From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing]

Require Export prosa.analysis.definitions.schedulability.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.analysis.abstract.search_space.
Require Export prosa.analysis.abstract.run_to_completion.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]

(** * Abstract Response-Time Analysis *)
(** In this module, we propose the general framework for response-time analysis (RTA) 
    of uni-processor scheduling of real-time tasks with arbitrary arrival models. *)
(** We prove that the maximum (with respect to the set of offsets) among the solutions
   of the response-time bound recurrence is a response time bound for [tsk]. Note that
   in this section we do not rely on any hypotheses about job sequentiality. *)
Section Abstract_RTA.
 
  (** Consider any type of tasks ... *)
  Context {Task : TaskType}.
  Context `{TaskCost Task}.
  Context `{TaskRunToCompletionThreshold Task}.

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

  (** Consider any kind of uni-service ideal processor state model. *)
  Context {PState : ProcessorState Job}.
  Hypothesis H_ideal_progress_proc_model : ideal_progress_proc_model PState.
  Hypothesis H_unit_service_proc_model : unit_service_proc_model PState.  
  
  (** 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 schedule of this arrival sequence...*)
  Variable sched : schedule PState.
  Hypothesis H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched arr_seq.
  
  (** ... where jobs do not execute before their arrival nor after completion. *) 
  Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.
  Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.

  (** Assume that the job costs are no larger than the task costs. *)
  Hypothesis H_valid_job_cost:
    arrivals_have_valid_job_costs arr_seq.
  
  (** Consider a task set ts... *)
  Variable ts : list Task.

  (** ... and a task [tsk] of 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.

  (** Let's define some local names for clarity. *)
  Let work_conserving := work_conserving arr_seq sched.
  Let busy_intervals_are_bounded_by := busy_intervals_are_bounded_by arr_seq sched tsk.
  Let job_interference_is_bounded_by := job_interference_is_bounded_by arr_seq sched tsk.
  
  (** Assume we are provided with abstract functions for interference and interfering workload. *)
  Variable interference : Job -> instant -> bool.
  Variable interfering_workload : Job -> instant -> duration.

  (** We assume that the scheduler is work-conserving. *)
  Hypothesis H_work_conserving : work_conserving interference interfering_workload.
  
  (** For simplicity, let's define some local names. *)
  Let cumul_interference := cumul_interference interference.
  Let cumul_interfering_workload := cumul_interfering_workload interfering_workload.
  Let busy_interval := busy_interval sched interference interfering_workload.
  Let response_time_bounded_by := task_response_time_bound arr_seq sched.
  
  (** Let L be a constant which bounds any busy interval of task [tsk]. *)
  Variable L : duration.
  Hypothesis H_busy_interval_exists:
    busy_intervals_are_bounded_by interference interfering_workload L.

  (** Next, assume that interference_bound_function is a bound on 
     the interference incurred by jobs of task [tsk]. *)
  Variable interference_bound_function : Task -> duration -> duration -> duration.
  Hypothesis H_job_interference_is_bounded:
    job_interference_is_bounded_by
      interference interfering_workload interference_bound_function.

  (** For simplicity, let's define a local name for the search space. *)
  Let is_in_search_space A := is_in_search_space tsk L interference_bound_function A.

  (** Consider any value [R] that upper-bounds the solution of each
      response-time recurrence, i.e., for any relative arrival time [A]
      in the search space, there exists a corresponding solution [F]
      such that [R >= F + (task_cost tsk - task_rtct tsk)]. *)
  Variable R: nat.
  Hypothesis H_R_is_maximum:
    forall A,
      is_in_search_space A -> 
      exists F,
        A + F >= task_rtct tsk
                + interference_bound_function tsk A (A + F) /\
        R >= F + (task_cost tsk - task_rtct tsk).

  (** In this section we show a detailed proof of the main theorem 
     that establishes that R is a response-time bound of task [tsk]. *) 
  Section ProofOfTheorem.

    (** Consider any job j of [tsk] with positive cost. *)
    Variable j: Job. 
    Hypothesis H_j_arrives: arrives_in arr_seq j.
    Hypothesis H_job_of_tsk: job_of_task tsk j. 
    Hypothesis H_job_cost_positive: job_cost_positive j.
    
    (** Assume we have a busy interval <<[t1, t2)>> of job j that is bounded by L. *)
    Variable t1 t2: instant.
    Hypothesis H_busy_interval: busy_interval j t1 t2.
    
    (** Let's define A as a relative arrival time of job j (with respect to time t1). *)
    Let A := job_arrival j - t1.

    (** In order to prove that R is a response-time bound of job j, we use hypothesis H_R_is_maximum. 
       Note that the relative arrival time (A) is not necessarily from the search space. However, 
       earlier we have proven that for any A there exists another [A_sp] from the search space that
       shares the same IBF value. Moreover, we've also shown that there exists an [F_sp] such that
       [F_sp] is a solution of the response time recurrence for parameter [A_sp]. Thus, despite the 
       fact that the relative arrival time may not lie in the search space, we can still use
       the assumption H_R_is_maximum. *)
    
    (** More formally, consider any [A_sp] and [F_sp] such that:.. *)
    Variable A_sp F_sp : duration.
    
    (** (a) [A_sp] is less than or equal to [A]... *)
    Hypothesis H_A_gt_Asp : A_sp <= A.
    
    (** (b) [interference_bound_function(A, x)] is equal to
       [interference_bound_function(A_sp, x)] for all [x] less than [L]... *)
    Hypothesis H_equivalent :
      are_equivalent_at_values_less_than
        (interference_bound_function tsk A)
        (interference_bound_function tsk A_sp) L.
    
    (** (c) [A_sp] is in the search space, ... *)
    Hypothesis H_Asp_is_in_search_space : is_in_search_space A_sp.
   
    (** (d) [A_sp + F_sp] is a solution of the response time recurrence... *)
    Hypothesis H_Asp_Fsp_fixpoint :
      A_sp + F_sp >= task_rtct tsk + interference_bound_function tsk A_sp (A_sp + F_sp).

    (** (e) and finally, [F_sp + (task_last - ε)] is no greater than R. *)
    Hypothesis H_R_gt_Fsp : R >= F_sp + (task_cost tsk - task_rtct tsk).

    (** In this section, we consider the case where the solution is so large 
       that the value of [t1 + A_sp + F_sp] goes beyond the busy interval. 
       Although this case may be impossible in some scenarios, it can be 
       easily proven, since any job that completes by the end of the busy
       interval remains completed. *)
    Section FixpointOutsideBusyInterval.

      (** By assumption, suppose that t2 is less than or equal to [t1 + A_sp + F_sp]. *)
      Hypothesis H_big_fixpoint_solution : t2 <= t1 + (A_sp + F_sp).

      (** Then we prove that [job_arrival j + R] is no less than [t2]. *)
      Lemma t2_le_arrival_plus_R:
        t2 <= job_arrival j + R.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
t2 <= job_arrival j + R
      Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
t2 <= job_arrival j + R
        move: H_busy_interval => [[/andP [GT LT] [QT1 NTQ]] QT2].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
t2 <= job_arrival j + R
        apply leq_trans with (t1 + (A_sp + F_sp)); first by done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
t1 + (A_sp + F_sp) <= job_arrival j + R
        apply leq_trans with (t1 + A + F_sp).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
t1 + (A_sp + F_sp) <= t1 + A + F_sp
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
t1 + A + F_sp <= job_arrival j + R
        {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
t1 + (A_sp + F_sp) <= t1 + A + F_sp by rewrite !addnA leq_add2r leq_add2l. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
t1 + A + F_sp <= job_arrival j + R
        rewrite /A subnKC; last by done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
job_arrival j + F_sp <= job_arrival j + R
        rewrite leq_add2l.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
F_sp <= R
          by apply leq_trans with (F_sp + (task_cost tsk - task_rtct tsk));
            first rewrite leq_addr.
      Qed.
      
      (** But since we know that the job is completed by the end of its busy interval, 
         we can show that job j is completed by [job arrival j + R]. *)
      Lemma job_completed_by_arrival_plus_R_1:
        completed_by sched j (job_arrival j + R).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
completed_by sched j (job_arrival j + R)
      Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
completed_by sched j (job_arrival j + R)
        move: H_busy_interval => [[/andP [GT LT] [QT1 NTQ]] QT2].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
completed_by sched j (job_arrival j + R)
        apply completion_monotonic with t2; try done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
t2 <= job_arrival j + R
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
completed_by sched j t2
        apply t2_le_arrival_plus_R.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_big_fixpoint_solution: t2 <= t1 + (A_sp + F_sp)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
completed_by sched j t2
        eapply job_completes_within_busy_interval; eauto 2.
      Qed.

    End FixpointOutsideBusyInterval.

    (** In this section, we consider the complementary case where
       [t1 + A_sp + F_sp] lies inside the busy interval. *)
    Section FixpointInsideBusyInterval.

      (** So, assume that [t1 + A_sp + F_sp] is less than t2. *) 
      Hypothesis H_small_fixpoint_solution : t1 + (A_sp + F_sp) < t2.

      (** Next, let's consider two other cases: *)
      (** CASE 1: the value of the fix-point is no less than the relative arrival time of job [j]. *)
      Section FixpointIsNoLessThanArrival.
        
        (** Suppose that [A_sp + F_sp] is no less than relative arrival of job [j]. *)
        Hypothesis H_fixpoint_is_no_less_than_relative_arrival_of_j : A <= A_sp + F_sp.
 
        (** In this section, we prove that the fact that job [j] is not completed by 
           time [job_arrival j + R] leads to a contradiction. Which in turn implies
           that the opposite is true -- job [j] completes by time [job_arrival j + R]. *)
        Section ProofByContradiction.
            
          (** Recall that by lemma "solution_for_A_exists" there is a solution [F] 
             of the response-time recurrence for the given relative arrival time [A] 
             (which is not necessarily from the search space). *)

          (** Thus, consider a constant [F] such that:.. *)
          Variable F : duration.
          (** (a) the sum of [A_sp] and [F_sp] is equal to the sum of [A] and [F]... *)
          Hypothesis H_Asp_Fsp_eq_A_F : A_sp + F_sp = A + F.
          (** (b) [F] is at mo1st [F_sp]... *)
          Hypothesis H_F_le_Fsp : F <= F_sp.
          (** (c) and [A + F] is a solution for the response-time recurrence for [A]. *)
          Hypothesis H_A_F_fixpoint:
            A + F >= task_rtct tsk + interference_bound_function tsk A (A + F).
 
          (** Next, we assume that job [j] is not completed by time [job_arrival j + R]. *)
          Hypothesis H_j_not_completed : ~~ completed_by sched j (job_arrival j + R).

          (** Some additional reasoning is required since the term [task_cost tsk - task_rtct tsk]
             does not necessarily bound the term [job_cost j - job_rtct j]. That is, a job can 
             have a small run-to-completion threshold, thereby becoming non-preemptive much earlier than guaranteed 
             according to task run-to-completion threshold, while simultaneously executing the last non-preemptive 
             segment that is longer than [task_cost tsk - task_rtct tsk] (e.g., this is possible 
             in the case of floating non-preemptive sections). 

             In this case we cannot directly apply lemma "j_receives_at_least_run_to_completion_threshold". Therefore
             we introduce two temporal notions of the last non-preemptive region of job j and an execution 
             optimism. We use these notions inside this proof, so we define them only locally. *)

          (** Let the last non-preemptive region of job [j] (last) be
             the difference between the cost of the job and the [j]'s
             run-to-completion threshold (i.e. [job_cost j - job_rtct j]). 
             We know that after j has reached its
             run-to-completion threshold, it will additionally be
             executed [job_last j] units of time. *)          
          Let job_last := job_cost j - job_rtct j.

          (** And let execution optimism (optimism) be the difference
             between the [tsk]'s run-to-completion threshold and the
             [j]'s run-to-completion threshold (i.e.  [task_rtct -
             job_rtct]).  Intuitively, optimism is how much earlier
             job j has received its run-to-completion threshold than
             it could at worst.  *)
          Let optimism := task_rtct tsk - job_rtct j.
          
          (** From lemma "j_receives_at_least_run_to_completion_threshold" 
              with parameters [progress_of_job := job_rtct j] and [delta :=
              (A + F) - optimism)] we know that service of [j] by time
              [t1 + (A + F) - optimism] is no less than [job_rtct
              j]. Hence, job [j] is completed by time [t1 + (A + F) -
              optimism + last]. *)
          Lemma j_is_completed_by_t1_A_F_optimist_last :
            completed_by sched j (t1 + (A + F - optimism) + job_last).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
completed_by sched j
  (t1 + (A + F - optimism) + job_last)
          Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
completed_by sched j
  (t1 + (A + F - optimism) + job_last)
            move: H_busy_interval => [[/andP [GT LT] _] _].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
completed_by sched j
  (t1 + (A + F - optimism) + job_last)
            have ESERV :=
              @j_receives_at_least_run_to_completion_threshold
                _ _  _  PState _ _  arr_seq sched interference interfering_workload
                _ j _ _  t1 t2 _ (job_rtct j) _ ((A + F) - optimism).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: forall (j0 : JobArrival Job)
  (j1 : JobCost Job),
ideal_progress_proc_model PState ->
unit_service_proc_model PState ->
definitions.work_conserving arr_seq sched
  interference interfering_workload ->
arrives_in arr_seq j ->
job_cost_positive j ->
definitions.busy_interval sched interference
  interfering_workload j t1 t2 ->
forall (j2 : JobCost Job)
  (j3 : JobPreemptable Job),
job_rtct j <= job_cost j ->
job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
completed_by sched j
  (t1 + (A + F - optimism) + job_last)
            specialize (ESERV JA JC).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: ideal_progress_proc_model PState ->
unit_service_proc_model PState ->
definitions.work_conserving arr_seq sched
  interference interfering_workload ->
arrives_in arr_seq j ->
job_cost_positive j ->
definitions.busy_interval sched interference
  interfering_workload j t1 t2 ->
forall (j0 : JobCost Job)
  (j1 : JobPreemptable Job),
job_rtct j <= job_cost j ->
job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
completed_by sched j
  (t1 + (A + F - optimism) + job_last)              
            feed_n 6  ESERV; eauto 2.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: forall (j0 : JobCost Job)
  (j1 : JobPreemptable Job),
job_rtct j <= job_cost j ->
job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
completed_by sched j
  (t1 + (A + F - optimism) + job_last)
            specialize (ESERV JC _).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j <= job_cost j ->
job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
completed_by sched j
  (t1 + (A + F - optimism) + job_last)
            feed_n 2 ESERV.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j <= job_cost j ->
job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
job_rtct j <= job_cost j
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
job_rtct j +
definitions.cumul_interference interference j t1
  (t1 + (A + F - optimism)) <= 
A + F - optimism
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j <=
service sched j (t1 + (A + F - optimism))
completed_by sched j
  (t1 + (A + F - optimism) + job_last)
            {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j <= job_cost j ->
job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
job_rtct j <= job_cost j eapply job_run_to_completion_threshold_le_job_cost; eauto. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
job_rtct j +
definitions.cumul_interference interference j t1
  (t1 + (A + F - optimism)) <= A + F - optimism
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j <=
service sched j (t1 + (A + F - optimism))
completed_by sched j
  (t1 + (A + F - optimism) + job_last)
            {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
job_rtct j +
definitions.cumul_interference interference j t1
  (t1 + (A + F - optimism)) <= A + F - optimism rewrite -{2}(leqRW H_A_F_fixpoint).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
job_rtct j +
definitions.cumul_interference interference j t1
  (t1 + (A + F - optimism)) <=
task_rtct tsk +
interference_bound_function tsk A (A + F) - optimism
              rewrite /definitions.cumul_interference.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
job_rtct j +
\sum_(t1 <= t < t1 + (A + F - optimism))
   interference j t <=
task_rtct tsk +
interference_bound_function tsk A (A + F) - optimism
              rewrite -[in X in _ <= X]addnBAC; last by rewrite leq_subr.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
job_rtct j +
\sum_(t1 <= t < t1 + (A + F - optimism))
   interference j t <=
task_rtct tsk - optimism +
interference_bound_function tsk A (A + F)
              rewrite {2}/optimism.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
job_rtct j +
\sum_(t1 <= t < t1 + (A + F - optimism))
   interference j t <=
task_rtct tsk - (task_rtct tsk - job_rtct j) +
interference_bound_function tsk A (A + F)
              rewrite subKn; last by apply H_valid_run_to_completion_threshold.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
job_rtct j +
\sum_(t1 <= t < t1 + (A + F - optimism))
   interference j t <=
job_rtct j + interference_bound_function tsk A (A + F)
              rewrite leq_add2l.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
\sum_(t1 <= t < t1 + (A + F - optimism))
   interference j t <=
interference_bound_function tsk A (A + F)
              apply leq_trans with (cumul_interference j t1 (t1 + (A + F))).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
\sum_(t1 <= t < t1 + (A + F - optimism))
   interference j t <=
cumul_interference j t1 (t1 + (A + F))
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
cumul_interference j t1 (t1 + (A + F)) <=
interference_bound_function tsk A (A + F)
              {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
\sum_(t1 <= t < t1 + (A + F - optimism))
   interference j t <=
cumul_interference j t1 (t1 + (A + F)) rewrite /cumul_interference /definitions.cumul_interference
                   [in X in _ <= X](@big_cat_nat _ _ _ (t1 + (A + F - optimism))) //=.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
\sum_(t1 <= t < t1 + (A + F - optimism))
   interference j t <=
\sum_(t1 <= i < t1 + (A + F - optimism))
   interference j i +
\sum_(t1 + (A + F - optimism) <= i < t1 + (A + F))
   interference j i
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
t1 <= t1 + (A + F - optimism)
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
t1 + (A + F - optimism) <= t1 + (A + F)
                all: by lia. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
cumul_interference j t1 (t1 + (A + F)) <=
interference_bound_function tsk A (A + F)
              {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
cumul_interference j t1 (t1 + (A + F)) <=
interference_bound_function tsk A (A + F) apply H_job_interference_is_bounded with t2; try done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
t1 + (A + F) < t2
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
~~ completed_by sched j (t1 + (A + F))
                -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
t1 + (A + F) < t2
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
~~ completed_by sched j (t1 + (A + F)) by rewrite -H_Asp_Fsp_eq_A_F.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
~~ completed_by sched j (t1 + (A + F))
                -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
~~ completed_by sched j (t1 + (A + F)) apply/negP; intros CONTR.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
CONTR: completed_by sched j (t1 + (A + F))
False
                  move: H_j_not_completed => /negP C; apply: C.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
CONTR: completed_by sched j (t1 + (A + F))
completed_by sched j (job_arrival j + R)
                  apply completion_monotonic with (t1 + (A + F)); try done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
CONTR: completed_by sched j (t1 + (A + F))
t1 + (A + F) <= job_arrival j + R
                  rewrite addnA subnKC // leq_add2l.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
CONTR: completed_by sched j (t1 + (A + F))
F <= R
                  apply leq_trans with F_sp; first by done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A + F - optimism)) <=
A + F - optimism ->
job_rtct j <=
service sched j (t1 + (A + F - optimism))
CONTR: completed_by sched j (t1 + (A + F))
F_sp <= R
                  by lia. } }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j <=
service sched j (t1 + (A + F - optimism))
completed_by sched j
  (t1 + (A + F - optimism) + job_last)
            apply: job_completes_after_reaching_run_to_completion_threshold; eauto 2.
          Qed.

          (** However, [t1 + (A + F) - optimism + last ≤ job_arrival j + R]! 
             To prove this fact we need a few auxiliary inequalities that are 
             needed because we use the truncated subtraction in our development.
             So, for example [a + (b - c) = a + b - c] only if [b ≥ c]. *) 
          Section AuxiliaryInequalities.
            
            (** Recall that we consider a busy interval of a job [j], and [j] has arrived [A] time units 
               after the beginning the busy interval. From basic properties of a busy interval it 
               follows that job [j] incurs interference at any time instant t ∈ <<[t1, t1 + A)>>. 
               Therefore [interference_bound_function(tsk, A, A + F)] is at least [A]. *)
            Lemma relative_arrival_le_interference_bound:
              A <= interference_bound_function tsk A (A + F).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
A <= interference_bound_function tsk A (A + F)
            Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
A <= interference_bound_function tsk A (A + F)
              move: H_j_not_completed; clear H_j_not_completed; move => /negP CONTRc.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
A <= interference_bound_function tsk A (A + F)
              move: (H_busy_interval) => [[/andP [GT LT] _] _].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
A <= interference_bound_function tsk A (A + F)
              apply leq_trans with (cumul_interference j t1 (t1 + (A+F))).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
A <= cumul_interference j t1 (t1 + (A + F))
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
cumul_interference j t1 (t1 + (A + F)) <=
interference_bound_function tsk A (A + F)
              {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
A <= cumul_interference j t1 (t1 + (A + F)) rewrite /cumul_interference.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
A <=
definitions.cumul_interference interference j t1
  (t1 + (A + F))
                apply leq_trans with
                  (\sum_(t1 <= t < t1 + A) interference j t);last by
                  rewrite /definitions.cumul_interference [in X in _ <= X](@big_cat_nat _ _ _ (t1 + A)) //=; try by lia.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
A <= \sum_(t1 <= t < t1 + A) interference j t 
                {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
A <= \sum_(t1 <= t < t1 + A) interference j t rewrite -{1}[A](sum_of_ones t1).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
\sum_(t1 <= x < t1 + A) 1 <=
\sum_(t1 <= t < t1 + A) interference j t
                  rewrite [in X in X <= _]big_nat_cond [in X in _ <= X]big_nat_cond.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
\sum_(t1 <= i < t1 + A | (t1 <= i < t1 + A) && true) 1 <=
\sum_(t1 <= i < t1 + A | (t1 <= i < t1 + A) && true)
   interference j i
                  rewrite leq_sum //.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
forall i : nat,
(t1 <= i < t1 + A) && true -> 0 < interference j i
                  move => t /andP [/andP [NEQ1 NEQ2] _].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t: nat
NEQ1: t1 <= t
NEQ2: t < t1 + A
0 < interference j t
                  rewrite lt0b.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t: nat
NEQ1: t1 <= t
NEQ2: t < t1 + A
interference j t
                  move: (H_work_conserving j t1 t2 t) => CONS.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t: nat
NEQ1: t1 <= t
NEQ2: t < t1 + A
CONS: arrives_in arr_seq j ->
0 < job_cost j ->
definitions.busy_interval sched interference
  interfering_workload j t1 t2 ->
t1 <= t < t2 ->
~ interference j t <-> scheduled_at sched j t
interference j t 
                  feed_n 4 CONS; try done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t: nat
NEQ1: t1 <= t
NEQ2: t < t1 + A
CONS: t1 <= t < t2 ->
~ interference j t <-> scheduled_at sched j t
t1 <= t < t2
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t: nat
NEQ1: t1 <= t
NEQ2: t < t1 + A
CONS: ~ interference j t <-> scheduled_at sched j t
interference j t
                  {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t: nat
NEQ1: t1 <= t
NEQ2: t < t1 + A
CONS: t1 <= t < t2 ->
~ interference j t <-> scheduled_at sched j t
t1 <= t < t2 apply/andP; split; first by done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t: nat
NEQ1: t1 <= t
NEQ2: t < t1 + A
CONS: t1 <= t < t2 ->
~ interference j t <-> scheduled_at sched j t
t < t2
                    by apply leq_trans with (t1 + A); [done | lia]. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t: nat
NEQ1: t1 <= t
NEQ2: t < t1 + A
CONS: ~ interference j t <-> scheduled_at sched j t
interference j t
                  move: CONS => [CONS1 _].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t: nat
NEQ1: t1 <= t
NEQ2: t < t1 + A
CONS1: ~ interference j t -> scheduled_at sched j t
interference j t
                  apply/negP; intros CONTR.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t: nat
NEQ1: t1 <= t
NEQ2: t < t1 + A
CONS1: ~ interference j t -> scheduled_at sched j t
CONTR: ~ interference j t
False
                  move: (CONS1 CONTR) => SCHED; clear CONS1 CONTR.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t: nat
NEQ1: t1 <= t
NEQ2: t < t1 + A
SCHED: scheduled_at sched j t
False
                  apply H_jobs_must_arrive_to_execute in SCHED.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t: nat
NEQ1: t1 <= t
NEQ2: t < t1 + A
SCHED: has_arrived j t
False
                  move: NEQ2; rewrite ltnNge; move => /negP NEQ2; apply: NEQ2.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t: nat
NEQ1: t1 <= t
SCHED: has_arrived j t
t1 + A <= t
                    by rewrite subnKC. } }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
cumul_interference j t1 (t1 + (A + F)) <=
interference_bound_function tsk A (A + F)
              {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
cumul_interference j t1 (t1 + (A + F)) <=
interference_bound_function tsk A (A + F) apply H_job_interference_is_bounded with t2; try done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t1 + (A + F) < t2
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
~~ completed_by sched j (t1 + (A + F))
                -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t1 + (A + F) < t2
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
~~ completed_by sched j (t1 + (A + F)) by rewrite -H_Asp_Fsp_eq_A_F.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
~~ completed_by sched j (t1 + (A + F))
                -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
~~ completed_by sched j (t1 + (A + F)) apply /negP; move =>  CONTR.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
CONTRc: ~ completed_by sched j (job_arrival j + R)
GT: t1 <= job_arrival j
LT: job_arrival j < t2
CONTR: completed_by sched j (t1 + (A + F))
False
                  apply: CONTRc.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
CONTR: completed_by sched j (t1 + (A + F))
completed_by sched j (job_arrival j + R) 
                  by apply completion_monotonic with (t1 + (A + F)); [lia | done]. }
            Qed.
            
            (** As two trivial corollaries, we show that 
               [tsk]'s run-to-completion threshold is at most [F_sp]... *)
            Corollary tsk_run_to_completion_threshold_le_Fsp :
              task_rtct tsk <= F_sp.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
task_rtct tsk <= F_sp
            Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
task_rtct tsk <= F_sp
              move: H_A_F_fixpoint => EQ.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
EQ: task_rtct tsk +
interference_bound_function tsk A (A + F) <=
A + F
task_rtct tsk <= F_sp
              have L1 := relative_arrival_le_interference_bound.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
EQ: task_rtct tsk +
interference_bound_function tsk A (A + F) <=
A + F
L1: A <= interference_bound_function tsk A (A + F)
task_rtct tsk <= F_sp 
              by lia.
            Qed.
            
            (** ... and optimism is at most [F]. *)
            Corollary optimism_le_F :
              optimism <= F.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
optimism <= F
            Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
optimism <= F
              move: H_A_F_fixpoint => EQ.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
EQ: task_rtct tsk +
interference_bound_function tsk A (A + F) <=
A + F
optimism <= F
              have L1 := relative_arrival_le_interference_bound.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
EQ: task_rtct tsk +
interference_bound_function tsk A (A + F) <=
A + F
L1: A <= interference_bound_function tsk A (A + F)
optimism <= F
              by lia.
            Qed.
            
          End AuxiliaryInequalities.

          (** Next we show that [t1 + (A + F) - optimism + last] is at most [job_arrival j + R], 
             which is easy to see from the following sequence of inequalities:

             [t1 + (A + F) - optimism + last]
             [≤ job_arrival j + (F - optimism) + job_last]
             [≤ job_arrival j + (F_sp - optimism) + job_last]
             [≤ job_arrival j + F_sp + (job_last - optimism)]
             [≤ job_arrival j + F_sp + job_cost j - task_rtct tsk]
             [≤ job_arrival j + F_sp + task_cost tsk - task_rtct tsk]
             [≤ job_arrival j + R]. *)
          Lemma t1_A_F_optimist_last_le_arrival_R :
            t1 + (A + F - optimism) + job_last <= job_arrival j + R.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
t1 + (A + F - optimism) + job_last <=
job_arrival j + R
          Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
t1 + (A + F - optimism) + job_last <=
job_arrival j + R
            move: (H_busy_interval) => [[/andP [GT LT] _] _].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t1 + (A + F - optimism) + job_last <=
job_arrival j + R
            have L1 := tsk_run_to_completion_threshold_le_Fsp.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
t1 + (A + F - optimism) + job_last <=
job_arrival j + R
            have L2 := optimism_le_F.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
t1 + (A + F - optimism) + job_last <=
job_arrival j + R
            apply leq_trans with (job_arrival j + (F - optimism) + job_last).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
t1 + (A + F - optimism) + job_last <=
job_arrival j + (F - optimism) + job_last
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
job_arrival j + (F - optimism) + job_last <=
job_arrival j + R
            {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
t1 + (A + F - optimism) + job_last <=
job_arrival j + (F - optimism) + job_last rewrite leq_add2r addnBA.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
t1 + (A + F) - optimism <=
job_arrival j + (F - optimism)
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
optimism <= A + F
              -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
t1 + (A + F) - optimism <=
job_arrival j + (F - optimism)
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
optimism <= A + F by rewrite /A !addnA subnKC // addnBA.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
optimism <= A + F
              -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
optimism <= A + F by apply leq_trans with F; last rewrite leq_addl.
            }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
job_arrival j + (F - optimism) + job_last <=
job_arrival j + R
            {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
job_arrival j + (F - optimism) + job_last <=
job_arrival j + R move: H_valid_run_to_completion_threshold => [PRT1 PRT2].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
PRT1: task_rtc_bounded_by_cost tsk
PRT2: job_respects_task_rtc arr_seq tsk
job_arrival j + (F - optimism) + job_last <=
job_arrival j + R 
              rewrite -addnA leq_add2l.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
PRT1: task_rtc_bounded_by_cost tsk
PRT2: job_respects_task_rtc arr_seq tsk
F - optimism + job_last <= R
              apply leq_trans with (F_sp - optimism + job_last ); first by rewrite leq_add2r leq_sub2r.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
PRT1: task_rtc_bounded_by_cost tsk
PRT2: job_respects_task_rtc arr_seq tsk
F_sp - optimism + job_last <= R 
              apply leq_trans with (F_sp + (task_cost tsk - task_rtct tsk)); last by done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
PRT1: task_rtc_bounded_by_cost tsk
PRT2: job_respects_task_rtc arr_seq tsk
F_sp - optimism + job_last <=
F_sp + (task_cost tsk - task_rtct tsk) 
              rewrite /optimism subnBA; last by apply PRT2.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
PRT1: task_rtc_bounded_by_cost tsk
PRT2: job_respects_task_rtc arr_seq tsk
F_sp + job_rtct j - task_rtct tsk + job_last <=
F_sp + (task_cost tsk - task_rtct tsk)
              rewrite -addnBAC // /job_last.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
PRT1: task_rtc_bounded_by_cost tsk
PRT2: job_respects_task_rtc arr_seq tsk
F_sp - task_rtct tsk + job_rtct j +
(job_cost j - job_rtct j) <=
F_sp + (task_cost tsk - task_rtct tsk)
              rewrite addnBA; last by eapply job_run_to_completion_threshold_le_job_cost; eauto 2.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
PRT1: task_rtc_bounded_by_cost tsk
PRT2: job_respects_task_rtc arr_seq tsk
F_sp - task_rtct tsk + job_rtct j + job_cost j -
job_rtct j <= F_sp + (task_cost tsk - task_rtct tsk)
              rewrite -addnBAC; last by lia.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
PRT1: task_rtc_bounded_by_cost tsk
PRT2: job_respects_task_rtc arr_seq tsk
F_sp - task_rtct tsk + job_rtct j - job_rtct j +
job_cost j <= F_sp + (task_cost tsk - task_rtct tsk)
              rewrite -addnBA // subnn addn0.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
PRT1: task_rtc_bounded_by_cost tsk
PRT2: job_respects_task_rtc arr_seq tsk
F_sp - task_rtct tsk + job_cost j <=
F_sp + (task_cost tsk - task_rtct tsk)
              rewrite addnBA; last by apply PRT1.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
PRT1: task_rtc_bounded_by_cost tsk
PRT2: job_respects_task_rtc arr_seq tsk
F_sp - task_rtct tsk + job_cost j <=
F_sp + task_cost tsk - task_rtct tsk
              rewrite addnBAC; last by done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
PRT1: task_rtc_bounded_by_cost tsk
PRT2: job_respects_task_rtc arr_seq tsk
F_sp + job_cost j - task_rtct tsk <=
F_sp + task_cost tsk - task_rtct tsk
              rewrite leq_sub2r // leq_add2l.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: task_rtct tsk <= F_sp
L2: optimism <= F
PRT1: task_rtc_bounded_by_cost tsk
PRT2: job_respects_task_rtc arr_seq tsk
job_cost j <= task_cost tsk
              by move: H_job_of_tsk => /eqP <-; apply H_valid_job_cost.
            }
          Qed.
          
          (** ... which contradicts the initial assumption about [j] is not 
             completed by time [job_arrival j + R]. *)
          Lemma j_is_completed_earlier_contradiction : False.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
False
          Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
False
            move: H_j_not_completed => /negP C; apply: C.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
F: duration
H_Asp_Fsp_eq_A_F: A_sp + F_sp = A + F
H_F_le_Fsp: F <= F_sp
H_A_F_fixpoint: task_rtct tsk +
interference_bound_function tsk A
  (A + F) <= 
A + F
H_j_not_completed: ~~
completed_by sched j
  (job_arrival j + R)
job_last:= job_cost j - job_rtct j: nat
optimism:= task_rtct tsk - job_rtct j: nat
completed_by sched j (job_arrival j + R)
            apply completion_monotonic with (t1 + ((A + F) - optimism) + job_last);
              auto using j_is_completed_by_t1_A_F_optimist_last, t1_A_F_optimist_last_le_arrival_R. 
          Qed.
          
        End ProofByContradiction.

        (** Putting everything together, we conclude that [j] is completed by [job_arrival j + R]. *) 
        Lemma job_completed_by_arrival_plus_R_2:
          completed_by sched j (job_arrival j + R).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
completed_by sched j (job_arrival j + R)
        Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
completed_by sched j (job_arrival j + R)
          move: H_busy_interval => [[/andP [GT LT] _] _].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
completed_by sched j (job_arrival j + R)
          have L1 := solution_for_A_exists
                       tsk L (fun tsk A R => task_rtct tsk
                                          + interference_bound_function tsk A R) A_sp F_sp.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: forall t : TaskRunToCompletionThreshold Task,
A_sp + F_sp < L ->
task_rtct tsk +
interference_bound_function tsk A_sp
  (A_sp + F_sp) <= 
A_sp + F_sp ->
forall A : duration,
A_sp <= A <= A_sp + F_sp ->
are_equivalent_at_values_less_than
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A R)
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A_sp R) L ->
exists F : nat,
  A_sp + F_sp = A + F /\
  F <= F_sp /\
  task_rtct tsk +
  interference_bound_function tsk A (A + F) <=
  A + F
completed_by sched j (job_arrival j + R)
          specialize (L1 _).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: A_sp + F_sp < L ->
task_rtct tsk +
interference_bound_function tsk A_sp
  (A_sp + F_sp) <= 
A_sp + F_sp ->
forall A : duration,
A_sp <= A <= A_sp + F_sp ->
are_equivalent_at_values_less_than
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A R)
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A_sp R) L ->
exists F : nat,
  A_sp + F_sp = A + F /\
  F <= F_sp /\
  task_rtct tsk +
  interference_bound_function tsk A (A + F) <=
  A + F
completed_by sched j (job_arrival j + R)
          feed_n 2 L1; try done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: A_sp + F_sp < L ->
task_rtct tsk +
interference_bound_function tsk A_sp
  (A_sp + F_sp) <= 
A_sp + F_sp ->
forall A : duration,
A_sp <= A <= A_sp + F_sp ->
are_equivalent_at_values_less_than
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A R)
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A_sp R) L ->
exists F : nat,
  A_sp + F_sp = A + F /\
  F <= F_sp /\
  task_rtct tsk +
  interference_bound_function tsk A (A + F) <=
  A + F
A_sp + F_sp < L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: forall A : duration,
A_sp <= A <= A_sp + F_sp ->
are_equivalent_at_values_less_than
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A R)
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A_sp R) L ->
exists F : nat,
  A_sp + F_sp = A + F /\
  F <= F_sp /\
  task_rtct tsk +
  interference_bound_function tsk A (A + F) <=
  A + F
completed_by sched j (job_arrival j + R)
          {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: A_sp + F_sp < L ->
task_rtct tsk +
interference_bound_function tsk A_sp
  (A_sp + F_sp) <= 
A_sp + F_sp ->
forall A : duration,
A_sp <= A <= A_sp + F_sp ->
are_equivalent_at_values_less_than
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A R)
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A_sp R) L ->
exists F : nat,
  A_sp + F_sp = A + F /\
  F <= F_sp /\
  task_rtct tsk +
  interference_bound_function tsk A (A + F) <=
  A + F
A_sp + F_sp < L move: (H_busy_interval_exists j H_j_arrives H_job_of_tsk H_job_cost_positive)
                => [t1' [t2' [BOUND BUSY]]].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: A_sp + F_sp < L ->
task_rtct tsk +
interference_bound_function tsk A_sp
  (A_sp + F_sp) <= 
A_sp + F_sp ->
forall A : duration,
A_sp <= A <= A_sp + F_sp ->
are_equivalent_at_values_less_than
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A R)
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A_sp R) L ->
exists F : nat,
  A_sp + F_sp = A + F /\
  F <= F_sp /\
  task_rtct tsk +
  interference_bound_function tsk A (A + F) <=
  A + F
t1', t2': nat
BOUND: t2' <= t1' + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1' t2'
A_sp + F_sp < L
            have EQ:= busy_interval_is_unique _ _ _ _ _ _ _ _ H_busy_interval BUSY.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: A_sp + F_sp < L ->
task_rtct tsk +
interference_bound_function tsk A_sp
  (A_sp + F_sp) <= 
A_sp + F_sp ->
forall A : duration,
A_sp <= A <= A_sp + F_sp ->
are_equivalent_at_values_less_than
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A R)
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A_sp R) L ->
exists F : nat,
  A_sp + F_sp = A + F /\
  F <= F_sp /\
  task_rtct tsk +
  interference_bound_function tsk A (A + F) <=
  A + F
t1', t2': nat
BOUND: t2' <= t1' + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1' t2'
EQ: t1 = t1' /\ t2 = t2'
A_sp + F_sp < L
            move : EQ => [EQ1 EQ2].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: A_sp + F_sp < L ->
task_rtct tsk +
interference_bound_function tsk A_sp
  (A_sp + F_sp) <= 
A_sp + F_sp ->
forall A : duration,
A_sp <= A <= A_sp + F_sp ->
are_equivalent_at_values_less_than
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A R)
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A_sp R) L ->
exists F : nat,
  A_sp + F_sp = A + F /\
  F <= F_sp /\
  task_rtct tsk +
  interference_bound_function tsk A (A + F) <=
  A + F
t1', t2': nat
BOUND: t2' <= t1' + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1' t2'
EQ1: t1 = t1'
EQ2: t2 = t2'
A_sp + F_sp < L
            subst t1' t2'; clear BUSY.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: A_sp + F_sp < L ->
task_rtct tsk +
interference_bound_function tsk A_sp
  (A_sp + F_sp) <= 
A_sp + F_sp ->
forall A : duration,
A_sp <= A <= A_sp + F_sp ->
are_equivalent_at_values_less_than
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A R)
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A_sp R) L ->
exists F : nat,
  A_sp + F_sp = A + F /\
  F <= F_sp /\
  task_rtct tsk +
  interference_bound_function tsk A (A + F) <=
  A + F
BOUND: t2 <= t1 + L
A_sp + F_sp < L
            by rewrite -(ltn_add2l t1); apply leq_trans with t2.
          }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: forall A : duration,
A_sp <= A <= A_sp + F_sp ->
are_equivalent_at_values_less_than
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A R)
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A_sp R) L ->
exists F : nat,
  A_sp + F_sp = A + F /\
  F <= F_sp /\
  task_rtct tsk +
  interference_bound_function tsk A (A + F) <=
  A + F
completed_by sched j (job_arrival j + R)
          specialize (L1 A); feed_n 2 L1; first by apply/andP; split.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: are_equivalent_at_values_less_than
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A R)
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A_sp R) L ->
exists F : nat,
  A_sp + F_sp = A + F /\
  F <= F_sp /\
  task_rtct tsk +
  interference_bound_function tsk A (A + F) <=
  A + F
are_equivalent_at_values_less_than
  (fun R : duration =>
   task_rtct tsk + interference_bound_function tsk A R)
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A_sp R) L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: exists F : nat,
  A_sp + F_sp = A + F /\
  F <= F_sp /\
  task_rtct tsk +
  interference_bound_function tsk A (A + F) <=
  A + F
completed_by sched j (job_arrival j + R) 
          +
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: are_equivalent_at_values_less_than
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A R)
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A_sp R) L ->
exists F : nat,
  A_sp + F_sp = A + F /\
  F <= F_sp /\
  task_rtct tsk +
  interference_bound_function tsk A (A + F) <=
  A + F
are_equivalent_at_values_less_than
  (fun R : duration =>
   task_rtct tsk + interference_bound_function tsk A R)
  (fun R : duration =>
   task_rtct tsk +
   interference_bound_function tsk A_sp R) L
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: exists F : nat,
  A_sp + F_sp = A + F /\
  F <= F_sp /\
  task_rtct tsk +
  interference_bound_function tsk A (A + F) <=
  A + F
completed_by sched j (job_arrival j + R) by intros x LTG; apply/eqP; rewrite eqn_add2l H_equivalent.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
L1: exists F : nat,
  A_sp + F_sp = A + F /\
  F <= F_sp /\
  task_rtct tsk +
  interference_bound_function tsk A (A + F) <=
  A + F
completed_by sched j (job_arrival j + R) 
            move: L1 => [F [EQSUM [F2LEF1 FIX2]]].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
F: nat
EQSUM: A_sp + F_sp = A + F
F2LEF1: F <= F_sp
FIX2: task_rtct tsk +
interference_bound_function tsk A (A + F) <=
A + F
completed_by sched j (job_arrival j + R)
            apply/negP; intros CONTRc; move: CONTRc => /negP CONTRc.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_no_less_than_relative_arrival_of_j: A <= A_sp + F_sp
GT: t1 <= job_arrival j
LT: job_arrival j < t2
F: nat
EQSUM: A_sp + F_sp = A + F
F2LEF1: F <= F_sp
FIX2: task_rtct tsk +
interference_bound_function tsk A (A + F) <=
A + F
CONTRc: ~~ completed_by sched j (job_arrival j + R)
False
            by eapply j_is_completed_earlier_contradiction in CONTRc; eauto 2.
        Qed.
        
      End FixpointIsNoLessThanArrival.

      (** CASE 2: the value of the fix-point is less than the relative arrival time of 
         job j (which turns out to be impossible, i.e. the solution of the response-time 
         recurrence is always equal to or greater than the relative arrival time). *)
      Section FixpointCannotBeSmallerThanArrival.

        (** Assume that [A_sp + F_sp] is less than A. *)
        Hypothesis H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A.

        (** Note that the relative arrival time of job j is less than L. *)
        Lemma relative_arrival_is_bounded: A < L.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
A < L
        Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
A < L
          rewrite /A.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
job_arrival j - t1 < L
          move: (H_busy_interval_exists j H_j_arrives H_job_of_tsk H_job_cost_positive) => [t1' [t2' [BOUND BUSY]]].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
t1', t2': nat
BOUND: t2' <= t1' + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1' t2'
job_arrival j - t1 < L
          have EQ:= busy_interval_is_unique _ _ _ _ _ _ _ _ H_busy_interval BUSY.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
t1', t2': nat
BOUND: t2' <= t1' + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1' t2'
EQ: t1 = t1' /\ t2 = t2'
job_arrival j - t1 < L destruct EQ as [EQ1 EQ2].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
t1', t2': nat
BOUND: t2' <= t1' + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1' t2'
EQ1: t1 = t1'
EQ2: t2 = t2'
job_arrival j - t1 < L
          subst t1' t2'; clear BUSY.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
BOUND: t2 <= t1 + L
job_arrival j - t1 < L
          apply leq_trans with (t2 - t1); last by rewrite leq_subLR.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
BOUND: t2 <= t1 + L
job_arrival j - t1 < t2 - t1
          move: (H_busy_interval)=> [[/andP [T1 T3] [_ _]] _].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
BOUND: t2 <= t1 + L
T1: t1 <= job_arrival j
T3: job_arrival j < t2
job_arrival j - t1 < t2 - t1
          by apply ltn_sub2r; first apply leq_ltn_trans with (job_arrival j).
        Qed.                  

        (** We can use [j_receives_at_least_run_to_completion_threshold] to prove that the service 
           received by j by time [t1 + (A_sp + F_sp)] is no less than run-to-completion threshold. *)
        Lemma service_of_job_ge_run_to_completion_threshold:
          service sched j (t1 + (A_sp + F_sp)) >= job_rtct j.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
job_rtct j <= service sched j (t1 + (A_sp + F_sp))
        Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
job_rtct j <= service sched j (t1 + (A_sp + F_sp))
          move: (H_busy_interval) => [[NEQ [QT1 NTQ]] QT2].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
job_rtct j <= service sched j (t1 + (A_sp + F_sp))
          move: (NEQ) => /andP [GT LT].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
job_rtct j <= service sched j (t1 + (A_sp + F_sp)) 
          move: (H_job_interference_is_bounded t1 t2 (A_sp + F_sp) j) => IB.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: arrives_in arr_seq j ->
job_of_task tsk j ->
definitions.busy_interval sched interference
  interfering_workload j t1 t2 ->
t1 + (A_sp + F_sp) < t2 ->
~~ completed_by sched j (t1 + (A_sp + F_sp)) ->
let offset := job_arrival j - t1 in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
job_rtct j <= service sched j (t1 + (A_sp + F_sp)) 
          feed_n 5 IB; try done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: ~~ completed_by sched j (t1 + (A_sp + F_sp)) ->
let offset := job_arrival j - t1 in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
~~ completed_by sched j (t1 + (A_sp + F_sp))
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: let offset := job_arrival j - t1 in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
job_rtct j <= service sched j (t1 + (A_sp + F_sp))
          {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: ~~ completed_by sched j (t1 + (A_sp + F_sp)) ->
let offset := job_arrival j - t1 in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
~~ completed_by sched j (t1 + (A_sp + F_sp)) apply/negP => COMPL.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: ~~ completed_by sched j (t1 + (A_sp + F_sp)) ->
let offset := job_arrival j - t1 in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
COMPL: completed_by sched j (t1 + (A_sp + F_sp))
False
            apply completion_monotonic with (t' := t1 + A) in COMPL; try done; last first.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: ~~ completed_by sched j (t1 + (A_sp + F_sp)) ->
let offset := job_arrival j - t1 in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
COMPL: completed_by sched j (t1 + (A_sp + F_sp))
t1 + (A_sp + F_sp) <= t1 + A
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: ~~ completed_by sched j (t1 + (A_sp + F_sp)) ->
let offset := job_arrival j - t1 in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
COMPL: completed_by sched j (t1 + A)
False
            {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: ~~ completed_by sched j (t1 + (A_sp + F_sp)) ->
let offset := job_arrival j - t1 in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
COMPL: completed_by sched j (t1 + (A_sp + F_sp))
t1 + (A_sp + F_sp) <= t1 + A by rewrite leq_add2l; apply ltnW. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: ~~ completed_by sched j (t1 + (A_sp + F_sp)) ->
let offset := job_arrival j - t1 in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
COMPL: completed_by sched j (t1 + A)
False
            {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: ~~ completed_by sched j (t1 + (A_sp + F_sp)) ->
let offset := job_arrival j - t1 in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
COMPL: completed_by sched j (t1 + A)
False rewrite /A subnKC in COMPL; last by done.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: ~~ completed_by sched j (t1 + (A_sp + F_sp)) ->
let offset := job_arrival j - t1 in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
COMPL: completed_by sched j (job_arrival j)
False
              move: COMPL; rewrite /completed_by leqNgt; move => /negP COMPL; apply: COMPL.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: ~~ completed_by sched j (t1 + (A_sp + F_sp)) ->
let offset := job_arrival j - t1 in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
service sched j (job_arrival j) < job_cost j 
              rewrite /service -(service_during_cat _ _ _ (job_arrival j)); last by apply/andP; split.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: ~~ completed_by sched j (t1 + (A_sp + F_sp)) ->
let offset := job_arrival j - t1 in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
service_during sched j 0 (job_arrival j) +
service_during sched j (job_arrival j) (job_arrival j) <
job_cost j
              rewrite (cumulative_service_before_job_arrival_zero) // add0n.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: ~~ completed_by sched j (t1 + (A_sp + F_sp)) ->
let offset := job_arrival j - t1 in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
service_during sched j (job_arrival j) (job_arrival j) <
job_cost j
                by rewrite /service_during big_geq //.
            } 
          }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: let offset := job_arrival j - t1 in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
job_rtct j <= service sched j (t1 + (A_sp + F_sp))
          rewrite -/A in IB.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: let offset := A in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
job_rtct j <= service sched j (t1 + (A_sp + F_sp))
          have ALTT := relative_arrival_is_bounded.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
IB: let offset := A in
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk offset
  (A_sp + F_sp)
ALTT: A < L
job_rtct j <= service sched j (t1 + (A_sp + F_sp))
          simpl in IB; rewrite H_equivalent in IB; last by apply ltn_trans with A.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ALTT: A < L
IB: definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk A_sp
  (A_sp + F_sp)
job_rtct j <= service sched j (t1 + (A_sp + F_sp))
          have ESERV :=
              @j_receives_at_least_run_to_completion_threshold
                _ _  _  PState _ _  arr_seq sched interference interfering_workload
                _ j _ _  t1 t2 _ (job_rtct j) _ (A_sp + F_sp).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ALTT: A < L
IB: definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk A_sp
  (A_sp + F_sp)
ESERV: forall (j0 : JobArrival Job)
  (j1 : JobCost Job),
ideal_progress_proc_model PState ->
unit_service_proc_model PState ->
definitions.work_conserving arr_seq sched
  interference interfering_workload ->
arrives_in arr_seq j ->
job_cost_positive j ->
definitions.busy_interval sched interference
  interfering_workload j t1 t2 ->
forall (j2 : JobCost Job)
  (j3 : JobPreemptable Job),
job_rtct j <= job_cost j ->
job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A_sp + F_sp)) <= 
A_sp + F_sp ->
job_rtct j <=
service sched j (t1 + (A_sp + F_sp))
job_rtct j <= service sched j (t1 + (A_sp + F_sp))
          specialize (ESERV JA JC).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ALTT: A < L
IB: definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk A_sp
  (A_sp + F_sp)
ESERV: ideal_progress_proc_model PState ->
unit_service_proc_model PState ->
definitions.work_conserving arr_seq sched
  interference interfering_workload ->
arrives_in arr_seq j ->
job_cost_positive j ->
definitions.busy_interval sched interference
  interfering_workload j t1 t2 ->
forall (j0 : JobCost Job)
  (j1 : JobPreemptable Job),
job_rtct j <= job_cost j ->
job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A_sp + F_sp)) <= 
A_sp + F_sp ->
job_rtct j <=
service sched j (t1 + (A_sp + F_sp))
job_rtct j <= service sched j (t1 + (A_sp + F_sp))
          feed_n 6 ESERV; eauto 2.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ALTT: A < L
IB: definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk A_sp
  (A_sp + F_sp)
ESERV: forall (j0 : JobCost Job)
  (j1 : JobPreemptable Job),
job_rtct j <= job_cost j ->
job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A_sp + F_sp)) <= 
A_sp + F_sp ->
job_rtct j <=
service sched j (t1 + (A_sp + F_sp))
job_rtct j <= service sched j (t1 + (A_sp + F_sp))
          specialize (ESERV JC _).
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ALTT: A < L
IB: definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk A_sp
  (A_sp + F_sp)
ESERV: job_rtct j <= job_cost j ->
job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A_sp + F_sp)) <= 
A_sp + F_sp ->
job_rtct j <=
service sched j (t1 + (A_sp + F_sp))
job_rtct j <= service sched j (t1 + (A_sp + F_sp))
          feed_n 2 ESERV; eauto using job_run_to_completion_threshold_le_job_cost.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ALTT: A < L
IB: definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <=
interference_bound_function tsk A_sp
  (A_sp + F_sp)
ESERV: job_rtct j +
definitions.cumul_interference interference j
  t1 (t1 + (A_sp + F_sp)) <= 
A_sp + F_sp ->
job_rtct j <=
service sched j (t1 + (A_sp + F_sp))
job_rtct j +
definitions.cumul_interference interference j t1
  (t1 + (A_sp + F_sp)) <= A_sp + F_sp
          by rewrite -{2}(leqRW H_Asp_Fsp_fixpoint) leq_add //; apply H_valid_run_to_completion_threshold.
          Qed.

        (** However, this is a contradiction. Since job [j] has not yet arrived, its service 
           is equal to [0]. However, run-to-completion threshold is always positive. *)
        Lemma relative_arrival_time_is_no_less_than_fixpoint:
          False.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
False
        Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
False
          move: (H_busy_interval) => [[NEQ [QT1 NTQ]] QT2].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
False
          move: (NEQ) => /andP [GT LT].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
False                    
          have ESERV := service_of_job_ge_run_to_completion_threshold.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
ESERV: job_rtct j <=
service sched j (t1 + (A_sp + F_sp))
False
          move: ESERV; rewrite leqNgt; move => /negP ESERV; apply: ESERV.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
service sched j (t1 + (A_sp + F_sp)) < job_rtct j
          rewrite /service cumulative_service_before_job_arrival_zero;
            eauto 5 using job_run_to_completion_threshold_positive.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t1 + (A_sp + F_sp) <= job_arrival j 
          rewrite -[X in _ <= X](@subnKC t1) //.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
H_j_arrives: arrives_in arr_seq j
H_job_of_tsk: job_of_task tsk j
H_job_cost_positive: job_cost_positive j
t1, t2: instant
H_busy_interval: busy_interval j t1 t2
A:= job_arrival j - t1: nat
A_sp, F_sp: duration
H_A_gt_Asp: A_sp <= A
H_equivalent: are_equivalent_at_values_less_than
  (interference_bound_function tsk A)
  (interference_bound_function tsk A_sp)
  L
H_Asp_is_in_search_space: is_in_search_space A_sp
H_Asp_Fsp_fixpoint: task_rtct tsk +
interference_bound_function tsk
  A_sp (A_sp + F_sp) <=
A_sp + F_sp
H_R_gt_Fsp: F_sp + (task_cost tsk - task_rtct tsk) <=
R
H_small_fixpoint_solution: t1 + (A_sp + F_sp) < t2
H_fixpoint_is_less_that_relative_arrival_of_j: A_sp + F_sp < A
NEQ: t1 <= job_arrival j < t2
QT1: quiet_time sched interference
  interfering_workload j t1
NTQ: forall t : nat,
t1 < t < t2 ->
~
quiet_time sched interference
  interfering_workload j t
QT2: quiet_time sched interference
  interfering_workload j t2
GT: t1 <= job_arrival j
LT: job_arrival j < t2
t1 + (A_sp + F_sp) <= t1 + (job_arrival j - t1)
            by rewrite -/A leq_add2l ltnW.
        Qed.

      End FixpointCannotBeSmallerThanArrival.

    End FixpointInsideBusyInterval.
    
  End ProofOfTheorem.

  (** Using the lemmas above, we prove that [R] is a response-time bound. *)  
  Theorem uniprocessor_response_time_bound:
    response_time_bounded_by tsk R.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
response_time_bounded_by tsk R 
  Proof.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
response_time_bounded_by tsk R 
    intros j ARR JOBtsk.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
job_response_time_bound sched j R unfold job_response_time_bound.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
completed_by sched j (job_arrival j + R)
    move: (posnP (@job_cost _ JC j)) => [ZERO|POS].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
ZERO: job_cost j = 0
completed_by sched j (job_arrival j + R)
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
completed_by sched j (job_arrival j + R)
    {
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
ZERO: job_cost j = 0
completed_by sched j (job_arrival j + R) by rewrite /completed_by ZERO. }
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
completed_by sched j (job_arrival j + R) 
    move: (H_busy_interval_exists j ARR JOBtsk POS) => [t1 [t2 [T2 BUSY]]].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
completed_by sched j (job_arrival j + R)
    move: (BUSY) => [[/andP [GE LT] _] QTt2].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
completed_by sched j (job_arrival j + R)
    have A2LTL := relative_arrival_is_bounded _ ARR JOBtsk POS _ _ BUSY.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2LTL: job_arrival j - t1 < L
completed_by sched j (job_arrival j + R)
    set (A2 := job_arrival j - t1) in *.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
completed_by sched j (job_arrival j + R)
    move: (representative_exists tsk _ interference_bound_function _ A2LTL) => [A1 [ALEA2 [EQΦ INSP]]].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
completed_by sched j (job_arrival j + R)
    move: (H_R_is_maximum _ INSP) => [F1 [FIX1 LE1]].
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
F1: nat
FIX1: task_rtct tsk +
interference_bound_function tsk A1 (A1 + F1) <=
A1 + F1
LE1: F1 + (task_cost tsk - task_rtct tsk) <= R
completed_by sched j (job_arrival j + R)
    destruct (t1 + (A1 + F1) >= t2) eqn:BIG.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
F1: nat
FIX1: task_rtct tsk +
interference_bound_function tsk A1 (A1 + F1) <=
A1 + F1
LE1: F1 + (task_cost tsk - task_rtct tsk) <= R
BIG: (t2 <= t1 + (A1 + F1)) = true
completed_by sched j (job_arrival j + R)
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
F1: nat
FIX1: task_rtct tsk +
interference_bound_function tsk A1 (A1 + F1) <=
A1 + F1
LE1: F1 + (task_cost tsk - task_rtct tsk) <= R
BIG: (t2 <= t1 + (A1 + F1)) = false
completed_by sched j (job_arrival j + R)
    -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
F1: nat
FIX1: task_rtct tsk +
interference_bound_function tsk A1 (A1 + F1) <=
A1 + F1
LE1: F1 + (task_cost tsk - task_rtct tsk) <= R
BIG: (t2 <= t1 + (A1 + F1)) = true
completed_by sched j (job_arrival j + R)
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
F1: nat
FIX1: task_rtct tsk +
interference_bound_function tsk A1 (A1 + F1) <=
A1 + F1
LE1: F1 + (task_cost tsk - task_rtct tsk) <= R
BIG: (t2 <= t1 + (A1 + F1)) = false
completed_by sched j (job_arrival j + R) eapply job_completed_by_arrival_plus_R_1; eauto 2.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
F1: nat
FIX1: task_rtct tsk +
interference_bound_function tsk A1 (A1 + F1) <=
A1 + F1
LE1: F1 + (task_cost tsk - task_rtct tsk) <= R
BIG: (t2 <= t1 + (A1 + F1)) = false
completed_by sched j (job_arrival j + R)
    -
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
F1: nat
FIX1: task_rtct tsk +
interference_bound_function tsk A1 (A1 + F1) <=
A1 + F1
LE1: F1 + (task_cost tsk - task_rtct tsk) <= R
BIG: (t2 <= t1 + (A1 + F1)) = false
completed_by sched j (job_arrival j + R) apply negbT in BIG; rewrite -ltnNge in BIG.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
F1: nat
FIX1: task_rtct tsk +
interference_bound_function tsk A1 (A1 + F1) <=
A1 + F1
LE1: F1 + (task_cost tsk - task_rtct tsk) <= R
BIG: t1 + (A1 + F1) < t2
completed_by sched j (job_arrival j + R)
      destruct (A2 <= A1 + F1) eqn:BOUND.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
F1: nat
FIX1: task_rtct tsk +
interference_bound_function tsk A1 (A1 + F1) <=
A1 + F1
LE1: F1 + (task_cost tsk - task_rtct tsk) <= R
BIG: t1 + (A1 + F1) < t2
BOUND: (A2 <= A1 + F1) = true
completed_by sched j (job_arrival j + R)
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
F1: nat
FIX1: task_rtct tsk +
interference_bound_function tsk A1 (A1 + F1) <=
A1 + F1
LE1: F1 + (task_cost tsk - task_rtct tsk) <= R
BIG: t1 + (A1 + F1) < t2
BOUND: (A2 <= A1 + F1) = false
completed_by sched j (job_arrival j + R) 
      +
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
F1: nat
FIX1: task_rtct tsk +
interference_bound_function tsk A1 (A1 + F1) <=
A1 + F1
LE1: F1 + (task_cost tsk - task_rtct tsk) <= R
BIG: t1 + (A1 + F1) < t2
BOUND: (A2 <= A1 + F1) = true
completed_by sched j (job_arrival j + R)
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
F1: nat
FIX1: task_rtct tsk +
interference_bound_function tsk A1 (A1 + F1) <=
A1 + F1
LE1: F1 + (task_cost tsk - task_rtct tsk) <= R
BIG: t1 + (A1 + F1) < t2
BOUND: (A2 <= A1 + F1) = false
completed_by sched j (job_arrival j + R) eapply job_completed_by_arrival_plus_R_2; eauto 2.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
F1: nat
FIX1: task_rtct tsk +
interference_bound_function tsk A1 (A1 + F1) <=
A1 + F1
LE1: F1 + (task_cost tsk - task_rtct tsk) <= R
BIG: t1 + (A1 + F1) < t2
BOUND: (A2 <= A1 + F1) = false
completed_by sched j (job_arrival j + R)
      +
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
F1: nat
FIX1: task_rtct tsk +
interference_bound_function tsk A1 (A1 + F1) <=
A1 + F1
LE1: F1 + (task_cost tsk - task_rtct tsk) <= R
BIG: t1 + (A1 + F1) < t2
BOUND: (A2 <= A1 + F1) = false
completed_by sched j (job_arrival j + R) apply negbT in BOUND; rewrite -ltnNge in BOUND.
Task: TaskType
H: TaskCost Task
H0: TaskRunToCompletionThreshold Task
Job: JobType
H1: JobTask Job Task
JA: JobArrival Job
JC: JobCost Job
H2: JobPreemptable Job
PState: ProcessorState Job
H_ideal_progress_proc_model: ideal_progress_proc_model
  PState
H_unit_service_proc_model: unit_service_proc_model
  PState
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 PState
H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence
  sched arr_seq
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute
  sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute
  sched
H_valid_job_cost: arrivals_have_valid_job_costs
  arr_seq
ts: seq Task
tsk: Task
H_tsk_in_ts: tsk \in ts
H_valid_preemption_model: valid_preemption_model
  arr_seq sched
H_valid_run_to_completion_threshold: valid_task_run_to_completion_threshold
  arr_seq tsk
work_conserving:= definitions.work_conserving arr_seq
  sched: (Job -> instant -> bool) ->
(Job -> instant -> duration) -> Prop
busy_intervals_are_bounded_by:= definitions.busy_intervals_are_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
nat -> Prop
job_interference_is_bounded_by:= definitions.job_interference_is_bounded_by
  arr_seq sched tsk: (Job ->
 instant -> bool) ->
(Job ->
 instant -> duration) ->
(Task ->
 duration ->
 duration -> duration) ->
Prop
interference: Job -> instant -> bool
interfering_workload: Job -> instant -> duration
H_work_conserving: work_conserving interference
  interfering_workload
cumul_interference:= definitions.cumul_interference
  interference: Job -> nat -> nat -> nat
cumul_interfering_workload:= definitions.cumul_interfering_workload
  interfering_workload: Job -> nat -> nat -> nat
busy_interval:= definitions.busy_interval sched
  interference interfering_workload: Job -> instant -> instant -> Prop
response_time_bounded_by:= task_response_time_bound
  arr_seq sched: Task -> duration -> Prop
L: duration
H_busy_interval_exists: busy_intervals_are_bounded_by
  interference
  interfering_workload L
interference_bound_function: Task ->
duration ->
duration -> duration
H_job_interference_is_bounded: job_interference_is_bounded_by
  interference
  interfering_workload
  interference_bound_function
is_in_search_space:= [eta search_space.is_in_search_space
       tsk L
       interference_bound_function]: nat -> Prop
R: nat
H_R_is_maximum: forall A : nat,
is_in_search_space A ->
exists F : nat,
  task_rtct tsk +
  interference_bound_function tsk A
    (A + F) <= 
  A + F /\
  F + (task_cost tsk - task_rtct tsk) <=
  R
j: Job
ARR: arrives_in arr_seq j
JOBtsk: job_of_task tsk j
POS: 0 < job_cost j
t1, t2: nat
T2: t2 <= t1 + L
BUSY: definitions.busy_interval sched interference
  interfering_workload j t1 t2
GE: t1 <= job_arrival j
LT: job_arrival j < t2
QTt2: quiet_time sched interference
  interfering_workload j t2
A2:= job_arrival j - t1: nat
A2LTL: A2 < L
A1: nat
ALEA2: A1 <= A2
EQΦ: are_equivalent_at_values_less_than
  (interference_bound_function tsk A2)
  (interference_bound_function tsk A1) L
INSP: search_space.is_in_search_space tsk L
  interference_bound_function A1
F1: nat
FIX1: task_rtct tsk +
interference_bound_function tsk A1 (A1 + F1) <=
A1 + F1
LE1: F1 + (task_cost tsk - task_rtct tsk) <= R
BIG: t1 + (A1 + F1) < t2
BOUND: A1 + F1 < A2
completed_by sched j (job_arrival j + R)
        exfalso; apply relative_arrival_time_is_no_less_than_fixpoint
                   with (j := j) (t1 := t1) (t2 := t2) (A_sp := A1) (F_sp := F1); auto.
  Qed.

End Abstract_RTA.