### fix spelling and markup issues uncovered by `hunspell`

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 ... ... @@ -123,7 +123,7 @@ Section Abstract_RTA. Variable t1 t2: instant. Hypothesis H_busy_interval: busy_interval j t1 t2. (** Let's define A as a relative arrival time of job j (with respect to time t1). *) (** Let's define [A] as a relative arrival time of job [j] (with respect to time [t1]). *) Let A := job_arrival j - t1. (** In order to prove that R is a response-time bound of job j, we use hypothesis H_R_is_maximum. ... ... @@ -164,7 +164,7 @@ Section Abstract_RTA. interval remains completed. *) Section FixpointOutsideBusyInterval. (** By assumption, suppose that t2 is less than or equal to [t1 + A_sp + F_sp]. *) (** By assumption, suppose that [t2] is less than or equal to [t1 + A_sp + F_sp]. *) Hypothesis H_big_fixpoint_solution : t2 <= t1 + (A_sp + F_sp). (** Then we prove that [job_arrival j + R] is no less than [t2]. *) ... ... @@ -198,7 +198,7 @@ Section Abstract_RTA. [t1 + A_sp + F_sp] lies inside the busy interval. *) Section FixpointInsideBusyInterval. (** So, assume that [t1 + A_sp + F_sp] is less than t2. *) (** So, assume that [t1 + A_sp + F_sp] is less than [t2]. *) Hypothesis H_small_fixpoint_solution : t1 + (A_sp + F_sp) < t2. (** Next, let's consider two other cases: *) ... ... @@ -221,7 +221,7 @@ Section Abstract_RTA. Variable F : duration. (** (a) the sum of [A_sp] and [F_sp] is equal to the sum of [A] and [F]... *) Hypothesis H_Asp_Fsp_eq_A_F : A_sp + F_sp = A + F. (** (b) [F] is at mo1st [F_sp]... *) (** (b) [F] is at most [F_sp]... *) Hypothesis H_F_le_Fsp : F <= F_sp. (** (c) and [A + F] is a solution for the response-time recurrence for [A]. *) Hypothesis H_A_F_fixpoint: ... ...
 ... ... @@ -220,7 +220,7 @@ Section Sequential_Abstract_RTA. Hypothesis H_j2_from_tsk: job_of_task tsk j2. Hypothesis H_j1_cost_positive: job_cost_positive j1. (** Consider the busy interval <<[t1, t2)>> of job j1. *) (** Consider the busy interval <<[t1, t2)>> of job [j1]. *) Variable t1 t2 : instant. Hypothesis H_busy_interval : busy_interval j1 t1 t2. ... ... @@ -241,7 +241,7 @@ Section Sequential_Abstract_RTA. Qed. (** Next we prove that if a job is pending after the beginning of the busy interval <<[t1, t2)>> then it arrives after t1. *) of the busy interval <<[t1, t2)>> then it arrives after [t1]. *) Lemma arrives_after_beginning_of_busy_interval: forall t, t1 <= t -> ... ... @@ -280,14 +280,14 @@ Section Sequential_Abstract_RTA. Variable t1 t2 : instant. Hypothesis H_busy_interval : busy_interval j t1 t2. (** Let's define A as a relative arrival time of job j (with respect to time t1). *) (** Let's define [A] as a relative arrival time of job [j] (with respect to time [t1]). *) Let A : duration := job_arrival j - t1. (** Consider an arbitrary time x ... *) Variable x : duration. (** ... such that (t1 + x) is inside the busy interval... *) (** ... such that [(t1 + x)] is inside the busy interval... *) Hypothesis H_inside_busy_interval : t1 + x < t2. (** ... and job j is not completed by time (t1 + x). *) (** ... and job [j] is not completed by time [(t1 + x)]. *) Hypothesis H_job_j_is_not_completed : ~~ completed_by sched j (t1 + x). (** In this section, we show that the cumulative interference of job j in the interval <<[t1, t1 + x)>> ... ... @@ -549,8 +549,8 @@ Section Sequential_Abstract_RTA. Qed. (** Finally, we show that the cumulative interference of job j in the interval <<[t1, t1 + x)>> is bounded by the sum of the task workload in the interval [t1, t1 + A + ε) and the cumulative interference of [j]'s task in the interval [t1, t1 + x). *) is bounded by the sum of the task workload in the interval <<[t1, t1 + A + ε)>> and the cumulative interference of [j]'s task in the interval <<[t1, t1 + x)>>. *) Lemma cumulative_job_interference_le_task_interference_bound: cumul_interference j t1 (t1 + x) <= (task_workload_between t1 (t1 + A + ε) - job_cost j) ... ...
 ... ... @@ -96,7 +96,7 @@ Section AbstractRTADefinitions. (forall t, t1 < t < t2 -> ~ quiet_time j t). (** Next, we say that an interval <<[t1, t2)>> is a busy interval iff [t1, t2) is a busy-interval prefix and t2 is a quiet time. *) <<[t1, t2)>> is a busy-interval prefix and [t2] is a quiet time. *) Definition busy_interval (j : Job) (t1 t2 : instant) := busy_interval_prefix j t1 t2 /\ quiet_time j t2. ... ...
 ... ... @@ -57,9 +57,9 @@ Section AbstractRTARunToCompletionThreshold. Hypothesis H_busy_interval : busy_interval j t1 t2. (** First, we prove that job [j] completes by the end of the busy interval. Note that the busy interval contains the execution of job j, in addition time instant t2 is a quiet time. Thus by the definition of a quiet time the job should be completed before time t2. *) Note that the busy interval contains the execution of job j, in addition time instant [t2] is a quiet time. Thus by the definition of a quiet time the job should be completed before time [t2]. *) Lemma job_completes_within_busy_interval: completed_by sched j t2. Proof. ... ... @@ -74,7 +74,7 @@ Section AbstractRTARunToCompletionThreshold. the total time where job [j] is scheduled inside the busy interval. *) Section InterferenceIsComplement. (** Consider any sub-interval <<[t, t + delta)>> inside the busy interval [t1, t2). *) (** Consider any sub-interval <<[t, t + delta)>> inside the busy interval <<[t1, t2)>>. *) Variables (t : instant) (delta : duration). Hypothesis H_greater_than_or_equal : t1 <= t. Hypothesis H_less_or_equal: t + delta <= t2. ... ...
 ... ... @@ -275,7 +275,7 @@ Section ArrivalSequencePrefix. Qed. (** Next, we prove that if a job belongs to the prefix (jobs_arrived_before t), then it arrives in the arrival [(jobs_arrived_before t)], then it arrives in the arrival sequence. *) Lemma in_arrivals_implies_arrived: forall j t1 t2, ... ... @@ -291,16 +291,16 @@ Section ArrivalSequencePrefix. Proof. by move=> t ? ?; exists t. Qed. (** Next, we prove that if a job belongs to the prefix (jobs_arrived_between t1 t2), then it indeed arrives between t1 and t2. *) [(jobs_arrived_between t1 t2)], then it indeed arrives between [t1] and [t2]. *) Lemma in_arrivals_implies_arrived_between: forall j t1 t2, j \in arrivals_between arr_seq t1 t2 -> arrived_between j t1 t2. Proof. by move=> ? ? ? /mem_bigcat_nat_exists[t0 [/job_arrival_at <-]]. Qed. (** Similarly, if a job belongs to the prefix (jobs_arrived_before t), then it indeed arrives before time t. *) (** Similarly, if a job belongs to the prefix [(jobs_arrived_before t)], then it indeed arrives before time [t]. *) Lemma in_arrivals_implies_arrived_before: forall j t, j \in arrivals_before arr_seq t -> ... ... @@ -308,7 +308,7 @@ Section ArrivalSequencePrefix. Proof. by move=> ? ? /in_arrivals_implies_arrived_between. Qed. (** Similarly, we prove that if a job from the arrival sequence arrives before t, then it belongs to the sequence (jobs_arrived_before t). *) before t, then it belongs to the sequence [(jobs_arrived_before t)]. *) Lemma arrived_between_implies_in_arrivals: forall j t1 t2, arrives_in arr_seq j -> ... ...
 ... ... @@ -94,7 +94,7 @@ Section Composition. Proof. move=> ?; exact: service_during_last_plus_before. Qed. (** Finally, we deconstruct the service received during an interval <<[t1, t3)>> into the service at a midpoint t2 and the service in the intervals before into the service at a midpoint [t2] and the service in the intervals before and after. *) Lemma service_split_at_point: forall t1 t2 t3, ... ...
 ... ... @@ -242,7 +242,7 @@ Section RequestBoundFunctions. (** Let [max_arrivals] be a family of valid arrival curves, i.e., for any task [tsk] in [ts] [max_arrival tsk] is (1) an arrival bound of [tsk], and (2) it is a monotonic function that equals 0 for the empty interval Δ = 0. *) that equals 0 for the empty interval [Δ = 0]. *) Context `{MaxArrivals Task}. Hypothesis H_valid_arrival_curve : valid_arrival_curve (max_arrivals tsk). Hypothesis H_is_arrival_curve : respects_max_arrivals arr_seq tsk (max_arrivals tsk). ... ... @@ -388,10 +388,10 @@ Section TotalRBFMonotonic. End TotalRBFMonotonic. (** ** RBFs Equal to Zero for Duration ε *) (** ** RBFs Equal to Zero for Duration [ε] *) (** In the following section, we derive simple properties that follow in the pathological case of an RBF that yields zero for duration ε. *) the pathological case of an RBF that yields zero for duration [ε]. *) Section DegenerateTotalRBFs. (** Consider a set of tasks characterized by WCETs and arrival curves ... *) ... ...
 ... ... @@ -45,7 +45,7 @@ Section FullyPreemptiveModel. by rewrite H1; compute. Qed. (** ... or ε when [job_cost j > 0]. *) (** ... or [ε] when [job_cost j > 0]. *) Lemma job_max_nps_is_ε: forall j, job_cost j > 0 -> ... ...
 ... ... @@ -47,7 +47,7 @@ Section TaskRTCThresholdFullyNonPreemptive. by rewrite H3; compute. Qed. (** ... and ε otherwise. *) (** ... and [ε] otherwise. *) Fact job_rtc_threshold_is_ε: forall j, job_cost j > 0 -> ... ...
 ... ... @@ -97,7 +97,7 @@ Section FindSwapCandidateFacts. (** 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. *) that it arrives not later than at time [t1]. *) Corollary fsc_found_job_arrival: forall j2, scheduled_at sched j2 (find_swap_candidate sched t1 j1) -> ... ...
 ... ... @@ -219,7 +219,7 @@ Section FSCWorkConservationLemmas. - by apply (non_idle_swap_maintains_work_conservation_t1 arr_seq _ _ _ j2). - case: (boolP(t == t2)) => [/eqP EQ'| /eqP NEQ']. + by apply (non_idle_swap_maintains_work_conservation_t2 arr_seq _ _ _ j1). + case: (boolP((t <= t1) || (t2 < t))) => [NOT_BET | BET]. (* t <> t2 *) + case: (boolP((t <= t1) || (t2 < t))) => [NOT_BET | BET]. * move: NOT_BET; move/orP => [] => NOT_BET. { by apply (non_idle_swap_maintains_work_conservation_LEQ_t1 arr_seq _ _ _ H_range _ _ H_not_idle t2_not_idle j). } { by apply (non_idle_swap_maintains_work_conservation_GT_t2 arr_seq _ _ _ H_range _ _ H_not_idle t2_not_idle j). } ... ...