Require Export prosa.model.schedule.priority_driven.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing]
Require Export prosa.analysis.abstract.ideal.iw_instantiation.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.analysis.facts.busy_interval.busy_interval.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]

(** * Abstract RTA for FP-schedulers with Bounded Priority Inversion *)
(** In this module we instantiate the Abstract Response-Time analysis
    (aRTA) to FP-schedulers for ideal uni-processor model of
    real-time tasks with arbitrary arrival models. *)

(** Given FP priority policy and an ideal uni-processor scheduler
    model, we can explicitly specify [interference],
    [interfering_workload], and [interference_bound_function]. In this
    settings, we can define natural notions of service, workload, busy
    interval, etc. The important feature of this instantiation is that
    we can induce the meaningful notion of priority
    inversion. However, we do not specify the exact cause of priority
    inversion (as there may be different reasons for this, like
    execution of a non-preemptive segment or blocking due to resource
    locking). We only assume that that a priority inversion is
    bounded. *)
Section AbstractRTAforFPwithArrivalCurves.

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

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

  (** Consider an FP policy that indicates a higher-or-equal priority relation,
     and assume that the relation is reflexive. Note that we do not relate
     the FP policy with the scheduler. However, we define functions for
     Interference and Interfering Workload that actively use the concept of
     priorities. We require the FP policy to be reflexive, so a job cannot
     cause lower-priority interference (i.e. priority inversion) to itself. *)
  Context `{FP_policy Task}.
  Hypothesis H_priority_is_reflexive : reflexive_priorities.

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

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

  (** ... allow for any work-bearing notion of job readiness, ... *)
  Context `{@JobReady Job (ideal.processor_state Job) Cost Arrival}.
  Hypothesis H_job_ready : work_bearing_readiness arr_seq sched.

  (** ... and assume that the schedule is valid.  *)
  Hypothesis H_sched_valid : valid_schedule sched arr_seq.

  (** ** Instantiation of Interference *)
  (** We say that job [j] incurs interference at time [t] iff it
      cannot execute due to a higher-or-equal-priority job being
      scheduled, or if it incurs a priority inversion. *)
  #[local] Instance ideal_jlfp_interference : Interference Job :=
    ideal_jlfp_interference arr_seq sched.

  (** ** Instantiation of Interfering Workload *)
  (** The interfering workload, in turn, is defined as the sum of the
      priority inversion function and interfering workload of jobs
      with higher or equal priority. *)
  #[local] Instance ideal_jlfp_interfering_workload : InterferingWorkload Job :=
    ideal_jlfp_interfering_workload arr_seq sched.

  (** Note that we differentiate between abstract and classical
      notions of work conserving schedule. *)
  Let work_conserving_ab := definitions.work_conserving arr_seq sched.
  Let work_conserving_cl := work_conserving.work_conserving arr_seq sched.

  (** We assume that the schedule is a work-conserving schedule in the
      _classical_ sense, and later prove that the hypothesis about
      abstract work-conservation also holds. *)
  Hypothesis H_work_conserving : work_conserving_cl.

  (** Assume we have sequential tasks, i.e, jobs from the
      same task execute in the order of their arrival. *)
  Hypothesis H_sequential_tasks : sequential_tasks arr_seq sched.

  (** Assume that a job cost cannot be larger than a task cost. *)
  Hypothesis H_valid_job_cost :
    arrivals_have_valid_job_costs arr_seq.

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

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

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

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

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

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

  (** For clarity, let's define some local names. *)
  Let job_pending_at := pending sched.
  Let job_scheduled_at := scheduled_at sched.
  Let job_completed_by := completed_by sched.
  Let job_backlogged_at := backlogged sched.
  Let response_time_bounded_by := task_response_time_bound arr_seq sched.

  (** We introduce [task_rbf] as an abbreviation of the task request
      bound function, which is defined as [task_cost(tsk) ×
      max_arrivals(tsk,Δ)]. *)
  Let task_rbf := task_request_bound_function tsk.

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

  (** Similarly, we define the request bound function of all tasks
      other than [tsk] with higher-or-equal priority (with respect to
      [tsk]). *)
  Let total_ohep_rbf :=
    total_ohep_request_bound_function_FP ts tsk.

  (** Assume that there exists a constant [priority_inversion_bound]
      that bounds the length of any priority inversion experienced by
      any job of [tsk].  Since we analyze only task [tsk], we ignore
      the lengths of priority inversions incurred by any other
      tasks. *)
  Variable priority_inversion_bound : duration.
  Hypothesis H_priority_inversion_is_bounded :
    priority_inversion_is_bounded_by_constant
      arr_seq sched tsk priority_inversion_bound.

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

  (** To reduce the time complexity of the analysis, recall the notion
      of search space. Intuitively, this corresponds to all
      "interesting" arrival offsets that the job under analysis might
      have with regard to the beginning of its busy-window. *)
  Definition is_in_search_space A := (A < L) && (task_rbf A != task_rbf (A + ε)).

  (** Let [R] be a value that upper-bounds the solution of each
      response-time recurrence, i.e., for any relative arrival time
      [A] in the search space, there exists a corresponding solution
      [F] such that [R >= F + (task cost - task lock-in service)]. *)
  Variable R : duration.
  Hypothesis H_R_is_maximum :
    forall (A : duration),
      is_in_search_space A ->
      exists (F : duration),
        A + F >= priority_inversion_bound
                + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk))
                + total_ohep_rbf (A + F) /\
        R >= F + (task_cost tsk - task_rtct tsk).

  (** Finally, we define the interference bound function
      ([IBF_other]). [IBF_other] bounds the interference if tasks are
      sequential. Since tasks are sequential, we exclude interference
      from other jobs of the same task. For FP, we define [IBF_other]
      as the sum of the priority interference bound and the
      higher-or-equal-priority workload. *)
  Let IBF_other (R : duration) := priority_inversion_bound + total_ohep_rbf R.

  (** ** Filling Out Hypotheses Of Abstract RTA Theorem *)
  (** In this section we prove that all preconditions necessary to use
      the abstract theorem are satisfied. *)
  Section FillingOutHypothesesOfAbstractRTATheorem.

    (** Recall that [L] is assumed to be a fixed point of the busy
        interval recurrence. Thanks to this fact, we can prove that
        every busy interval (according to the concrete definition) is
        bounded. In addition, we know that the conventional concept of
        busy interval and the one obtained from the abstract
        definition (with the interference and interfering workload)
        coincide. Thus, it follows that any busy interval (in the
        abstract sense) is bounded. *)
    Lemma instantiated_busy_intervals_are_bounded :
      busy_intervals_are_bounded_by arr_seq sched tsk L.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
busy_intervals_are_bounded_by arr_seq sched tsk L
    Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
busy_intervals_are_bounded_by arr_seq sched tsk L
      move => j ARR TSK POS.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
exists t1 t2 : nat,
  t1 <= job_arrival j < t2 /\
  t2 <= t1 + L /\
  definitions.busy_interval sched j t1 t2
      edestruct (exists_busy_interval) with (delta := L) as [t1 [t2 [T1 [T2 BI]]]]; rt_eauto.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
forall t : instant,
priority_inversion_bound +
workload_of_higher_or_equal_priority_jobs j
  (arrivals_between arr_seq t (t + L)) <= L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T1: t1 <= job_arrival j < t2
T2: t2 <= t1 + L
BI: busy_interval arr_seq sched j t1 t2
exists t1 t2 : nat,
  t1 <= job_arrival j < t2 /\
  t2 <= t1 + L /\
  definitions.busy_interval sched j t1 t2
      {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
forall t : instant,
priority_inversion_bound +
workload_of_higher_or_equal_priority_jobs j
  (arrivals_between arr_seq t (t + L)) <= L intros; rewrite {2}H_fixed_point leq_add //.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t: instant
workload_of_higher_or_equal_priority_jobs j
  (arrivals_between arr_seq t (t + L)) <=
total_hep_rbf L
        rewrite /workload_of_higher_or_equal_priority_jobs /total_hep_rbf
          /total_hep_request_bound_function_FP
          /workload_of_jobs /hep_job /FP_to_JLFP.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t: instant
\sum_(j0 <- arrivals_between arr_seq t (t + L) | hep_task
                                                 (job_task
                                                 j0)
                                                 (job_task
                                                 j))
   job_cost j0 <=
\sum_(tsk_other <- ts | hep_task tsk_other tsk)
   task_request_bound_function tsk_other L
        move: (TSK) =>  /eqP ->.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t: instant
\sum_(j <- arrivals_between arr_seq t (t + L) | hep_task
                                                 (job_task
                                                 j)
                                                 tsk)
   job_cost j <=
\sum_(tsk_other <- ts | hep_task tsk_other tsk)
   task_request_bound_function tsk_other L
        by apply: sum_of_jobs_le_sum_rbf; eauto. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T1: t1 <= job_arrival j < t2
T2: t2 <= t1 + L
BI: busy_interval arr_seq sched j t1 t2
exists t1 t2 : nat,
  t1 <= job_arrival j < t2 /\
  t2 <= t1 + L /\
  definitions.busy_interval sched j t1 t2
      exists t1, t2; split; first by done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T1: t1 <= job_arrival j < t2
T2: t2 <= t1 + L
BI: busy_interval arr_seq sched j t1 t2
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2 split; first by done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T1: t1 <= job_arrival j < t2
T2: t2 <= t1 + L
BI: busy_interval arr_seq sched j t1 t2
definitions.busy_interval sched j t1 t2
      eapply instantiated_busy_interval_equivalent_busy_interval; rt_eauto.
    Qed.

    (** Next, we prove that [IBF_other] is indeed an interference
        bound.

        Recall that in module abstract_seq_RTA hypothesis
        [task_interference_is_bounded_by] expects to receive a
        function that maps some task [tsk], the relative arrival time of
        a job [j] of task [tsk], and the length of the interval to the
        maximum amount of interference.

        However, in this module we analyze only one task -- [tsk],
        therefore it is “hard-coded” inside the interference bound
        function [IBF_other]. Moreover, in case of a model with fixed
        priorities, interference that some job [j] incurs from
        higher-or-equal priority jobs does not depend on the relative
        arrival time of job [j]. Therefore, in order for the
        [IBF_other] signature to match the required signature in
        module [abstract_seq_RTA], we wrap the [IBF_other] function in
        a function that accepts, but simply ignores, the task and the
        relative arrival time. *)
    Lemma instantiated_task_interference_is_bounded :
      task_interference_is_bounded_by arr_seq sched tsk (fun t A R => IBF_other R).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
task_interference_is_bounded_by arr_seq sched tsk
  (fun=> (fun=> [eta IBF_other]))
    Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
task_interference_is_bounded_by arr_seq sched tsk
  (fun=> (fun=> [eta IBF_other]))
      intros ? ? ? ? ARR TSK ? NCOMPL BUSY; simpl.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: definitions.busy_interval sched j t1 t2
cumul_task_interference arr_seq sched tsk t2 t1
  (t1 + R0) <= IBF_other R0
      move: (posnP (@job_cost _ Cost j)) => [ZERO|POS].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: definitions.busy_interval sched j t1 t2
ZERO: job_cost j = 0
cumul_task_interference arr_seq sched tsk t2 t1
  (t1 + R0) <= IBF_other R0
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: definitions.busy_interval sched j t1 t2
POS: 0 < job_cost j
cumul_task_interference arr_seq sched tsk t2 t1
  (t1 + R0) <= 
IBF_other R0
      {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: definitions.busy_interval sched j t1 t2
ZERO: job_cost j = 0
cumul_task_interference arr_seq sched tsk t2 t1
  (t1 + R0) <= IBF_other R0 by exfalso; rewrite /completed_by ZERO in  NCOMPL. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: definitions.busy_interval sched j t1 t2
POS: 0 < job_cost j
cumul_task_interference arr_seq sched tsk t2 t1
  (t1 + R0) <= IBF_other R0
      eapply instantiated_busy_interval_equivalent_busy_interval in BUSY; rt_eauto.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
cumul_task_interference arr_seq sched tsk t2 t1
  (t1 + R0) <= IBF_other R0
      rewrite /ideal_jlfp_interference; erewrite cumulative_task_interference_split; rt_eauto; last first.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
j \in arrivals_before arr_seq t2
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
  (t1 + R0) +
cumulative_another_task_hep_job_interference arr_seq
  sched j t1 
  (t1 + R0) <= 
IBF_other R0
      {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
j \in arrivals_before arr_seq t2 by move: BUSY => [[_ [_ [_ /andP [GE LT]]]] _]; eapply arrived_between_implies_in_arrivals; rt_eauto. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
  (t1 + R0) +
cumulative_another_task_hep_job_interference arr_seq
  sched j t1 (t1 + R0) <= IBF_other R0
      rewrite /IBF_other leq_add; try done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
  (t1 + R0) <= priority_inversion_bound
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
cumulative_another_task_hep_job_interference arr_seq
  sched j t1 
  (t1 + R0) <= 
total_ohep_rbf R0
      {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
  (t1 + R0) <= priority_inversion_bound apply leq_trans with (cumulative_priority_inversion arr_seq sched j t1 (t1 + R0)); first by done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
  (t1 + R0) <= priority_inversion_bound
        apply leq_trans with (cumulative_priority_inversion arr_seq sched j t1 t2); last first.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1 t2 <=
priority_inversion_bound
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
  (t1 + R0) <=
cumulative_priority_inversion arr_seq sched j t1 t2
        {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1 t2 <=
priority_inversion_bound by apply H_priority_inversion_is_bounded; rt_eauto; move: BUSY => [PREF QT2]. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
  (t1 + R0) <=
cumulative_priority_inversion arr_seq sched j t1 t2
        rewrite [X in _ <= X](@big_cat_nat _ _ _ (t1 + R0)) //=.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
  (t1 + R0) <=
\sum_(t1 <= i < t1 + R0)
   priority_inversion_dec arr_seq sched j i +
\sum_(t1 + R0 <= i < t2)
   priority_inversion_dec arr_seq sched j i
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
t1 <= t1 + R0
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
t1 + R0 <= t2
        -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
  (t1 + R0) <=
\sum_(t1 <= i < t1 + R0)
   priority_inversion_dec arr_seq sched j i +
\sum_(t1 + R0 <= i < t2)
   priority_inversion_dec arr_seq sched j i
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
t1 <= t1 + R0
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
t1 + R0 <= t2 by rewrite leq_addr.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
t1 <= t1 + R0
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
t1 + R0 <= t2
        -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
t1 <= t1 + R0
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
t1 + R0 <= t2 by rewrite leq_addr.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
t1 + R0 <= t2
        -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
t1 + R0 <= t2 by rewrite ltnW.
      }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
cumulative_another_task_hep_job_interference arr_seq
  sched j t1 (t1 + R0) <= total_ohep_rbf R0
      {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
cumulative_another_task_hep_job_interference arr_seq
  sched j t1 (t1 + R0) <= total_ohep_rbf R0 erewrite cumulative_i_thep_eq_service_of_othep; rt_eauto;
          last by unfold quiet_time; move: BUSY => [[_ [T1 T2]] _].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
service_of_another_task_hep_job arr_seq sched j t1
  (t1 + R0) <= total_ohep_rbf R0
        apply: leq_trans.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
service_of_another_task_hep_job arr_seq sched j t1
  (t1 + R0) <= ?Goal
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
?Goal <= total_ohep_rbf R0
        {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
service_of_another_task_hep_job arr_seq sched j t1
  (t1 + R0) <= ?Goal apply service_of_jobs_le_workload; first apply ideal_proc_model_provides_unit_service.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
completed_jobs_dont_execute sched
          by apply (valid_schedule_implies_completed_jobs_dont_execute sched arr_seq). }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
workload_of_jobs (another_task_hep_job^~ j)
  (arrivals_between arr_seq t1 (t1 + R0)) <=
total_ohep_rbf R0
        {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
workload_of_jobs (another_task_hep_job^~ j)
  (arrivals_between arr_seq t1 (t1 + R0)) <=
total_ohep_rbf R0 rewrite /workload_of_jobs /total_ohep_rbf /total_ohep_request_bound_function_FP.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + R0) | 
another_task_hep_job j0 j) job_cost j0 <=
\sum_(tsk_other <- ts | hep_task tsk_other tsk &&
                        (tsk_other != tsk))
   task_request_bound_function tsk_other R0
          rewrite /another_task_hep_job /hep_job /FP_to_JLFP.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + R0) | 
hep_task (job_task j0) (job_task j) &&
(job_task j0 != job_task j)) job_cost j0 <=
\sum_(tsk_other <- ts | hep_task tsk_other tsk &&
                        (tsk_other != tsk))
   task_request_bound_function tsk_other R0
          set (pred_task tsk_other := hep_task tsk_other tsk && (tsk_other != tsk)).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
pred_task:= fun tsk_other : Task =>
hep_task tsk_other tsk &&
(tsk_other != tsk): Task -> bool
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + R0) | 
hep_task (job_task j0) (job_task j) &&
(job_task j0 != job_task j)) job_cost j0 <=
\sum_(tsk_other <- ts | hep_task tsk_other tsk &&
                        (tsk_other != tsk))
   task_request_bound_function tsk_other R0
          rewrite (eq_big (fun j=> pred_task (job_task j)) job_cost) //;
            last by move=> j'; rewrite /pred_task; move: TSK => /eqP ->.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
pred_task:= fun tsk_other : Task =>
hep_task tsk_other tsk &&
(tsk_other != tsk): Task -> bool
\sum_(i <- arrivals_between arr_seq t1 (t1 + R0) | 
pred_task (job_task i)) job_cost i <=
\sum_(tsk_other <- ts | hep_task tsk_other tsk &&
                        (tsk_other != tsk))
   task_request_bound_function tsk_other R0
          erewrite (eq_big pred_task); [|by done|by move=> tsk'; eauto].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
R0, t1, t2: nat
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
H6: t1 + R0 < t2
NCOMPL: ~~ completed_by sched j (t1 + R0)
BUSY: busy_interval arr_seq sched j t1 t2
POS: 0 < job_cost j
pred_task:= fun tsk_other : Task =>
hep_task tsk_other tsk &&
(tsk_other != tsk): Task -> bool
\sum_(i <- arrivals_between arr_seq t1 (t1 + R0) | 
pred_task (job_task i)) job_cost i <=
\sum_(i <- ts | pred_task i)
   (task_request_bound_function^~ R0) i
          by apply: sum_of_jobs_le_sum_rbf; eauto. } }
    Qed.

    (** Finally, we show that there exists a solution for the
        response-time recurrence. *)
    Section SolutionOfResponseTimeRecurrenceExists.

      (** Consider any job [j] of [tsk]. *)
      Variable j : Job.
      Hypothesis H_j_arrives : arrives_in arr_seq j.
      Hypothesis H_job_of_tsk : job_of_task tsk j.
      Hypothesis H_job_cost_positive: job_cost_positive j.

      (** Given any job [j] of task [tsk] that arrives exactly [A]
          units after the beginning of the busy interval, the bound of
          the total interference incurred by [j] within an interval of
          length [Δ] is equal to [task_rbf (A + ε) - task_cost tsk +
          IBF_other Δ]. *)
      Let total_interference_bound tsk A Δ :=
        task_rbf (A + ε) - task_cost tsk + IBF_other Δ.

      (** Next, consider any [A] from the search space (in the
          abstract sense). *)
      Variable A : duration.
      Hypothesis H_A_is_in_abstract_search_space :
        search_space.is_in_search_space tsk L total_interference_bound A.

      (** We prove that [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
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
is_in_search_space A
      Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
is_in_search_space A
        move: H_A_is_in_abstract_search_space => [INSP | [/andP [POSA LTL] [x [LTx INSP2]]]].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
INSP: A = 0
is_in_search_space A
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
is_in_search_space A
        -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
INSP: A = 0
is_in_search_space A
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
is_in_search_space A rewrite INSP.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
INSP: A = 0
is_in_search_space 0
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
is_in_search_space A
          apply/andP; split; first by done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
INSP: A = 0
task_rbf 0 != task_rbf (0 + ε)
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
is_in_search_space A
          rewrite neq_ltn; apply/orP; left.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
INSP: A = 0
task_rbf 0 < task_rbf (0 + ε)
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
is_in_search_space A
          rewrite {1}/task_rbf; erewrite task_rbf_0_zero; eauto 2; try done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
INSP: A = 0
0 < task_rbf (0 + ε)
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
is_in_search_space A
          rewrite add0n /task_rbf; apply leq_trans with (task_cost tsk).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
INSP: A = 0
0 < task_cost tsk
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
INSP: A = 0
task_cost tsk <= task_request_bound_function tsk ε
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
is_in_search_space A
          +
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
INSP: A = 0
0 < task_cost tsk
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
INSP: A = 0
task_cost tsk <= task_request_bound_function tsk ε
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
is_in_search_space A by apply leq_trans with (job_cost j); eauto 2; move: (H_job_of_tsk) => /eqP <-; eauto 2.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
INSP: A = 0
task_cost tsk <= task_request_bound_function tsk ε
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
is_in_search_space A
          +
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
INSP: A = 0
task_cost tsk <= task_request_bound_function tsk ε
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
is_in_search_space A by eapply task_rbf_1_ge_task_cost; rt_eauto.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
is_in_search_space A
        -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
is_in_search_space A apply/andP; split; first by done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
task_rbf A != task_rbf (A + ε)
          apply/negP; intros EQ; move: EQ => /eqP EQ.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
EQ: task_rbf A = task_rbf (A + ε)
False
          apply INSP2.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
EQ: task_rbf A = task_rbf (A + ε)
total_interference_bound tsk (A - ε) x =
total_interference_bound tsk A x
          unfold total_interference_bound in *.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  (fun (tsk : Task)
     (A : nat)
     (Δ : duration)
   =>
   task_rbf (A + ε) -
   task_cost tsk +
   IBF_other Δ) A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: task_rbf (A - ε + ε) - task_cost tsk +
IBF_other x <>
task_rbf (A + ε) - task_cost tsk + IBF_other x
EQ: task_rbf A = task_rbf (A + ε)
task_rbf (A - ε + ε) - task_cost tsk + IBF_other x =
task_rbf (A + ε) - task_cost tsk + IBF_other x
          rewrite subn1 addn1 prednK; last by done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  (fun (tsk : Task)
     (A : nat)
     (Δ : duration)
   =>
   task_rbf (A + ε) -
   task_cost tsk +
   IBF_other Δ) A
POSA: 0 < A
LTL: A < L
x: nat
LTx: x < L
INSP2: task_rbf (A - ε + ε) - task_cost tsk +
IBF_other x <>
task_rbf (A + ε) - task_cost tsk + IBF_other x
EQ: task_rbf A = task_rbf (A + ε)
task_rbf A - task_cost tsk + IBF_other x =
task_rbf (A + ε) - task_cost tsk + IBF_other x
            by rewrite -EQ.
      Qed.

      (** Then, there exists a solution for the response-time
          recurrence (in the abstract sense). *)
      Corollary correct_search_space :
        exists (F : duration),
          A + F >= task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) + IBF_other (A + F) /\
          R >= F + (task_cost tsk - task_rtct tsk).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
exists F : duration,
  task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) +
  IBF_other (A + F) <= A + F /\
  F + (task_cost tsk - task_rtct tsk) <= R
      Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
exists F : duration,
  task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) +
  IBF_other (A + F) <= A + F /\
  F + (task_cost tsk - task_rtct tsk) <= R
        move: (H_R_is_maximum A) => FIX.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
FIX: is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <= R
exists F : duration,
  task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) +
  IBF_other (A + F) <= A + F /\
  F + (task_cost tsk - task_rtct tsk) <= R
        feed FIX; first by apply A_is_in_concrete_search_space.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
FIX: exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <= R
exists F : duration,
  task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) +
  IBF_other (A + F) <= A + F /\
  F + (task_cost tsk - task_rtct tsk) <= R
        move: FIX => [F [FIX NEQ]].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
F: duration
FIX: priority_inversion_bound +
(task_rbf (A + ε) -
 (task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <= 
A + F
NEQ: F + (task_cost tsk - task_rtct tsk) <= R
exists F : duration,
  task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) +
  IBF_other (A + F) <= A + F /\
  F + (task_cost tsk - task_rtct tsk) <= R
        exists F; split; last by done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
F: duration
FIX: priority_inversion_bound +
(task_rbf (A + ε) -
 (task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <= 
A + F
NEQ: F + (task_cost tsk - task_rtct tsk) <= R
task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) +
IBF_other (A + F) <= A + F
        rewrite -{2}(leqRW FIX).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
total_interference_bound:= fun (tsk : Task) 
  (A : nat) 
  (Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk + 
IBF_other Δ: Task ->
nat -> duration -> nat
A: duration
H_A_is_in_abstract_search_space: search_space.is_in_search_space
  tsk L
  total_interference_bound
  A
F: duration
FIX: priority_inversion_bound +
(task_rbf (A + ε) -
 (task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <= 
A + F
NEQ: F + (task_cost tsk - task_rtct tsk) <= R
task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) +
IBF_other (A + F) <=
priority_inversion_bound +
(task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F)
          by rewrite addnA [_ + priority_inversion_bound]addnC -!addnA.
      Qed.

    End SolutionOfResponseTimeRecurrenceExists.

  End FillingOutHypothesesOfAbstractRTATheorem.


  (** ** Final Theorem *)
  (** Based on the properties established above, we apply the abstract
      analysis framework to infer that [R] is a response-time bound
      for [tsk]. *)
  Theorem uniprocessor_response_time_bound_fp :
    response_time_bounded_by tsk R.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
response_time_bounded_by tsk R
  Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
response_time_bounded_by tsk R
    intros js ARRs TSKs.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
job_response_time_bound sched js R
    move: (posnP (@job_cost _ Cost js)) => [ZERO|POS].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
ZERO: job_cost js = 0
job_response_time_bound sched js R
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
job_response_time_bound sched js R
    {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
ZERO: job_cost js = 0
job_response_time_bound sched js R by rewrite /job_response_time_bound /completed_by ZERO. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
job_response_time_bound sched js R
    eapply uniprocessor_response_time_bound_seq; rt_eauto.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
definitions.work_conserving arr_seq sched
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
interference_and_workload_consistent_with_sequential_tasks
  arr_seq sched tsk
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
busy_intervals_are_bounded_by arr_seq sched tsk ?L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
task_interference_is_bounded_by arr_seq sched tsk
  ?task_interference_bound_function
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
forall A : duration,
search_space.is_in_search_space tsk 
  ?L
  (fun (tsk0 : Task) (A0 Δ : duration) =>
   task_request_bound_function tsk (A0 + ε) -
   task_cost tsk0 +
   ?task_interference_bound_function tsk0 A0 Δ) A ->
exists F : duration,
  task_request_bound_function tsk (A + ε) -
  (task_cost tsk - task_rtct tsk) +
  ?task_interference_bound_function tsk A (A + F) <=
  A + F /\ 
  F + (task_cost tsk - task_rtct tsk) <= R
    -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
definitions.work_conserving arr_seq sched
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
interference_and_workload_consistent_with_sequential_tasks
  arr_seq sched tsk
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
busy_intervals_are_bounded_by arr_seq sched tsk ?L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
task_interference_is_bounded_by arr_seq sched tsk
  ?task_interference_bound_function
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
forall A : duration,
search_space.is_in_search_space tsk 
  ?L
  (fun (tsk0 : Task) (A0 Δ : duration) =>
   task_request_bound_function tsk (A0 + ε) -
   task_cost tsk0 +
   ?task_interference_bound_function tsk0 A0 Δ) A ->
exists F : duration,
  task_request_bound_function tsk (A + ε) -
  (task_cost tsk - task_rtct tsk) +
  ?task_interference_bound_function tsk A (A + F) <=
  A + F /\ 
  F + (task_cost tsk - task_rtct tsk) <= R by eapply instantiated_i_and_w_are_coherent_with_schedule; rt_eauto.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
interference_and_workload_consistent_with_sequential_tasks
  arr_seq sched tsk
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
busy_intervals_are_bounded_by arr_seq sched tsk ?L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
task_interference_is_bounded_by arr_seq sched tsk
  ?task_interference_bound_function
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
forall A : duration,
search_space.is_in_search_space tsk 
  ?L
  (fun (tsk0 : Task) (A0 Δ : duration) =>
   task_request_bound_function tsk (A0 + ε) -
   task_cost tsk0 +
   ?task_interference_bound_function tsk0 A0 Δ) A ->
exists F : duration,
  task_request_bound_function tsk (A + ε) -
  (task_cost tsk - task_rtct tsk) +
  ?task_interference_bound_function tsk A (A + F) <=
  A + F /\ 
  F + (task_cost tsk - task_rtct tsk) <= R
    -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
interference_and_workload_consistent_with_sequential_tasks
  arr_seq sched tsk
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
busy_intervals_are_bounded_by arr_seq sched tsk ?L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
task_interference_is_bounded_by arr_seq sched tsk
  ?task_interference_bound_function
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
forall A : duration,
search_space.is_in_search_space tsk 
  ?L
  (fun (tsk0 : Task) (A0 Δ : duration) =>
   task_request_bound_function tsk (A0 + ε) -
   task_cost tsk0 +
   ?task_interference_bound_function tsk0 A0 Δ) A ->
exists F : duration,
  task_request_bound_function tsk (A + ε) -
  (task_cost tsk - task_rtct tsk) +
  ?task_interference_bound_function tsk A (A + F) <=
  A + F /\ 
  F + (task_cost tsk - task_rtct tsk) <= R by eapply instantiated_interference_and_workload_consistent_with_sequential_tasks; rt_eauto.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
busy_intervals_are_bounded_by arr_seq sched tsk ?L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
task_interference_is_bounded_by arr_seq sched tsk
  ?task_interference_bound_function
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
forall A : duration,
search_space.is_in_search_space tsk 
  ?L
  (fun (tsk0 : Task) (A0 Δ : duration) =>
   task_request_bound_function tsk (A0 + ε) -
   task_cost tsk0 +
   ?task_interference_bound_function tsk0 A0 Δ) A ->
exists F : duration,
  task_request_bound_function tsk (A + ε) -
  (task_cost tsk - task_rtct tsk) +
  ?task_interference_bound_function tsk A (A + F) <=
  A + F /\ 
  F + (task_cost tsk - task_rtct tsk) <= R
    -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
busy_intervals_are_bounded_by arr_seq sched tsk ?L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
task_interference_is_bounded_by arr_seq sched tsk
  ?task_interference_bound_function
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
forall A : duration,
search_space.is_in_search_space tsk 
  ?L
  (fun (tsk0 : Task) (A0 Δ : duration) =>
   task_request_bound_function tsk (A0 + ε) -
   task_cost tsk0 +
   ?task_interference_bound_function tsk0 A0 Δ) A ->
exists F : duration,
  task_request_bound_function tsk (A + ε) -
  (task_cost tsk - task_rtct tsk) +
  ?task_interference_bound_function tsk A (A + F) <=
  A + F /\ 
  F + (task_cost tsk - task_rtct tsk) <= R by apply instantiated_busy_intervals_are_bounded.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
task_interference_is_bounded_by arr_seq sched tsk
  ?task_interference_bound_function
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
forall A : duration,
search_space.is_in_search_space tsk L
  (fun (tsk0 : Task) (A0 Δ : duration) =>
   task_request_bound_function tsk (A0 + ε) -
   task_cost tsk0 +
   ?task_interference_bound_function tsk0 A0 Δ) A ->
exists F : duration,
  task_request_bound_function tsk (A + ε) -
  (task_cost tsk - task_rtct tsk) +
  ?task_interference_bound_function tsk A (A + F) <=
  A + F /\ 
  F + (task_cost tsk - task_rtct tsk) <= R
    -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
task_interference_is_bounded_by arr_seq sched tsk
  ?task_interference_bound_function
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
forall A : duration,
search_space.is_in_search_space tsk L
  (fun (tsk0 : Task) (A0 Δ : duration) =>
   task_request_bound_function tsk (A0 + ε) -
   task_cost tsk0 +
   ?task_interference_bound_function tsk0 A0 Δ) A ->
exists F : duration,
  task_request_bound_function tsk (A + ε) -
  (task_cost tsk - task_rtct tsk) +
  ?task_interference_bound_function tsk A (A + F) <=
  A + F /\ 
  F + (task_cost tsk - task_rtct tsk) <= R by apply instantiated_task_interference_is_bounded.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
forall A : duration,
search_space.is_in_search_space tsk L
  (fun (tsk0 : Task) (A0 Δ : duration) =>
   task_request_bound_function tsk (A0 + ε) -
   task_cost tsk0 +
   (fun=> (fun=> [eta IBF_other])) tsk0 A0 Δ) A ->
exists F : duration,
  task_request_bound_function tsk (A + ε) -
  (task_cost tsk - task_rtct tsk) +
  (fun=> (fun=> [eta IBF_other])) tsk A (A + F) <=
  A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
    -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
H2: JobPreemptable Job
H3: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
sched: schedule (ideal.processor_state Job)
H4: JobReady Job (ideal.processor_state Job)
H_job_ready: work_bearing_readiness arr_seq sched
H_sched_valid: valid_schedule sched arr_seq
work_conserving_ab:= definitions.work_conserving
  arr_seq sched: Prop
work_conserving_cl:= work_conserving arr_seq sched: Prop
H_work_conserving: work_conserving_cl
H_sequential_tasks: sequential_tasks arr_seq sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
job_pending_at:= pending sched: Job -> instant -> bool
job_scheduled_at:= scheduled_at sched: Job -> instant -> bool
job_completed_by:= completed_by sched: Job -> instant -> bool
job_backlogged_at:= backlogged sched: Job -> instant -> bool
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
task_rbf:= task_request_bound_function tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts
  tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP
  ts tsk: duration -> nat
priority_inversion_bound: duration
H_priority_inversion_is_bounded: priority_inversion_is_bounded_by_constant
  arr_seq sched tsk
  priority_inversion_bound
L: duration
H_L_positive: 0 < L
H_fixed_point: L =
priority_inversion_bound +
total_hep_rbf L
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  priority_inversion_bound +
  (task_rbf (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
IBF_other:= fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
js: Job
ARRs: arrives_in arr_seq js
TSKs: job_of_task tsk js
POS: 0 < job_cost js
forall A : duration,
search_space.is_in_search_space tsk L
  (fun (tsk0 : Task) (A0 Δ : duration) =>
   task_request_bound_function tsk (A0 + ε) -
   task_cost tsk0 +
   (fun=> (fun=> [eta IBF_other])) tsk0 A0 Δ) A ->
exists F : duration,
  task_request_bound_function tsk (A + ε) -
  (task_cost tsk - task_rtct tsk) +
  (fun=> (fun=> [eta IBF_other])) tsk A (A + F) <=
  A + F /\ F + (task_cost tsk - task_rtct tsk) <= R by eapply correct_search_space; eauto 2.
  Qed.

End AbstractRTAforFPwithArrivalCurves.