clean up in util/unit_growth.v

* rename [step_function] to [unit_growth]
* restructure file

analysis/facts/behavior/service.v

@@ -197,13 +197,13 @@ Section UnitService.
   Qed.
   
 
-  Section ServiceIsAStepFunction.
+  Section ServiceIsUnitGrowthFunction.
 
-    (** We show that the service received by any job [j] is a step function. *)
-    Lemma service_is_a_step_function:
-      is_step_function (service sched j).
+    (** We show that the service received by any job [j] is a unit growth function. *)
+    Lemma service_is_unit_growth_function:
+      unit_growth_function (service sched j).
     Proof.
-      rewrite /is_step_function => t.
+      rewrite /unit_growth_function => t.
       rewrite addn1 -service_last_plus_before leq_add2l.
       by apply service_at_most_one.
     Qed.
@@ -224,14 +224,14 @@ Section UnitService.
         service sched j t' = s.
     Proof.
       feed (exists_intermediate_point (service sched j));
-        [by apply service_is_a_step_function | intros EX].
+        [by apply service_is_unit_growth_function | intros EX].
       feed (EX 0 t); first by done.
       feed (EX s); first by rewrite /service /service_during big_geq //.
       by move: EX => /= [x_mid EX]; exists x_mid.
     Qed.
 
-  End ServiceIsAStepFunction.
-
+  End ServiceIsUnitGrowthFunction.
+  
 End UnitService.
 
 (** We establish a basic fact about the monotonicity of service. *)

classic/model/schedule/uni/schedule.v

@@ -492,15 +492,15 @@ Module UniprocessorSchedule.
           
       End OnlyOneJobScheduled.
 
-      Section ServiceIsAStepFunction.
+      Section ServiceIsUnitGrowthFunction.
 
         (* First, we show that the service received by any job j
-           is a step function. *)
-        Lemma service_is_a_step_function:
+           is a unit growth funciton. *)
+        Lemma service_is_unit_growth_function:
           forall j,
-            is_step_function (service sched j).
+            unit_growth_function (service sched j).
         Proof.
-          unfold is_step_function, service, service_during; intros j t.
+          unfold unit_growth_function, service, service_during; intros j t.
           rewrite addn1 big_nat_recr //=.
           by apply leq_add; last by apply leq_b1.
         Qed.
@@ -522,14 +522,14 @@ Module UniprocessorSchedule.
             service sched j t0 = s0.
         Proof.
           feed (exists_intermediate_point (service sched j));
-            [by apply service_is_a_step_function | intros EX].
+            [by apply service_is_unit_growth_function | intros EX].
           feed (EX 0 t); first by done.
           feed (EX s0);
             first by rewrite /service /service_during big_geq //. 
           by move: EX => /= [x_mid EX]; exists x_mid.
         Qed.
 
-      End ServiceIsAStepFunction.
+      End ServiceIsUnitGrowthFunction.
 
       Section ScheduledAtEarlierTime.
 
@@ -641,4 +641,4 @@ Module UniprocessorSchedule.
   
   End Schedule.
  
-End UniprocessorSchedule.
\ No newline at end of file
+End UniprocessorSchedule.

classic/model/schedule/uni/susp/last_execution.v

@@ -335,7 +335,7 @@ Module LastExecution.
           rename H_jobs_must_arrive_to_execute into ARR.
           move: H_j_has_completed => COMP.
           feed (exists_intermediate_point (service sched j));
-            first by apply service_is_a_step_function.
+            first by apply service_is_unit_growth_function.
           move => EX; feed (EX (job_arrival j) t).
           { feed (cumulative_service_implies_scheduled sched j 0 t).
             apply leq_ltn_trans with (n := s); first by done.
@@ -408,4 +408,4 @@ Module LastExecution.
       
   End TimeAfterLastExecution.
 
-End LastExecution.
\ No newline at end of file
+End LastExecution.

classic/util/step_function.v

-Require Export prosa.util.step_function.
+Require Export prosa.util.unit_growth.
 
 Require Import prosa.classic.util.tactics prosa.classic.util.notation prosa.classic.util.induction.
 From mathcomp Require Import ssreflect ssrbool eqtype ssrnat.

util/all.v

@@ -7,7 +7,7 @@ Require Export prosa.util.nat.
 Require Export prosa.util.ssrlia.
 Require Export prosa.util.sum.
 Require Export prosa.util.seqset.
-Require Export prosa.util.step_function.
+Require Export prosa.util.unit_growth.
 Require Export prosa.util.epsilon.
 Require Export prosa.util.search_arg.
 Require Export prosa.util.rel.

util/step_function.v → util/unit_growth.v

 Require Import prosa.util.tactics prosa.util.notation.
 From mathcomp Require Import ssreflect ssrbool eqtype ssrnat.
 
-Section StepFunction.
+(** We say that a function [f] is a unit growth function iff for any
+    time instant [t] it holds that [f (t + 1) <= f t + 1]. *)
+Definition unit_growth_function (f : nat -> nat) :=
+  forall t, f (t + 1) <= f t + 1.
 
-  Section Defs.
+(** In this section, we prove a few useful lemmas about unit growth functions. *)
+Section Lemmas.
 
-    (* We say that a function [f]... *)
-    Variable f : nat -> nat.
+  (** Let [f] be any unit growth function over natural numbers. *)
+  Variable f : nat -> nat.
+  Hypothesis H_unit_growth_function : unit_growth_function f.
 
-    (* ...is a step function iff the following holds. *)
-    Definition is_step_function :=
-      forall t, f (t + 1) <= f t + 1.
+  (** In the following section, we prove a result similar to the
+      intermediate value theorem for continuous functions. *)
+  Section ExistsIntermediateValue.
 
-  End Defs.
+    (** Consider any interval [x1, x2]. *)
+    Variable x1 x2 : nat.
+    Hypothesis H_is_interval : x1 <= x2.
 
-  Section Lemmas.
+    (** Let [t] be any value such that [f x1 <= y < f x2]. *)
+    Variable y : nat.
+    Hypothesis H_between : f x1 <= y < f x2.
 
-    (* Let [f] be any step function over natural numbers. *)
-    Variable f : nat -> nat.
-    Hypothesis H_step_function : is_step_function f.
-
-    (* In this section, we prove a result similar to the intermediate
-       value theorem for continuous functions. *)
-    Section ExistsIntermediateValue.
-
-      (* Consider any interval [x1, x2]. *)
-      Variable x1 x2 : nat.
-      Hypothesis H_is_interval : x1 <= x2.
-
-      (* Let [t] be any value such that [f x1 <= y < f x2]. *)
-      Variable y : nat.
-      Hypothesis H_between : f x1 <= y < f x2.
-
-      (* Then, we prove that there exists an intermediate point [x_mid] such that
-         [f x_mid = y]. *)
-      Lemma exists_intermediate_point :
-        exists x_mid, x1 <= x_mid < x2 /\ f x_mid = y.
-      Proof.
-        rename H_is_interval into INT, H_step_function into STEP, H_between into BETWEEN.
-        move: x2 INT BETWEEN; clear x2.
-        suff DELTA:
-          forall delta,
-            f x1 <= y < f (x1 + delta) ->
-            exists x_mid, x1 <= x_mid < x1 + delta /\ f x_mid = y.
-        { move => x2 LE /andP [GEy LTy].
-          exploit (DELTA (x2 - x1));
-            first by apply/andP; split; last by rewrite addnBA // addKn.
+    (** Then, we prove that there exists an intermediate point [x_mid] such that [f x_mid = y]. *)
+    Lemma exists_intermediate_point :
+      exists x_mid, x1 <= x_mid < x2 /\ f x_mid = y.
+    Proof.
+      rename H_is_interval into INT, H_unit_growth_function into UNIT, H_between into BETWEEN.
+      move: x2 INT BETWEEN; clear x2.
+      suff DELTA:
+        forall delta,
+          f x1 <= y < f (x1 + delta) ->
+          exists x_mid, x1 <= x_mid < x1 + delta /\ f x_mid = y.
+      { move => x2 LE /andP [GEy LTy].
+        exploit (DELTA (x2 - x1));
+          first by apply/andP; split; last by rewrite addnBA // addKn.
           by rewrite addnBA // addKn.
+      }
+      induction delta.
+      { rewrite addn0; move => /andP [GE0 LT0].
+        by apply (leq_ltn_trans GE0) in LT0; rewrite ltnn in LT0.
+      }
+      { move => /andP [GT LT].
+        specialize (UNIT (x1 + delta)); rewrite leq_eqVlt in UNIT.
+        have LE: y <= f (x1 + delta).
+        { move: UNIT => /orP [/eqP EQ | UNIT]; first by rewrite !addn1 in EQ; rewrite addnS EQ ltnS in LT.
+          rewrite [X in _ < X]addn1 ltnS in UNIT.
+          apply: (leq_trans _ UNIT).
+          by rewrite addn1 -addnS ltnW.
+        } clear UNIT LT.
+        rewrite leq_eqVlt in LE.
+        move: LE => /orP [/eqP EQy | LT].
+        { exists (x1 + delta); split; last by rewrite EQy.
+          by apply/andP; split; [apply leq_addr | rewrite addnS].
         }
-        induction delta.
-        { rewrite addn0; move => /andP [GE0 LT0].
-          by apply (leq_ltn_trans GE0) in LT0; rewrite ltnn in LT0.
-        }
-        { move => /andP [GT LT].
-          specialize (STEP (x1 + delta)); rewrite leq_eqVlt in STEP.
-          have LE: y <= f (x1 + delta).
-          { move: STEP => /orP [/eqP EQ | STEP];
-              first by rewrite !addn1 in EQ; rewrite addnS EQ ltnS in LT.
-            rewrite [X in _ < X]addn1 ltnS in STEP.
-            apply: (leq_trans _ STEP).
-            by rewrite addn1 -addnS ltnW.
-          } clear STEP LT.
-          rewrite leq_eqVlt in LE.
-          move: LE => /orP [/eqP EQy | LT].
-          { exists (x1 + delta); split; last by rewrite EQy.
-            by apply/andP; split; [apply leq_addr | rewrite addnS].
-          }
-          { feed (IHdelta); first by apply/andP; split.
-            move: IHdelta => [x_mid [/andP [GE0 LT0] EQ0]].
-            exists x_mid; split; last by done.
-            apply/andP; split; first by done.
-            by apply: (leq_trans LT0); rewrite addnS.
-          }  
-        }
-      Qed.
-
-    End ExistsIntermediateValue.
+        { feed (IHdelta); first by apply/andP; split.
+          move: IHdelta => [x_mid [/andP [GE0 LT0] EQ0]].
+          exists x_mid; split; last by done.
+          apply/andP; split; first by done.
+          by apply: (leq_trans LT0); rewrite addnS.
+        }  
+      }
+    Qed.
 
-  End Lemmas.
+  End ExistsIntermediateValue.
 
-  (* In this section, we prove an analogue of the intermediate
-     value theorem, but for predicates of natural numbers. *) 
+  (** In this section, we, again, prove an analogue of the
+      intermediate value theorem, but for predicates over natural
+      numbers. *) 
   Section ExistsIntermediateValuePredicates. 
 
-    (* Let [P] be any predicate on natural numbers. *)
+    (** Let [P] be any predicate on natural numbers. *)
     Variable P : nat -> bool.
 
-    (* Consider a time interval [t1,t2] such that ... *)
+    (** Consider a time interval <<[t1,t2]>> such that ... *)
     Variables t1 t2 : nat.
     Hypothesis H_t1_le_t2 : t1 <= t2.
 
-    (* ... [P] doesn't hold for [t1] ... *)
+    (** ... [P] doesn't hold for [t1] ... *)
     Hypothesis H_not_P_at_t1 : ~~ P t1.
 
-    (* ... but holds for [t2]. *)
+    (** ... but holds for [t2]. *)
     Hypothesis H_P_at_t2 : P t2.
     
-    (* Then we prove that within time interval [t1,t2] there exists time 
-       instant [t] such that [t] is the first time instant when [P] holds. *)
+    (** Then we prove that within time interval <<[t1,t2]>> there exists
+        time instant [t] such that [t] is the first time instant when
+        [P] holds. *)
     Lemma exists_first_intermediate_point :
       exists t, (t1 < t <= t2) /\ (forall x, t1 <= x < t -> ~~ P x) /\ P t.
     Proof.
@@ -122,6 +113,6 @@ Section StepFunction.
       by move: NEQ2; rewrite ltnNge; move => /negP NEQ2.
     Qed.
     
-  End ExistsIntermediateValuePredicates.  
+  End ExistsIntermediateValuePredicates.
 
-End StepFunction.
+End Lemmas.