Add Request Bound Functions

So that we can link them to arrival curves of Network Calculus in NCCoq as discussed in https://gitlab.mpi-sws.org/RT-PROOFS/project-administration/-/issues/23

The link with Network Calculus arrival curves can be found here : https://gitlab.mpi-sws.org/proux/nc-coq/-/tree/prosa-link/link

model/task/arrival/request_bound_functions.v

+Require Export prosa.util.rel.
+Require Export prosa.model.task.arrivals.
+
+(** * Request Bound Functions (RBF) *)
+
+(** In the following, we define the notion of Request Bound Functions
+    (RBF), which can be used to reason about the job cost arrivals. In
+    contrast to arrival curves which constrain the number of arrivals
+    per time interval, request bound functions bound the sum of costs
+    of arriving jobs. *)
+
+(** ** Task Parameters for the Request Bound Functions *)
+
+(** The request bound functions give an upper bound and, optionally, a
+    lower bound on the cost of new job arrivals during any given
+    interval. *)
+
+(** We let [max_request_bound tsk Δ] denote a bound on the maximum
+    cost of arrivals of jobs of task [tsk] in any interval of length
+    [Δ]. *)
+Class MaxRequestBound (Task : TaskType) := max_request_bound : Task -> duration -> work.
+
+(** Conversely, we let [min_request_bound tsk Δ] denote a bound on the
+    minimum cost of arrivals of jobs of task [tsk] in any interval of
+    length [Δ]. *)
+Class MinRequestBound (Task : TaskType) := min_request_bound : Task -> duration -> work.
+
+(** ** Parameter Semantics *)
+
+(** In the following, we precisely define the semantics of the request
+    bound functions. *)
+Section RequestBoundFunctions.
+
+  (** Consider any type of tasks ... *)
+  Context {Task : TaskType}.
+
+  (**  ... and any type of jobs associated with these tasks. *)
+  Context {Job : JobType}.
+  Context `{JobTask Job Task}.
+  Context `{JobCost Job}.
+
+  (** Consider any job arrival sequence. *)
+  Variable arr_seq : arrival_sequence Job.
+
+  (** *** Definition of Request Bound Functions *)
+
+  (** First, what constitutes a valid request bound function for a task? *)
+  Section RequestBoundFunctions.
+
+    (** We say that a given bound [request_bound] is a valid request
+        bound function iff [request_bound] is a monotonic function
+        that equals 0 for the empty interval [delta = 0]. *)
+    Definition valid_request_bound_function (request_bound : duration -> work) :=
+      request_bound 0 = 0 /\
+      monotone request_bound leq.
+
+    (** We say that [request_bound] is an upper request bound for task
+        [tsk] iff, for any interval <<[t1, t2)>>, [request_bound (t2 -
+        t1)] bounds the sum of costs of jobs of [tsk] that arrive in
+        that interval. *)
+    Definition respects_max_request_bound (tsk : Task) (max_request_bound : duration -> work) :=
+      forall (t1 t2 : instant),
+        t1 <= t2 ->
+        cost_of_task_arrivals arr_seq tsk t1 t2 <= max_request_bound (t2 - t1).
+
+    (** We analogously define the lower request bound. *)
+    Definition respects_min_request_bound (tsk : Task) (min_request_bound : duration -> work) :=
+      forall (t1 t2 : instant),
+        t1 <= t2 ->
+        min_request_bound (t2 - t1) <= cost_of_task_arrivals arr_seq tsk t1 t2.
+
+  End RequestBoundFunctions.
+
+End RequestBoundFunctions.
+
+
+(** ** Model Validity *)
+
+(** Based on the just-established semantics, we define the properties of a
+    valid request bound model. *)
+Section RequestBoundFunctionsModel.
+
+  (** Consider any type of tasks ... *)
+  Context {Task : TaskType}.
+
+  (**  ... and any type of jobs associated with these tasks. *)
+  Context {Job : JobType}.
+  Context `{JobTask Job Task}.
+  Context `{JobCost Job}.
+
+  (** Consider any job arrival sequence... *)
+  Variable arr_seq : arrival_sequence Job.
+
+  (** ...and all kinds of request bounds. *)
+  Context `{MaxRequestBound Task}
+          `{MinRequestBound Task}.
+
+  (** Let [ts] be an arbitrary task set. *)
+  Variable ts : TaskSet Task.
+
+  (** We say that [request_bound] is a valid arrival curve for a task set
+      if it is valid for any task in the task set *)
+  Definition valid_taskset_request_bound_function (request_bound : Task -> duration -> work) :=
+    forall (tsk : Task), tsk \in ts -> valid_request_bound_function (request_bound tsk).
+
+  (** Finally, we lift the per-task semantics of the respective
+      request bound functions to the entire task set.  *)
+
+  Definition taskset_respects_max_request_bound :=
+    forall (tsk : Task), tsk \in ts -> respects_max_request_bound arr_seq tsk (max_request_bound tsk).
+
+  Definition taskset_respects_min_request_bound :=
+    forall (tsk : Task), tsk \in ts -> respects_min_request_bound arr_seq tsk (min_request_bound tsk).
+
+End RequestBoundFunctionsModel.