From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing]
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.results.fixed_priority.rta.bounded_nps.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.analysis.facts.preemption.rtc_threshold.floating.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.analysis.facts.readiness.sequential.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]

(** * RTA for Model with Floating Non-Preemptive Regions *)
(** In this module we prove the RTA theorem for floating non-preemptive regions FP model. *)

(** ** Setup and Assumptions *)

Section RTAforFloatingModelwithArrivalCurves.

  (** We assume ideal uni-processor schedules. *)
  #[local] Existing Instance ideal.processor_state.

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

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

  (** We assume that jobs are limited-preemptive. *)
  #[local] Existing Instance limited_preemptive_job_model.

  (** 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.

  (** Assume we have the model with floating non-preemptive regions.
      I.e., for each task only the length of the maximal non-preemptive
      segment is known _and_ each job level is divided into a number of
      non-preemptive segments by inserting preemption points. *)
  Context `{JobPreemptionPoints Job}
          `{TaskMaxNonpreemptiveSegment Task}.
  Hypothesis H_valid_task_model_with_floating_nonpreemptive_regions:
    valid_model_with_floating_nonpreemptive_regions arr_seq.

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

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

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

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

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

  (** Recall that we assume sequential readiness. *)
  #[local] Instance sequential_readiness : JobReady _ _ :=
    sequential_ready_instance arr_seq.

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

  (** Consider an FP policy that indicates a higher-or-equal priority relation,
      and assume that the relation is reflexive and transitive. *)
  Context {FP :FP_policy Task}.
  Hypothesis H_priority_is_reflexive : reflexive_priorities.
  Hypothesis H_priority_is_transitive : transitive_priorities.

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

  (** ... and the schedule respects the scheduling policy. *)
  Hypothesis H_respects_policy : respects_FP_policy_at_preemption_point arr_seq sched FP.

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

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

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

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

  (** ... and the request bound function of all tasks with higher
      priority other than task [tsk]. *)
  Let total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk.

  (** Next, we define a bound for the priority inversion caused by tasks of lower priority. *)
  Let blocking_bound :=
    \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
     (task_max_nonpreemptive_segment tsk_other - ε).

  (** Let L be any positive fixed point of the busy interval recurrence, determined by
      the sum of blocking and higher-or-equal-priority workload. *)
  Variable L : duration.
  Hypothesis H_L_positive : L > 0.
  Hypothesis H_fixed_point : L = blocking_bound + total_hep_rbf L.

  (** ** Response-Time Bound *)

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

  (** Next, consider any value R, and assume that for any given
      arrival A from search space there is a solution of the
      response-time bound recurrence which is bounded by R. *)
  Variable R : duration.
  Hypothesis H_R_is_maximum:
    forall (A : duration),
      is_in_search_space A ->
      exists  (F : duration),
        A + F >= blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) /\
        R >= F.

  (** Now, we can reuse the results for the abstract model with
      bounded nonpreemptive segments to establish a response-time
      bound for the more concrete model with floating nonpreemptive
      regions.  *)

  Let response_time_bounded_by := task_response_time_bound arr_seq sched.

  Theorem uniprocessor_response_time_bound_fp_with_floating_nonpreemptive_regions:
    response_time_bounded_by tsk R.
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
response_time_bounded_by tsk R
  Proof.
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
response_time_bounded_by tsk R
    move: (H_valid_task_model_with_floating_nonpreemptive_regions) => [LIMJ JMLETM].
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
response_time_bounded_by tsk R
    move: (LIMJ) => [BEG [END _]].
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
response_time_bounded_by tsk R
    eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments.
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
reflexive_priorities
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
transitive_priorities
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
valid_arrival_sequence arr_seq
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
work_bearing_readiness arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
valid_schedule sched arr_seq
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
work_conserving arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
respects_FP_policy_at_preemption_point arr_seq sched
  FP
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
sequential_tasks arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
all_jobs_from_taskset arr_seq ?ts
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
arrivals_have_valid_job_costs arr_seq
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
valid_taskset_arrival_curve ?ts max_arrivals
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
taskset_respects_max_arrivals arr_seq ?ts
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
tsk \in ?ts
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
valid_preemption_model arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
valid_task_run_to_completion_threshold arr_seq tsk
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
0 < ?L
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
?L =
bounded_nps.blocking_bound ?ts tsk +
total_hep_request_bound_function_FP ?ts tsk ?L
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
forall A : duration,
bounded_pi.is_in_search_space tsk ?L A ->
exists F : duration,
  bounded_nps.blocking_bound ?ts tsk +
  (task_request_bound_function tsk (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_request_bound_function_FP ?ts tsk (A + F) <=
  A + F /\ 
  F + (task_cost tsk - task_rtct tsk) <= R
    all: rt_eauto.
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
work_bearing_readiness arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
sequential_tasks arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
forall A : duration,
bounded_pi.is_in_search_space tsk L A ->
exists F : duration,
  bounded_nps.blocking_bound ts tsk +
  (task_request_bound_function tsk (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_request_bound_function_FP ts tsk (A + F) <=
  A + F /\ 
  F + (task_cost tsk - task_rtct tsk) <= R
    -
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
work_bearing_readiness arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
sequential_tasks arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
forall A : duration,
bounded_pi.is_in_search_space tsk L A ->
exists F : duration,
  bounded_nps.blocking_bound ts tsk +
  (task_request_bound_function tsk (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_request_bound_function_FP ts tsk (A + F) <=
  A + F /\ 
  F + (task_cost tsk - task_rtct tsk) <= R by apply sequential_readiness_implies_work_bearing_readiness; rt_eauto.
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
sequential_tasks arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
forall A : duration,
bounded_pi.is_in_search_space tsk L A ->
exists F : duration,
  bounded_nps.blocking_bound ts tsk +
  (task_request_bound_function tsk (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_request_bound_function_FP ts tsk (A + F) <=
  A + F /\ 
  F + (task_cost tsk - task_rtct tsk) <= R
    -
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
sequential_tasks arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
forall A : duration,
bounded_pi.is_in_search_space tsk L A ->
exists F : duration,
  bounded_nps.blocking_bound ts tsk +
  (task_request_bound_function tsk (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_request_bound_function_FP ts tsk (A + F) <=
  A + F /\ 
  F + (task_cost tsk - task_rtct tsk) <= R by apply sequential_readiness_implies_sequential_tasks; rt_eauto.
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
forall A : duration,
bounded_pi.is_in_search_space tsk L A ->
exists F : duration,
  bounded_nps.blocking_bound ts tsk +
  (task_request_bound_function tsk (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_request_bound_function_FP ts tsk (A + F) <=
  A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
    -
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
forall A : duration,
bounded_pi.is_in_search_space tsk L A ->
exists F : duration,
  bounded_nps.blocking_bound ts tsk +
  (task_request_bound_function tsk (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_request_bound_function_FP ts tsk (A + F) <=
  A + F /\ F + (task_cost tsk - task_rtct tsk) <= R intros A SP.
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
A: duration
SP: bounded_pi.is_in_search_space tsk L A
exists F : duration,
  bounded_nps.blocking_bound ts tsk +
  (task_request_bound_function tsk (A + ε) -
   (task_cost tsk - task_rtct tsk)) +
  total_ohep_request_bound_function_FP ts tsk (A + F) <=
  A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
      rewrite subnn subn0.
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
A: duration
SP: bounded_pi.is_in_search_space tsk L A
exists F : duration,
  bounded_nps.blocking_bound ts tsk +
  task_request_bound_function tsk (A + ε) +
  total_ohep_request_bound_function_FP ts tsk (A + F) <=
  A + F /\ F + 0 <= R
      destruct (H_R_is_maximum _ SP) as [F [EQ LE]].
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence
  arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions
  arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset
  arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model
  arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point
  arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf 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
blocking_bound:= \max_(tsk_other <- ts | 
~~ hep_task tsk_other tsk)
   (task_max_nonpreemptive_segment
      tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk
  L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration,
is_in_search_space A ->
exists F : duration,
  blocking_bound + task_rbf (A + ε) +
  total_ohep_rbf (A + F) <= 
  A + F /\ 
  F <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points
  arr_seq
END: end_of_execution_in_preemption_points arr_seq
A: duration
SP: bounded_pi.is_in_search_space tsk L A
F: duration
EQ: blocking_bound + task_rbf (A + ε) +
total_ohep_rbf (A + F) <= 
A + F
LE: F <= R
exists F : duration,
  bounded_nps.blocking_bound ts tsk +
  task_request_bound_function tsk (A + ε) +
  total_ohep_request_bound_function_FP ts tsk (A + F) <=
  A + F /\ F + 0 <= R
      by exists F; rewrite addn0; split.
  Qed.

End RTAforFloatingModelwithArrivalCurves.