R (ep_task_intf_interval tsk_i A)|
     x)
      have [leq_x|gtn_x] := leqP `|Num.max 0%R (ep_task_intf_interval tsk_i A)| x.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
tsk_i: Task
TS: tsk_i \in ts
OTHER: ep_task tsk tsk_i && (tsk_i != tsk)
WL: ep_task_intf_interval tsk_i (A - 1) =
ep_task_intf_interval tsk_i A
leq_x: `|Num.max 0%R (ep_task_intf_interval tsk_i A)| <=
x
task_request_bound_function tsk_i
  (minn
     `|Num.max 0%R
         (ep_task_intf_interval tsk_i (A - 1))| x) =
task_request_bound_function tsk_i
  `|Num.max 0%R (ep_task_intf_interval tsk_i A)|
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
tsk_i: Task
TS: tsk_i \in ts
OTHER: ep_task tsk tsk_i && (tsk_i != tsk)
WL: ep_task_intf_interval tsk_i (A - 1) =
ep_task_intf_interval tsk_i A
gtn_x: x <
`|Num.max 0%R (ep_task_intf_interval tsk_i A)|
task_request_bound_function tsk_i
  (minn
     `|Num.max 0%R
        (ep_task_intf_interval tsk_i (A - 1))| x) =
task_request_bound_function tsk_i x
      -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
tsk_i: Task
TS: tsk_i \in ts
OTHER: ep_task tsk tsk_i && (tsk_i != tsk)
WL: ep_task_intf_interval tsk_i (A - 1) =
ep_task_intf_interval tsk_i A
leq_x: `|Num.max 0%R (ep_task_intf_interval tsk_i A)| <=
x
task_request_bound_function tsk_i
  (minn
     `|Num.max 0%R
         (ep_task_intf_interval tsk_i (A - 1))| x) =
task_request_bound_function tsk_i
  `|Num.max 0%R (ep_task_intf_interval tsk_i A)| by rewrite WL (minn_idPl leq_x).
      -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
PI: priority_inversion_bound (A - 1) =
priority_inversion_bound A
RBF: task_request_bound_function tsk A =
task_request_bound_function tsk (A + 1)
tsk_i: Task
TS: tsk_i \in ts
OTHER: ep_task tsk tsk_i && (tsk_i != tsk)
WL: ep_task_intf_interval tsk_i (A - 1) =
ep_task_intf_interval tsk_i A
gtn_x: x <
`|Num.max 0%R (ep_task_intf_interval tsk_i A)|
task_request_bound_function tsk_i
  (minn
     `|Num.max 0%R
         (ep_task_intf_interval tsk_i (A - 1))| x) =
task_request_bound_function tsk_i x by rewrite WL (minn_idPr (ltnW gtn_x)).
    Qed.

  End ConcreteSearchSpace.

  (** ** H. The Response-Time Bound [R] *)

  (** Finally, we define the response-time bound [R] as the maximum offset by
      which any job [j] of task [tsk] has completed. *)
  Variable R : duration.
  Hypothesis H_R_is_maximum :
    forall (A : duration),
      is_in_search_space A ->
      exists (F : duration),
        A + F >= priority_inversion_bound A
                + bound_on_total_ep_workload A (A + F)
                + total_hp_rbf (A + F)
                + (task_request_bound_function tsk (A + ε)
                - (task_cost tsk - task_rtct tsk))
          /\ R >= F + (task_cost tsk - task_rtct tsk).

  Section ResponseTimeReccurence.

    (** To rule out pathological cases with the [H_R_is_maximum]
        equation (such as [task_cost tsk] being greater than [task_rbf
        (A + ε)]), we assume that the arrival curve is
        non-pathological. *)
    Hypothesis H_task_cost_pos : 0 < task_cost tsk.
    Hypothesis H_arrival_curve_pos : 0 < max_arrivals tsk ε.

    (** We have established that if [A] is in the abstract search then it is in
        the concrete search space, too.  We also know by assumption that,
        if [A] is in the concrete search space, then there exists an [R] that
        satisfies [H_R_is_maximum]. *)

    (** Using these facts, here we prove that if, [A] is in the abstract search space, ... *)
    Let is_in_search_space := search_space.is_in_search_space L total_interference_bound.

    (** ... then there exists a solution to the response-time equation as stated
            in the aRTA.  *)
    Corollary response_time_recurrence_solution_exists :
      forall A,
        is_in_search_space A ->
        exists F,
          A + F >= task_request_bound_function tsk (A + ε)
                  - (task_cost tsk - task_rtct tsk)
                  + task_IBF A (A + F)
          /\ R >= F + (task_cost tsk - task_rtct tsk).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
AbstractRTAforELFwithArrivalCurves.is_in_search_space
  A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
is_in_search_space:= search_space.is_in_search_space L
  total_interference_bound: nat -> Prop
forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_request_bound_function tsk (A + 1) -
  (task_cost tsk - task_rtct tsk) + task_IBF A (A + F) <=
  A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
    Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
AbstractRTAforELFwithArrivalCurves.is_in_search_space
  A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
is_in_search_space:= search_space.is_in_search_space L
  total_interference_bound: nat -> Prop
forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_request_bound_function tsk (A + 1) -
  (task_cost tsk - task_rtct tsk) + task_IBF A (A + F) <=
  A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
      move => A0 IN.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
AbstractRTAforELFwithArrivalCurves.is_in_search_space
  A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
is_in_search_space:= search_space.is_in_search_space L
  total_interference_bound: nat -> Prop
A0: nat
IN: is_in_search_space A0
exists F : nat,
  task_request_bound_function tsk (A0 + 1) -
  (task_cost tsk - task_rtct tsk) +
  task_IBF A0 (A0 + F) <= A0 + F /\
  F + (task_cost tsk - task_rtct tsk) <= R
      have [|F [FIX NEQ]] := H_R_is_maximum A0;
        first by apply: A_is_in_concrete_search_space.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
AbstractRTAforELFwithArrivalCurves.is_in_search_space
  A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
is_in_search_space:= search_space.is_in_search_space L
  total_interference_bound: nat -> Prop
A0: nat
IN: is_in_search_space A0
F: duration
FIX: priority_inversion_bound A0 +
bound_on_total_ep_workload A0 (A0 + F) +
total_hp_rbf (A0 + F) +
(task_request_bound_function tsk (A0 + 1) -
 (task_cost tsk - task_rtct tsk)) <= 
A0 + F
NEQ: F + (task_cost tsk - task_rtct tsk) <= R
exists F : nat,
  task_request_bound_function tsk (A0 + 1) -
  (task_cost tsk - task_rtct tsk) +
  task_IBF A0 (A0 + F) <= A0 + F /\
  F + (task_cost tsk - task_rtct tsk) <= R
      exists F; split =>//.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
AbstractRTAforELFwithArrivalCurves.is_in_search_space
  A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
H_task_cost_pos: 0 < task_cost tsk
H_arrival_curve_pos: 0 < max_arrivals tsk 1
is_in_search_space:= search_space.is_in_search_space L
  total_interference_bound: nat -> Prop
A0: nat
IN: is_in_search_space A0
F: duration
FIX: priority_inversion_bound A0 +
bound_on_total_ep_workload A0 (A0 + F) +
total_hp_rbf (A0 + F) +
(task_request_bound_function tsk (A0 + 1) -
 (task_cost tsk - task_rtct tsk)) <= 
A0 + F
NEQ: F + (task_cost tsk - task_rtct tsk) <= R
task_request_bound_function tsk (A0 + 1) -
(task_cost tsk - task_rtct tsk) + task_IBF A0 (A0 + F) <=
A0 + F
      by rewrite -{2}(leqRW FIX) addnC.
    Qed.

  End ResponseTimeReccurence.

  (** ** I. Soundness of the Response-Time Bound *)

  (** Finally, we prove that [R] is a bound on the response time of the task [tsk]. *)
  Theorem uniprocessor_response_time_bound_elf :
    task_response_time_bound arr_seq sched tsk R.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
task_response_time_bound arr_seq sched tsk R
  Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
task_response_time_bound arr_seq sched tsk R
    move => j ARRs TSKs.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARRs: arrives_in arr_seq j
TSKs: job_of_task tsk j
job_response_time_bound sched j R
    have [ZERO|POS] := posnP (job_cost j);
      first by rewrite /job_response_time_bound /completed_by ZERO.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARRs: arrives_in arr_seq j
TSKs: job_of_task tsk j
POS: 0 < job_cost j
job_response_time_bound sched j R
    eapply uniprocessor_response_time_bound_seq with (L := L) => //.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARRs: arrives_in arr_seq j
TSKs: job_of_task tsk j
POS: 0 < job_cost j
work_conserving arr_seq sched
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARRs: arrives_in arr_seq j
TSKs: job_of_task tsk j
POS: 0 < job_cost j
interference_and_workload_consistent_with_sequential_tasks
  arr_seq sched tsk
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARRs: arrives_in arr_seq j
TSKs: job_of_task tsk j
POS: 0 < job_cost j
busy_intervals_are_bounded_by arr_seq sched tsk L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARRs: arrives_in arr_seq j
TSKs: job_of_task tsk j
POS: 0 < job_cost j
task_interference_is_bounded_by arr_seq sched tsk
  ?task_IBF
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARRs: arrives_in arr_seq j
TSKs: job_of_task tsk j
POS: 0 < job_cost j
forall A : duration,
search_space.is_in_search_space L
  (fun A0 Δ : duration =>
   task_request_bound_function tsk (A0 + 1) -
   task_cost tsk + 
   ?task_IBF A0 Δ) A ->
exists F : duration,
  task_request_bound_function tsk (A + 1) -
  (task_cost tsk - task_rtct tsk) +
  ?task_IBF A (A + F) <= 
  A + F /\ 
  F + (task_cost tsk - task_rtct tsk) <= R
    -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARRs: arrives_in arr_seq j
TSKs: job_of_task tsk j
POS: 0 < job_cost j
work_conserving arr_seq sched exact: instantiated_i_and_w_are_coherent_with_schedule.
    -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARRs: arrives_in arr_seq j
TSKs: job_of_task tsk j
POS: 0 < job_cost j
interference_and_workload_consistent_with_sequential_tasks
  arr_seq sched tsk exact: instantiated_interference_and_workload_consistent_with_sequential_tasks.
    -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARRs: arrives_in arr_seq j
TSKs: job_of_task tsk j
POS: 0 < job_cost j
busy_intervals_are_bounded_by arr_seq sched tsk L exact: instantiated_busy_intervals_are_bounded.
    -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARRs: arrives_in arr_seq j
TSKs: job_of_task tsk j
POS: 0 < job_cost j
task_interference_is_bounded_by arr_seq sched tsk
  ?task_IBF exact: instantiated_task_interference_is_bounded.
    -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
H1: TaskMaxNonpreemptiveSegment Task
H2: MaxArrivals Task
H3: PriorityPoint Task
Job: JobType
H4: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H5: JobPreemptable Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_task_set: uniq ts
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
FP: FP_policy Task
H_reflexive_priorities: reflexive_task_priorities FP
H_transitive_priorities: transitive_task_priorities
  FP
H_total_priorities: total_task_priorities FP
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point
  arr_seq sched 
  (ELF FP)
H_work_conserving: work_conserving.work_conserving
  arr_seq sched
is_lower_priority:= fun j' : Job =>
hp_task tsk (job_task j'): Job -> bool
priority_inversion_lp_tasks_bound: duration
H_priority_inversion_from_lp_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_lower_priority
  (constant
     priority_inversion_lp_tasks_bound)
is_equal_priority:= fun j' : Job =>
ep_task tsk (job_task j'): Job -> bool
priority_inversion_ep_tasks_bound: duration ->
duration
H_priority_inversion_from_ep_tasks_is_bounded: priority_inversion_cond_is_bounded_by
  arr_seq sched
  tsk
  is_equal_priority
  priority_inversion_ep_tasks_bound
blocking_relevant:= fun tsk_o : Task =>
(0 < max_arrivals tsk_o 1) &&
(0 < task_cost tsk_o): Task -> bool
is_ep_causing_intf:= fun (j : Job) 
  (t1 : instant)
  (tsk_other : Task) =>
(0 <=
 (job_arrival j)%:R - t1%:R +
 task_priority_point tsk -
 task_priority_point tsk_other)%R: Job -> instant -> Task -> bool
ep_task_blocking_relevant:= fun (tsk_other : Task)
  (j : Job) 
  (t1 : instant) =>
ep_task tsk_other tsk &&
~~
is_ep_causing_intf j t1
  tsk_other &&
blocking_relevant tsk_other: Task ->
Job -> instant -> bool
H_priority_inversion_from_ep_tasks_concrete_bound: forall (j : Job)
  (t1 : nat),
job_task j = tsk ->
priority_inversion_ep_tasks_bound
  (job_arrival j -
   t1) <=
\max_(i <- ts | 
ep_task_blocking_relevant
  i j t1)
   task_cost i
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_lp_tasks_bound +
total_hep_rbf L
total_interference_bound:= fun A Δ : duration =>
task_request_bound_function
  tsk (A + 1) -
task_cost tsk + 
task_IBF A Δ: duration -> duration -> nat
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound A +
  bound_on_total_ep_workload A
    (A + F) + 
  total_hp_rbf (A + F) +
  (task_request_bound_function tsk
     (A + 1) -
   (task_cost tsk - task_rtct tsk)) <=
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARRs: arrives_in arr_seq j
TSKs: job_of_task tsk j
POS: 0 < job_cost j
forall A : duration,
search_space.is_in_search_space L
  (fun A0 Δ : duration =>
   task_request_bound_function tsk (A0 + 1) -
   task_cost tsk + task_IBF A0 Δ) A ->
exists F : duration,
  task_request_bound_function tsk (A + 1) -
  (task_cost tsk - task_rtct tsk) + task_IBF A (A + F) <=
  A + F /\ F + (task_cost tsk - task_rtct tsk) <= R exact: response_time_recurrence_solution_exists.
  Qed.

End AbstractRTAforELFwithArrivalCurves.