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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]
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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 "[ rel _ _ | _ ]" was already used in scope fun_scope. [notation-overridden,parsing]
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Notation "[ rel _ _ in _ & _ | _ ]" was already used in scope fun_scope. [notation-overridden,parsing]
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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]
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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]
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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]
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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]
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]
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]
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]
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]
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]
(** This file contains the main argument of the EDF optimality proof, starting with an analysis of the individual functions that drive the piece-wise transformation of a given reference schedule in an EDF schedule, and ending with proofs of individual properties of the obtained EDF schedule. *) (** We start by analyzing the helper function [find_swap_candidate], which is a problem-specific wrapper around [search_arg]. *) Section FindSwapCandidateFacts. (** We assume the classic (i.e., Liu & Layland) model of readiness without jitter or self-suspensions, wherein pending jobs are always ready. *) #[local] Existing Instance basic_ready_instance. (** For any given type of jobs... *) Context {Job : JobType} `{JobCost Job} `{JobDeadline Job} `{JobArrival Job}. (** ...consider an ideal uniprocessor schedule... *) Variable sched: schedule (ideal.processor_state Job). (** ...that is well-behaved (i.e., in which jobs execute only after having arrived and only if they are not yet complete). *) Hypothesis H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched. Hypothesis H_completed_jobs_dont_execute: completed_jobs_dont_execute sched. (** Suppose we are given a job [j1]... *) Variable j1: Job. (** ...and a point in time [t1]... *) Variable t1: instant. (** ...at which [j1] is scheduled... *) Hypothesis H_not_idle: scheduled_at sched j1 t1. (** ...and that is before its deadline. *) Hypothesis H_deadline_not_missed: t1 < job_deadline j1. (** First, we observe that under these assumptions the processor state at time [t1] is "relevant" according to the notion of relevance underlying the EDF transformation, namely [relevant_pstate]. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

relevant_pstate t1 (sched t1)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

relevant_pstate t1 (sched t1)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

scheduled_at sched j1 t1 -> relevant_pstate t1 (sched t1)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

relevant_pstate t1 (Some j1)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

has_arrived j1 t1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
SCHED_ARR: scheduled_at sched j1 t1 -> has_arrived j1 t1

has_arrived j1 t1
now apply SCHED_ARR. Qed. (** Since [t1] is relevant, we conclude that a search for a relevant state succeeds (if nothing else, it finds [t1]). *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

exists t : nat, search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

exists t : nat, search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

exists x : nat, t1 <= x < job_deadline j1 /\ relevant_pstate t1 (sched x)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

t1 <= t1 < job_deadline j1 /\ relevant_pstate t1 (sched t1)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

t1 <= t1 < job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
relevant_pstate t1 (sched t1)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

t1 <= t1 < job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
relevant_pstate t1 (sched t1)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

relevant_pstate t1 (sched t1)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

relevant_pstate t1 (sched t1)
by apply t1_relevant. Qed. (** For rewriting purposes, we observe that the [search_arg] operation within [find_swap_candidate] yields the final result of [find_swap_candidate]. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some (find_swap_candidate sched t1 j1)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some (find_swap_candidate sched t1 j1)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
t: nat
FOUND: search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some t

search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some (find_swap_candidate sched t1 j1)
now rewrite /find_swap_candidate FOUND. Qed. (** There is a job that is scheduled at the time that [find_swap_candidate] returns, and that job arrives no later than at time [t1]. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

exists j' : Job, scheduled_at sched j' (find_swap_candidate sched t1 j1) /\ job_arrival j' <= t1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

exists j' : Job, scheduled_at sched j' (find_swap_candidate sched t1 j1) /\ job_arrival j' <= t1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
t: nat
FOUND: search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some t

exists j' : Job, scheduled_at sched j' (find_swap_candidate sched t1 j1) /\ job_arrival j' <= t1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
t: nat
FOUND: search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some t

relevant_pstate t1 (sched t) -> exists j' : Job, scheduled_at sched j' (find_swap_candidate sched t1 j1) /\ job_arrival j' <= t1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
t: nat
FOUND: search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some t

match sched t with | Some j' => job_arrival j' <= t1 | None => false end -> exists j' : Job, scheduled_at sched j' (find_swap_candidate sched t1 j1) /\ job_arrival j' <= t1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
t: nat
FOUND: search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some t
j': Job
SCHED_t: sched t = Some j'
ARR_j': job_arrival j' <= t1

exists j' : Job, scheduled_at sched j' (find_swap_candidate sched t1 j1) /\ job_arrival j' <= t1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
t: nat
FOUND: search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some t
j': Job
SCHED_t: sched t = Some j'
ARR_j': job_arrival j' <= t1

scheduled_at sched j' (find_swap_candidate sched t1 j1) /\ job_arrival j' <= t1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
t: nat
FOUND: search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some t
j': Job
SCHED_t: sched t = Some j'
ARR_j': job_arrival j' <= t1

scheduled_at sched j' (find_swap_candidate sched t1 j1)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
t: nat
FOUND: search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some t
j': Job
SCHED_t: sched t = Some j'
ARR_j': job_arrival j' <= t1

scheduled_at sched j' t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
t: nat
FOUND: search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some t
j': Job
SCHED_t: sched t = Some j'
ARR_j': job_arrival j' <= t1

sched t == Some j'
now apply /eqP. Qed. (** Since we are considering a uniprocessor model, only one job is scheduled at a time. Hence once we know that a job is scheduled at the time that [find_swap_candidate] returns, we can conclude that it arrives not later than at time t1. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

forall j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t1 j1) -> job_arrival j2 <= t1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

forall j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t1 j1) -> job_arrival j2 <= t1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2: Job
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t1 j1)

job_arrival j2 <= t1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2: Job
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t1 j1)
j': Job
SCHED_j': scheduled_at sched j' (find_swap_candidate sched t1 j1)
ARR: job_arrival j' <= t1

job_arrival j2 <= t1
now rewrite -(ideal_proc_model_is_a_uniprocessor_model _ _ _ _ SCHED_j' SCHED_j2). Qed. (** We observe that [find_swap_candidate] returns a value within a known finite interval. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

t1 <= find_swap_candidate sched t1 j1 < job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

t1 <= find_swap_candidate sched t1 j1 < job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some (find_swap_candidate sched t1 j1) -> t1 <= find_swap_candidate sched t1 j1 < job_deadline j1
by apply search_arg_in_range. Qed. (** For convenience, since we often only require the lower bound on the interval, we re-state it as a corollary. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

t1 <= find_swap_candidate sched t1 j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

t1 <= find_swap_candidate sched t1 j1
by move: fsc_range => /andP [LE _]. Qed. (** The following lemma is a key step of the overall proof: the job scheduled at the time found by [find_swap_candidate] has the property that it has a deadline that is no later than that of any other job in the window given by time [t1] and the deadline of the job scheduled at time [t1]. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

forall j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t1 j1) -> forall (j : Job) (t : nat), t1 <= t < job_deadline j1 -> scheduled_at sched j t -> job_arrival j <= t1 -> job_deadline j2 <= job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

forall j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t1 j1) -> forall (j : Job) (t : nat), t1 <= t < job_deadline j1 -> scheduled_at sched j t -> job_arrival j <= t1 -> job_deadline j2 <= job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2: Job
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t1 j1)
j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
SCHED_j: scheduled_at sched j t
ARR_j: job_arrival j <= t1

job_deadline j2 <= job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2: Job
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t1 j1)
j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
SCHED_j: scheduled_at sched j t
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline

job_deadline j2 <= job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2: Job
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t1 j1)
j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
SCHED_j: scheduled_at sched j t
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline
TRANS: transitive (T:=ideal.processor_state Job) earlier_deadline

job_deadline j2 <= job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2: Job
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t1 j1)
j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
SCHED_j: scheduled_at sched j t
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline
TRANS: transitive (T:=ideal.processor_state Job) earlier_deadline
REFL: reflexive (T:=ideal.processor_state Job) earlier_deadline

job_deadline j2 <= job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2, j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline
TRANS: transitive (T:=ideal.processor_state Job) earlier_deadline
REFL: reflexive (T:=ideal.processor_state Job) earlier_deadline

scheduled_at sched j t -> scheduled_at sched j2 (find_swap_candidate sched t1 j1) -> job_deadline j2 <= job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2, j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline
TRANS: transitive (T:=ideal.processor_state Job) earlier_deadline
REFL: reflexive (T:=ideal.processor_state Job) earlier_deadline
SCHED_j: sched t = Some j
SCHED_j2: sched (find_swap_candidate sched t1 j1) = Some j2

job_deadline j2 <= job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2, j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline
TRANS: transitive (T:=ideal.processor_state Job) earlier_deadline
REFL: reflexive (T:=ideal.processor_state Job) earlier_deadline
SCHED_j: sched t = Some j
SCHED_j2: sched (find_swap_candidate sched t1 j1) = Some j2

earlier_deadline (sched (find_swap_candidate sched t1 j1)) (sched t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2, j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline
TRANS: transitive (T:=ideal.processor_state Job) earlier_deadline
REFL: reflexive (T:=ideal.processor_state Job) earlier_deadline
SCHED_j: sched t = Some j
SCHED_j2: sched (find_swap_candidate sched t1 j1) = Some j2
ED: earlier_deadline (sched (find_swap_candidate sched t1 j1)) (sched t)
job_deadline j2 <= job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2, j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline
TRANS: transitive (T:=ideal.processor_state Job) earlier_deadline
REFL: reflexive (T:=ideal.processor_state Job) earlier_deadline
SCHED_j: sched t = Some j
SCHED_j2: sched (find_swap_candidate sched t1 j1) = Some j2

earlier_deadline (sched (find_swap_candidate sched t1 j1)) (sched t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2, j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline
TRANS: transitive (T:=ideal.processor_state Job) earlier_deadline
REFL: reflexive (T:=ideal.processor_state Job) earlier_deadline
SCHED_j: sched t = Some j
SCHED_j2: sched (find_swap_candidate sched t1 j1) = Some j2
MIN: forall y : nat, t1 <= y < job_deadline j1 -> relevant_pstate t1 (sched y) -> earlier_deadline (sched (find_swap_candidate sched t1 j1)) (sched y)

earlier_deadline (sched (find_swap_candidate sched t1 j1)) (sched t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2, j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline
TRANS: transitive (T:=ideal.processor_state Job) earlier_deadline
REFL: reflexive (T:=ideal.processor_state Job) earlier_deadline
SCHED_j: sched t = Some j
SCHED_j2: sched (find_swap_candidate sched t1 j1) = Some j2
MIN: forall y : nat, t1 <= y < job_deadline j1 -> relevant_pstate t1 (sched y) -> earlier_deadline (sched (find_swap_candidate sched t1 j1)) (sched y)

t1 <= t < job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2, j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline
TRANS: transitive (T:=ideal.processor_state Job) earlier_deadline
REFL: reflexive (T:=ideal.processor_state Job) earlier_deadline
SCHED_j: sched t = Some j
SCHED_j2: sched (find_swap_candidate sched t1 j1) = Some j2
MIN: forall y : nat, t1 <= y < job_deadline j1 -> relevant_pstate t1 (sched y) -> earlier_deadline (sched (find_swap_candidate sched t1 j1)) (sched y)
relevant_pstate t1 (sched t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2, j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline
TRANS: transitive (T:=ideal.processor_state Job) earlier_deadline
REFL: reflexive (T:=ideal.processor_state Job) earlier_deadline
SCHED_j: sched t = Some j
SCHED_j2: sched (find_swap_candidate sched t1 j1) = Some j2
MIN: forall y : nat, t1 <= y < job_deadline j1 -> relevant_pstate t1 (sched y) -> earlier_deadline (sched (find_swap_candidate sched t1 j1)) (sched y)

t1 <= t < job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2, j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline
TRANS: transitive (T:=ideal.processor_state Job) earlier_deadline
REFL: reflexive (T:=ideal.processor_state Job) earlier_deadline
SCHED_j: sched t = Some j
SCHED_j2: sched (find_swap_candidate sched t1 j1) = Some j2
MIN: forall y : nat, t1 <= y < job_deadline j1 -> relevant_pstate t1 (sched y) -> earlier_deadline (sched (find_swap_candidate sched t1 j1)) (sched y)
relevant_pstate t1 (sched t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2, j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline
TRANS: transitive (T:=ideal.processor_state Job) earlier_deadline
REFL: reflexive (T:=ideal.processor_state Job) earlier_deadline
SCHED_j: sched t = Some j
SCHED_j2: sched (find_swap_candidate sched t1 j1) = Some j2
MIN: forall y : nat, t1 <= y < job_deadline j1 -> relevant_pstate t1 (sched y) -> earlier_deadline (sched (find_swap_candidate sched t1 j1)) (sched y)

relevant_pstate t1 (sched t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2, j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline
TRANS: transitive (T:=ideal.processor_state Job) earlier_deadline
REFL: reflexive (T:=ideal.processor_state Job) earlier_deadline
SCHED_j: sched t = Some j
SCHED_j2: sched (find_swap_candidate sched t1 j1) = Some j2
MIN: forall y : nat, t1 <= y < job_deadline j1 -> relevant_pstate t1 (sched y) -> earlier_deadline (sched (find_swap_candidate sched t1 j1)) (sched y)

relevant_pstate t1 (sched t)
by rewrite /relevant_pstate SCHED_j.
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2, j: Job
t: nat
t1_le_t: t1 <= t
t_lt_dl1: t < job_deadline j1
ARR_j: job_arrival j <= t1
TOTAL: total (T:=ideal.processor_state Job) earlier_deadline
TRANS: transitive (T:=ideal.processor_state Job) earlier_deadline
REFL: reflexive (T:=ideal.processor_state Job) earlier_deadline
SCHED_j: sched t = Some j
SCHED_j2: sched (find_swap_candidate sched t1 j1) = Some j2
ED: earlier_deadline (sched (find_swap_candidate sched t1 j1)) (sched t)

job_deadline j2 <= job_deadline j
now move: ED; rewrite /earlier_deadline /oapp SCHED_j SCHED_j2. Qed. (** As a special case of the above lemma, we observe that the job scheduled at the time given by [find_swap_candidate] in particular has a deadline no later than the job scheduled at time [t1]. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

forall j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t1 j1) -> job_deadline j2 <= job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1

forall j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t1 j1) -> job_deadline j2 <= job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2: Job
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t1 j1)

job_deadline j2 <= job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2: Job
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t1 j1)

t1 <= t1 < job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2: Job
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t1 j1)
job_arrival j1 <= t1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2: Job
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t1 j1)

t1 <= t1 < job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2: Job
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t1 j1)
job_arrival j1 <= t1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2: Job
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t1 j1)

job_arrival j1 <= t1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
j1: Job
t1: instant
H_not_idle: scheduled_at sched j1 t1
H_deadline_not_missed: t1 < job_deadline j1
j2: Job
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t1 j1)

job_arrival j1 <= t1
by apply H_jobs_must_arrive_to_execute. Qed. End FindSwapCandidateFacts. (** In the next section, we analyze properties of [make_edf_at], which we abbreviate as "[mea]" in the following. *) Section MakeEDFAtFacts. (** For any given type of jobs... *) Context {Job : JobType} `{JobCost Job} `{JobDeadline Job} `{JobArrival Job}. (** ...consider an ideal uniprocessor schedule... *) Variable sched: schedule (ideal.processor_state Job). (** ...that is well-behaved... *) Hypothesis H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched. Hypothesis H_completed_jobs_dont_execute: completed_jobs_dont_execute sched. (** ...and in which no scheduled job misses a deadline. *) Hypothesis H_no_deadline_misses: all_deadlines_met sched. (** Since we will require this fact repeatedly, we briefly observe that, since no scheduled job misses its deadline, if a job is scheduled at some time [t], then its deadline is later than [t]. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched

forall (j : Job) (t : instant), scheduled_at sched j t -> t < job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched

forall (j : Job) (t : instant), scheduled_at sched j t -> t < job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
j: Job
t: instant
SCHED: scheduled_at sched j t

t < job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
j: Job
t: instant
SCHED: scheduled_at sched j t

job_meets_deadline sched j
exact: (H_no_deadline_misses _ t). Qed. (** We analyze [make_edf_at] applied to an arbitrary point in time, which we denote [t_edf] in the following. *) Variable t_edf: instant. (** For brevity, let [sched'] denote the schedule obtained from [make_edf_at] applied to [sched] at time [t_edf]. *) Let sched' := make_edf_at sched t_edf. (** First, we observe that in [sched'] jobs still don't execute past completion. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)

completed_jobs_dont_execute sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)

completed_jobs_dont_execute sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
IDEAL: ideal_progress_proc_model (ideal.processor_state Job)

completed_jobs_dont_execute sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
IDEAL: ideal_progress_proc_model (ideal.processor_state Job)
UNIT: unit_service_proc_model (ideal.processor_state Job)

completed_jobs_dont_execute sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
IDEAL: ideal_progress_proc_model (ideal.processor_state Job)
UNIT: unit_service_proc_model (ideal.processor_state Job)

completed_jobs_dont_execute match sched t_edf with | Some j => swapped sched t_edf (find_swap_candidate sched t_edf j) | None => sched end
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
IDEAL: ideal_progress_proc_model (ideal.processor_state Job)
UNIT: unit_service_proc_model (ideal.processor_state Job)
j_orig: Job
SCHED: sched t_edf = Some j_orig

completed_jobs_dont_execute (swapped sched t_edf (find_swap_candidate sched t_edf j_orig))
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
IDEAL: ideal_progress_proc_model (ideal.processor_state Job)
UNIT: unit_service_proc_model (ideal.processor_state Job)
j_orig: Job
SCHED: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf

completed_jobs_dont_execute (swapped sched t_edf (find_swap_candidate sched t_edf j_orig))
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
IDEAL: ideal_progress_proc_model (ideal.processor_state Job)
UNIT: unit_service_proc_model (ideal.processor_state Job)
j_orig: Job
SCHED: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf

t_edf <= find_swap_candidate sched t_edf j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
IDEAL: ideal_progress_proc_model (ideal.processor_state Job)
UNIT: unit_service_proc_model (ideal.processor_state Job)
j_orig: Job
SCHED: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf

t_edf < job_deadline j_orig
now apply scheduled_job_in_sched_has_later_deadline. Qed. (** Importantly, [make_edf_at] does not introduce any deadline misses, which is a crucial step in the EDF optimality argument. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)

all_deadlines_met sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)

all_deadlines_met sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t

job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t

job_meets_deadline match sched t_edf with | Some j => swapped sched t_edf (find_swap_candidate sched t_edf j) | None => sched end j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
SCHED_orig: sched t_edf = None

job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
job_meets_deadline (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
SCHED_orig: sched t_edf = None

job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
SCHED_orig: sched t_edf = None

scheduled_at sched j t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED_orig: sched t_edf = None

scheduled_at sched' j t -> scheduled_at sched j t
now rewrite /sched' /make_edf_at SCHED_orig.
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig

job_meets_deadline (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig

job_meets_deadline (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf

job_meets_deadline (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig

job_meets_deadline (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig

t_edf <= find_swap_candidate sched t_edf j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
forall j1 j2 : Job, scheduled_at sched j1 t_edf -> scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) -> job_deadline j2 <= job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
forall j1 : Job, scheduled_at sched j1 t_edf -> exists j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) /\ find_swap_candidate sched t_edf j_orig < job_deadline j2
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig

t_edf <= find_swap_candidate sched t_edf j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
forall j1 j2 : Job, scheduled_at sched j1 t_edf -> scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) -> job_deadline j2 <= job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
forall j1 : Job, scheduled_at sched j1 t_edf -> exists j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) /\ find_swap_candidate sched t_edf j_orig < job_deadline j2
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig

forall j1 j2 : Job, scheduled_at sched j1 t_edf -> scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) -> job_deadline j2 <= job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
forall j1 : Job, scheduled_at sched j1 t_edf -> exists j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) /\ find_swap_candidate sched t_edf j_orig < job_deadline j2
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig

forall j1 j2 : Job, scheduled_at sched j1 t_edf -> scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) -> job_deadline j2 <= job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
forall j1 : Job, scheduled_at sched j1 t_edf -> exists j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) /\ find_swap_candidate sched t_edf j_orig < job_deadline j2
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
j1, j2: Job
SCHED_j1: scheduled_at sched j1 t_edf
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig)

job_deadline j2 <= job_deadline j1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
forall j1 : Job, scheduled_at sched j1 t_edf -> exists j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) /\ find_swap_candidate sched t_edf j_orig < job_deadline j2
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
j1, j2: Job
SCHED_j1: scheduled_at sched j1 t_edf
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig)

t_edf <= t_edf < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
j1, j2: Job
SCHED_j1: scheduled_at sched j1 t_edf
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig)
job_arrival j1 <= t_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
forall j1 : Job, scheduled_at sched j1 t_edf -> exists j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) /\ find_swap_candidate sched t_edf j_orig < job_deadline j2
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
j1, j2: Job
SCHED_j1: scheduled_at sched j1 t_edf
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig)

t_edf <= t_edf < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
j1, j2: Job
SCHED_j1: scheduled_at sched j1 t_edf
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig)
job_arrival j1 <= t_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
forall j1 : Job, scheduled_at sched j1 t_edf -> exists j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) /\ find_swap_candidate sched t_edf j_orig < job_deadline j2
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
j1, j2: Job
SCHED_j1: scheduled_at sched j1 t_edf
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig)

job_arrival j1 <= t_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
forall j1 : Job, scheduled_at sched j1 t_edf -> exists j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) /\ find_swap_candidate sched t_edf j_orig < job_deadline j2
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
j1, j2: Job
SCHED_j1: scheduled_at sched j1 t_edf
SCHED_j2: scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig)

job_arrival j1 <= t_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
forall j1 : Job, scheduled_at sched j1 t_edf -> exists j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) /\ find_swap_candidate sched t_edf j_orig < job_deadline j2
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig

forall j1 : Job, scheduled_at sched j1 t_edf -> exists j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) /\ find_swap_candidate sched t_edf j_orig < job_deadline j2
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig

forall j1 : Job, scheduled_at sched j1 t_edf -> exists j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) /\ find_swap_candidate sched t_edf j_orig < job_deadline j2
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
j1: Job
SCHED_j1: scheduled_at sched j1 t_edf

exists j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) /\ find_swap_candidate sched t_edf j_orig < job_deadline j2
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
j1: Job
SCHED_j1: scheduled_at sched j1 t_edf
j': Job
SCHED_j': scheduled_at sched j' (find_swap_candidate sched t_edf j_orig)
ARR_j': job_arrival j' <= t_edf

exists j2 : Job, scheduled_at sched j2 (find_swap_candidate sched t_edf j_orig) /\ find_swap_candidate sched t_edf j_orig < job_deadline j2
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
j1: Job
SCHED_j1: scheduled_at sched j1 t_edf
j': Job
SCHED_j': scheduled_at sched j' (find_swap_candidate sched t_edf j_orig)
ARR_j': job_arrival j' <= t_edf

scheduled_at sched j' (find_swap_candidate sched t_edf j_orig) /\ find_swap_candidate sched t_edf j_orig < job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
j1: Job
SCHED_j1: scheduled_at sched j1 t_edf
j': Job
SCHED_j': scheduled_at sched j' (find_swap_candidate sched t_edf j_orig)
ARR_j': job_arrival j' <= t_edf

find_swap_candidate sched t_edf j_orig < job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig

job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig

job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig

exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
EX: exists t' : instant, scheduled_at sched j t'
job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig

exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig

scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t
now move: SCHED; rewrite /sched' /make_edf_at SCHED_orig.
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
EX: exists t' : instant, scheduled_at sched j t'

job_meets_deadline sched j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
t': instant
SCHED_t': scheduled_at sched j t'

job_meets_deadline sched j
now apply H_no_deadline_misses with (t := t'). } Qed. (** As a result, we may conclude that any job scheduled at a time t has a deadline later than t. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)

forall (j : Job) (t : instant), scheduled_at sched' j t -> t < job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)

forall (j : Job) (t : instant), scheduled_at sched' j t -> t < job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t

t < job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t

completed_jobs_dont_execute sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t

completed_jobs_dont_execute sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t

job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t

job_meets_deadline sched' j
by apply mea_no_deadline_misses with (t := t). Qed. (** Next comes a big step in the optimality proof: we observe that [make_edf_at] indeed ensures that [EDF_at] holds at time [t_edf] in [sched']. As this is a larger argument, we proceed by case analysis and first establish a couple of helper lemmas in the following section. *) Section GuaranteeCaseAnalysis. (** Let [j_orig] denote the job scheduled in [sched] at time [t_edf], let [j_edf] denote the job scheduled in [sched'] at time [t_edf], and let [j'] denote any job scheduled in [sched'] at some time [t'] after [t_edf]... *) Variable j_orig j_edf j': Job. Variable t': instant. Hypothesis H_t_edf_le_t' : t_edf <= t'. Hypothesis H_sched_orig: scheduled_at sched j_orig t_edf. Hypothesis H_sched_edf: scheduled_at sched' j_edf t_edf. Hypothesis H_sched': scheduled_at sched' j' t'. (** ... and that arrives before time [t_edf]. *) Hypothesis H_arrival_j' : job_arrival j' <= t_edf. (** We begin by observing three simple facts that will be used repeatedly in the case analysis. *) (** First, the deadline of [j_orig] is later than [t_edf]. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

t_edf < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

t_edf < job_deadline j_orig
by apply (scheduled_job_in_sched_has_later_deadline j_orig t_edf H_sched_orig). Qed. (** Second, by the definition of [sched'], [j_edf] is scheduled in [sched] at the time returned by [find_swap_candidate]. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

scheduled_at sched j_orig t_edf -> sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig

sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig

scheduled_at sched' j_edf t_edf -> sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig

scheduled_at (fun t : instant => if find_swap_candidate sched t_edf j_orig == t then sched t_edf else if t_edf == t then sched (find_swap_candidate sched t_edf j_orig) else sched t) j_edf t_edf -> sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig

(if find_swap_candidate sched t_edf j_orig == t_edf then sched t_edf else if t_edf == t_edf then sched (find_swap_candidate sched t_edf j_orig) else sched t_edf) == Some j_edf -> sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
FSC: (find_swap_candidate sched t_edf j_orig == t_edf) = true

sched t_edf == Some j_edf -> sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
FSC: (find_swap_candidate sched t_edf j_orig == t_edf) = false
(if t_edf == t_edf then sched (find_swap_candidate sched t_edf j_orig) else sched t_edf) == Some j_edf -> sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
FSC: (find_swap_candidate sched t_edf j_orig == t_edf) = true

sched t_edf == Some j_edf -> sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
FSC: (find_swap_candidate sched t_edf j_orig == t_edf) = false
(if t_edf == t_edf then sched (find_swap_candidate sched t_edf j_orig) else sched t_edf) == Some j_edf -> sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
FSC: (find_swap_candidate sched t_edf j_orig == t_edf) = false

(if t_edf == t_edf then sched (find_swap_candidate sched t_edf j_orig) else sched t_edf) == Some j_edf -> sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
FSC: (find_swap_candidate sched t_edf j_orig == t_edf) = false

(if t_edf == t_edf then sched (find_swap_candidate sched t_edf j_orig) else sched t_edf) == Some j_edf -> sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
by rewrite ifT // => /eqP. Qed. (** Third, the deadline of [j_edf] is no later than the deadline of [j_orig]. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

job_deadline j_edf <= job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

job_deadline j_edf <= job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

t_edf < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
scheduled_at sched j_edf (find_swap_candidate sched t_edf j_orig)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

t_edf < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
scheduled_at sched j_edf (find_swap_candidate sched t_edf j_orig)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

scheduled_at sched j_edf (find_swap_candidate sched t_edf j_orig)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

scheduled_at sched j_edf (find_swap_candidate sched t_edf j_orig)
by rewrite scheduled_at_def mea_guarantee_fsc_is_j_edf //=. Qed. (** With the setup in place, we are now ready to begin the case analysis. *) (** First, we consider the simpler case where [t'] is no earlier than the deadline of [j_orig]. This case is simpler because [t'] being no earlier than [j_orig]'s deadline implies that [j'] has deadline no earlier than [j_orig] (since no scheduled job in [sched] misses a deadline), which in turn has a deadline no earlier than [j_edf]. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

job_deadline j_orig <= t' -> job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

job_deadline j_orig <= t' -> job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
DL_j': t' < job_deadline j'
BOUND_t': job_deadline j_orig <= t'

job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
DL_j': t' < job_deadline j'
BOUND_t': job_deadline j_orig <= t'

job_deadline j_orig <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
DL_j': t' < job_deadline j'
BOUND_t': job_deadline j_orig <= t'

t' <= job_deadline j'
now apply ltnW. Qed. (** Next, we consider the more difficult case, where [t'] is before the deadline of [j_orig]. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

t' < job_deadline j_orig -> job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

t' < job_deadline j_orig -> job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

scheduled_at sched j_orig t_edf -> t' < job_deadline j_orig -> job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig

job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf

job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf

exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
EX: exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf

exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
EQ: t_edf = t'

exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
EQ: t_edf = t'

exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
EQ: t_edf = t'

scheduled_at sched j' (find_swap_candidate sched t_edf j_orig) /\ t_edf <= find_swap_candidate sched t_edf j_orig < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
EQ: t_edf = t'

scheduled_at sched j' (find_swap_candidate sched t_edf j_orig)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t': instant
j_orig: Job
SCHED: sched t' = Some j_orig
j': Job
H_arrival_j': job_arrival j' <= t'
sched':= make_edf_at sched t': schedule (ideal.processor_state Job)
H_sched': scheduled_at sched' j' t'
j_edf: Job
H_sched_edf: scheduled_at sched' j_edf t'
H_sched_orig: scheduled_at sched j_orig t'
H_t_edf_le_t': t' <= t'
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t' j_orig) = Some j_edf

scheduled_at sched j' (find_swap_candidate sched t' j_orig)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t': instant
j_orig: Job
SCHED: sched t' = Some j_orig
j': Job
H_arrival_j': job_arrival j' <= t'
sched':= make_edf_at sched t': schedule (ideal.processor_state Job)
H_sched': scheduled_at sched' j' t'
j_edf: Job
H_sched_edf: scheduled_at sched' j_edf t'
H_sched_orig: scheduled_at sched j_orig t'
H_t_edf_le_t': t' <= t'
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t' j_orig) = Some j_edf

scheduled_at sched j_edf (find_swap_candidate sched t' j_orig)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'

exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'

exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
EQ': find_swap_candidate sched t_edf j_orig = t'

exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
NEQ': find_swap_candidate sched t_edf j_orig <> t'
exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
EQ': find_swap_candidate sched t_edf j_orig = t'

exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
NEQ': find_swap_candidate sched t_edf j_orig <> t'
exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
EQ': find_swap_candidate sched t_edf j_orig = t'

scheduled_at sched j' t_edf /\ t_edf <= t_edf < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
NEQ': find_swap_candidate sched t_edf j_orig <> t'
exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
EQ': find_swap_candidate sched t_edf j_orig = t'

scheduled_at sched j' t_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
NEQ': find_swap_candidate sched t_edf j_orig <> t'
exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
EQ': find_swap_candidate sched t_edf j_orig = t'

scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' (find_swap_candidate sched t_edf j_orig)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
NEQ': find_swap_candidate sched t_edf j_orig <> t'
exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
EQ': find_swap_candidate sched t_edf j_orig = t'

scheduled_at sched' j' t' -> scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' (find_swap_candidate sched t_edf j_orig)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
NEQ': find_swap_candidate sched t_edf j_orig <> t'
exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
EQ': find_swap_candidate sched t_edf j_orig = t'

scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t' -> scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' (find_swap_candidate sched t_edf j_orig)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
NEQ': find_swap_candidate sched t_edf j_orig <> t'
exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
NEQ': find_swap_candidate sched t_edf j_orig <> t'

exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf <> t'
NEQ': find_swap_candidate sched t_edf j_orig <> t'

exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf != t'
NEQ': find_swap_candidate sched t_edf j_orig != t'

exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf != t'
NEQ': find_swap_candidate sched t_edf j_orig != t'

scheduled_at sched j' t' /\ t_edf <= t' < job_deadline j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf != t'
NEQ': find_swap_candidate sched t_edf j_orig != t'

scheduled_at sched j' t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf != t'
NEQ': find_swap_candidate sched t_edf j_orig != t'

scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
NEQ: t_edf != t'
NEQ': find_swap_candidate sched t_edf j_orig != t'

scheduled_at sched' j' t' -> scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
now rewrite /sched' /make_edf_at SCHED.
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
EX: exists x : instant, scheduled_at sched j' x /\ t_edf <= x < job_deadline j_orig

job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
t'': instant
SCHED'': scheduled_at sched j' t''
RANGE: t_edf <= t'' < job_deadline j_orig

job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_orig, j_edf, j': Job
t': instant
H_t_edf_le_t': t_edf <= t'
H_sched_orig: scheduled_at sched j_orig t_edf
H_sched_edf: scheduled_at sched' j_edf t_edf
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
SCHED: sched t_edf = Some j_orig
BOUND_t': t' < job_deadline j_orig
FSC: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf
t'': instant
SCHED'': scheduled_at sched j' t''
RANGE: t_edf <= t'' < job_deadline j_orig

scheduled_at sched j_edf (find_swap_candidate sched t_edf j_orig)
now rewrite scheduled_at_def FSC //=. Qed. End GuaranteeCaseAnalysis. (** Finally, putting the preceding cases together, we obtain the result that [make_edf_at] establishes [EDF_at] at time [t_edf]. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)

EDF_at sched' t_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)

EDF_at sched' t_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_edf: Job
H_sched_edf: scheduled_at sched' j_edf t_edf
t': instant
j': Job
t_edf_le_t': t_edf <= t'
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf

job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_edf: Job
H_sched_edf: scheduled_at sched' j_edf t_edf
t': instant
j': Job
t_edf_le_t': t_edf <= t'
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
j_orig: Job
SCHED: sched t_edf = Some j_orig

job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_edf: Job
H_sched_edf: scheduled_at sched' j_edf t_edf
t': instant
j': Job
t_edf_le_t': t_edf <= t'
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
j_orig: Job
SCHED: sched t_edf = Some j_orig
H_sched: scheduled_at sched j_orig t_edf

job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_edf: Job
H_sched_edf: scheduled_at sched' j_edf t_edf
t': instant
j': Job
t_edf_le_t': t_edf <= t'
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
j_orig: Job
SCHED: sched t_edf = Some j_orig
H_sched: scheduled_at sched j_orig t_edf

t' < job_deadline j_orig -> job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_edf: Job
H_sched_edf: scheduled_at sched' j_edf t_edf
t': instant
j': Job
t_edf_le_t': t_edf <= t'
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
j_orig: Job
SCHED: sched t_edf = Some j_orig
H_sched: scheduled_at sched j_orig t_edf
~~ (t' < job_deadline j_orig) -> job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_edf: Job
H_sched_edf: scheduled_at sched' j_edf t_edf
t': instant
j': Job
t_edf_le_t': t_edf <= t'
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
j_orig: Job
SCHED: sched t_edf = Some j_orig
H_sched: scheduled_at sched j_orig t_edf

t' < job_deadline j_orig -> job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_edf: Job
H_sched_edf: scheduled_at sched' j_edf t_edf
t': instant
j': Job
t_edf_le_t': t_edf <= t'
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
j_orig: Job
SCHED: sched t_edf = Some j_orig
H_sched: scheduled_at sched j_orig t_edf
~~ (t' < job_deadline j_orig) -> job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_edf: Job
H_sched_edf: scheduled_at sched' j_edf t_edf
t': instant
j': Job
t_edf_le_t': t_edf <= t'
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
j_orig: Job
SCHED: sched t_edf = Some j_orig
H_sched: scheduled_at sched j_orig t_edf

~~ (t' < job_deadline j_orig) -> job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_edf: Job
H_sched_edf: scheduled_at sched' j_edf t_edf
t': instant
j': Job
t_edf_le_t': t_edf <= t'
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
j_orig: Job
SCHED: sched t_edf = Some j_orig
H_sched: scheduled_at sched j_orig t_edf

~~ (t' < job_deadline j_orig) -> job_deadline j_edf <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j_edf: Job
H_sched_edf: scheduled_at sched' j_edf t_edf
t': instant
j': Job
t_edf_le_t': t_edf <= t'
H_sched': scheduled_at sched' j' t'
H_arrival_j': job_arrival j' <= t_edf
j_orig: Job
SCHED: sched t_edf = Some j_orig
H_sched: scheduled_at sched j_orig t_edf
BOUND_t': job_deadline j_orig <= t'

job_deadline j_edf <= job_deadline j'
now apply: (mea_guarantee_case_t'_past_deadline j_orig j_edf j' t'). Qed. (** We observe that [make_edf_at] maintains the property that jobs must arrive to execute. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)

jobs_must_arrive_to_execute sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)

jobs_must_arrive_to_execute sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant

scheduled_at sched' j t -> has_arrived j t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant

scheduled_at match sched t_edf with | Some j => swapped sched t_edf (find_swap_candidate sched t_edf j) | None => sched end j t -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig

scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf

scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig

scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig

(if find_swap_candidate sched t_edf j_orig == t then sched t_edf else if t_edf == t then sched (find_swap_candidate sched t_edf j_orig) else sched t) == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
EQ: find_swap_candidate sched t_edf j_orig = t

sched t_edf == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
(if t_edf == t then sched (find_swap_candidate sched t_edf j_orig) else sched t) == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
EQ: find_swap_candidate sched t_edf j_orig = t

sched t_edf == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
(if t_edf == t then sched (find_swap_candidate sched t_edf j_orig) else sched t) == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
EQ: find_swap_candidate sched t_edf j_orig = t
j_is_orig: Some j_orig = Some j

job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
(if t_edf == t then sched (find_swap_candidate sched t_edf j_orig) else sched t) == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
EQ: find_swap_candidate sched t_edf j_orig = t
j_is_orig: Some j_orig = Some j

job_arrival j_orig <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
(if t_edf == t then sched (find_swap_candidate sched t_edf j_orig) else sched t) == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
EQ: find_swap_candidate sched t_edf j_orig = t
j_is_orig: Some j_orig = Some j

job_arrival j_orig <= t_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
EQ: find_swap_candidate sched t_edf j_orig = t
j_is_orig: Some j_orig = Some j
t_edf <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
(if t_edf == t then sched (find_swap_candidate sched t_edf j_orig) else sched t) == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
EQ: find_swap_candidate sched t_edf j_orig = t
j_is_orig: Some j_orig = Some j

job_arrival j_orig <= t_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
EQ: find_swap_candidate sched t_edf j_orig = t
j_is_orig: Some j_orig = Some j
t_edf <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
(if t_edf == t then sched (find_swap_candidate sched t_edf j_orig) else sched t) == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
EQ: find_swap_candidate sched t_edf j_orig = t
j_is_orig: Some j_orig = Some j

t_edf <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
(if t_edf == t then sched (find_swap_candidate sched t_edf j_orig) else sched t) == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
EQ: find_swap_candidate sched t_edf j_orig = t
j_is_orig: Some j_orig = Some j

t_edf <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
(if t_edf == t then sched (find_swap_candidate sched t_edf j_orig) else sched t) == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t

(if t_edf == t then sched (find_swap_candidate sched t_edf j_orig) else sched t) == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t

(if t_edf == t then sched (find_swap_candidate sched t_edf j_orig) else sched t) == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
EQ': t_edf = t

sched (find_swap_candidate sched t_edf j_orig) == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
NEQ': t_edf <> t
sched t == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
EQ': t_edf = t

sched (find_swap_candidate sched t_edf j_orig) == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
NEQ': t_edf <> t
sched t == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
EQ': t_edf = t
SCHED_j: sched (find_swap_candidate sched t_edf j_orig) == Some j

job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
NEQ': t_edf <> t
sched t == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
EQ': t_edf = t
SCHED_j: sched (find_swap_candidate sched t_edf j_orig) == Some j
ARR_j: job_arrival j <= t_edf

job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
NEQ': t_edf <> t
sched t == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
NEQ': t_edf <> t

sched t == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
NEQ': t_edf <> t

sched t == Some j -> job_arrival j <= t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED': scheduled_at sched j_orig t_edf
DL_orig: t_edf < job_deadline j_orig
NEQ: find_swap_candidate sched t_edf j_orig <> t
NEQ': t_edf <> t
SCHED_j: sched t == Some j

job_arrival j <= t
by apply H_jobs_must_arrive_to_execute; rewrite scheduled_at_def. Qed. (** We connect the fact that a job is scheduled in [sched'] to the fact that it must be scheduled somewhere in [sched], too, since [make_edf_at] does not introduce any new jobs. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)

forall (j : Job) (t : instant), scheduled_at sched' j t -> exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)

forall (j : Job) (t : instant), scheduled_at sched' j t -> exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)

forall (j : Job) (t : instant), scheduled_at match sched t_edf with | Some j0 => swapped sched t_edf (find_swap_candidate sched t_edf j0) | None => sched end j t -> exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED_j: scheduled_at match sched t_edf with | Some j => swapped sched t_edf (find_swap_candidate sched t_edf j) | None => sched end j t

exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED_j: scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t

exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig
SCHED_j: scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t

scheduled_at (swapped sched ?t1 ?t2) j ?t
now exact SCHED_j. Qed. (** Conversely, if a job is scheduled in [sched], it is also scheduled somewhere in [sched'] since [make_edf_at] does not lose any jobs. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)

forall (j : Job) (t : instant), scheduled_at sched j t -> exists t' : instant, scheduled_at sched' j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)

forall (j : Job) (t : instant), scheduled_at sched j t -> exists t' : instant, scheduled_at sched' j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED_j: scheduled_at sched j t

exists t' : instant, scheduled_at sched' j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED_j: scheduled_at sched j t

exists t' : instant, scheduled_at match sched t_edf with | Some j => swapped sched t_edf (find_swap_candidate sched t_edf j) | None => sched end j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED_j: scheduled_at sched j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig

exists t' : instant, scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED_j: scheduled_at sched j t
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig

scheduled_at sched j ?t
now exact SCHED_j. Qed. (** Next, we observe that if all jobs in [sched] come from a given arrival sequence, then that's still the case in [sched'], too. *) Section ArrivalSequence. (** For given arrival sequence,... *) Variable arr_seq: arrival_sequence Job. (** ...if all jobs in [sched] come from the arrival sequence,... *) Hypothesis H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq. (** ...then all jobs in [sched'] do, too. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq

jobs_come_from_arrival_sequence sched' arr_seq
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq

jobs_come_from_arrival_sequence sched' arr_seq
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq

jobs_come_from_arrival_sequence match sched t_edf with | Some j => swapped sched t_edf (find_swap_candidate sched t_edf j) | None => sched end arr_seq
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq
j_orig: Job
SCHED_orig: sched t_edf = Some j_orig

jobs_come_from_arrival_sequence (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) arr_seq
now apply swapped_jobs_come_from_arrival_sequence. Qed. End ArrivalSequence. (** For the final claim, assume that [EDF_at] already holds everywhere prior to time [t_edf], i.e., that [sched] consists of an EDF prefix. *) Hypothesis H_EDF_prefix: forall t, t < t_edf -> EDF_at sched t. (** We establish a key property of [make_edf_at]: not only does it ensure [EDF_at] at time [t_edf], it also maintains the fact that the schedule has an EDF prefix prior to time [t_edf]. In other words, it grows the EDF prefix by one time unit. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t

forall t : nat, t <= t_edf -> EDF_at sched' t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t

forall t : nat, t <= t_edf -> EDF_at sched' t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat

t <= t_edf -> EDF_at sched' t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf

EDF_at sched' t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf

EDF_at match sched t_edf with | Some j => swapped sched t_edf (find_swap_candidate sched t_edf j) | None => sched end t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig

EDF_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
SCHED_j: scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t

job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
SCHED_j: scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf

job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
SCHED_j: scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf

t < find_swap_candidate sched t_edf j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
SCHED_j: scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
SCHED_j: scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf

t < find_swap_candidate sched t_edf j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
SCHED_j: scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf

t_edf <= find_swap_candidate sched t_edf j_orig
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
SCHED_j: scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf

t_edf < job_deadline j_orig
now apply scheduled_job_in_sched_has_later_deadline.
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
SCHED_j: scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig

job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig

scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t -> job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig

scheduled_at sched j t -> job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t

job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t

EDF_at sched t -> job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t
EDF: forall j : Job, scheduled_at sched j t -> forall (t' : instant) (j' : Job), t <= t' -> scheduled_at sched j' t' -> job_arrival j' <= t -> job_deadline j <= job_deadline j'

job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t
EDF: forall j : Job, scheduled_at sched j t -> forall (t' : instant) (j' : Job), t <= t' -> scheduled_at sched j' t' -> job_arrival j' <= t -> job_deadline j <= job_deadline j'

scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t' -> job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t
EDF: forall j : Job, scheduled_at sched j t -> forall (t' : instant) (j' : Job), t <= t' -> scheduled_at sched j' t' -> job_arrival j' <= t -> job_deadline j <= job_deadline j'
SCHED_j'_orig: scheduled_at sched j' t'

job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t
EDF: forall j : Job, scheduled_at sched j t -> forall (t' : instant) (j' : Job), t <= t' -> scheduled_at sched j' t' -> job_arrival j' <= t -> job_deadline j <= job_deadline j'
EQ: t' = t_edf
SCHED_j'_orig: scheduled_at sched j' (find_swap_candidate sched t_edf j_orig)
job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t
EDF: forall j : Job, scheduled_at sched j t -> forall (t' : instant) (j' : Job), t <= t' -> scheduled_at sched j' t' -> job_arrival j' <= t -> job_deadline j <= job_deadline j'
EQ: t' = find_swap_candidate sched t_edf j_orig
SCHED_j'_orig: scheduled_at sched j' t_edf
job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t
EDF: forall j : Job, scheduled_at sched j t -> forall (t' : instant) (j' : Job), t <= t' -> scheduled_at sched j' t' -> job_arrival j' <= t -> job_deadline j <= job_deadline j'
SCHED_j'_orig: scheduled_at sched j' t'

job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t
EDF: forall j : Job, scheduled_at sched j t -> forall (t' : instant) (j' : Job), t <= t' -> scheduled_at sched j' t' -> job_arrival j' <= t -> job_deadline j <= job_deadline j'
EQ: t' = t_edf
SCHED_j'_orig: scheduled_at sched j' (find_swap_candidate sched t_edf j_orig)
job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t
EDF: forall j : Job, scheduled_at sched j t -> forall (t' : instant) (j' : Job), t <= t' -> scheduled_at sched j' t' -> job_arrival j' <= t -> job_deadline j <= job_deadline j'
EQ: t' = find_swap_candidate sched t_edf j_orig
SCHED_j'_orig: scheduled_at sched j' t_edf
job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t
EDF: forall j : Job, scheduled_at sched j t -> forall (t' : instant) (j' : Job), t <= t' -> scheduled_at sched j' t' -> job_arrival j' <= t -> job_deadline j <= job_deadline j'
EQ: t' = t_edf
SCHED_j'_orig: scheduled_at sched j' (find_swap_candidate sched t_edf j_orig)

job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t
EDF: forall j : Job, scheduled_at sched j t -> forall (t' : instant) (j' : Job), t <= t' -> scheduled_at sched j' t' -> job_arrival j' <= t -> job_deadline j <= job_deadline j'
EQ: t' = find_swap_candidate sched t_edf j_orig
SCHED_j'_orig: scheduled_at sched j' t_edf
job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t
EDF: forall j : Job, scheduled_at sched j t -> forall (t' : instant) (j' : Job), t <= t' -> scheduled_at sched j' t' -> job_arrival j' <= t -> job_deadline j <= job_deadline j'
EQ: t' = t_edf
SCHED_j'_orig: scheduled_at sched j' (find_swap_candidate sched t_edf j_orig)

job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t
EDF: forall j : Job, scheduled_at sched j t -> forall (t' : instant) (j' : Job), t <= t' -> scheduled_at sched j' t' -> job_arrival j' <= t -> job_deadline j <= job_deadline j'
EQ: t' = find_swap_candidate sched t_edf j_orig
SCHED_j'_orig: scheduled_at sched j' t_edf
job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t
EDF: forall j : Job, scheduled_at sched j t -> forall (t' : instant) (j' : Job), t <= t' -> scheduled_at sched j' t' -> job_arrival j' <= t -> job_deadline j <= job_deadline j'
EQ: t' = find_swap_candidate sched t_edf j_orig
SCHED_j'_orig: scheduled_at sched j' t_edf

job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
t_edf: instant
sched':= make_edf_at sched t_edf: schedule (ideal.processor_state Job)
H_EDF_prefix: forall t : nat, t < t_edf -> EDF_at sched t
t: nat
LT: t < t_edf
j_orig: Job
SCHED_edf: sched t_edf = Some j_orig
j: Job
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j' t'
ARR_j': job_arrival j' <= t
SCHED_edf': scheduled_at sched j_orig t_edf
LT_t_fsc: t < find_swap_candidate sched t_edf j_orig
SCHED_j: scheduled_at sched j t
EDF: forall j : Job, scheduled_at sched j t -> forall (t' : instant) (j' : Job), t <= t' -> scheduled_at sched j' t' -> job_arrival j' <= t -> job_deadline j <= job_deadline j'
EQ: t' = find_swap_candidate sched t_edf j_orig
SCHED_j'_orig: scheduled_at sched j' t_edf

job_deadline j <= job_deadline j'
by apply EDF with (t' := t_edf) => //; apply ltnW. Qed. End MakeEDFAtFacts. (** In the following section, we establish properties of [edf_transform_prefix]. *) Section EDFPrefixFacts. (** For any given type of jobs... *) Context {Job : JobType} `{JobCost Job} `{JobDeadline Job} `{JobArrival Job}. (** ...consider an ideal uniprocessor schedule... *) Variable sched: schedule (ideal.processor_state Job). (** ...that is well-behaved... *) Hypothesis H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched. Hypothesis H_completed_jobs_dont_execute: completed_jobs_dont_execute sched. (** ...and in which no scheduled job misses a deadline. *) Hypothesis H_no_deadline_misses: all_deadlines_met sched. (** Consider any point in time, denoted [horizon], and... *) Variable horizon: instant. (** ...let [sched'] denote the schedule obtained by transforming [sched] up to the horizon. *) Let sched' := edf_transform_prefix sched horizon. (** To start, we observe that [sched'] is still well-behaved and without deadline misses. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)

completed_jobs_dont_execute sched' /\ jobs_must_arrive_to_execute sched' /\ all_deadlines_met sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)

completed_jobs_dont_execute sched' /\ jobs_must_arrive_to_execute sched' /\ all_deadlines_met sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)

completed_jobs_dont_execute (prefix_map sched make_edf_at horizon) /\ jobs_must_arrive_to_execute (prefix_map sched make_edf_at horizon) /\ all_deadlines_met (prefix_map sched make_edf_at horizon)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)

forall (sched : schedule (ideal.processor_state Job)) (t : instant), completed_jobs_dont_execute sched /\ jobs_must_arrive_to_execute sched /\ all_deadlines_met sched -> completed_jobs_dont_execute (make_edf_at sched t) /\ jobs_must_arrive_to_execute (make_edf_at sched t) /\ all_deadlines_met (make_edf_at sched t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
sched'': schedule (ideal.processor_state Job)
t: instant
COMP: completed_jobs_dont_execute sched''
ARR: jobs_must_arrive_to_execute sched''
DL_MET: all_deadlines_met sched''

completed_jobs_dont_execute (make_edf_at sched'' t) /\ jobs_must_arrive_to_execute (make_edf_at sched'' t) /\ all_deadlines_met (make_edf_at sched'' t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
sched'': schedule (ideal.processor_state Job)
t: instant
COMP: completed_jobs_dont_execute sched''
ARR: jobs_must_arrive_to_execute sched''
DL_MET: all_deadlines_met sched''

completed_jobs_dont_execute (make_edf_at sched'' t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
sched'': schedule (ideal.processor_state Job)
t: instant
COMP: completed_jobs_dont_execute sched''
ARR: jobs_must_arrive_to_execute sched''
DL_MET: all_deadlines_met sched''
jobs_must_arrive_to_execute (make_edf_at sched'' t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
sched'': schedule (ideal.processor_state Job)
t: instant
COMP: completed_jobs_dont_execute sched''
ARR: jobs_must_arrive_to_execute sched''
DL_MET: all_deadlines_met sched''
all_deadlines_met (make_edf_at sched'' t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
sched'': schedule (ideal.processor_state Job)
t: instant
COMP: completed_jobs_dont_execute sched''
ARR: jobs_must_arrive_to_execute sched''
DL_MET: all_deadlines_met sched''

completed_jobs_dont_execute (make_edf_at sched'' t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
sched'': schedule (ideal.processor_state Job)
t: instant
COMP: completed_jobs_dont_execute sched''
ARR: jobs_must_arrive_to_execute sched''
DL_MET: all_deadlines_met sched''
jobs_must_arrive_to_execute (make_edf_at sched'' t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
sched'': schedule (ideal.processor_state Job)
t: instant
COMP: completed_jobs_dont_execute sched''
ARR: jobs_must_arrive_to_execute sched''
DL_MET: all_deadlines_met sched''
all_deadlines_met (make_edf_at sched'' t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
sched'': schedule (ideal.processor_state Job)
t: instant
COMP: completed_jobs_dont_execute sched''
ARR: jobs_must_arrive_to_execute sched''
DL_MET: all_deadlines_met sched''

jobs_must_arrive_to_execute (make_edf_at sched'' t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
sched'': schedule (ideal.processor_state Job)
t: instant
COMP: completed_jobs_dont_execute sched''
ARR: jobs_must_arrive_to_execute sched''
DL_MET: all_deadlines_met sched''
all_deadlines_met (make_edf_at sched'' t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
sched'': schedule (ideal.processor_state Job)
t: instant
COMP: completed_jobs_dont_execute sched''
ARR: jobs_must_arrive_to_execute sched''
DL_MET: all_deadlines_met sched''

jobs_must_arrive_to_execute (make_edf_at sched'' t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
sched'': schedule (ideal.processor_state Job)
t: instant
COMP: completed_jobs_dont_execute sched''
ARR: jobs_must_arrive_to_execute sched''
DL_MET: all_deadlines_met sched''
all_deadlines_met (make_edf_at sched'' t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
sched'': schedule (ideal.processor_state Job)
t: instant
COMP: completed_jobs_dont_execute sched''
ARR: jobs_must_arrive_to_execute sched''
DL_MET: all_deadlines_met sched''

all_deadlines_met (make_edf_at sched'' t)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
sched'': schedule (ideal.processor_state Job)
t: instant
COMP: completed_jobs_dont_execute sched''
ARR: jobs_must_arrive_to_execute sched''
DL_MET: all_deadlines_met sched''

all_deadlines_met (make_edf_at sched'' t)
apply mea_no_deadline_misses => //. Qed. (** Because it is needed frequently, we extract the second clause of the above conjunction as a corollary. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)

jobs_must_arrive_to_execute sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)

jobs_must_arrive_to_execute sched'
by move: edf_prefix_well_formedness => [_ [ARR _]]. Qed. (** We similarly observe that the absence of deadline misses implies that any scheduled job must have a deadline at a time later then when it is scheduled. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)

forall (j : Job) (t : instant), scheduled_at sched' j t -> t < job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)

forall (j : Job) (t : instant), scheduled_at sched' j t -> t < job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t

t < job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
COMP: completed_jobs_dont_execute sched'
ARR: jobs_must_arrive_to_execute sched'
DL_MET: all_deadlines_met sched'

t < job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED: scheduled_at sched' j t
COMP: completed_jobs_dont_execute sched'
ARR: jobs_must_arrive_to_execute sched'
DL_MET: all_deadlines_met sched'

job_meets_deadline sched' j
exact: (DL_MET j t). Qed. (** Since no jobs are lost or added to the schedule by [edf_transform_prefix], we if a job is scheduled in the transformed schedule, then it is also scheduled at some point in the original schedule. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)

forall (j : Job) (t : instant), scheduled_at sched' j t -> exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)

forall (j : Job) (t : instant), scheduled_at sched' j t -> exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)

forall (j : Job) (t : instant), scheduled_at (prefix_map sched make_edf_at horizon) j t -> exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
j: Job

forall t : instant, scheduled_at (prefix_map sched make_edf_at horizon) j t -> exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
j: Job

forall (sched0 : schedule (ideal.processor_state Job)) (t : instant), (forall t0 : instant, scheduled_at sched0 j t0 -> exists t' : instant, scheduled_at sched j t') -> forall t0 : instant, scheduled_at (make_edf_at sched0 t) j t0 -> exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
j: Job
sched'': schedule (ideal.processor_state Job)
t'': instant
EX: forall t : instant, scheduled_at sched'' j t -> exists t' : instant, scheduled_at sched j t'
t''': instant
SCHED_mea: scheduled_at (make_edf_at sched'' t'') j t'''

exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
j: Job
sched'': schedule (ideal.processor_state Job)
t'': instant
EX: forall t : instant, scheduled_at sched'' j t -> exists t' : instant, scheduled_at sched j t'
t''': instant
SCHED_mea: scheduled_at (make_edf_at sched'' t'') j t'''
t'''': instant
SCHED'''': scheduled_at sched'' j t''''

exists t' : instant, scheduled_at sched j t'
now apply: (EX t'''' SCHED''''). Qed. (** Conversely, if a job is scheduled in the original schedule, it is also scheduled at some point in the transformed schedule. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)

forall (j : Job) (t : instant), scheduled_at sched j t -> exists t' : instant, scheduled_at sched' j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)

forall (j : Job) (t : instant), scheduled_at sched j t -> exists t' : instant, scheduled_at sched' j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED_j: scheduled_at sched j t

exists t' : instant, scheduled_at sched' j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED_j: scheduled_at sched j t

exists t' : instant, scheduled_at (prefix_map sched make_edf_at horizon) j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED_j: scheduled_at sched j t

forall (sched : schedule (ideal.processor_state Job)) (t : instant), (exists t' : instant, scheduled_at sched j t') -> exists t' : instant, scheduled_at (make_edf_at sched t) j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED_j: scheduled_at sched j t
schedX: schedule (ideal.processor_state Job)
tx, t': instant
SCHEDX_j: scheduled_at schedX j t'

exists t' : instant, scheduled_at (make_edf_at schedX tx) j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
j: Job
t: instant
SCHED_j: scheduled_at sched j t
schedX: schedule (ideal.processor_state Job)
tx, t': instant
SCHEDX_j: scheduled_at schedX j t'

scheduled_at schedX j ?t
now exact SCHEDX_j. Qed. (** Next, we note that [edf_transform_prefix] maintains the property that all jobs stem from a given arrival sequence. *) Section ArrivalSequence. (** For any arrival sequence,... *) Variable arr_seq: arrival_sequence Job. (** ...if all jobs in the original schedule come from the arrival sequence,... *) Hypothesis H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq. (** ...then all jobs in the transformed schedule still come from the same arrival sequence. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq

jobs_come_from_arrival_sequence sched' arr_seq
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq

jobs_come_from_arrival_sequence sched' arr_seq
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq

jobs_come_from_arrival_sequence (prefix_map sched make_edf_at horizon) arr_seq
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq

forall (sched : schedule (ideal.processor_state Job)) (t : instant), jobs_come_from_arrival_sequence sched arr_seq -> jobs_come_from_arrival_sequence (make_edf_at sched t) arr_seq
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq
schedX: schedule (ideal.processor_state Job)
t: instant
ARR: jobs_come_from_arrival_sequence schedX arr_seq

jobs_come_from_arrival_sequence (make_edf_at schedX t) arr_seq
now apply mea_jobs_come_from_arrival_sequence. Qed. End ArrivalSequence. (** We establish the key property of [edf_transform_prefix]: that it indeed ensures that the resulting schedule ensures the EDF invariant up to the given [horizon]. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)

forall t : nat, t < horizon -> EDF_at sched' t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)

forall t : nat, t < horizon -> EDF_at sched' t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon

EDF_at sched' t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon

EDF_at (prefix_map sched make_edf_at horizon) t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon

forall (sched : schedule (ideal.processor_state Job)) (t_ref : instant), completed_jobs_dont_execute sched /\ jobs_must_arrive_to_execute sched /\ all_deadlines_met sched -> completed_jobs_dont_execute (make_edf_at sched t_ref) /\ jobs_must_arrive_to_execute (make_edf_at sched t_ref) /\ all_deadlines_met (make_edf_at sched t_ref)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
forall (sched : schedule (ideal.processor_state Job)) (t_ref : nat), completed_jobs_dont_execute sched /\ jobs_must_arrive_to_execute sched /\ all_deadlines_met sched -> (forall t' : nat, t' < t_ref -> EDF_at sched t') -> forall t' : nat, t' <= t_ref -> EDF_at (make_edf_at sched t_ref) t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon

forall (sched : schedule (ideal.processor_state Job)) (t_ref : instant), completed_jobs_dont_execute sched /\ jobs_must_arrive_to_execute sched /\ all_deadlines_met sched -> completed_jobs_dont_execute (make_edf_at sched t_ref) /\ jobs_must_arrive_to_execute (make_edf_at sched t_ref) /\ all_deadlines_met (make_edf_at sched t_ref)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
forall (sched : schedule (ideal.processor_state Job)) (t_ref : nat), completed_jobs_dont_execute sched /\ jobs_must_arrive_to_execute sched /\ all_deadlines_met sched -> (forall t' : nat, t' < t_ref -> EDF_at sched t') -> forall t' : nat, t' <= t_ref -> EDF_at (make_edf_at sched t_ref) t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
schedX: schedule (ideal.processor_state Job)
t_ref: instant
COMP: completed_jobs_dont_execute schedX
ARR: jobs_must_arrive_to_execute schedX
DL: all_deadlines_met schedX

completed_jobs_dont_execute (make_edf_at schedX t_ref) /\ jobs_must_arrive_to_execute (make_edf_at schedX t_ref) /\ all_deadlines_met (make_edf_at schedX t_ref)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
forall (sched : schedule (ideal.processor_state Job)) (t_ref : nat), completed_jobs_dont_execute sched /\ jobs_must_arrive_to_execute sched /\ all_deadlines_met sched -> (forall t' : nat, t' < t_ref -> EDF_at sched t') -> forall t' : nat, t' <= t_ref -> EDF_at (make_edf_at sched t_ref) t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
schedX: schedule (ideal.processor_state Job)
t_ref: instant
COMP: completed_jobs_dont_execute schedX
ARR: jobs_must_arrive_to_execute schedX
DL: all_deadlines_met schedX

completed_jobs_dont_execute (make_edf_at schedX t_ref)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
schedX: schedule (ideal.processor_state Job)
t_ref: instant
COMP: completed_jobs_dont_execute schedX
ARR: jobs_must_arrive_to_execute schedX
DL: all_deadlines_met schedX
jobs_must_arrive_to_execute (make_edf_at schedX t_ref)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
schedX: schedule (ideal.processor_state Job)
t_ref: instant
COMP: completed_jobs_dont_execute schedX
ARR: jobs_must_arrive_to_execute schedX
DL: all_deadlines_met schedX
all_deadlines_met (make_edf_at schedX t_ref)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
forall (sched : schedule (ideal.processor_state Job)) (t_ref : nat), completed_jobs_dont_execute sched /\ jobs_must_arrive_to_execute sched /\ all_deadlines_met sched -> (forall t' : nat, t' < t_ref -> EDF_at sched t') -> forall t' : nat, t' <= t_ref -> EDF_at (make_edf_at sched t_ref) t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
schedX: schedule (ideal.processor_state Job)
t_ref: instant
COMP: completed_jobs_dont_execute schedX
ARR: jobs_must_arrive_to_execute schedX
DL: all_deadlines_met schedX

completed_jobs_dont_execute (make_edf_at schedX t_ref)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
schedX: schedule (ideal.processor_state Job)
t_ref: instant
COMP: completed_jobs_dont_execute schedX
ARR: jobs_must_arrive_to_execute schedX
DL: all_deadlines_met schedX
jobs_must_arrive_to_execute (make_edf_at schedX t_ref)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
schedX: schedule (ideal.processor_state Job)
t_ref: instant
COMP: completed_jobs_dont_execute schedX
ARR: jobs_must_arrive_to_execute schedX
DL: all_deadlines_met schedX
all_deadlines_met (make_edf_at schedX t_ref)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
forall (sched : schedule (ideal.processor_state Job)) (t_ref : nat), completed_jobs_dont_execute sched /\ jobs_must_arrive_to_execute sched /\ all_deadlines_met sched -> (forall t' : nat, t' < t_ref -> EDF_at sched t') -> forall t' : nat, t' <= t_ref -> EDF_at (make_edf_at sched t_ref) t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
schedX: schedule (ideal.processor_state Job)
t_ref: instant
COMP: completed_jobs_dont_execute schedX
ARR: jobs_must_arrive_to_execute schedX
DL: all_deadlines_met schedX

jobs_must_arrive_to_execute (make_edf_at schedX t_ref)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
schedX: schedule (ideal.processor_state Job)
t_ref: instant
COMP: completed_jobs_dont_execute schedX
ARR: jobs_must_arrive_to_execute schedX
DL: all_deadlines_met schedX
all_deadlines_met (make_edf_at schedX t_ref)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
forall (sched : schedule (ideal.processor_state Job)) (t_ref : nat), completed_jobs_dont_execute sched /\ jobs_must_arrive_to_execute sched /\ all_deadlines_met sched -> (forall t' : nat, t' < t_ref -> EDF_at sched t') -> forall t' : nat, t' <= t_ref -> EDF_at (make_edf_at sched t_ref) t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
schedX: schedule (ideal.processor_state Job)
t_ref: instant
COMP: completed_jobs_dont_execute schedX
ARR: jobs_must_arrive_to_execute schedX
DL: all_deadlines_met schedX

jobs_must_arrive_to_execute (make_edf_at schedX t_ref)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
schedX: schedule (ideal.processor_state Job)
t_ref: instant
COMP: completed_jobs_dont_execute schedX
ARR: jobs_must_arrive_to_execute schedX
DL: all_deadlines_met schedX
all_deadlines_met (make_edf_at schedX t_ref)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
forall (sched : schedule (ideal.processor_state Job)) (t_ref : nat), completed_jobs_dont_execute sched /\ jobs_must_arrive_to_execute sched /\ all_deadlines_met sched -> (forall t' : nat, t' < t_ref -> EDF_at sched t') -> forall t' : nat, t' <= t_ref -> EDF_at (make_edf_at sched t_ref) t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
schedX: schedule (ideal.processor_state Job)
t_ref: instant
COMP: completed_jobs_dont_execute schedX
ARR: jobs_must_arrive_to_execute schedX
DL: all_deadlines_met schedX

all_deadlines_met (make_edf_at schedX t_ref)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
forall (sched : schedule (ideal.processor_state Job)) (t_ref : nat), completed_jobs_dont_execute sched /\ jobs_must_arrive_to_execute sched /\ all_deadlines_met sched -> (forall t' : nat, t' < t_ref -> EDF_at sched t') -> forall t' : nat, t' <= t_ref -> EDF_at (make_edf_at sched t_ref) t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
schedX: schedule (ideal.processor_state Job)
t_ref: instant
COMP: completed_jobs_dont_execute schedX
ARR: jobs_must_arrive_to_execute schedX
DL: all_deadlines_met schedX

all_deadlines_met (make_edf_at schedX t_ref)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
forall (sched : schedule (ideal.processor_state Job)) (t_ref : nat), completed_jobs_dont_execute sched /\ jobs_must_arrive_to_execute sched /\ all_deadlines_met sched -> (forall t' : nat, t' < t_ref -> EDF_at sched t') -> forall t' : nat, t' <= t_ref -> EDF_at (make_edf_at sched t_ref) t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon

forall (sched : schedule (ideal.processor_state Job)) (t_ref : nat), completed_jobs_dont_execute sched /\ jobs_must_arrive_to_execute sched /\ all_deadlines_met sched -> (forall t' : nat, t' < t_ref -> EDF_at sched t') -> forall t' : nat, t' <= t_ref -> EDF_at (make_edf_at sched t_ref) t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon

forall (sched : schedule (ideal.processor_state Job)) (t_ref : nat), completed_jobs_dont_execute sched /\ jobs_must_arrive_to_execute sched /\ all_deadlines_met sched -> (forall t' : nat, t' < t_ref -> EDF_at sched t') -> forall t' : nat, t' <= t_ref -> EDF_at (make_edf_at sched t_ref) t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
horizon: instant
sched':= edf_transform_prefix sched horizon: schedule (ideal.processor_state Job)
t: nat
IN_PREFIX: t < horizon
schedX: schedule (ideal.processor_state Job)
t_ref: nat
COMP: completed_jobs_dont_execute schedX
ARR: jobs_must_arrive_to_execute schedX
DL: all_deadlines_met schedX

(forall t' : nat, t' < t_ref -> EDF_at schedX t') -> forall t' : nat, t' <= t_ref -> EDF_at (make_edf_at schedX t_ref) t'
now apply mea_EDF_widen. Qed. End EDFPrefixFacts. (** Finally, we observe that [edf_transform_prefix] is prefix-stable, which allows us to replace an earlier horizon with a later horizon. Note: this is in a separate section because we need [edf_prefix_jobs_must_arrive] generalized for any schedule. *) Section EDFPrefixInclusion. (** For any given type of jobs... *) Context {Job : JobType} `{JobCost Job} `{JobDeadline Job} `{JobArrival Job}. (** ...consider an ideal uniprocessor schedule... *) Variable sched: schedule (ideal.processor_state Job). (** ...that is well-behaved... *) Hypothesis H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched. Hypothesis H_completed_jobs_dont_execute: completed_jobs_dont_execute sched. (** ...and in which no scheduled job misses a deadline. *) Hypothesis H_no_deadline_misses: all_deadlines_met sched.
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched

forall h1 h2 : nat, h1 <= h2 -> identical_prefix (edf_transform_prefix sched h1) (edf_transform_prefix sched h2) h1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched

forall h1 h2 : nat, h1 <= h2 -> identical_prefix (edf_transform_prefix sched h1) (edf_transform_prefix sched h2) h1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2: nat
LE_h1_h2: h1 <= h2

identical_prefix (edf_transform_prefix sched h1) (edf_transform_prefix sched h2) h1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2: nat
LE_h1_h2: h1 <= h2
t: nat
LT_t_h1: t < h1

edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2: nat
LE_h1_h2: h1 <= h2.+1
t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t

edf_transform_prefix sched h1 t = edf_transform_prefix sched h2.+1 t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2, t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t

h1 <= h2.+1 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2.+1 t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2, t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t
LT: h1 < h2.+1

edf_transform_prefix sched h1 t = edf_transform_prefix sched h2.+1 t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2, t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t

h1 < h2.+1 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2.+1 t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2, t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t
LE_h1_h2: h1 <= h2

edf_transform_prefix sched h1 t = edf_transform_prefix sched h2.+1 t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2, t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t
LE_h1_h2: h1 <= h2

edf_transform_prefix sched h2 t = make_edf_at (prefix_map sched make_edf_at h2) h2 t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2, t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t
LE_h1_h2: h1 <= h2

edf_transform_prefix sched h2 t = match prefix_map sched make_edf_at h2 h2 with | Some j => swapped (prefix_map sched make_edf_at h2) h2 (find_swap_candidate (prefix_map sched make_edf_at h2) h2 j) | None => prefix_map sched make_edf_at h2 end t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2, t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t
LE_h1_h2: h1 <= h2
j: Job
SCHED: prefix_map sched make_edf_at h2 h2 = Some j

edf_transform_prefix sched h2 t = swapped (prefix_map sched make_edf_at h2) h2 (find_swap_candidate (prefix_map sched make_edf_at h2) h2 j) t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2, t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t
LE_h1_h2: h1 <= h2
j: Job
SCHED: prefix_map sched make_edf_at h2 h2 = Some j

h2 <= find_swap_candidate (edf_transform_prefix sched h2) h2 j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2, t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t
LE_h1_h2: h1 <= h2
j: Job
SCHED: prefix_map sched make_edf_at h2 h2 = Some j
SCHED_j: scheduled_at (edf_transform_prefix sched h2) j h2

h2 <= find_swap_candidate (edf_transform_prefix sched h2) h2 j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2, t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t
LE_h1_h2: h1 <= h2
j: Job
SCHED: prefix_map sched make_edf_at h2 h2 = Some j
SCHED_j: scheduled_at (edf_transform_prefix sched h2) j h2

jobs_must_arrive_to_execute (edf_transform_prefix sched h2)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2, t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t
LE_h1_h2: h1 <= h2
j: Job
SCHED: prefix_map sched make_edf_at h2 h2 = Some j
SCHED_j: scheduled_at (edf_transform_prefix sched h2) j h2
h2 < job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2, t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t
LE_h1_h2: h1 <= h2
j: Job
SCHED: prefix_map sched make_edf_at h2 h2 = Some j
SCHED_j: scheduled_at (edf_transform_prefix sched h2) j h2

jobs_must_arrive_to_execute (edf_transform_prefix sched h2)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2, t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t
LE_h1_h2: h1 <= h2
j: Job
SCHED: prefix_map sched make_edf_at h2 h2 = Some j
SCHED_j: scheduled_at (edf_transform_prefix sched h2) j h2
h2 < job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2, t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t
LE_h1_h2: h1 <= h2
j: Job
SCHED: prefix_map sched make_edf_at h2 h2 = Some j
SCHED_j: scheduled_at (edf_transform_prefix sched h2) j h2

h2 < job_deadline j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
h1, h2, t: nat
LT_t_h1: t < h1
IHh2: h1 <= h2 -> edf_transform_prefix sched h1 t = edf_transform_prefix sched h2 t
LE_h1_h2: h1 <= h2
j: Job
SCHED: prefix_map sched make_edf_at h2 h2 = Some j
SCHED_j: scheduled_at (edf_transform_prefix sched h2) j h2

h2 < job_deadline j
try ( apply edf_prefix_scheduled_job_has_later_deadline with (sched0 := sched) (horizon := h2) => // ) || apply edf_prefix_scheduled_job_has_later_deadline with (sched := sched) (horizon := h2) => //. Qed. End EDFPrefixInclusion. (** In the following section, we finally establish properties of the overall EDF transformation[edf_transform]. *) Section EDFTransformFacts. (** For any given type of jobs... *) Context {Job : JobType} `{JobCost Job} `{JobDeadline Job} `{JobArrival Job}. (** ...consider an ideal uniprocessor schedule... *) Variable sched: schedule (ideal.processor_state Job). (** ...that is well-behaved... *) Hypothesis H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched. Hypothesis H_completed_jobs_dont_execute: completed_jobs_dont_execute sched. (** ...and in which no scheduled job misses a deadline. *) Hypothesis H_no_deadline_misses: all_deadlines_met sched. (** In the following, let [sched_edf] denote the EDF schedule obtained by transforming the given reference schedule. *) Let sched_edf := edf_transform sched. (** We begin with a simple lemma relating [sched_edf] to its definition that allows us to easily look at any finite prefix of the EDF-transformed scheduled. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job

forall h : instant, identical_prefix sched_edf (edf_transform_prefix sched h) h
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job

forall h : instant, identical_prefix sched_edf (edf_transform_prefix sched h) h
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
h: instant

identical_prefix sched_edf (edf_transform_prefix sched h) h
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
h: instant
t: nat
LT_h: t < h

edf_transform_prefix sched t.+1 t = edf_transform_prefix sched h t
now apply edf_prefix_inclusion. Qed. (** From this, we move on to the defining property of the transformation: the resulting schedule is actually an EDF schedule. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job

EDF_schedule sched_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job

EDF_schedule sched_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
t: instant

EDF_at sched_edf t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
t: instant
j: Job
SCHED_j: scheduled_at sched_edf j t
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at sched_edf j' t'
ARR_j': job_arrival j' <= t

job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
t: instant
j: Job
SCHED_j: scheduled_at sched_edf j t
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at sched_edf j' t'
ARR_j': job_arrival j' <= t
S:= edf_transform_prefix sched t'.+1: schedule (ideal.processor_state Job)

job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
t: instant
j: Job
SCHED_j: scheduled_at sched_edf j t
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at sched_edf j' t'
ARR_j': job_arrival j' <= t
S:= edf_transform_prefix sched t'.+1: schedule (ideal.processor_state Job)
EDF: EDF_at S t

job_deadline j <= job_deadline j'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
t: instant
j: Job
SCHED_j: scheduled_at sched_edf j t
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at sched_edf j' t'
ARR_j': job_arrival j' <= t
S:= edf_transform_prefix sched t'.+1: schedule (ideal.processor_state Job)
EDF: EDF_at S t

scheduled_at S j t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
t: instant
j: Job
SCHED_j: scheduled_at sched_edf j t
t': instant
j': Job
LE_t_t': t <= t'
SCHED_j': scheduled_at sched_edf j' t'
ARR_j': job_arrival j' <= t
S:= edf_transform_prefix sched t'.+1: schedule (ideal.processor_state Job)
EDF: EDF_at S t

identical_prefix sched_edf S t'.+1
now apply edf_finite_prefix. Qed. (** Next, we observe that completed jobs still don't execute in the resulting EDF schedule. This observation is needed to establish that the resulting EDF schedule is valid. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job

completed_jobs_dont_execute sched_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job

completed_jobs_dont_execute sched_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant

scheduled_at sched_edf j t -> service sched_edf j t < job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
S:= edf_transform_prefix sched t.+1: schedule (ideal.processor_state Job)

scheduled_at sched_edf j t -> service sched_edf j t < job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
S:= edf_transform_prefix sched t.+1: schedule (ideal.processor_state Job)

scheduled_at S j t -> service sched_edf j t < job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
S:= edf_transform_prefix sched t.+1: schedule (ideal.processor_state Job)

scheduled_at S j t -> service S j t < job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
S:= edf_transform_prefix sched t.+1: schedule (ideal.processor_state Job)
COMP: completed_jobs_dont_execute (edf_transform_prefix sched t.+1)

scheduled_at S j t -> service S j t < job_cost j
now apply COMP. Qed. (** Similarly, we observe that no job is scheduled prior to its arrival. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job

jobs_must_arrive_to_execute sched_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job

jobs_must_arrive_to_execute sched_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant

scheduled_at sched_edf j t -> has_arrived j t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant

scheduled_at (fun t : instant => edf_transform_prefix sched t.+1 t) j t -> has_arrived j t
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
ARR: jobs_must_arrive_to_execute (edf_transform_prefix sched t.+1)

scheduled_at (fun t : instant => edf_transform_prefix sched t.+1 t) j t -> has_arrived j t
now apply ARR. Qed. (** We next establish the second key property: in the transformed EDF schedule, no scheduled job misses a deadline. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job

all_deadlines_met sched_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job

all_deadlines_met sched_edf
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant

scheduled_at sched_edf j t -> job_meets_deadline sched_edf j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant

scheduled_at (edf_transform sched) j t -> completed_by (edf_transform sched) j (job_deadline j)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant

scheduled_at (edf_transform sched) j t -> completed_by (edf_transform sched) j t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant

scheduled_at (edf_transform sched) j t -> completed_by (edf_transform_prefix sched t_dl.+1) j t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t

scheduled_at (edf_transform sched) j t -> completed_by (edf_transform_prefix sched t_dl.+1) j t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t

scheduled_at (edf_transform_prefix sched t_dl.+1) j t -> completed_by (edf_transform_prefix sched t_dl.+1) j t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t
identical_prefix (edf_transform sched) (edf_transform_prefix sched t_dl.+1) t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t
t < t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t

scheduled_at (edf_transform_prefix sched t_dl.+1) j t -> completed_by (edf_transform_prefix sched t_dl.+1) j t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t
identical_prefix (edf_transform sched) (edf_transform_prefix sched t_dl.+1) t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t
t < t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t
DL: all_deadlines_met (edf_transform_prefix sched t_dl.+1)

scheduled_at (edf_transform_prefix sched t_dl.+1) j t -> completed_by (edf_transform_prefix sched t_dl.+1) j t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t
identical_prefix (edf_transform sched) (edf_transform_prefix sched t_dl.+1) t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t
t < t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t

identical_prefix (edf_transform sched) (edf_transform_prefix sched t_dl.+1) t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t
t < t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t

identical_prefix (edf_transform sched) (edf_transform_prefix sched t_dl.+1) t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t
t < t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t

t < t_dl
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
t_dl:= job_deadline j: instant
SCHED_AT: scheduled_at (edf_transform sched) j t

t < t_dl
try ( by apply edf_prefix_scheduled_job_has_later_deadline with (sched0 := sched) (horizon := t.+1) ) || by apply edf_prefix_scheduled_job_has_later_deadline with (sched := sched) (horizon := t.+1). Qed. (** We observe that no new jobs are introduced: any job scheduled in the EDF schedule were also present in the reference schedule. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job

forall (j : Job) (t : instant), scheduled_at sched_edf j t -> exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job

forall (j : Job) (t : instant), scheduled_at sched_edf j t -> exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant

scheduled_at sched_edf j t -> exists t' : instant, scheduled_at sched j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant

scheduled_at (edf_transform_prefix sched t.+1) j t -> exists t' : instant, scheduled_at sched j t'
by apply edf_prefix_job_scheduled. Qed. (** Conversely, we observe that no jobs are lost: any job scheduled in the reference schedule is also present in the EDF schedule. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job

forall (j : Job) (t : instant), scheduled_at sched j t -> exists t' : instant, scheduled_at sched_edf j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job

forall (j : Job) (t : instant), scheduled_at sched j t -> exists t' : instant, scheduled_at sched_edf j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
SCHED_j: scheduled_at sched j t

exists t' : instant, scheduled_at sched_edf j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
SCHED_j: scheduled_at sched j t
EX: exists t' : instant, scheduled_at (edf_transform_prefix sched (job_deadline j)) j t'

exists t' : instant, scheduled_at sched_edf j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
SCHED_j: scheduled_at sched j t
t': instant
SCHED': scheduled_at (edf_transform_prefix sched (job_deadline j)) j t'

exists t' : instant, scheduled_at sched_edf j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
SCHED_j: scheduled_at sched j t
t': instant
SCHED': scheduled_at (edf_transform_prefix sched (job_deadline j)) j t'

scheduled_at sched_edf j t'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
SCHED_j: scheduled_at sched j t
t': instant
SCHED': scheduled_at (edf_transform_prefix sched (job_deadline j)) j t'

edf_transform_prefix sched t'.+1 t' == Some j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
j: Job
t: instant
SCHED_j: scheduled_at sched j t
t': instant
SCHED': scheduled_at (edf_transform_prefix sched (job_deadline j)) j t'

t' < job_deadline j
try ( now apply edf_prefix_scheduled_job_has_later_deadline with (sched0 := sched) (horizon := job_deadline j) ) || now apply edf_prefix_scheduled_job_has_later_deadline with (sched := sched) (horizon := job_deadline j). Qed. (** Next, we note that [edf_transform] maintains the property that all jobs stem from a given arrival sequence. *) Section ArrivalSequence. (** For any arrival sequence,... *) Variable arr_seq: arrival_sequence Job. (** ...if all jobs in the original schedule come from the arrival sequence,... *) Hypothesis H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq. (** ...then all jobs in the transformed EDF schedule still come from the same arrival sequence. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq

jobs_come_from_arrival_sequence sched_edf arr_seq
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq

jobs_come_from_arrival_sequence sched_edf arr_seq
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq

jobs_come_from_arrival_sequence (fun t : instant => edf_transform_prefix sched t.+1 t) arr_seq
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq
j: Job
t: instant

scheduled_at (fun t : instant => edf_transform_prefix sched t.+1 t) j t -> arrives_in arr_seq j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
sched: schedule (ideal.processor_state Job)
H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute: completed_jobs_dont_execute sched
H_no_deadline_misses: all_deadlines_met sched
sched_edf:= edf_transform sched: instant -> ideal.processor_state Job
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq
j: Job
t: instant

scheduled_at (edf_transform_prefix sched t.+1) j t -> arrives_in arr_seq j
now apply (edf_prefix_jobs_come_from_arrival_sequence sched t.+1 arr_seq H_from_arr_seq). Qed. End ArrivalSequence. End EDFTransformFacts. (** Finally, we state the theorems that jointly make up the EDF optimality claim. *) Section Optimality. (** We assume the classic (i.e., Liu & Layland) model of readiness without jitter or self-suspensions, wherein pending jobs are always ready. *) #[local] Existing Instance basic_ready_instance. (** For any given type of jobs... *) Context {Job : JobType} `{JobCost Job} `{JobDeadline Job} `{JobArrival Job}. (** ... consider an arbitrary valid job arrival sequence ... *) Variable arr_seq: arrival_sequence Job. Hypothesis H_arr_seq_valid: valid_arrival_sequence arr_seq. (** ... and an ideal uniprocessor schedule... *) Variable sched: schedule (ideal.processor_state Job). (** ... that corresponds to the given arrival sequence. *) Hypothesis H_sched_valid: valid_schedule sched arr_seq. (** In the following, let [equivalent_edf_schedule] denote the schedule that results from the EDF transformation. *) Let equivalent_edf_schedule := edf_transform sched. Section AllDeadlinesMet. (** Suppose no job scheduled in the given reference schedule misses a deadline. *) Hypothesis H_no_deadline_misses: all_deadlines_met sched. (** Then the resulting EDF schedule is a valid schedule for the given arrival sequence... *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses: all_deadlines_met sched

valid_schedule equivalent_edf_schedule arr_seq
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses: all_deadlines_met sched

valid_schedule equivalent_edf_schedule arr_seq
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses: all_deadlines_met sched

jobs_must_be_ready_to_execute equivalent_edf_schedule
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses: all_deadlines_met sched

jobs_must_arrive_to_execute equivalent_edf_schedule
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses: all_deadlines_met sched
completed_jobs_dont_execute equivalent_edf_schedule
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses: all_deadlines_met sched

jobs_must_arrive_to_execute equivalent_edf_schedule
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses: all_deadlines_met sched
completed_jobs_dont_execute equivalent_edf_schedule
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses: all_deadlines_met sched

completed_jobs_dont_execute equivalent_edf_schedule
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses: all_deadlines_met sched

completed_jobs_dont_execute equivalent_edf_schedule
by apply edf_transform_completed_jobs_dont_execute; rt_eauto. Qed. (** ...and no scheduled job misses its deadline. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses: all_deadlines_met sched

all_deadlines_met equivalent_edf_schedule
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses: all_deadlines_met sched

all_deadlines_met equivalent_edf_schedule
by apply edf_transform_deadlines_met; rt_eauto. Qed. End AllDeadlinesMet. (** Next, we strengthen the above "no deadline misses" claim by relating it not just to all scheduled jobs, but to all jobs in the given arrival sequence. *) Section AllDeadlinesOfArrivalsMet. (** Suppose no job that's part of the arrival sequence misses a deadline in the given reference schedule. *) Hypothesis H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched. (** Then no job that's part of the arrival sequence misses a deadline in the EDF schedule, either. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched

all_deadlines_of_arrivals_met arr_seq equivalent_edf_schedule
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched

all_deadlines_of_arrivals_met arr_seq equivalent_edf_schedule
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j

job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched

job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched

job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched

job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: (job_cost j == 0) = true

job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: (job_cost j == 0) = false
job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: (job_cost j == 0) = true

job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: (job_cost j == 0) = false
job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: job_cost j = 0

job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: (job_cost j == 0) = false
job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: job_cost j = 0

0 <= service equivalent_edf_schedule j (job_deadline j)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: (job_cost j == 0) = false
job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: (job_cost j == 0) = false

job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: (job_cost j == 0) = false

job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: (job_cost j == 0) = false
NONZERO: 0 < job_cost j

job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: (job_cost j == 0) = false
NONZERO: 0 < job_cost j

job_meets_deadline sched j -> job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: (job_cost j == 0) = false
NONZERO: 0 < job_cost j
COMP_j: completed_by sched j (job_deadline j)

job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: (job_cost j == 0) = false
NONZERO: 0 < job_cost j
COMP_j: completed_by sched j (job_deadline j)
t': nat
SCHED': scheduled_at sched j t'

job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: (job_cost j == 0) = false
NONZERO: 0 < job_cost j
COMP_j: completed_by sched j (job_deadline j)
t': nat
SCHED': scheduled_at sched j t'
NO_MISSES: all_deadlines_met sched

job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: (job_cost j == 0) = false
NONZERO: 0 < job_cost j
COMP_j: completed_by sched j (job_deadline j)
t': nat
SCHED': scheduled_at sched j t'
NO_MISSES: all_deadlines_met sched
t'': instant
SCHED'': scheduled_at (edf_transform sched) j t''

job_meets_deadline equivalent_edf_schedule j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
arr_seq: arrival_sequence Job
H_arr_seq_valid: valid_arrival_sequence arr_seq
sched: schedule (ideal.processor_state Job)
H_sched_valid: valid_schedule sched arr_seq
equivalent_edf_schedule:= edf_transform sched: instant -> ideal.processor_state Job
H_no_deadline_misses_of_arrivals: all_deadlines_of_arrivals_met arr_seq sched
j: Job
ARR_j: arrives_in arr_seq j
COME: jobs_come_from_arrival_sequence sched arr_seq
READY: jobs_must_be_ready_to_execute sched
ARR: jobs_must_arrive_to_execute sched
COMP: completed_jobs_dont_execute sched
COST: (job_cost j == 0) = false
NONZERO: 0 < job_cost j
COMP_j: completed_by sched j (job_deadline j)
t': nat
SCHED': scheduled_at sched j t'
NO_MISSES: all_deadlines_met sched
t'': instant
SCHED'': scheduled_at (edf_transform sched) j t''
DL_MET: all_deadlines_met equivalent_edf_schedule

job_meets_deadline equivalent_edf_schedule j
by apply: (DL_MET j t'' SCHED''). Qed. End AllDeadlinesOfArrivalsMet. End Optimality.