bertogna_fp_comp.v 30.1 KB
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Add LoadPath "../../" as rt.
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Require Import rt.util.Vbase rt.util.lemmas rt.util.divround.
Require Import rt.analysis.basic.bertogna_fp_theory.
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Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq fintype bigop div path.
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Module ResponseTimeIterationFP.

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  Import ResponseTimeAnalysisFP.
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  (* In this section, we define the algorithm of Bertogna and Cirinei's
     response-time analysis for FP scheduling. *)
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  Section Analysis.
    
    Context {sporadic_task: eqType}.
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    Variable task_cost: sporadic_task -> time.
    Variable task_period: sporadic_task -> time.
    Variable task_deadline: sporadic_task -> time.
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    (* As input for each iteration of the algorithm, we consider pairs
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       of tasks and computed response-time bounds. *)
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    Let task_with_response_time := (sporadic_task * time)%type.
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    Context {Job: eqType}.
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    Variable job_cost: Job -> time.
    Variable job_deadline: Job -> time.
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    Variable job_task: Job -> sporadic_task.

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    (* Consider a platform with num_cpus processors, ... *)
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    Variable num_cpus: nat.
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    (* ..., and priorities based on an FP policy. *)
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    Variable higher_priority: FP_policy sporadic_task.
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    (* Next we define the fixed-point iteration for computing
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       Bertogna's response-time bound of a task set. *)
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    (* First, given a sequence of pairs R_prev = <..., (tsk_hp, R_hp)> of
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       response-time bounds for the higher-priority tasks, we define an
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       iteration that computes the response-time bound of the current task:
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           R_tsk (0) = task_cost tsk
           R_tsk (step + 1) =  f (R step),
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       where f is the response-time recurrence, step is the number of iterations,
       and R_tsk (0) is the initial state. *)
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    Definition per_task_rta (tsk: sporadic_task)
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                            (R_prev: seq task_with_response_time) (step: nat) :=
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      iter step
        (fun t => task_cost tsk +
                  div_floor
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                    (total_interference_bound_fp task_cost task_period tsk
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                                                R_prev t higher_priority)
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                    num_cpus)
        (task_cost tsk).

    (* To ensure that the iteration converges, we will apply per_task_rta
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       a "sufficient" number of times: task_deadline tsk - task_cost tsk + 1.
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       This corresponds to the time complexity of the iteration. *)
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    Definition max_steps (tsk: sporadic_task) := task_deadline tsk - task_cost tsk + 1.
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    (* Next we compute the response-time bounds for the entire task set.
       Since high-priority tasks may not be schedulable, we allow the
       computation to fail.
       Thus, given the response-time bound of previous tasks, we either
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       (a) append the computed response-time bound (tsk, R) of the current task
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           to the list of pairs, or,
       (b) return None if the response-time analysis failed. *)
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    Definition fp_bound_of_task hp_pairs tsk :=
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      if hp_pairs is Some rt_bounds then
        let R := per_task_rta tsk rt_bounds (max_steps tsk) in
          if R <= task_deadline tsk then
            Some (rcons rt_bounds (tsk, R))
          else None
      else None.

    (* The response-time analysis for a given task set is defined
       as a left-fold (reduce) based on the function above.
       This either returns a list of task and response-time bounds, or None. *)
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    Definition fp_claimed_bounds (ts: taskset_of sporadic_task) :=
      foldl fp_bound_of_task (Some [::]) ts.
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    (* The schedulability test simply checks if we got a list of
       response-time bounds (i.e., if the computation did not fail). *)
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    Definition fp_schedulable (ts: taskset_of sporadic_task) :=
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      fp_claimed_bounds ts != None.
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    (* In the following section, we prove several helper lemmas about the
       list of response-time bounds. The results seem trivial, but must be proven
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       nonetheless since the list of response-time bounds is computed with
       a specific algorithm and there are no lemmas in the library for that. *)
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    Section SimpleLemmas.
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      (* First, we show that the first component of the computed list is the set of tasks. *)
      Lemma fp_claimed_bounds_unzip :
        forall ts hp_bounds, 
          fp_claimed_bounds ts = Some hp_bounds ->
          unzip1 hp_bounds = ts.
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      Proof.
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        unfold fp_claimed_bounds in *; intros ts.
        induction ts using last_ind; first by destruct hp_bounds.
        {
          intros hp_bounds SOME.
          destruct (lastP hp_bounds) as [| hp_bounds'].
          {
            rewrite -cats1 foldl_cat /= in SOME.
            unfold fp_bound_of_task at 1 in SOME; simpl in *; desf.
            by destruct l.
          }
          rewrite -cats1 foldl_cat /= in SOME.
          unfold fp_bound_of_task at 1 in SOME; simpl in *; desf.
          move: H0 => /eqP EQSEQ.
          rewrite eqseq_rcons in EQSEQ.
          move: EQSEQ => /andP [/eqP SUBST /eqP EQSEQ]; subst.
          unfold unzip1; rewrite map_rcons; f_equal.
          by apply IHts.
        }
      Qed.
      
      (* Next, we show that some properties of the analysis are preserved for the
         prefixes of the list: (a) the tasks do not change, (b) R <= deadline,
         (c) R is computed using the response-time equation, ... *) 
      Lemma fp_claimed_bounds_rcons :
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        forall ts' hp_bounds tsk1 tsk2 R,
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          (fp_claimed_bounds (rcons ts' tsk1) = Some (rcons hp_bounds (tsk2, R)) ->
           (fp_claimed_bounds ts' = Some hp_bounds /\
            tsk1 = tsk2 /\
            R = per_task_rta tsk1 hp_bounds (max_steps tsk1) /\
            R <= task_deadline tsk1)).
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      Proof.
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        intros ts hp_bounds tsk tsk' R.
        rewrite -cats1.
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        unfold fp_claimed_bounds in *.
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        rewrite foldl_cat /=.
        unfold fp_bound_of_task at 1; simpl; desf.
        intros EQ; inversion EQ; move: EQ H0 => _ /eqP EQ.
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        rewrite eqseq_rcons in EQ.
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        move: EQ => /andP [/eqP EQ /eqP RESP].
        by inversion RESP; repeat split; subst.
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      Qed.
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      (* ..., which implies that any prefix of the computation is the computation
         of the prefix. *)
      Lemma fp_claimed_bounds_take :
        forall ts hp_bounds i,
          fp_claimed_bounds ts = Some hp_bounds ->
          i <= size hp_bounds ->
          fp_claimed_bounds (take i ts) = Some (take i hp_bounds).
      Proof.                                                        
        intros ts hp_bounds i SOME LTi.
        have UNZIP := fp_claimed_bounds_unzip ts hp_bounds SOME.
        rewrite <- UNZIP in *.
        rewrite -[hp_bounds]take_size /unzip1 map_take in SOME.
        fold (unzip1 hp_bounds) in *; clear UNZIP.
        rewrite leq_eqVlt in LTi.
        move: LTi => /orP [/eqP EQ | LTi]; first by subst.
        remember (size hp_bounds) as len; apply eq_leq in Heqlen.
        induction len; first by rewrite ltn0 in LTi.
        {
          assert (TAKElen: fp_claimed_bounds (take len (unzip1 (hp_bounds))) =
                             Some (take len (hp_bounds))).
          {
            assert (exists p, p \in hp_bounds).
            {
              destruct hp_bounds; first by rewrite ltn0 in Heqlen.
              by exists t; rewrite in_cons eq_refl orTb.
            } destruct H as [[tsk R] _].
             rewrite (take_nth tsk) in SOME; last by rewrite size_map.
            rewrite (take_nth (tsk,R)) in SOME; last by done.
            destruct (nth (tsk, R) hp_bounds len) as [tsk_len R_len].
            by apply fp_claimed_bounds_rcons in SOME; des.
          }
          rewrite ltnS leq_eqVlt in LTi.
          move: LTi => /orP [/eqP EQ | LESS]; first by subst.
          apply ltnW in Heqlen.
          by specialize (IHlen Heqlen TAKElen LESS).
        }
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      Qed.
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      (* If the analysis suceeds, the computed response-time bounds are no larger
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         than the deadlines ... *)
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      Lemma fp_claimed_bounds_le_deadline :
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        forall ts' rt_bounds tsk R,
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          fp_claimed_bounds ts' = Some rt_bounds ->
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          (tsk, R) \in rt_bounds ->
          R <= task_deadline tsk.
      Proof.
        intros ts; induction ts as [| ts' tsk_lst] using last_ind.
        {
          intros rt_bounds tsk R SOME IN.
          by inversion SOME; subst; rewrite in_nil in IN.
        }
        {
          intros rt_bounds tsk_i R SOME IN.
          destruct (lastP rt_bounds) as [|rt_bounds (tsk_lst', R_lst)];
            first by rewrite in_nil in IN.
          rewrite mem_rcons in_cons in IN; move: IN => /orP IN.
          destruct IN as [LAST | FRONT].
          {
            move: LAST => /eqP LAST.
            rewrite -cats1 in SOME.
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            unfold fp_claimed_bounds in *.
            rewrite foldl_cat /= in SOME.
            unfold fp_bound_of_task in SOME.
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            desf; rename H0 into EQ.
            move: EQ => /eqP EQ.
            rewrite eqseq_rcons in EQ.
            move: EQ => /andP [_ /eqP EQ].
            inversion EQ; subst.
            by apply Heq0.
          }
          {
            apply IHts with (rt_bounds := rt_bounds); last by ins.
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            by apply fp_claimed_bounds_rcons in SOME; des.
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          }
        }
      Qed.
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      (* ... and no smaller than the task costs. *)
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      Lemma fp_claimed_bounds_ge_cost :
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        forall ts' rt_bounds tsk R,
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          fp_claimed_bounds ts' = Some rt_bounds ->
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          (tsk, R) \in rt_bounds ->
          R >= task_cost tsk.
      Proof.
        intros ts; induction ts as [| ts' tsk_lst] using last_ind.
        {
          intros rt_bounds tsk R SOME IN.
          by inversion SOME; subst; rewrite in_nil in IN.
        }
        {
          intros rt_bounds tsk_i R SOME IN.
          destruct (lastP rt_bounds) as [|rt_bounds (tsk_lst', R_lst)];
            first by rewrite in_nil in IN.
          rewrite mem_rcons in_cons in IN; move: IN => /orP IN.
          destruct IN as [LAST | FRONT].
          {
            move: LAST => /eqP LAST.
            rewrite -cats1 in SOME.
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            unfold fp_claimed_bounds in *.
            rewrite foldl_cat /= in SOME.
            unfold fp_bound_of_task in SOME.
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            desf; rename H0 into EQ.
            move: EQ => /eqP EQ.
            rewrite eqseq_rcons in EQ.
            move: EQ => /andP [_ /eqP EQ].
            inversion EQ; subst.
            by destruct (max_steps tsk_lst');
              [by apply leqnn | by apply leq_addr].
          }
          {
            apply IHts with (rt_bounds := rt_bounds); last by ins.
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            by apply fp_claimed_bounds_rcons in SOME; des.
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          }
        }
      Qed.

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      (* Short lemma about unfolding the iteration one step. *)
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      Lemma per_task_rta_fold :
        forall tsk rt_bounds,
          task_cost tsk +
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           div_floor (total_interference_bound_fp task_cost task_period tsk rt_bounds
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                     (per_task_rta tsk rt_bounds (max_steps tsk)) higher_priority) num_cpus
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          = per_task_rta tsk rt_bounds (max_steps tsk).+1.
      Proof.
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          by done.
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      Qed.
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    End SimpleLemmas.

    (* In this section, we prove that if the task set is sorted by priority,
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       the tasks in fp_claimed_bounds are interfering tasks.  *)
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    Section HighPriorityTasks.

      (* Consider a list of previous tasks and a task tsk to be analyzed. *)
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      Variable ts: taskset_of sporadic_task.
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      (* Assume that the task set doesn't contain duplicates and is sorted by priority, ... *)
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      Hypothesis H_task_set_is_a_set: uniq ts.
      Hypothesis H_task_set_is_sorted: sorted higher_priority ts.
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      (* ...the priority order is strict (<), ...*)
      Hypothesis H_priority_irreflexive: irreflexive higher_priority.
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      Hypothesis H_priority_transitive: transitive higher_priority.
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      Hypothesis H_priority_antissymetric: antisymmetric higher_priority.
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      (* ... and that the response-time analysis succeeds. *)
      Variable hp_bounds: seq task_with_response_time.
      Variable R: time.
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      Hypothesis H_analysis_succeeds: fp_claimed_bounds ts = Some hp_bounds.

      (* Let's refer to tasks by index. *)
      Variable elem: sporadic_task.
      Let TASK := nth elem ts.
                    
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      (* Then, the tasks in the prefix of fp_claimed_bounds are exactly interfering tasks
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         under FP scheduling.*)
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      Lemma fp_claimed_bounds_interf:
        forall idx,
          idx < size ts ->
          [seq tsk_hp <- ts | fp_can_interfere_with higher_priority (TASK idx) tsk_hp] = take idx ts.
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      Proof.
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        rename H_task_set_is_sorted into SORT,
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               H_task_set_is_a_set into UNIQ,
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               H_priority_antissymetric into ANTI,
               H_priority_irreflexive into IRR.
        induction idx.
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        {
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          intros LT.
          destruct ts as [| tsk0 ts']; [by done | simpl in SORT].
          unfold fp_can_interfere_with; rewrite /= eq_refl andbF.
          apply eq_trans with (y := filter pred0 ts');
            last by apply filter_pred0.
          apply eq_in_filter; red; intros x INx; rewrite /TASK /=.
          destruct (x != tsk0) eqn:SAME; rewrite ?andbT ?andbF //.
          apply negbTE; apply/negP; unfold not; intro HP.
          move: SAME => /eqP SAME; apply SAME; clear SAME.
          apply ANTI; apply/andP; split; first by done.
          apply order_path_min in SORT; last by done.
          by move: SORT => /allP SORT; apply SORT.
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        }
        {
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          intros LT.
          generalize LT; intro LT'; apply ltSnm in LT.
          feed IHidx; first by done.
          rewrite -filter_idx_le_takeS //.
          apply eq_in_filter; red; intros x INx.
          unfold fp_can_interfere_with.
          generalize INx; intro SUBST; apply nth_index with (x0 := elem) in SUBST.
          rewrite -SUBST; clear SUBST.
          rewrite index_uniq; [ | by rewrite index_mem | by done].
          apply/idP/idP.
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          {
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            move => /andP [HP DIFF].
            unfold TASK in *.
            apply sorted_uniq_rel_implies_le_idx in HP; try (by done);
              last by rewrite index_mem.
            by rewrite leq_eqVlt in HP; move: HP => /orP [/eqP SAME | LESS];
                first by rewrite SAME eq_refl in DIFF.
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          }
          {
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            intros LEidx; apply/andP; split;
              first by apply sorted_lt_idx_implies_rel.
            apply/eqP; red; intro BUG.
            eapply f_equal with (f := fun x => index x ts) in BUG.
            rewrite nth_index in BUG; last by done.
            rewrite BUG in LEidx.
            by rewrite index_uniq // ltnn in LEidx.
          }
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        }
      Qed.
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    End HighPriorityTasks.

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    (* In this section, we show that the fixed-point iteration converges. *)
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    Section Convergence.

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      (* Consider any set of higher-priority tasks. *)
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      Variable ts_hp: taskset_of sporadic_task.

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      (* Assume that the response-time analysis succeeds for the higher-priority tasks. *)
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      Variable rt_bounds: seq task_with_response_time.
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      Hypothesis H_test_succeeds: fp_claimed_bounds ts_hp = Some rt_bounds.
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      (* Consider any task tsk to be analyzed, ... *)
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      Variable tsk: sporadic_task.

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      (* ... and assume all tasks have valid parameters. *)
      Hypothesis H_valid_task_parameters:
        valid_sporadic_taskset task_cost task_period task_deadline (rcons ts_hp tsk).

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      (* To simplify, let f denote the fixed-point iteration. *)
      Let f := per_task_rta tsk rt_bounds.

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      (* Assume that f (max_steps tsk) is no larger than the deadline. *)
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      Hypothesis H_no_larger_than_deadline: f (max_steps tsk) <= task_deadline tsk.

      (* First, we show that f is monotonically increasing. *)
      Lemma bertogna_fp_comp_f_monotonic :
        forall x1 x2, x1 <= x2 -> f x1 <= f x2.
      Proof.
        unfold valid_sporadic_taskset, is_valid_sporadic_task in *.
        rename H_test_succeeds into SOME,
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               H_valid_task_parameters into VALID.
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        intros x1 x2 LEx; unfold f, per_task_rta.
        apply fun_mon_iter_mon; [by ins | by ins; apply leq_addr |].
        clear LEx x1 x2; intros x1 x2 LEx.
        unfold div_floor, total_interference_bound_fp.
        rewrite big_seq_cond [\sum_(i <- _ | let '(tsk_other, _) := i in
                                 _ && (tsk_other != tsk))_]big_seq_cond.
        rewrite leq_add2l leq_div2r // leq_sum //.

        intros i; destruct (i \in rt_bounds) eqn:HP; last by rewrite andFb.
        destruct i as [i R]; intros _.
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        have GE_COST := fp_claimed_bounds_ge_cost ts_hp rt_bounds i R SOME.
        have UNZIP := fp_claimed_bounds_unzip ts_hp rt_bounds SOME.
        assert (IN: i \in ts_hp).
        {
          by rewrite -UNZIP; apply/mapP; exists (i,R).
        }
        unfold interference_bound_generic; simpl.
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        rewrite leq_min; apply/andP; split.
        {
          apply leq_trans with (n := W task_cost task_period i R x1); first by apply geq_minl.
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            exploit (VALID i); [by rewrite mem_rcons in_cons IN orbT | ins; des].
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            by apply W_monotonic; try (by ins);
              [by apply GE_COST | by apply leqnn].
        }
        {
          apply leq_trans with (n := x1 - task_cost tsk + 1); first by apply geq_minr.
          by rewrite leq_add2r leq_sub2r //.
        }
      Qed.

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      (* If the iteration converged at an earlier step, it remains as a fixed point. *)
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      Lemma bertogna_fp_comp_f_converges_early :
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        (exists k, k <= max_steps tsk /\ f k = f k.+1) ->
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        f (max_steps tsk) = f (max_steps tsk).+1.
      Proof.
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        by intros EX; des; apply iter_fix with (k := k).
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      Qed.

      (* Else, we derive a contradiction. *)
      Section DerivingContradiction.

        (* Assume instead that the iteration continued to diverge. *)
        Hypothesis H_keeps_diverging:
          forall k,
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            k <= max_steps tsk -> f k != f k.+1.
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        (* By monotonicity, it follows that the value always increases. *)
        Lemma bertogna_fp_comp_f_increases :
          forall k,
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            k <= max_steps tsk ->
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            f k < f k.+1.
        Proof.
          intros k LT.
          rewrite ltn_neqAle; apply/andP; split.
            by apply H_keeps_diverging.
            by apply bertogna_fp_comp_f_monotonic, leqnSn.
        Qed.

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        (* In the end, the response-time bound must exceed the deadline. Contradiction! *)
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        Lemma bertogna_fp_comp_rt_grows_too_much :
          forall k,
            k <= max_steps tsk ->
            f k > k + task_cost tsk - 1.
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        Proof.
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          rename H_valid_task_parameters into TASK_PARAMS.
          unfold valid_sporadic_taskset, is_valid_sporadic_task in *; des.
          exploit (TASK_PARAMS tsk);
            [by rewrite mem_rcons in_cons eq_refl orTb | intro PARAMS; des].
          induction k.
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          {
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            intros _; rewrite add0n -addn1 subh1;
              first by rewrite -addnBA // subnn addn0 /= leqnn.
            by apply PARAMS.
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          }
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          {
            intros LT.
            specialize (IHk (ltnW LT)).
            apply leq_ltn_trans with (n := f k);
              last by apply bertogna_fp_comp_f_increases, ltnW.
            rewrite -addn1 -addnA [1 + _]addnC addnA -addnBA // subnn addn0.
            rewrite -(ltn_add2r 1) in IHk.
            rewrite subh1 in IHk; last first.
            {
              apply leq_trans with (n := task_cost tsk); last by apply leq_addl.
              by apply PARAMS.
            }
            by rewrite -addnBA // subnn addn0 addn1 ltnS in IHk.
          }  
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        Qed.

      End DerivingContradiction.
      
      (* Using the lemmas above, we prove the convergence of the iteration after max_steps. *)
      Lemma per_task_rta_converges:
        f (max_steps tsk) = f (max_steps tsk).+1.
      Proof.
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        rename H_no_larger_than_deadline into LE,
               H_valid_task_parameters into TASK_PARAMS.
        unfold valid_sporadic_taskset, is_valid_sporadic_task in *; des.
       
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        (* Either f converges by the deadline or not. *)
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        destruct ([exists k in 'I_(max_steps tsk).+1, f k == f k.+1]) eqn:EX.
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        {
          move: EX => /exists_inP EX; destruct EX as [k _ ITERk].
          apply bertogna_fp_comp_f_converges_early.
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          by exists k; split; [by rewrite -ltnS; apply ltn_ord | by apply/eqP].
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        }

        (* If not, then we reach a contradiction *)
        apply negbT in EX; rewrite negb_exists_in in EX.
        move: EX => /forall_inP EX.
        rewrite leqNgt in LE; move: LE => /negP LE.
        exfalso; apply LE.
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        have TOOMUCH := bertogna_fp_comp_rt_grows_too_much _ (max_steps tsk).
        exploit TOOMUCH; [| by apply leqnn |].
        {
          intros k LEk; rewrite -ltnS in LEk.
          by exploit (EX (Ordinal LEk)); [by done | intro DIFF].
        }
        unfold max_steps at 1.
        exploit (TASK_PARAMS tsk);
          [by rewrite mem_rcons in_cons eq_refl orTb | intro PARAMS; des].
        rewrite -addnA [1 + _]addnC addnA -addnBA // subnn addn0.
        rewrite subh1; last by apply PARAMS2.
        by rewrite -addnBA // subnn addn0.
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      Qed.
      
    End Convergence.
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    Section MainProof.
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      (* Consider a task set ts. *)
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      Variable ts: taskset_of sporadic_task.
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      (* Assume that higher_priority is a total strict order (<).
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         TODO: it doesn't have to be total over the universe of tasks,
         but only within the task set. However, to weaken this hypothesis
         we need to re-prove some lemmas from ssreflect. *)
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      Hypothesis H_irreflexive: irreflexive higher_priority.
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      Hypothesis H_transitive: transitive higher_priority.
      Hypothesis H_unique_priorities: antisymmetric higher_priority.
      Hypothesis H_total: total higher_priority.
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      (* Assume the task set has no duplicates, ... *)
      Hypothesis H_ts_is_a_set: uniq ts.

      (* ...all tasks have valid parameters, ... *)
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      Hypothesis H_valid_task_parameters:
        valid_sporadic_taskset task_cost task_period task_deadline ts.
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      (* ...restricted deadlines, ...*)
      Hypothesis H_restricted_deadlines:
        forall tsk, tsk \in ts -> task_deadline tsk <= task_period tsk.

      (* ...and tasks are ordered by increasing priorities. *)
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      Hypothesis H_sorted_ts: sorted higher_priority ts.
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      (* Next, consider any arrival sequence such that...*)
      Context {arr_seq: arrival_sequence Job}.

     (* ...all jobs come from task set ts, ...*)
      Hypothesis H_all_jobs_from_taskset:
        forall (j: JobIn arr_seq), job_task j \in ts.
      
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      (* ...jobs have valid parameters,...*)
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      Hypothesis H_valid_job_parameters:
        forall (j: JobIn arr_seq),
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          valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
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      (* ... and satisfy the sporadic task model.*)
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      Hypothesis H_sporadic_tasks:
        sporadic_task_model task_period arr_seq job_task.
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      (* Then, consider any platform with at least one CPU such that...*)
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      Variable sched: schedule num_cpus arr_seq.
      Hypothesis H_at_least_one_cpu :
        num_cpus > 0.

      (* ...jobs only execute after they arrived and no longer
         than their execution costs,... *)
      Hypothesis H_jobs_must_arrive_to_execute:
        jobs_must_arrive_to_execute sched.
      Hypothesis H_completed_jobs_dont_execute:
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        completed_jobs_dont_execute job_cost sched.
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      (* ...and do not execute in parallel (required by the workload bound). *)
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      Hypothesis H_no_parallelism:
        jobs_dont_execute_in_parallel sched.

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      (* Assume that the scheduler is work-conserving and enforces the FP policy. *)
      Hypothesis H_work_conserving: work_conserving job_cost sched.
      Hypothesis H_enforces_FP_policy:
        enforces_FP_policy job_cost job_task sched higher_priority.
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      Let no_deadline_missed_by_task (tsk: sporadic_task) :=
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        task_misses_no_deadline job_cost job_deadline job_task sched tsk.
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      Let no_deadline_missed_by_job :=
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        job_misses_no_deadline job_cost job_deadline sched.
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      Let response_time_bounded_by (tsk: sporadic_task) :=
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        is_response_time_bound_of_task job_cost job_task tsk sched.
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      (* In the following theorem, we prove that any response-time bound contained
         in fp_claimed_bounds is safe. The proof follows by induction on the task set:
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           Induction hypothesis: assume all higher-priority tasks have safe response-time bounds.
           Inductive step: we prove that the response-time bound of the current task is safe.
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         Note that the inductive step is a direct application of the main Theorem from
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         bertogna_fp_theory.v. *)
      Theorem fp_analysis_yields_response_time_bounds :
        forall tsk R,
          (tsk, R) \In fp_claimed_bounds ts ->
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          response_time_bounded_by tsk R.
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      Proof.
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        rename H_valid_job_parameters into JOBPARAMS, H_valid_task_parameters into TASKPARAMS.
        unfold valid_sporadic_taskset in *.
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        intros tsk R MATCH.
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        assert (SOME: exists hp_bounds, fp_claimed_bounds ts = Some hp_bounds /\
                                        (tsk, R) \in hp_bounds).
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        {
          destruct (fp_claimed_bounds ts); last by done.
          by exists l; split.
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        } clear MATCH; des; rename SOME0 into IN.

        have UNZIP := fp_claimed_bounds_unzip ts hp_bounds SOME.
        
        set elem := (tsk,R).
        move: IN => /(nthP elem) [idx LTidx EQ].
        set NTH := fun k => nth elem hp_bounds k.
        set TASK := fun k => (NTH k).1.
        set RESP := fun k => (NTH k).2.
        cut (response_time_bounded_by (TASK idx) (RESP idx));
          first by unfold TASK, RESP, NTH; rewrite EQ.
        clear EQ.

        assert (PAIR: forall idx, (TASK idx, RESP idx) = NTH idx).
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        {
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          by intros i; unfold TASK, RESP; destruct (NTH i).
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        }
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        assert (SUBST: forall i, i < size hp_bounds -> TASK i = nth tsk ts i).
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        {
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          by intros i LTi; rewrite /TASK /NTH -UNZIP (nth_map elem) //.
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        }
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        assert (SIZE: size hp_bounds = size ts).
        {
          by rewrite -UNZIP size_map.
        }

        induction idx as [idx IH'] using strong_ind.

        assert (IH: forall tsk_hp R_hp, (tsk_hp, R_hp) \in take idx hp_bounds -> response_time_bounded_by tsk_hp R_hp).
        {
          intros tsk_hp R_hp INhp.
          move: INhp => /(nthP elem) [k LTk EQ].
          rewrite size_take LTidx in LTk.
          rewrite nth_take in EQ; last by done.
          cut (response_time_bounded_by (TASK k) (RESP k));
            first by unfold TASK, RESP, NTH; rewrite EQ.
          by apply IH'; try (by done); apply (ltn_trans LTk).
        } clear IH'.

        unfold response_time_bounded_by in *.

        exploit (fp_claimed_bounds_rcons (take idx ts) (take idx hp_bounds) (TASK idx) (TASK idx) (RESP idx)).
        {
          by rewrite PAIR SUBST // -2?take_nth -?SIZE // (fp_claimed_bounds_take _ hp_bounds).
        }
        intros [_ [_ [REC DL]]].

        apply bertogna_cirinei_response_time_bound_fp with
              (task_cost0 := task_cost) (task_period0 := task_period)
              (task_deadline0 := task_deadline) (job_deadline0 := job_deadline) (tsk0 := (TASK idx))
              (job_task0 := job_task) (ts0 := ts) (hp_bounds0 := take idx hp_bounds)
              (higher_eq_priority := higher_priority); try (by done).
        {
          cut (NTH idx \in hp_bounds); [intros IN | by apply mem_nth].
          by rewrite -UNZIP; apply/mapP; exists (TASK idx, RESP idx); rewrite PAIR.
        }
        {
          unfold unzip1 in *; rewrite map_take UNZIP SUBST //.
          by apply fp_claimed_bounds_interf with (hp_bounds := hp_bounds); rewrite -?SIZE.
        }
        {
          intros hp_tsk R_hp IN; apply mem_take in IN.
          by apply fp_claimed_bounds_ge_cost with (ts' := ts) (rt_bounds := hp_bounds).
        }
        {
          intros hp_tsk R_hp IN; apply mem_take in IN.
          by apply fp_claimed_bounds_le_deadline with (ts' := ts) (rt_bounds := hp_bounds).
        }
        {
          rewrite REC per_task_rta_fold.
          apply per_task_rta_converges with (ts_hp := take idx ts);
            [by apply fp_claimed_bounds_take; try (by apply ltnW) | | by rewrite -REC ].
          rewrite SUBST // -take_nth -?SIZE //.
          by intros i IN; eapply TASKPARAMS, mem_take, IN.
        }
      Qed.
      
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      (* Therefore, if the schedulability test suceeds, ...*)
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      Hypothesis H_test_succeeds: fp_schedulable ts.
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      (*..., no task misses its deadline. *)
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      Theorem taskset_schedulable_by_fp_rta :
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        forall tsk, tsk \in ts -> no_deadline_missed_by_task tsk.
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      Proof.
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        unfold no_deadline_missed_by_task, task_misses_no_deadline,
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               job_misses_no_deadline, completed,
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               fp_schedulable, valid_sporadic_job in *.
        rename H_valid_job_parameters into JOBPARAMS.
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        move => tsk INtsk j JOBtsk.
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        have RLIST := (fp_analysis_yields_response_time_bounds).
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        have UNZIP := (fp_claimed_bounds_unzip ts).
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        have DL := (fp_claimed_bounds_le_deadline ts).
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        destruct (fp_claimed_bounds ts) as [rt_bounds |]; last by ins.
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        feed (UNZIP rt_bounds); first by done.
        assert (EX: exists R, (tsk, R) \in rt_bounds).
        {
          rewrite -UNZIP in INtsk; move: INtsk => /mapP EX.
          by destruct EX as [p]; destruct p as [tsk' R]; simpl in *; subst tsk'; exists R.
        } des.
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        exploit (RLIST tsk R); [by ins | by apply JOBtsk | intro COMPLETED].
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        exploit (DL rt_bounds tsk R); [by ins | by ins | clear DL; intro DL].
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        rewrite eqn_leq; apply/andP; split; first by apply cumulative_service_le_job_cost.
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        apply leq_trans with (n := service sched j (job_arrival j + R)); last first.
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        {
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          unfold valid_sporadic_taskset, is_valid_sporadic_task in *.
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          apply extend_sum; rewrite // leq_add2l.
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          specialize (JOBPARAMS j); des; rewrite JOBPARAMS1.
          by rewrite JOBtsk.
        }
        rewrite leq_eqVlt; apply/orP; left; rewrite eq_sym.
        by apply COMPLETED.
      Qed.

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      (* For completeness, since all jobs of the arrival sequence
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         are spawned by the task set, we also conclude that no job in
         the schedule misses its deadline. *)
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      Theorem jobs_schedulable_by_fp_rta :
        forall (j: JobIn arr_seq), no_deadline_missed_by_job j.
      Proof.
        intros j.
        have SCHED := taskset_schedulable_by_fp_rta.
        unfold no_deadline_missed_by_task, task_misses_no_deadline in *.
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        apply SCHED with (tsk := job_task j); last by done.
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        by apply H_all_jobs_from_taskset.
      Qed.
      
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    End MainProof.
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  End Analysis.

End ResponseTimeIterationFP.