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

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

(** ** Setup and Assumptions *)

Section RTAforFullyNonPreemptiveFPModelwithArrivalCurves.

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

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

  (** 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_cost 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) - (task_cost tsk - ε))
                + total_ohep_rbf (A + F) /\
        R >= F + (task_cost tsk - ε).

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

  Let response_time_bounded_by := task_response_time_bound arr_seq sched.

  Theorem uniprocessor_response_time_bound_fully_nonpreemptive_fp:
    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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
response_time_bounded_by tsk R
  Proof.
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
response_time_bounded_by tsk R
    move: (posnP (@task_cost _ H tsk)) => [ZERO|POS].
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
response_time_bounded_by tsk R
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
response_time_bounded_by tsk R
    {
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
response_time_bounded_by tsk R intros j ARR TSK.
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
job_response_time_bound sched j R
      have ZEROj: job_cost j = 0.
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
job_cost j = 0
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
ZEROj: job_cost j = 0
job_response_time_bound sched j R
      {
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
job_cost j = 0 move: (H_valid_job_cost j ARR) => NEQ.
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
NEQ: valid_job_cost j
job_cost j = 0
        rewrite /valid_job_cost in NEQ.
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
NEQ: job_cost j <= task_cost (job_task j)
job_cost j = 0
        move: TSK => /eqP -> in NEQ.
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
NEQ: job_cost j <= task_cost tsk
job_cost j = 0
        rewrite ZERO in NEQ.
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
NEQ: job_cost j <= 0
job_cost j = 0
        by apply/eqP; rewrite -leqn0.
      }
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
ZERO: task_cost tsk = 0
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
ZEROj: job_cost j = 0
job_response_time_bound sched j R
      by rewrite /job_response_time_bound /completed_by ZEROj.
    }
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
response_time_bounded_by tsk R
    try ( eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments with
        (L0 := L) ) ||
    eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments with
        (L := 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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
valid_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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
work_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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
valid_schedule sched arr_seq
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
valid_model_with_bounded_nonpreemptive_segments
  arr_seq sched
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
work_conserving arr_seq sched
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
respects_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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
all_jobs_from_taskset arr_seq ?ts
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
arrivals_have_valid_job_costs arr_seq
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
valid_taskset_arrival_curve ?ts max_arrivals
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
taskset_respects_max_arrivals arr_seq ?ts
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
tsk \in ?ts
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
valid_preemption_model arr_seq sched
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
valid_task_run_to_completion_threshold arr_seq tsk
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
0 < L
Task: TaskType
H: TaskCost Task
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
L =
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
forall A : duration,
bounded_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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
work_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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
work_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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
sequential_tasks arr_seq sched by apply sequential_readiness_implies_work_bearing_readiness; rt_auto.
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
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
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
H3: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts
  max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals
  arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_nonpreemptive_sched: nonpreemptive_schedule sched
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_cost 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) -
   (task_cost tsk - ε)) +
  total_ohep_rbf (A + F) <= 
  A + F /\
  F + (task_cost tsk - ε) <= R
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
POS: 0 < task_cost tsk
sequential_tasks arr_seq sched by apply sequential_readiness_implies_sequential_tasks; rt_auto.
  Qed.

End RTAforFullyNonPreemptiveFPModelwithArrivalCurves.