Library prosa.analysis.facts.transform.edf_opt

From mathcomp Require Import ssrnat ssrbool fintype.
Require Export prosa.model.schedule.edf.
Require Export prosa.analysis.definitions.schedulability.
Require Export prosa.analysis.transform.edf_trans.
Require Export prosa.analysis.facts.transform.swaps.
Require Export prosa.analysis.facts.readiness.basic.

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.
Throughout this file, we assume ideal uniprocessor schedules.
Require Import prosa.model.processor.ideal.
Throughout this file, we assume the basic (i.e., Liu & Layland) readiness model.
Require Import prosa.model.readiness.basic.

We start by analyzing the helper function find_swap_candidate, which is a problem-specific wrapper around search_arg.
Section FindSwapCandidateFacts.

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.
  Lemma t1_relevant: relevant_pstate t1 (sched t1).
  Proof.
    move: H_not_idle. rewrite scheduled_at_def ⇒ /eqP →.
    rewrite /relevant_pstate -/(has_arrived j1 t1).
    move: (H_jobs_must_arrive_to_execute j1 t1) ⇒ SCHED_ARR.
      by apply SCHED_ARR.
  Qed.

Since t1 is relevant, we conclude that a search for a relevant state succeeds (if nothing else, it finds t1).
  Lemma fsc_search_successful:
    ∃ t, search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some t.
  Proof.
    apply search_arg_not_none.
    ∃ t1. split.
    - by apply /andP; split.
    - 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.
  Corollary fsc_search_result:
    search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some (find_swap_candidate sched t1 j1).
  Proof.
    move: fsc_search_successful ⇒ [t FOUND].
    by 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.
  Lemma fsc_not_idle:
    ∃ j', (scheduled_at sched j' (find_swap_candidate sched t1 j1))
               ∧ job_arrival j' ≤ t1.
  Proof.
    move: fsc_search_successful ⇒ [t FOUND].
    move: (search_arg_pred _ _ _ _ _ _ FOUND).
    rewrite /relevant_pstate.
    destruct (sched t) as [j'|] eqn:SCHED_t ⇒ // ARR_j'.
    ∃ j'. split ⇒ //.
    rewrite /find_swap_candidate FOUND.
    rewrite scheduled_at_def //.
      by 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.
  Corollary fsc_found_job_arrival:
    ∀ j2,
      scheduled_at sched j2 (find_swap_candidate sched t1 j1) →
      job_arrival j2 ≤ t1.
  Proof.
    move⇒ j2 SCHED_j2.
    move: fsc_not_idle ⇒ [j' [SCHED_j' ARR]].
    by 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.
  Lemma fsc_range:
    t1 ≤ find_swap_candidate sched t1 j1 < job_deadline j1.
  Proof. move: fsc_search_result. 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.
  Corollary fsc_range1:
    t1 ≤ find_swap_candidate sched t1 j1.
  Proof. 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.
  Lemma fsc_found_job_deadline:
    ∀ j2,
      scheduled_at sched j2 (find_swap_candidate sched t1 j1) →
      ∀ j t,
        t1 ≤ t < job_deadline j1 →
        scheduled_at sched j t →
        job_arrival j ≤ t1 →
        job_deadline j2 ≤ job_deadline j.
  Proof.
    move⇒ j2 SCHED_j2 j t /andP [t1_le_t t_lt_dl1] SCHED_j ARR_j.
    have TOTAL: total earlier_deadline
      by rewrite /earlier_deadline ⇒ s1 s2; apply leq_total.
    have TRANS: transitive earlier_deadline
      by rewrite /earlier_deadline ⇒ s1 s2 s3; apply leq_trans.
    have REFL: reflexive earlier_deadline
      by rewrite /earlier_deadline ⇒ s; apply leqnn.
    move: SCHED_j SCHED_j2. rewrite !scheduled_at_def ⇒ /eqP SCHED_j /eqP SCHED_j2.
    have ED: earlier_deadline (sched (find_swap_candidate sched t1 j1)) (sched t).
    {
      move: (search_arg_extremum _ _ _ REFL TRANS TOTAL _ _ _ fsc_search_result) ⇒ MIN.
      apply (MIN t).
      - by apply /andP; split.
      - by rewrite /relevant_pstate SCHED_j.
    }
    by 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.
  Corollary fsc_no_later_deadline:
    ∀ j2,
      scheduled_at sched j2 (find_swap_candidate sched t1 j1) →
      job_deadline j2 ≤ job_deadline j1.
  Proof.
    move⇒ j2 SCHED_j2.
    apply fsc_found_job_deadline with (t := t1) ⇒ //.
    - by apply /andP; split.
    - 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.
  Fact scheduled_job_in_sched_has_later_deadline:
    ∀ j t,
      scheduled_at sched j t →
      job_deadline j > t.
  Proof.
    move⇒ j t SCHED.
    apply (scheduled_at_implies_later_deadline sched) ⇒ //.
    - by apply ideal_proc_model_ensures_ideal_progress.
    - by apply (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.
  Lemma mea_completed_jobs:
    completed_jobs_dont_execute sched'.
  Proof.
    have IDEAL := @ideal_proc_model_ensures_ideal_progress Job.
    have UNIT := @ideal_proc_model_provides_unit_service Job.
    rewrite /sched' /make_edf_at.
    destruct (sched t_edf) as [j_orig|] eqn:SCHED; last by done.
    have SCHED': scheduled_at sched j_orig t_edf
      by rewrite scheduled_at_def; apply /eqP.
    apply swapped_completed_jobs_dont_execute ⇒ //.
    apply fsc_range1 ⇒ //.
    by 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.
  Lemma mea_no_deadline_misses:
    all_deadlines_met sched'.
  Proof.
    move⇒ j t SCHED.
    rewrite /sched' /make_edf_at.
    destruct (sched t_edf) as [j_orig|] eqn:SCHED_orig; last first.
    {
      apply (H_no_deadline_misses _ t).
      move: SCHED.
      by rewrite /sched' /make_edf_at SCHED_orig.
    }
    {
      have SCHED': scheduled_at sched j_orig t_edf
        by rewrite scheduled_at_def; apply /eqP.
      move: (scheduled_job_in_sched_has_later_deadline _ _ SCHED') ⇒ DL_orig.
      apply edf_swap_no_deadline_misses_introduced ⇒ //.
      - by apply ideal_proc_model_ensures_ideal_progress.
      - by apply fsc_range1 ⇒ //.
      - move⇒ j1 j2 SCHED_j1 SCHED_j2.
        apply: (fsc_found_job_deadline sched _ j_orig t_edf _ _ _ _ _ t_edf) ⇒ //.
        + by apply /andP; split.
        + by apply H_jobs_must_arrive_to_execute.
      - move⇒ j1 SCHED_j1.
        move: (fsc_not_idle sched H_jobs_must_arrive_to_execute j_orig t_edf SCHED' DL_orig) ⇒ [j' [SCHED_j' ARR_j']].
        ∃ j'. split ⇒ //.
        by apply scheduled_job_in_sched_has_later_deadline.
      - have EX: (∃ t', scheduled_at sched j t').
        {
          apply swap_job_scheduled with (t1 := t_edf) (t2 := find_swap_candidate sched t_edf j_orig) (t0 := t).
          by move: SCHED; rewrite /sched' /make_edf_at SCHED_orig.
        }
        move: EX ⇒ [t' SCHED_t'].
        by 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.
  Corollary mea_scheduled_job_has_later_deadline:
    ∀ j t,
      scheduled_at sched' j t →
      job_deadline j > t.
  Proof.
    move⇒ j t SCHED.
    apply (scheduled_at_implies_later_deadline sched') ⇒ //.
    - by apply mea_completed_jobs.
    - by apply ideal_proc_model_ensures_ideal_progress.
    - 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.
    Fact mea_guarantee_dl_orig: t_edf < job_deadline j_orig.
    Proof. 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.
    Fact mea_guarantee_fsc_is_j_edf: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf.
    Proof.
      move: (H_sched_orig). rewrite scheduled_at_def ⇒ /eqP SCHED.
      move: (H_sched_edf). rewrite /sched' /make_edf_at /swapped /replace_at {1}SCHED //=.
      rewrite scheduled_at_def.
      destruct (find_swap_candidate sched t_edf j_orig == t_edf) eqn:FSC.
      - by move: FSC ⇒ /eqP → /eqP.
      - by rewrite ifT // ⇒ /eqP.
    Qed.

Third, the deadline of j_edf is no later than the deadline of j_orig.
    Fact mea_guarantee_deadlines: job_deadline j_edf ≤ job_deadline j_orig.
    Proof.
      apply: (fsc_no_later_deadline sched _ _ t_edf) ⇒ //.
      - by exact mea_guarantee_dl_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.
    Lemma mea_guarantee_case_t'_past_deadline:
      job_deadline j_orig ≤ t' →
      job_deadline j_edf ≤ job_deadline j'.
    Proof.
      move: (mea_scheduled_job_has_later_deadline j' t' H_sched') ⇒ DL_j' BOUND_t'.
      apply leq_trans with (n := job_deadline j_orig) ⇒ // ;
        first by exact mea_guarantee_deadlines.
      apply leq_trans with (n := t') ⇒ //.
      by apply ltnW.
    Qed.

Next, we consider the more difficult case, where t' is before the deadline of j_orig.
    Lemma mea_guarantee_case_t'_before_deadline:
      t' < job_deadline j_orig →
      job_deadline j_edf ≤ job_deadline j'.
    Proof.
      move: (H_sched_orig). rewrite scheduled_at_def ⇒ /eqP SCHED BOUND_t'.
      move: (mea_guarantee_fsc_is_j_edf) ⇒ FSC.
      have EX: (∃ x, scheduled_at sched j' x ∧ t_edf ≤ x < job_deadline j_orig).
      {
        case: (boolP(t_edf == t')) ⇒ [/eqP EQ| /eqP NEQ].
        - ∃ (find_swap_candidate sched t_edf j_orig).
          split; last by apply fsc_range ⇒ //; exact mea_guarantee_dl_orig.
          subst. rewrite -(ideal_proc_model_is_a_uniprocessor_model _ _ _ _ H_sched_edf H_sched').
          by rewrite scheduled_at_def FSC //=.
        - case: (boolP(find_swap_candidate sched t_edf j_orig == t')) ⇒ [/eqP EQ' | /eqP NEQ'].
          + ∃ t_edf.
            split; last by apply /andP; split ⇒ //; exact mea_guarantee_dl_orig.
            rewrite -(swap_job_scheduled_t2 _ _ (find_swap_candidate sched t_edf j_orig) _).
            move: H_sched'. rewrite /sched' /make_edf_at SCHED.
            by rewrite EQ'.
          + move: NEQ NEQ' ⇒ /eqP NEQ /eqP NEQ'. ∃ t'.
            split; last by apply /andP; split.
            rewrite -(swap_job_scheduled_other_times _ t_edf (find_swap_candidate sched t_edf j_orig)) //.
            move: H_sched'.
            by rewrite /sched' /make_edf_at SCHED.
      }
      move: EX ⇒ [t'' [SCHED'' RANGE]].
      apply: (fsc_found_job_deadline sched _ j_orig t_edf _ _ _ _ _ t'') ⇒ // ;
        first by exact mea_guarantee_dl_orig.
      by 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.
  Lemma make_edf_at_guarantee:
    EDF_at sched' t_edf.
  Proof.
    move⇒ j_edf H_sched_edf t' j' t_edf_le_t' H_sched' H_arrival_j'.
    destruct (sched t_edf) as [j_orig|] eqn:SCHED;
      last by move: (H_sched_edf); rewrite /sched' /make_edf_at scheduled_at_def ⇒ /eqP; rewrite !SCHED.
    have H_sched: scheduled_at sched j_orig t_edf
      by rewrite scheduled_at_def; apply /eqP.
    case: (boolP (t' < job_deadline j_orig)).
    - by apply mea_guarantee_case_t'_before_deadline.
    - rewrite -leqNgt ⇒ BOUND_t'.
      by 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.
  Lemma mea_jobs_must_arrive:
    jobs_must_arrive_to_execute sched'.
  Proof.
    move⇒ j t.
    rewrite /has_arrived /sched' /make_edf_at.
    destruct (sched t_edf) as [j_orig|] eqn:SCHED_orig;
      last by move⇒ SCHED; by apply H_jobs_must_arrive_to_execute.
    have SCHED': scheduled_at sched j_orig t_edf
      by rewrite scheduled_at_def; apply /eqP.
    move: (scheduled_job_in_sched_has_later_deadline j_orig t_edf SCHED') ⇒ DL_orig.
    rewrite scheduled_at_def /swapped /replace_at.
    case: (boolP((find_swap_candidate sched t_edf j_orig) == t)) ⇒ [/eqP EQ| /eqP NEQ].
    - rewrite SCHED_orig ⇒ /eqP j_is_orig.
      injection j_is_orig ⇒ <-.
      apply leq_trans with (n := t_edf).
      + by apply H_jobs_must_arrive_to_execute.
      + by rewrite -EQ; apply fsc_range1.
    - case (boolP(t_edf == t)) ⇒ [/eqP EQ'| /eqP NEQ'].
      + move⇒ SCHED_j.
        have ARR_j: job_arrival j ≤ t_edf by apply fsc_found_job_arrival with (sched0 := sched) (j1 := j_orig) ⇒ //; rewrite scheduled_at_def.
        by rewrite -EQ'.
      + move⇒ SCHED_j.
        apply H_jobs_must_arrive_to_execute.
        by rewrite scheduled_at_def /scheduled_in /pstate_instance.
  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.
  Lemma mea_job_scheduled:
    ∀ j t,
      scheduled_at sched' j t →
      ∃ t', scheduled_at sched j t'.
  Proof.
    rewrite /sched' /make_edf_at.
    move⇒ j t SCHED_j.
    destruct (sched t_edf) as [j_orig|] eqn:SCHED_orig; last by ∃ t.
    eapply swap_job_scheduled.
    by 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.
  Lemma mea_job_scheduled':
    ∀ j t,
      scheduled_at sched j t →
      ∃ t', scheduled_at sched' j t'.
  Proof.
    move⇒ j t SCHED_j.
    rewrite /sched' /make_edf_at.
    destruct (sched t_edf) as [j_orig|] eqn:SCHED_orig;
      last by ∃ t.
    eapply swap_job_scheduled_original.
    by 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.
    Lemma mea_jobs_come_from_arrival_sequence:
      jobs_come_from_arrival_sequence sched' arr_seq.
    Proof.
      rewrite /sched' /make_edf_at.
      destruct (sched t_edf) as [j_orig|] eqn:SCHED_orig;
        last by done.
      by 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: ∀ 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.
  Lemma mea_EDF_widen:
    ∀ t, t ≤ t_edf → EDF_at sched' t.
  Proof.
    move⇒ t.
    rewrite leq_eqVlt ⇒ /orP [/eqP EQ|LT] ;
      first by rewrite EQ; apply make_edf_at_guarantee.
    rewrite /sched' /make_edf_at.
    destruct (sched t_edf) as [j_orig|] eqn:SCHED_edf; last by apply H_EDF_prefix.
    move⇒ j SCHED_j t' j' LE_t_t' SCHED_j' ARR_j'.
    have SCHED_edf': scheduled_at sched j_orig t_edf
      by rewrite scheduled_at_def; apply /eqP.
    have LT_t_fsc: t < find_swap_candidate sched t_edf j_orig.
    {
      apply ltn_leq_trans with (n := t_edf) ⇒ //.
      apply fsc_range1 ⇒ //.
      by apply scheduled_job_in_sched_has_later_deadline.
    }
    move: SCHED_j.
    have ->: scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t = scheduled_at sched j t
      by apply swap_job_scheduled_other_times; [move: LT | move: LT_t_fsc]; rewrite ltn_neqAle ⇒ /andP [NEQ _]; rewrite eq_sym.
    move ⇒ SCHED_j.
    move: (H_EDF_prefix t LT). rewrite /EDF_at ⇒ EDF.
    move: (SCHED_j').
    move: (swap_job_scheduled_cases _ _ _ _ _ SCHED_j') ⇒ [->|[[EQ ->]|[EQ ->]]] SCHED_j'_orig.
    - by apply EDF with (t' := t').
    - by apply EDF with (t' := (find_swap_candidate sched t_edf j_orig)) ⇒ //; apply ltnW.
    - 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.
  Lemma edf_prefix_well_formedness:
    completed_jobs_dont_execute sched'
    ∧
    jobs_must_arrive_to_execute sched'
    ∧
    all_deadlines_met sched'.
  Proof.
    rewrite /sched' /edf_transform_prefix.
    apply prefix_map_property_invariance; last by split.
    move⇒ sched'' t [COMP [ARR DL_MET]].
    split; last split.
    - apply mea_completed_jobs ⇒ //.
    - apply mea_jobs_must_arrive ⇒ //.
    - apply mea_no_deadline_misses ⇒ //.
  Qed.

Because it is needed frequently, we extract the second clause of the above conjunction as a corollary.
  Corollary edf_prefix_jobs_must_arrive:
    jobs_must_arrive_to_execute sched'.
  Proof. 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.
  Corollary edf_prefix_scheduled_job_has_later_deadline:
    ∀ j t,
      scheduled_at sched' j t →
      job_deadline j > t.
  Proof.
    move⇒ j t SCHED.
    move: edf_prefix_well_formedness ⇒ [COMP [ARR DL_MET]].
    apply (scheduled_at_implies_later_deadline sched') ⇒ //.
    - by apply ideal_proc_model_ensures_ideal_progress.
    - by apply (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.
  Lemma edf_prefix_job_scheduled:
    ∀ j t,
      scheduled_at sched' j t →
      ∃ t', scheduled_at sched j t'.
  Proof.
    rewrite /sched' /edf_transform_prefix.
    move⇒ j.
    apply prefix_map_property_invariance;
      last by move⇒ t SCHED; ∃ t.
    move⇒ sched'' t'' EX t''' SCHED_mea.
    move: (mea_job_scheduled _ _ _ _ SCHED_mea) ⇒ [t'''' SCHED''''].
      by 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.
  Lemma edf_prefix_job_scheduled':
    ∀ j t,
      scheduled_at sched j t →
      ∃ t', scheduled_at sched' j t'.
  Proof.
    move⇒ j t SCHED_j.
    rewrite /sched' /edf_transform_prefix.
    apply prefix_map_property_invariance; last by ∃ t.
    move⇒ schedX tx [t' SCHEDX_j].
    eapply mea_job_scheduled'.
    by 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.
    Lemma edf_prefix_jobs_come_from_arrival_sequence:
      jobs_come_from_arrival_sequence sched' arr_seq.
    Proof.
      rewrite /sched' /edf_transform_prefix.
      apply prefix_map_property_invariance; last by done.
      move ⇒ schedX t ARR.
      by 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.
  Lemma edf_prefix_guarantee:
    ∀ t,
      t < horizon →
      EDF_at sched' t.
  Proof.
    move⇒ t IN_PREFIX.
    rewrite /sched' /edf_transform_prefix.
    apply prefix_map_pointwise_property
      with (Q := EDF_at)
           (P := (fun sched ⇒ completed_jobs_dont_execute sched
                               ∧
                               jobs_must_arrive_to_execute sched
                               ∧
                               all_deadlines_met sched))=> //.
    - move⇒ schedX t_ref [COMP [ARR DL]].
      split; last split.
      + by apply mea_completed_jobs ⇒ //.
      + by apply mea_jobs_must_arrive ⇒ //.
      + by apply mea_no_deadline_misses ⇒ //.
    - move⇒ schedX t_ref [COMP [ARR DL]].
      by 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.

  Lemma edf_prefix_inclusion:
    ∀ h1 h2,
      h1 ≤ h2 →
      ∀ t,
        t < h1 →
        (edf_transform_prefix sched h1) t = (edf_transform_prefix sched h2) t.
  Proof.
    move⇒ h1 h2 LE_h1_h2 t LT_t_h1.
    induction h2; first by move: (ltn_leq_trans LT_t_h1 LE_h1_h2).
    move: LE_h1_h2. rewrite leq_eqVlt ⇒ /orP [/eqP ->|LT]; first by done.
    move: LT. rewrite ltnS ⇒ LE_h1_h2.
    rewrite [RHS]/edf_transform_prefix /prefix_map -/prefix_map IHh2 //.
    rewrite {1}/make_edf_at.
    destruct (prefix_map sched make_edf_at h2 h2) as [j|] eqn:SCHED; last by done.
    rewrite -(swap_before_invariant _ h2 (find_swap_candidate (edf_transform_prefix sched h2) h2 j)) // ;
      last by apply ltn_leq_trans with (n := h1).
    have SCHED_j: scheduled_at (edf_transform_prefix sched h2) j h2
      by rewrite scheduled_at_def /edf_transform_prefix; apply /eqP.
    apply fsc_range1 ⇒ //.
    - by apply edf_prefix_jobs_must_arrive.
    - apply edf_prefix_scheduled_job_has_later_deadline with (sched0 := sched) (horizon := h2) ⇒ //.
  Qed.

End EDFPrefixInclusion.

In the following section, we finally establish properties of the overall EDF transformationedf_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 the first key property: the resulting schedule is actually an EDF schedule.
  Theorem edf_transform_ensures_edf:
    EDF_schedule sched_edf.
  Proof.
    rewrite /EDF_schedule /sched_edf /edf_transform ⇒ t.
    rewrite /EDF_at //= ⇒ j SCHED_j t' j' LE_t_t' SCHED_j' ARR_j'.
    move: SCHED_j.
    rewrite scheduled_at_def.
    have ->: edf_transform_prefix sched t.+1 t = edf_transform_prefix sched t'.+1 t
      by apply edf_prefix_inclusion.
    rewrite -scheduled_at_def.
    move⇒ SCHED_j.
    move: (edf_prefix_guarantee sched H_jobs_must_arrive_to_execute H_completed_jobs_dont_execute H_no_deadline_misses t'.+1 t) ⇒ EDF.
    feed EDF; first by done.
    by apply (EDF j SCHED_j t' j' LE_t_t').
  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.
  Lemma edf_transform_completed_jobs_dont_execute:
    completed_jobs_dont_execute sched_edf.
  Proof.
    move⇒ j t.
    rewrite /sched_edf /edf_transform /service /service_during.
    have ->: \sum_(0 ≤ t0 < t) service_at (fun t1 : instant ⇒ edf_transform_prefix sched t1.+1 t1) j t0 =
             \sum_(0 ≤ t0 < t) service_at (edf_transform_prefix sched t.+1) j t0.
    {
      apply eq_big_nat ⇒ t' /andP [_ LT_t].
      rewrite /service_at.
      rewrite (edf_prefix_inclusion _ _ _ _ t'.+1 t.+1) ⇒ //.
      by apply ltn_trans with (n := t).
    }
    set S := (edf_transform_prefix sched t.+1).
    rewrite -/(service_during S j 0 t) -/(service S j t) {}/S.
    move: (edf_prefix_well_formedness sched H_jobs_must_arrive_to_execute H_completed_jobs_dont_execute H_no_deadline_misses t.+1) ⇒ [COMP _].
    by apply COMP.
  Qed.

Similarly, we observe that no job is scheduled prior to its arrival.
  Lemma edf_transform_jobs_must_arrive:
    jobs_must_arrive_to_execute sched_edf.
  Proof.
    move⇒ j t.
    rewrite /sched_edf /edf_transform.
    move: (edf_prefix_well_formedness sched H_jobs_must_arrive_to_execute H_completed_jobs_dont_execute H_no_deadline_misses t.+1) ⇒ [_ [ARR _]].
    by apply ARR.
  Qed.

We next establish the second key property: in the transformed EDF schedule, no scheduled job misses a deadline.
  Theorem edf_transform_deadlines_met:
    all_deadlines_met sched_edf.
  Proof.
    move⇒ j t.
    rewrite /sched_edf /edf_transform /job_meets_deadline /completed_by /service /service_during ⇒ SCHED.
    have LT_t_dl: t < job_deadline j
      by apply edf_prefix_scheduled_job_has_later_deadline with (sched0 := sched) (horizon := t.+1).
    set t_dl := (job_deadline j).
    have ->: \sum_(0 ≤ t0 < t_dl) service_at (fun t1 : instant ⇒ edf_transform_prefix sched t1.+1 t1) j t0 =
             \sum_(0 ≤ t0 < t_dl) service_at (edf_transform_prefix sched t_dl) j t0.
    {
      apply eq_big_nat ⇒ t' /andP [_ LT_t].
      rewrite /service_at.
      by rewrite (edf_prefix_inclusion _ _ _ _ t'.+1 t_dl).
    }
    move: SCHED. rewrite scheduled_at_def (edf_prefix_inclusion _ _ _ _ t.+1 t_dl) ⇒ // SCHED.
    move: (edf_prefix_well_formedness sched H_jobs_must_arrive_to_execute H_completed_jobs_dont_execute H_no_deadline_misses t_dl) ⇒ [_ [_ DL]]. move: DL.
    rewrite /all_deadlines_met /job_meets_deadline /completed_by /service /service_during ⇒ DL.
    apply: (DL j t)=>//=.
      by rewrite scheduled_at_def.
  Qed.

We observe that no new jobs are introduced: any job scheduled in the EDF schedule were also present in the reference schedule.
  Lemma edf_transform_job_scheduled:
    ∀ j t, scheduled_at sched_edf j t → ∃ t', scheduled_at sched j t'.
  Proof.
    move⇒ j t.
    rewrite /sched_edf /edf_transform {1}scheduled_at_def -scheduled_at_def.
    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.
  Lemma edf_transform_job_scheduled':
    ∀ j t, scheduled_at sched j t → ∃ t', scheduled_at sched_edf j t'.
  Proof.
    move⇒ j t SCHED_j.
    have EX: ∃ t', scheduled_at (edf_transform_prefix sched (job_deadline j)) j t'
      by apply edf_prefix_job_scheduled' with (t0 := t).
    move: EX ⇒ [t' SCHED'].
    ∃ t'.
    rewrite /sched_edf /edf_transform scheduled_at_def.
    rewrite (edf_prefix_inclusion _ _ _ _ t'.+1 (job_deadline j)) -?scheduled_at_def⇒ //.
    by apply edf_prefix_scheduled_job_has_later_deadline with (sched0 := 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.
    Lemma edf_transform_jobs_come_from_arrival_sequence:
      jobs_come_from_arrival_sequence sched_edf arr_seq.
    Proof.
      rewrite /sched_edf /edf_transform.
      move⇒ j t.
      rewrite scheduled_at_def - scheduled_at_def.
      by 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.
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...
    Theorem edf_schedule_is_valid:
      valid_schedule equivalent_edf_schedule arr_seq.
    Proof.
      move: H_sched_valid ⇒ [COME READY].
      rewrite /valid_schedule; split;
        first by apply edf_transform_jobs_come_from_arrival_sequence.
      have ARR := jobs_must_arrive_to_be_ready sched READY.
      have COMP := completed_jobs_are_not_ready sched READY.
      apply basic_readiness_compliance.
      - by apply edf_transform_jobs_must_arrive.
      - by apply edf_transform_completed_jobs_dont_execute.
    Qed.

...and no scheduled job misses its deadline.
    Theorem edf_schedule_meets_all_deadlines:
      all_deadlines_met equivalent_edf_schedule.
    Proof.
      move: H_sched_valid ⇒ [COME READY].
      have ARR := jobs_must_arrive_to_be_ready sched READY.
      have COMP := completed_jobs_are_not_ready sched READY.
      by apply edf_transform_deadlines_met.
    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.
    Theorem edf_schedule_meets_all_deadlines_wrt_arrivals:
      all_deadlines_of_arrivals_met arr_seq equivalent_edf_schedule.
    Proof.
      move⇒ j ARR_j.
      move: H_sched_valid ⇒ [COME READY].
      have ARR := jobs_must_arrive_to_be_ready sched READY.
      have COMP := completed_jobs_are_not_ready sched READY.
      destruct (job_cost j == 0) eqn:COST.
      - move: COST ⇒ /eqP COST.
        rewrite /job_meets_deadline /completed_by COST.
        by apply leq0n.
      - move: (neq0_lt0n COST) ⇒ NONZERO.
        move: (H_no_deadline_misses_of_arrivals j ARR_j). rewrite {1}/job_meets_deadline ⇒ COMP_j.
        move: (completed_implies_scheduled_before sched j NONZERO ARR (job_deadline j) COMP_j) ⇒ [t' [_ SCHED']].
        move: (all_deadlines_met_in_valid_schedule arr_seq sched COME H_no_deadline_misses_of_arrivals) ⇒ NO_MISSES.
        move: (edf_transform_job_scheduled' sched ARR COMP NO_MISSES j t' SCHED') ⇒ [t'' SCHED''].
        move: (edf_schedule_meets_all_deadlines NO_MISSES) ⇒ DL_MET.
        by apply: (DL_MET j t'' SCHED'').
    Qed.

  End AllDeadlinesOfArrivalsMet.

End Optimality.