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(** * RTA for Model with Floating Non-Preemptive Regions *) (** In this module we prove the RTA theorem for floating non-preemptive regions FP model. *) (** ** Setup and Assumptions *) Section RTAforFloatingModelwithArrivalCurves. (** We assume ideal uni-processor schedules. *) #[local] Existing Instance ideal.processor_state. (** Consider any type of tasks ... *) Context {Task : TaskType}. Context `{TaskCost Task}. (** ... and any type of jobs associated with these tasks. *) Context {Job : JobType}. Context `{JobTask Job Task}. Context `{JobArrival Job}. Context `{JobCost Job}. (** We assume that jobs are limited-preemptive. *) #[local] Existing Instance limited_preemptive_job_model. (** Consider any arrival sequence with consistent, non-duplicate arrivals. *) Variable arr_seq : arrival_sequence Job. Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq. (** Assume we have the model with floating non-preemptive regions. I.e., for each task only the length of the maximal non-preemptive segment is known _and_ each job level is divided into a number of non-preemptive segments by inserting preemption points. *) Context `{JobPreemptionPoints Job} `{TaskMaxNonpreemptiveSegment Task}. Hypothesis H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq. (** Consider an arbitrary task set ts, ... *) Variable ts : list Task. (** ... assume that all jobs come from the task set, ... *) Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts. (** ... and the cost of a job cannot be larger than the task cost. *) Hypothesis H_valid_job_cost: arrivals_have_valid_job_costs arr_seq. (** Let max_arrivals be a family of valid arrival curves, i.e., for any task [tsk] in ts [max_arrival tsk] is (1) an arrival bound of [tsk], and (2) it is a monotonic function that equals [0] for the empty interval [delta = 0]. *) Context `{MaxArrivals Task}. Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals. Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts. (** Let [tsk] be any task in ts that is to be analyzed. *) Variable tsk : Task. Hypothesis H_tsk_in_ts : tsk \in ts. (** Recall that we assume sequential readiness. *) #[local] Instance sequential_readiness : JobReady _ _ := sequential_ready_instance arr_seq. (** Next, consider any valid ideal uni-processor schedule with with limited preemptions of this arrival sequence ... *) Variable sched : schedule (ideal.processor_state Job). Hypothesis H_sched_valid : valid_schedule sched arr_seq. Hypothesis H_schedule_with_limited_preemptions : schedule_respects_preemption_model arr_seq sched. (** Consider an FP policy that indicates a higher-or-equal priority relation, and assume that the relation is reflexive and transitive. *) Context {FP :FP_policy Task}. Hypothesis H_priority_is_reflexive : reflexive_priorities. Hypothesis H_priority_is_transitive : transitive_priorities. (** Next, we assume that the schedule is a work-conserving schedule... *) Hypothesis H_work_conserving : work_conserving arr_seq sched. (** ... and the schedule respects the scheduling policy. *) Hypothesis H_respects_policy : respects_FP_policy_at_preemption_point arr_seq sched FP. (** ** Total Workload and Length of Busy Interval *) (** We introduce the abbreviation [rbf] for the task request bound function, which is defined as [task_cost(T) × max_arrivals(T,Δ)] for a task T. *) Let rbf := task_request_bound_function. (** Next, we introduce [task_rbf] as an abbreviation for the task request bound function of task [tsk]. *) Let task_rbf := rbf tsk. (** Using the sum of individual request bound functions, we define the request bound function of all tasks with higher priority ... *) Let total_hep_rbf := total_hep_request_bound_function_FP ts tsk. (** ... and the request bound function of all tasks with higher priority other than task [tsk]. *) Let total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk. (** Next, we define a bound for the priority inversion caused by tasks of lower priority. *) Let blocking_bound := \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε). (** Let L be any positive fixed point of the busy interval recurrence, determined by the sum of blocking and higher-or-equal-priority workload. *) Variable L : duration. Hypothesis H_L_positive : L > 0. Hypothesis H_fixed_point : L = blocking_bound + total_hep_rbf L. (** ** Response-Time Bound *) (** To reduce the time complexity of the analysis, recall the notion of search space. *) Let is_in_search_space := is_in_search_space tsk L. (** Next, consider any value R, and assume that for any given arrival A from search space there is a solution of the response-time bound recurrence which is bounded by R. *) Variable R : duration. Hypothesis H_R_is_maximum: forall (A : duration), is_in_search_space A -> exists (F : duration), A + F >= blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) /\ R >= F. (** Now, we can reuse the results for the abstract model with bounded nonpreemptive segments to establish a response-time bound for the more concrete model with floating nonpreemptive regions. *) Let response_time_bounded_by := task_response_time_bound arr_seq sched.
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop

response_time_bounded_by tsk R
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop

response_time_bounded_by tsk R
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq

response_time_bounded_by tsk R
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq

response_time_bounded_by tsk R
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq

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

work_bearing_readiness arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq
sequential_tasks arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq
forall A : duration, bounded_pi.is_in_search_space tsk L A -> exists F : duration, bounded_nps.blocking_bound ts tsk + (task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) + total_ohep_request_bound_function_FP ts tsk (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq

work_bearing_readiness arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq
sequential_tasks arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq
forall A : duration, bounded_pi.is_in_search_space tsk L A -> exists F : duration, bounded_nps.blocking_bound ts tsk + (task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) + total_ohep_request_bound_function_FP ts tsk (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq

sequential_tasks arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq
forall A : duration, bounded_pi.is_in_search_space tsk L A -> exists F : duration, bounded_nps.blocking_bound ts tsk + (task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) + total_ohep_request_bound_function_FP ts tsk (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq

sequential_tasks arr_seq sched
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq
forall A : duration, bounded_pi.is_in_search_space tsk L A -> exists F : duration, bounded_nps.blocking_bound ts tsk + (task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) + total_ohep_request_bound_function_FP ts tsk (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq

forall A : duration, bounded_pi.is_in_search_space tsk L A -> exists F : duration, bounded_nps.blocking_bound ts tsk + (task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) + total_ohep_request_bound_function_FP ts tsk (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq

forall A : duration, bounded_pi.is_in_search_space tsk L A -> exists F : duration, bounded_nps.blocking_bound ts tsk + (task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) + total_ohep_request_bound_function_FP ts tsk (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq
A: duration
SP: bounded_pi.is_in_search_space tsk L A

exists F : duration, bounded_nps.blocking_bound ts tsk + (task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) + total_ohep_request_bound_function_FP ts tsk (A + F) <= A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq
A: duration
SP: bounded_pi.is_in_search_space tsk L A

exists F : duration, bounded_nps.blocking_bound ts tsk + task_request_bound_function tsk (A + ε) + total_ohep_request_bound_function_FP ts tsk (A + F) <= A + F /\ F + 0 <= R
Task: TaskType
H: TaskCost Task
Job: JobType
H0: JobTask Job Task
H1: JobArrival Job
H2: JobCost Job
arr_seq: arrival_sequence Job
H_valid_arrival_sequence: valid_arrival_sequence arr_seq
H3: JobPreemptionPoints Job
H4: TaskMaxNonpreemptiveSegment Task
H_valid_task_model_with_floating_nonpreemptive_regions: valid_model_with_floating_nonpreemptive_regions arr_seq
ts: seq Task
H_all_jobs_from_taskset: all_jobs_from_taskset arr_seq ts
H_valid_job_cost: arrivals_have_valid_job_costs arr_seq
H5: MaxArrivals Task
H_valid_arrival_curve: valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve: taskset_respects_max_arrivals arr_seq ts
tsk: Task
H_tsk_in_ts: tsk \in ts
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
H_schedule_with_limited_preemptions: schedule_respects_preemption_model arr_seq sched
FP: FP_policy Task
H_priority_is_reflexive: reflexive_priorities
H_priority_is_transitive: transitive_priorities
H_work_conserving: work_conserving arr_seq sched
H_respects_policy: respects_FP_policy_at_preemption_point arr_seq sched FP
rbf:= task_request_bound_function: Task -> duration -> nat
task_rbf:= rbf tsk: duration -> nat
total_hep_rbf:= total_hep_request_bound_function_FP ts tsk: duration -> nat
total_ohep_rbf:= total_ohep_request_bound_function_FP ts tsk: duration -> nat
blocking_bound:= \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk) (task_max_nonpreemptive_segment tsk_other - ε): nat
L: duration
H_L_positive: 0 < L
H_fixed_point: L = blocking_bound + total_hep_rbf L
is_in_search_space:= bounded_pi.is_in_search_space tsk L: nat -> bool
R: duration
H_R_is_maximum: forall A : duration, is_in_search_space A -> exists F : duration, blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F /\ F <= R
response_time_bounded_by:= task_response_time_bound arr_seq sched: Task -> duration -> Prop
LIMJ: valid_limited_preemptions_job_model arr_seq
JMLETM: job_respects_task_max_np_segment arr_seq
BEG: beginning_of_execution_in_preemption_points arr_seq
END: end_of_execution_in_preemption_points arr_seq
A: duration
SP: bounded_pi.is_in_search_space tsk L A
F: duration
EQ: blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) <= A + F
LE: F <= R

exists F : duration, bounded_nps.blocking_bound ts tsk + task_request_bound_function tsk (A + ε) + total_ohep_request_bound_function_FP ts tsk (A + F) <= A + F /\ F + 0 <= R
by exists F; rewrite addn0; split. Qed. End RTAforFloatingModelwithArrivalCurves.