small clean up in util/step_function.v

util/step_function.v

@@ -5,8 +5,8 @@ Section StepFunction.
 
   Section Defs.
 
-    (* We say that a function f... *)
-    Variable f: nat -> nat.
+    (* We say that a function [f]... *)
+    Variable f : nat -> nat.
 
     (* ...is a step function iff the following holds. *)
     Definition is_step_function :=
@@ -16,25 +16,25 @@ Section StepFunction.
 
   Section Lemmas.
 
-    (* Let f be any step function over natural numbers. *)
-    Variable f: nat -> nat.
-    Hypothesis H_step_function: is_step_function f.
+    (* 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.
+      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.
+      (* 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:
+      (* 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.
@@ -46,11 +46,11 @@ Section StepFunction.
         { move => x2 LE /andP [GEy LTy].
           exploit (DELTA (x2 - x1));
             first by apply/andP; split; last by rewrite addnBA // addKn.
-            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.
+          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.
@@ -59,18 +59,18 @@ Section StepFunction.
               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.
+            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; [by apply leq_addr | by rewrite addnS].
+            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.
+            by apply: (leq_trans LT0); rewrite addnS.
           }  
         }
       Qed.
@@ -83,22 +83,22 @@ Section StepFunction.
      value theorem, but for predicates of 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 ... *)
     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. *)
-    Lemma exists_first_intermediate_point:
+       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.
       have EX: exists x, P x && (t1 < x <= t2).
@@ -106,7 +106,7 @@ Section StepFunction.
         apply/andP; split; first by done.
         apply/andP; split; last by done.
         move: H_t1_le_t2; rewrite leq_eqVlt; move => /orP [/eqP EQ | NEQ1]; last by done.
-          by exfalso; subst t2; move: H_not_P_at_t1 => /negP NPt1. 
+        by exfalso; subst t2; move: H_not_P_at_t1 => /negP NPt1. 
       }
       have MIN := ex_minnP EX.
       move: MIN => [x /andP [Px /andP [LT1 LT2]] MIN]; clear EX.
@@ -116,12 +116,12 @@ Section StepFunction.
       { apply/andP; split; first by done.
         apply/andP; split.
         - move: NEQ1. rewrite leq_eqVlt; move => /orP [/eqP EQ | NEQ1]; last by done.
-            by exfalso; subst y; move: H_not_P_at_t1 => /negP NPt1. 
+          by exfalso; subst y; move: H_not_P_at_t1 => /negP NPt1. 
         - by apply ltnW, leq_trans with x.
       }
-        by move: NEQ2; rewrite ltnNge; move => /negP NEQ2.
+      by move: NEQ2; rewrite ltnNge; move => /negP NEQ2.
     Qed.
     
   End ExistsIntermediateValuePredicates.  
 
-End StepFunction.
+End StepFunction.
\ No newline at end of file