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(** * RTA for EDF with Bounded Non-Preemptive Segments *) (** In this section we instantiate the Abstract RTA for EDF-schedulers with Bounded Priority Inversion to EDF-schedulers for ideal uni-processor model of real-time tasks with arbitrary arrival models _and_ bounded non-preemptive segments. *) (** Recall that Abstract RTA for EDF-schedulers with Bounded Priority Inversion does not specify the cause of priority inversion. In this section, we prove that the priority inversion caused by execution of non-preemptive segments is bounded. Thus the Abstract RTA for EDF-schedulers is applicable to this instantiation. *) Section RTAforEDFwithBoundedNonpreemptiveSegmentsWithArrivalCurves. (** Consider any type of tasks ... *) Context {Task : TaskType}. Context `{TaskCost Task}. Context `{TaskDeadline Task}. Context `{TaskRunToCompletionThreshold Task}. Context `{TaskMaxNonpreemptiveSegment Task}. (** ... and any type of jobs associated with these tasks. *) Context {Job : JobType}. Context `{JobTask Job Task}. Context `{Arrival : JobArrival Job}. Context `{Cost : JobCost Job}. (** We assume the classic (i.e., Liu & Layland) model of readiness without jitter or self-suspensions, wherein pending jobs are always ready. *) #[local] Existing Instance basic_ready_instance. (** For clarity, let's denote the relative deadline of a task as [D]. *) Let D tsk := task_deadline tsk. (** Consider the EDF policy that indicates a higher-or-equal priority relation. Note that we do not relate the EDF policy with the scheduler. However, we define functions for Interference and Interfering Workload that actively use the concept of priorities. *) Let EDF := EDF Job. (** Consider any arrival sequence with consistent, non-duplicate arrivals. *) Variable arr_seq : arrival_sequence Job. Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq. Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq. (** Next, consider any valid ideal uni-processor schedule of this arrival sequence ... *) Variable sched : schedule (ideal.processor_state Job). Hypothesis H_sched_valid : valid_schedule sched arr_seq. (** In addition, we assume the existence of a function mapping jobs to their preemption points ... *) Context `{JobPreemptable Job}. (** ... and assume that it defines a valid preemption model with bounded non-preemptive segments. *) Hypothesis H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched. (** Next, we assume that the schedule is a work-conserving schedule... *) Hypothesis H_work_conserving : work_conserving arr_seq sched. (** ... and the schedule respects the policy defined by the [job_preemptable] function (i.e., jobs have bounded non-preemptive segments). *) Hypothesis H_respects_policy : respects_JLFP_policy_at_preemption_point arr_seq sched EDF. (** Consider an arbitrary task set ts, ... *) Variable ts : list Task. (** ... assume that all jobs come from the task set, ... *) Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts. (** ... and the cost of a job cannot be larger than the task cost. *) Hypothesis H_valid_job_cost: arrivals_have_valid_job_costs arr_seq. (** Let max_arrivals be a family of valid arrival curves, i.e., for any task [tsk] in ts [max_arrival tsk] is (1) an arrival bound of [tsk], and (2) it is a monotonic function that equals 0 for the empty interval delta = 0. *) Context `{MaxArrivals Task}. Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals. Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts. (** Let [tsk] be any task in ts that is to be analyzed. *) Variable tsk : Task. Hypothesis H_tsk_in_ts : tsk \in ts. (** Consider a valid preemption model... *) Hypothesis H_valid_preemption_model: valid_preemption_model arr_seq sched. (** ...and a valid task run-to-completion threshold function. That is, [task_rtct tsk] is (1) no bigger than [tsk]'s cost, (2) for any job of task [tsk] [job_rtct] is bounded by [task_rtct]. *) Hypothesis H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk. (** We introduce as an abbreviation [rbf] for the task request bound function, which is defined as [task_cost(T) × max_arrivals(T,Δ)] for a task T. *) Let rbf := task_request_bound_function. (** Next, we introduce [task_rbf] as an abbreviation for the task request bound function of task [tsk]. *) Let task_rbf := rbf tsk. (** Using the sum of individual request bound functions, we define the request bound function of all tasks (total request bound function). *) Let total_rbf := total_request_bound_function ts. (** Next, we define an upper bound on interfering workload received from jobs of other tasks with higher-than-or-equal priority. *) Let bound_on_total_hep_workload A Δ := \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn ((A + ε) + D tsk - D tsk_o) Δ). (** Let's define some local names for clarity. *) Let max_length_of_priority_inversion := max_length_of_priority_inversion arr_seq. Let response_time_bounded_by := task_response_time_bound arr_seq sched. (** For a job with the relative arrival offset [A] within its busy window, we define the following blocking bound. Only other tasks that potentially release non-zero-cost jobs are relevant, so we define a predicate to exclude pathological cases. *) Definition blocking_relevant (tsk_o : Task) := (max_arrivals tsk_o ε > 0) && (task_cost tsk_o > 0). Definition blocking_bound (A : duration) := \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk_o > D tsk + A)) (task_max_nonpreemptive_segment tsk_o - ε). (** ** Search Space *) (** If priority inversion is caused exclusively by non-preemptive sections, then we do not need to consider the priority-inversion bound in the search space. Hence we define the following search space, which refines the more general [bounded_pi.is_in_search_space] for our specific setting. *) Definition is_in_search_space (L A : duration) := (A < L) && (task_rbf_changes_at tsk A || bound_on_total_hep_workload_changes_at ts tsk A). (** For the following proof, we exploit the fact that the blocking bound is monotonically decreasing in [A], which we note here. *)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop

forall A1 A2 : nat, A1 <= A2 -> blocking_bound A2 <= blocking_bound A1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop

forall A1 A2 : nat, A1 <= A2 -> blocking_bound A2 <= blocking_bound A1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A1, A2: nat
LEQ: A1 <= A2

blocking_bound A2 <= blocking_bound A1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A1, A2: nat
LEQ: A1 <= A2

\max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + A2 < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε) <= \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + A1 < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A1, A2: nat
LEQ: A1 <= A2
tsk_o: Task
IN: tsk_o \in ts
OTHER: 0 < max_arrivals tsk_o ε
LT: 0 < task_cost tsk_o
ARR: D tsk + A2 < D tsk_o

blocking_relevant tsk_o && (D tsk + A1 < D tsk_o)
by repeat (apply /andP; split) => //; lia. Qed. (** To use the refined search space with the abstract theorem, we must show that it still includes all relevant points. To this end, we first observe that a step in the blocking bound implies the existence of a task that could release a job with an absolute deadline equal to the absolute deadline of the job under analysis. *)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop

forall A : duration, priority_inversion_changes_at blocking_bound A -> exists tsk_o : Task, (tsk_o \in ts) && blocking_relevant tsk_o && (tsk_o != tsk) && (D tsk_o == D tsk + A)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop

forall A : duration, priority_inversion_changes_at blocking_bound A -> exists tsk_o : Task, (tsk_o \in ts) && blocking_relevant tsk_o && (tsk_o != tsk) && (D tsk_o == D tsk + A)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A: duration

priority_inversion_changes_at blocking_bound A -> exists tsk_o : Task, (tsk_o \in ts) && blocking_relevant tsk_o && (tsk_o != tsk) && (D tsk_o == D tsk + A)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A: duration
NEQ: blocking_bound (A - ε) != blocking_bound A

exists tsk_o : Task, (tsk_o \in ts) && blocking_relevant tsk_o && (tsk_o != tsk) && (D tsk_o == D tsk + A)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A: duration
NEQ: blocking_bound (A - ε) != blocking_bound A
LEQ: blocking_bound A <= blocking_bound (A - ε)

exists tsk_o : Task, (tsk_o \in ts) && blocking_relevant tsk_o && (tsk_o != tsk) && (D tsk_o == D tsk + A)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A: duration
NEQ: blocking_bound (A - ε) != blocking_bound A
LEQ: blocking_bound A <= blocking_bound (A - ε)
LT: blocking_bound A < blocking_bound (A - ε)

exists tsk_o : Task, (tsk_o \in ts) && blocking_relevant tsk_o && (tsk_o != tsk) && (D tsk_o == D tsk + A)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A: duration
LT: \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + A < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε) < \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + (A - ε) < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε)

exists tsk_o : Task, (tsk_o \in ts) && blocking_relevant tsk_o && (tsk_o != tsk) && (D tsk_o == D tsk + A)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A: duration
LT: \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + A < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε) < \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + (A - ε) < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε)
tsk_o: Task
IN: tsk_o \in ts
NOT: ~~ (blocking_relevant tsk_o && (D tsk + A < D tsk_o))
HOLDS: blocking_relevant tsk_o && (D tsk + (A - ε) < D tsk_o)

exists tsk_o : Task, (tsk_o \in ts) && blocking_relevant tsk_o && (tsk_o != tsk) && (D tsk_o == D tsk + A)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A: duration
LT: \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + A < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε) < \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + (A - ε) < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε)
tsk_o: Task
IN: tsk_o \in ts
NOT: ~~ (blocking_relevant tsk_o && (D tsk + A < D tsk_o))
REL: blocking_relevant tsk_o
LTeps: D tsk + (A - ε) < D tsk_o

exists tsk_o : Task, (tsk_o \in ts) && blocking_relevant tsk_o && (tsk_o != tsk) && (D tsk_o == D tsk + A)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A: duration
LT: \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + A < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε) < \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + (A - ε) < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε)
tsk_o: Task
IN: tsk_o \in ts
NOT: ~~ (blocking_relevant tsk_o && (D tsk + A < D tsk_o))
REL: blocking_relevant tsk_o
LTeps: D tsk + (A - ε) < D tsk_o

D tsk_o == D tsk + A
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A: duration
LT: \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + A < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε) < \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + (A - ε) < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε)
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
LTeps: D tsk + (A - ε) < D tsk_o

~~ (D tsk + A < D tsk_o) -> D tsk_o == D tsk + A
by move: LTeps; rewrite /ε => LTeps; lia. Qed. (** With the above setup in place, we can show that the search space defined above by [is_in_search_space] covers the the more abstract search space defined by [bounded_pi.is_in_search_space]. *)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop

forall A L : duration, bounded_pi.is_in_search_space ts tsk blocking_bound L A -> is_in_search_space L A
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop

forall A L : duration, bounded_pi.is_in_search_space ts tsk blocking_bound L A -> is_in_search_space L A
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A || task_rbf_changes_at tsk A || bound_on_total_hep_workload_changes_at ts tsk A

is_in_search_space L A
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A || task_rbf_changes_at tsk A || bound_on_total_hep_workload_changes_at ts tsk A

task_rbf_changes_at tsk A \/ bound_on_total_hep_workload_changes_at ts tsk A
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A

bound_on_total_hep_workload_changes_at ts tsk A
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
OTHER: tsk_o != tsk
EQ: D tsk_o == D tsk + A

bound_on_total_hep_workload_changes_at ts tsk A
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
OTHER: tsk_o != tsk
EQ: D tsk_o == D tsk + A

has (fun tsko : Task => (tsk != tsko) && (task_request_bound_function tsko (A + task_deadline tsk - task_deadline tsko) != task_request_bound_function tsko (A + ε + task_deadline tsk - task_deadline tsko))) ts
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
OTHER: tsk_o != tsk
EQ: D tsk_o == D tsk + A

(tsk != tsk_o) && (task_request_bound_function tsk_o (A + task_deadline tsk - task_deadline tsk_o) != task_request_bound_function tsk_o (A + ε + task_deadline tsk - task_deadline tsk_o))
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
OTHER: tsk_o != tsk
EQ: D tsk_o == D tsk + A

task_request_bound_function tsk_o (A + task_deadline tsk - task_deadline tsk_o) != task_request_bound_function tsk_o (A + ε + task_deadline tsk - task_deadline tsk_o)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
OTHER: tsk_o != tsk

D tsk_o == D tsk + A -> task_request_bound_function tsk_o (A + task_deadline tsk - task_deadline tsk_o) != task_request_bound_function tsk_o (A + ε + task_deadline tsk - task_deadline tsk_o)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
OTHER: tsk_o != tsk
EQ: task_deadline tsk_o = task_deadline tsk + A

task_request_bound_function tsk_o (A + task_deadline tsk - task_deadline tsk_o) != task_request_bound_function tsk_o (A + ε + task_deadline tsk - task_deadline tsk_o)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
REL: blocking_relevant tsk_o
OTHER: tsk_o != tsk
EQ: task_deadline tsk_o = task_deadline tsk + A

task_cost tsk_o * max_arrivals tsk_o (A + task_deadline tsk - (task_deadline tsk + A)) != task_cost tsk_o * max_arrivals tsk_o (A + ε + task_deadline tsk - (task_deadline tsk + A))
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
OTHER: tsk_o != tsk
EQ: task_deadline tsk_o = task_deadline tsk + A
ARRIVES: 0 < max_arrivals tsk_o ε
COST: 0 < task_cost tsk_o

task_cost tsk_o * max_arrivals tsk_o (A + task_deadline tsk - (task_deadline tsk + A)) != task_cost tsk_o * max_arrivals tsk_o (A + ε + task_deadline tsk - (task_deadline tsk + A))
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
OTHER: tsk_o != tsk
EQ: task_deadline tsk_o = task_deadline tsk + A
ARRIVES: 0 < max_arrivals tsk_o ε
COST: 0 < task_cost tsk_o

max_arrivals tsk_o (A + task_deadline tsk - (task_deadline tsk + A)) != max_arrivals tsk_o (A + ε + task_deadline tsk - (task_deadline tsk + A))
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
OTHER: tsk_o != tsk
EQ: task_deadline tsk_o = task_deadline tsk + A
ARRIVES: 0 < max_arrivals tsk_o ε
COST: 0 < task_cost tsk_o

max_arrivals tsk_o 0 != max_arrivals tsk_o (A + ε + task_deadline tsk - (task_deadline tsk + A))
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
A, L: duration
BOUND: A < L
STEP: priority_inversion_changes_at blocking_bound A
tsk_o: Task
IN: tsk_o \in ts
OTHER: tsk_o != tsk
EQ: task_deadline tsk_o = task_deadline tsk + A
ARRIVES: 0 < max_arrivals tsk_o ε
COST: 0 < task_cost tsk_o

max_arrivals tsk_o 0 != max_arrivals tsk_o ε
by move: (H_valid_arrival_curve tsk_o IN) => [-> _]; lia. Qed. (** ** Priority inversion is bounded *) (** In this section, we prove that a priority inversion for task [tsk] is bounded by the maximum length of non-preemptive segments among the tasks with lower priority. *) Section PriorityInversionIsBounded. (** First, we observe that the maximum non-preemptive segment length of any task that releases a job with an earlier absolute deadline (w.r.t. a given job [j]) and non-zero execution cost upper-bounds the maximum possible length of priority inversion (of said job [j]). *)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop

forall (j : Job) (t1 : instant), max_length_of_priority_inversion j t1 <= \max_(j_lp <- arrivals_between arr_seq 0 t1 | ~~ EDF j_lp j && (0 < job_cost j_lp)) (task_max_nonpreemptive_segment (job_task j_lp) - ε)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop

forall (j : Job) (t1 : instant), max_length_of_priority_inversion j t1 <= \max_(j_lp <- arrivals_between arr_seq 0 t1 | ~~ EDF j_lp j && (0 < job_cost j_lp)) (task_max_nonpreemptive_segment (job_task j_lp) - ε)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1: instant

max_length_of_priority_inversion j t1 <= \max_(j_lp <- arrivals_between arr_seq 0 t1 | ~~ EDF j_lp j && (0 < job_cost j_lp)) (task_max_nonpreemptive_segment (job_task j_lp) - ε)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1: instant

\max_(j_lp <- arrivals_before arr_seq t1 | ~~ hep_job j_lp j && (0 < job_cost j_lp)) (job_max_nonpreemptive_segment j_lp - ε) <= \max_(j_lp <- arrivals_between arr_seq 0 t1 | ~~ EDF j_lp j && (0 < job_cost j_lp)) (task_max_nonpreemptive_segment (job_task j_lp) - ε)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1: instant
j': Job
JINB: j' \in arrivals_before arr_seq t1
NOTHEP: ~~ hep_job j' j && (0 < job_cost j')

job_max_nonpreemptive_segment j' - ε <= task_max_nonpreemptive_segment (job_task j') - ε
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1: instant
j': Job
JINB: j' \in arrivals_before arr_seq t1
NOTHEP: ~~ hep_job j' j && (0 < job_cost j')

job_max_nonpreemptive_segment j' <= task_max_nonpreemptive_segment (job_task j')
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1: instant
j': Job
JINB: arrives_in arr_seq j'
NOTHEP: ~~ hep_job j' j && (0 < job_cost j')

job_max_nonpreemptive_segment j' <= task_max_nonpreemptive_segment (job_task j')
by apply H_valid_model_with_bounded_nonpreemptive_segments. Qed. (** Second, we prove that the maximum length of a priority inversion of a given job [j] is indeed bounded by defined the blocking bound. *)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop

forall (j : Job) (t1 t2 : instant), arrives_in arr_seq j -> job_of_task tsk j -> busy_interval_prefix arr_seq sched j t1 t2 -> max_length_of_priority_inversion j t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop

forall (j : Job) (t1 t2 : instant), arrives_in arr_seq j -> job_of_task tsk j -> busy_interval_prefix arr_seq sched j t1 t2 -> max_length_of_priority_inversion j t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
BUSY: busy_interval_prefix arr_seq sched j t1 t2

priority_inversion.max_length_of_priority_inversion arr_seq j t1 <= \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + (job_arrival j - t1) < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2

priority_inversion.max_length_of_priority_inversion arr_seq j t1 <= \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + (job_arrival j - t1) < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2

\max_(j_lp <- arrivals_between arr_seq 0 t1 | ~~ EDF j_lp j && (0 < job_cost j_lp)) (task_max_nonpreemptive_segment (job_task j_lp) - ε) <= \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + (job_arrival j - t1) < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
JINB: j' \in arrivals_between arr_seq 0 t1
NOTHEP: ~~ EDF j' j && (0 < job_cost j')

task_max_nonpreemptive_segment (job_task j') - ε <= \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + (job_arrival j - t1) < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
JINB: j' \in arrivals_between arr_seq 0 t1
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'

task_max_nonpreemptive_segment (job_task j') - ε <= \max_(tsk_o <- ts | blocking_relevant tsk_o && (D tsk + (job_arrival j - t1) < D tsk_o)) (task_max_nonpreemptive_segment tsk_o - ε)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
JINB: j' \in arrivals_between arr_seq 0 t1
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'

blocking_relevant (job_task j') && (D tsk + (job_arrival j - t1) < D (job_task j'))
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
JINB: arrived_between j' 0 t1
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'

blocking_relevant (job_task j') && (D tsk + (job_arrival j - t1) < D (job_task j'))
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1

blocking_relevant (job_task j') && (D tsk + (job_arrival j - t1) < D (job_task j'))
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1

D tsk + (job_arrival j - t1) < D (job_task j')
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
0 < task_cost (job_task j')
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
0 < max_arrivals (job_task j') ε
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1

D tsk + (job_arrival j - t1) < D (job_task j')
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
NOTHEP: (job_deadline j < job_deadline j') && (0 < job_cost j')

D tsk + (job_arrival j - t1) < D (job_task j')
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
NOTHEP: (job_deadline j < job_deadline j') && (0 < job_cost j')

D (job_task j) + (job_arrival j - t1) < D (job_task j')
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
NOTHEP: (job_deadline j < job_deadline j') && (0 < job_cost j')
ARRLE: job_arrival j' < job_arrival j

D (job_task j) + (job_arrival j - t1) < D (job_task j')
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
ARRLE: job_arrival j' < job_arrival j

(job_arrival j + task_deadline (job_task j) < job_arrival j' + task_deadline (job_task j')) && (0 < job_cost j') -> task_deadline (job_task j) + (job_arrival j - t1) < task_deadline (job_task j')
by lia.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1

0 < task_cost (job_task j')
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
0 < max_arrivals (job_task j') ε
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1

0 < task_cost (job_task j')
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
NZ: 0 < job_cost j'

0 < task_cost (job_task j')
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
NZ: 0 < job_cost j'

job_cost j' <= task_cost (job_task j') -> 0 < task_cost (job_task j')
by lia.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1

0 < max_arrivals (job_task j') ε
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1

0 < max_arrivals (job_task j') ε
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1

arrives_in ?Goal ?Goal1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
job_of_task (job_task j') ?Goal1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
respects_max_arrivals ?Goal (job_task j') (max_arrivals (job_task j'))
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1

arrives_in ?Goal ?Goal1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
job_of_task (job_task j') ?Goal1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
respects_max_arrivals ?Goal (job_task j') (max_arrivals (job_task j'))
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1

job_of_task (job_task j') j'
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
respects_max_arrivals arr_seq (job_task j') (max_arrivals (job_task j'))
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1

job_of_task (job_task j') j'
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1
respects_max_arrivals arr_seq (job_task j') (max_arrivals (job_task j'))
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1

respects_max_arrivals arr_seq (job_task j') (max_arrivals (job_task j'))
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
t1, t2: instant
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
TT: t1 < t2
QT: quiet_time arr_seq sched j t1
GE: t1 <= job_arrival j
LT: job_arrival j < t2
j': Job
NOTHEP: ~~ EDF j' j && (0 < job_cost j')
ARR': arrives_in arr_seq j'
TJ': job_arrival j' < t1

respects_max_arrivals arr_seq (job_task j') (max_arrivals (job_task j'))
by apply H_is_arrival_curve, H_all_jobs_from_taskset, ARR'. } Qed. (** Using the lemma above, we prove that the priority inversion of the task is bounded by the maximum length of a nonpreemptive section of lower-priority tasks. *)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop

priority_inversion_is_bounded_by arr_seq sched tsk blocking_bound
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop

priority_inversion_is_bounded_by arr_seq sched tsk blocking_bound
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j

cumulative_priority_inversion sched j t1 t2 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: t2 - t1 <= blocking_bound (job_arrival j - t1)

cumulative_priority_inversion sched j t1 t2 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
cumulative_priority_inversion sched j t1 t2 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: t2 - t1 <= blocking_bound (job_arrival j - t1)

cumulative_priority_inversion sched j t1 t2 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: t2 - t1 <= blocking_bound (job_arrival j - t1)

cumulative_priority_inversion sched j t1 t2 <= t2 - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: t2 - t1 <= blocking_bound (job_arrival j - t1)

\sum_(t1 <= t < t2) match sched t with | Some jlp => ~~ hep_job jlp j | None => false end <= t2 - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: t2 - t1 <= blocking_bound (job_arrival j - t1)

\sum_(t1 <= t < t2) match sched t with | Some jlp => ~~ hep_job jlp j | None => false end <= \sum_(t1 <= i < t2) 1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: t2 - t1 <= blocking_bound (job_arrival j - t1)

forall i : nat, true -> match sched i with | Some jlp => ~~ hep_job jlp j | None => false end <= 1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: t2 - t1 <= blocking_bound (job_arrival j - t1)
t: nat

forall a : Job, ~~ hep_job a j <= 1
by intros s; destruct (hep_job s j).
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1

cumulative_priority_inversion sched j t1 t2 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
NEQ2: t1 <= ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

cumulative_priority_inversion sched j t1 t2 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

cumulative_priority_inversion sched j t1 t2 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

cumulative_priority_inversion sched j t1 t2 <= cumulative_priority_inversion sched j t1 ppt
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

cumulative_priority_inversion sched j t1 t2 <= cumulative_priority_inversion sched j t1 ppt
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

\sum_(t1 <= t < t2) match sched t with | Some jlp => ~~ hep_job jlp j | None => false end <= \sum_(t1 <= t < ppt) match sched t with | Some jlp => ~~ hep_job jlp j | None => false end
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

ppt <= t2
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
\sum_(t1 <= i < ppt) match sched i with | Some jlp => ~~ hep_job jlp j | None => false end + \sum_(ppt <= i < t2) match sched i with | Some jlp => ~~ hep_job jlp j | None => false end <= \sum_(t1 <= t < ppt) match sched t with | Some jlp => ~~ hep_job jlp j | None => false end
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

ppt <= t2
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
NEQ: t1 + blocking_bound (job_arrival j - t1) < t2

ppt <= t2
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
NEQ: t1 + blocking_bound (job_arrival j - t1) < t2

ppt <= t1 + blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
NEQ: t1 + blocking_bound (job_arrival j - t1) < t2

t1 + max_length_of_priority_inversion j t1 <= t1 + blocking_bound (job_arrival j - t1)
by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

\sum_(t1 <= i < ppt) match sched i with | Some jlp => ~~ hep_job jlp j | None => false end + \sum_(ppt <= i < t2) match sched i with | Some jlp => ~~ hep_job jlp j | None => false end <= \sum_(t1 <= t < ppt) match sched t with | Some jlp => ~~ hep_job jlp j | None => false end
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

\sum_(ppt <= i < t2) match sched i with | Some jlp => ~~ hep_job jlp j | None => false end == 0
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2

match sched t with | Some jlp => ~~ hep_job jlp j | None => false end = 0
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
s: Job
SCHED: sched t = Some s

~~ hep_job s j = 0
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
s: Job
SCHED: sched t = Some s

exists pr_t : instant, preemption_time sched pr_t /\ t1 <= pr_t <= t1 + (ppt - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
s: Job
SCHED: sched t = Some s
t1 + (ppt - t1) <= t < t2
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
s: Job
SCHED: sched t = Some s
j_hp: Job
ARRB: arrived_between j_hp t1 t.+1
HP: hep_job j_hp j
SCHEDHP: scheduled_at sched j_hp t
~~ hep_job s j = 0
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
s: Job
SCHED: sched t = Some s

exists pr_t : instant, preemption_time sched pr_t /\ t1 <= pr_t <= t1 + (ppt - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
s: Job
SCHED: sched t = Some s

preemption_time sched ppt
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
s: Job
SCHED: sched t = Some s
t1 <= ppt <= t1 + (ppt - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
s: Job
SCHED: sched t = Some s

t1 <= ppt <= t1 + (ppt - t1)
by rewrite subnKC //; apply/andP; split.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
s: Job
SCHED: sched t = Some s

t1 + (ppt - t1) <= t < t2
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
s: Job
SCHED: sched t = Some s
j_hp: Job
ARRB: arrived_between j_hp t1 t.+1
HP: hep_job j_hp j
SCHEDHP: scheduled_at sched j_hp t
~~ hep_job s j = 0
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
s: Job
SCHED: sched t = Some s

t1 + (ppt - t1) <= t < t2
by rewrite subnKC //; apply/andP; split.
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
s: Job
SCHED: sched t = Some s
j_hp: Job
ARRB: arrived_between j_hp t1 t.+1
HP: hep_job j_hp j
SCHEDHP: scheduled_at sched j_hp t

~~ hep_job s j = 0
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
s: Job
SCHED: sched t = Some s
j_hp: Job
ARRB: arrived_between j_hp t1 t.+1
HP: hep_job j_hp j
SCHEDHP: scheduled_at sched j_hp t

hep_job s j
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
s: Job
SCHED: sched t = Some s
j_hp: Job
ARRB: arrived_between j_hp t1 t.+1
HP: hep_job j_hp j
SCHEDHP: scheduled_at sched j_hp t

s = j_hp
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat
GEt: ppt <= t
LTt: t < t2
s, j_hp: Job
ARRB: arrived_between j_hp t1 t.+1
HP: hep_job j_hp j
SCHEDHP: scheduled_at sched j_hp t
SCHED: scheduled_at sched s t

s = j_hp
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

cumulative_priority_inversion sched j t1 ppt <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

\sum_(t1 <= t < ppt) match sched t with | Some jlp => ~~ hep_job jlp j | None => false end <= ppt - t1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

\sum_(t1 <= t < ppt) match sched t with | Some jlp => ~~ hep_job jlp j | None => false end <= \sum_(t1 <= i < ppt) 1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

forall i : nat, true -> match sched i with | Some jlp => ~~ hep_job jlp j | None => false end <= 1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t: nat

forall a : Job, ~~ hep_job a j <= 1
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

ppt - t1 <= blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

ppt <= t1 + blocking_bound (job_arrival j - t1)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
j: Job
ARR: arrives_in arr_seq j
TSK: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: instant
PREF: busy_interval_prefix arr_seq sched j t1 t2
T: t1 <= job_arrival j
NEQ: blocking_bound (job_arrival j - t1) < t2 - t1
ppt: instant
PPT: preemption_time sched ppt
GE: t1 <= ppt
LE: ppt <= t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1

t1 + max_length_of_priority_inversion j t1 <= t1 + blocking_bound (job_arrival j - t1)
by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2. Qed. End PriorityInversionIsBounded. (** ** Response-Time Bound *) (** In this section, we prove that the maximum among the solutions of the response-time bound recurrence is a response-time bound for [tsk]. *) Section ResponseTimeBound. (** Let L be any positive fixed point of the busy interval recurrence. *) Variable L : duration. Hypothesis H_L_positive : L > 0. Hypothesis H_fixed_point : L = total_rbf L. (** Consider any value [R], and assume that for any given arrival offset [A] in the search space, there is a solution of the response-time bound recurrence which is bounded by [R]. *) Variable R : duration. Hypothesis H_R_is_maximum: forall (A : duration), is_in_search_space L A -> exists (F : duration), A + F >= blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) /\ R >= F + (task_cost tsk - task_rtct tsk). (** Then, using the results for the general RTA for EDF-schedulers, we establish a response-time bound for the more concrete model of bounded nonpreemptive segments. Note that in case of the general RTA for EDF-schedulers, we just _assume_ that the priority inversion is bounded. In this module we provide the preemption model with bounded nonpreemptive segments and _prove_ that the priority inversion is bounded. *)
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R

response_time_bounded_by tsk R
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R

response_time_bounded_by tsk R
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R

sequential_tasks arr_seq sched
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
priority_inversion_is_bounded_by arr_seq sched tsk ?priority_inversion_bound
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
forall A : duration, bounded_pi.is_in_search_space ts tsk ?priority_inversion_bound L A -> exists F : duration, ?priority_inversion_bound A + (task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) + \sum_(tsk_o <- ts | tsk_o != tsk) task_request_bound_function tsk_o (minn (A + ε + task_deadline tsk - task_deadline tsk_o) (A + F)) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R

sequential_tasks arr_seq sched
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
priority_inversion_is_bounded_by arr_seq sched tsk ?priority_inversion_bound
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
forall A : duration, bounded_pi.is_in_search_space ts tsk ?priority_inversion_bound L A -> exists F : duration, ?priority_inversion_bound A + (task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) + \sum_(tsk_o <- ts | tsk_o != tsk) task_request_bound_function tsk_o (minn (A + ε + task_deadline tsk - task_deadline tsk_o) (A + F)) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R

work_bearing_readiness arr_seq sched
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
priority_inversion_is_bounded_by arr_seq sched tsk ?priority_inversion_bound
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
forall A : duration, bounded_pi.is_in_search_space ts tsk ?priority_inversion_bound L A -> exists F : duration, ?priority_inversion_bound A + (task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) + \sum_(tsk_o <- ts | tsk_o != tsk) task_request_bound_function tsk_o (minn (A + ε + task_deadline tsk - task_deadline tsk_o) (A + F)) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R

priority_inversion_is_bounded_by arr_seq sched tsk ?priority_inversion_bound
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
forall A : duration, bounded_pi.is_in_search_space ts tsk ?priority_inversion_bound L A -> exists F : duration, ?priority_inversion_bound A + (task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) + \sum_(tsk_o <- ts | tsk_o != tsk) task_request_bound_function tsk_o (minn (A + ε + task_deadline tsk - task_deadline tsk_o) (A + F)) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R

priority_inversion_is_bounded_by arr_seq sched tsk ?priority_inversion_bound
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
forall A : duration, bounded_pi.is_in_search_space ts tsk ?priority_inversion_bound L A -> exists F : duration, ?priority_inversion_bound A + (task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) + \sum_(tsk_o <- ts | tsk_o != tsk) task_request_bound_function tsk_o (minn (A + ε + task_deadline tsk - task_deadline tsk_o) (A + F)) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R

forall A : duration, bounded_pi.is_in_search_space ts tsk blocking_bound L A -> exists F : duration, blocking_bound A + (task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) + \sum_(tsk_o <- ts | tsk_o != tsk) task_request_bound_function tsk_o (minn (A + ε + task_deadline tsk - task_deadline tsk_o) (A + F)) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R

forall A : duration, bounded_pi.is_in_search_space ts tsk blocking_bound L A -> exists F : duration, blocking_bound A + (task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) + \sum_(tsk_o <- ts | tsk_o != tsk) task_request_bound_function tsk_o (minn (A + ε + task_deadline tsk - task_deadline tsk_o) (A + F)) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
Task: TaskType
H: TaskCost Task
H0: TaskDeadline Task
H1: TaskRunToCompletionThreshold Task
H2: TaskMaxNonpreemptiveSegment Task
Job: JobType
H3: JobTask Job Task
Arrival: JobArrival Job
Cost: JobCost Job
D:= [eta task_deadline]: Task -> duration
EDF:= edf.EDF Job: JLFP_policy Job
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
H_arr_seq_is_a_set: arrival_sequence_uniq arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H4: JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments: valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_JLFP_policy_at_preemption_point arr_seq sched EDF
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold arr_seq tsk
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_rbf:= total_request_bound_function ts: duration -> nat
bound_on_total_hep_workload:= fun A Δ : nat => \sum_(tsk_o <- ts | tsk_o != tsk) rbf tsk_o (minn (A + ε + D tsk - D tsk_o) Δ): nat -> nat -> nat
max_length_of_priority_inversion:= priority_inversion.max_length_of_priority_inversion arr_seq: Job -> instant -> nat
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
L: duration
H_L_positive: 0 < L
H_fixed_point: L = total_rbf L
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space L A -> exists F : duration, blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk)) + bound_on_total_hep_workload A (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
A: duration
BPI_SP: bounded_pi.is_in_search_space ts tsk blocking_bound L A

exists F : duration, blocking_bound A + (task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) + \sum_(tsk_o <- ts | tsk_o != tsk) task_request_bound_function tsk_o (minn (A + ε + task_deadline tsk - task_deadline tsk_o) (A + F)) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
by apply H_R_is_maximum, search_space_inclusion. Qed. End ResponseTimeBound. End RTAforEDFwithBoundedNonpreemptiveSegmentsWithArrivalCurves.