remove trailing whitespace

This was done using `sed --in-place 's/[[:space:]]\+$//' theories/*.v`.

theories/base.v

@@ -1000,7 +1000,7 @@ Hint Mode UpClose - ! : typeclass_instances.
 Notation "↑ x" := (up_close x) (at level 20, format "↑ x").
 
 (** * Monadic operations *)
-(** We define operational type classes for the monadic operations bind, join 
+(** We define operational type classes for the monadic operations bind, join
 and fmap. We use these type classes merely for convenient overloading of
 notations and do not formalize any theory on monads (we do not even define a
 class with the monad laws). *)
@@ -1106,7 +1106,7 @@ Arguments alter {_ _ _ _} _ !_ !_ / : simpl nomatch, assert.
 
 (** The function [partial_alter f k m] should update the value at key [k] using the
 function [f], which is called with the original value at key [k] or [None]
-if [k] is not a member of [m]. The value at [k] should be deleted if [f] 
+if [k] is not a member of [m]. The value at [k] should be deleted if [f]
 yields [None]. *)
 Class PartialAlter (K A M : Type) :=
   partial_alter: (option A → option A) → K → M → M.

theories/boolset.v

@@ -24,7 +24,7 @@ Proof.
   - by intros x y; rewrite <-(bool_decide_spec (x = y)).
   - split. apply orb_prop_elim. apply orb_prop_intro.
   - split. apply andb_prop_elim. apply andb_prop_intro.
-  - intros X Y x; unfold elem_of, boolset_elem_of; simpl. 
+  - intros X Y x; unfold elem_of, boolset_elem_of; simpl.
     destruct (boolset_car X x), (boolset_car Y x); simpl; tauto.
   - done.
 Qed.

theories/fin_maps.v

@@ -41,7 +41,7 @@ Class FinMap K M `{FMap M, ∀ A, Lookup K A (M A), ∀ A, Empty (M A), ∀ A,
   elem_of_map_to_list {A} (m : M A) i x :
     (i,x) ∈ map_to_list m ↔ m !! i = Some x;
   lookup_omap {A B} (f : A → option B) (m : M A) i :
-    omap f m !! i = m !! i ≫= f; 
+    omap f m !! i = m !! i ≫= f;
   lookup_merge {A B C} (f : option A → option B → option C)
       `{!DiagNone f} (m1 : M A) (m2 : M B) i :
     merge f m1 m2 !! i = f (m1 !! i) (m2 !! i)

theories/gmultiset.v

@@ -50,7 +50,7 @@ Section definitions.
 
   Global Instance gmultiset_dom : Dom (gmultiset A) (gset A) := λ X,
     let (X) := X in dom _ X.
-End definitions. 
+End definitions.
 
 Typeclasses Opaque gmultiset_elem_of gmultiset_subseteq.
 Typeclasses Opaque gmultiset_elements gmultiset_size gmultiset_empty.

theories/hashset.v

@@ -29,7 +29,7 @@ Next Obligation.
 Qed.
 Global Program Instance hashset_union: Union (hashset hash) := λ m1 m2,
   let (m1,Hm1) := m1 in let (m2,Hm2) := m2 in
-  Hashset (union_with (λ l k, Some (list_union l k)) m1 m2) _. 
+  Hashset (union_with (λ l k, Some (list_union l k)) m1 m2) _.
 Next Obligation.
   intros _ _ m1 Hm1 m2 Hm2 n l'; rewrite lookup_union_with_Some.
   intros [[??]|[[??]|(l&k&?&?&?)]]; simplify_eq/=; auto.

theories/lexico.v

@@ -141,7 +141,7 @@ Instance sig_lexico_po `{Lexico A, !StrictOrder (@lexico A _)}
   (P : A → Prop) `{∀ x, ProofIrrel (P x)} : StrictOrder (@lexico (sig P) _).
 Proof.
   unfold lexico, sig_lexico. split.
-  - intros [x ?] ?. by apply (irreflexivity lexico x). 
+  - intros [x ?] ?. by apply (irreflexivity lexico x).
   - intros [x1 ?] [x2 ?] [x3 ?] ??. by trans x2.
 Qed.
 Instance sig_lexico_trichotomy `{Lexico A, tA : !TrichotomyT (@lexico A _)}

theories/list.v

@@ -401,7 +401,7 @@ used by [positives_flatten]. *)
 Definition positives_unflatten (p : positive) : option (list positive) :=
   positives_unflatten_go p [] 1.
 
-(** [seqZ m n] generates the sequence [m], [m + 1], ..., [m + n - 1] 
+(** [seqZ m n] generates the sequence [m], [m + 1], ..., [m + n - 1]
 over integers, provided [0 ≤ n]. If [n < 0], then the range is empty. **)
 Definition seqZ (m len: Z) : list Z :=
   (λ i: nat, Z.add i m) <$> (seq 0 (Z.to_nat len)).
@@ -4144,7 +4144,7 @@ Section positives_flatten_unflatten.
       rewrite 2!(assoc_L (++)).
       reflexivity.
   Qed.
-  
+
   Lemma positives_unflatten_go_app p suffix xs acc :
     positives_unflatten_go (suffix ++ Preverse (Pdup p)) xs acc =
     positives_unflatten_go suffix xs (acc ++ p).
@@ -4161,7 +4161,7 @@ Section positives_flatten_unflatten.
       reflexivity.
     - reflexivity.
   Qed.
-  
+
   Lemma positives_unflatten_flatten_go suffix xs acc :
     positives_unflatten_go (suffix ++ positives_flatten_go xs 1) acc 1 =
     positives_unflatten_go suffix (xs ++ acc) 1.
@@ -4178,7 +4178,7 @@ Section positives_flatten_unflatten.
       rewrite (left_id_L 1 (++)).
       reflexivity.
   Qed.
-  
+
   Lemma positives_unflatten_flatten xs :
     positives_unflatten (positives_flatten xs) = Some xs.
   Proof.
@@ -4191,7 +4191,7 @@ Section positives_flatten_unflatten.
     rewrite (right_id_L [] (++)%list).
     reflexivity.
   Qed.
-  
+
   Lemma positives_flatten_app xs ys :
     positives_flatten (xs ++ ys) = positives_flatten xs ++ positives_flatten ys.
   Proof.
@@ -4205,7 +4205,7 @@ Section positives_flatten_unflatten.
       rewrite (assoc_L (++)).
       reflexivity.
   Qed.
-  
+
   Lemma positives_flatten_cons x xs :
     positives_flatten (x :: xs) = 1~1~0 ++ Preverse (Pdup x) ++ positives_flatten xs.
   Proof.
@@ -4214,7 +4214,7 @@ Section positives_flatten_unflatten.
     rewrite (assoc_L (++)).
     reflexivity.
   Qed.
-  
+
   Lemma positives_flatten_suffix (l k : list positive) :
     l `suffix_of` k → ∃ q, positives_flatten k = q ++ positives_flatten l.
   Proof.
@@ -4222,7 +4222,7 @@ Section positives_flatten_unflatten.
     exists (positives_flatten l').
     apply positives_flatten_app.
   Qed.
-  
+
   Lemma positives_flatten_suffix_eq p1 p2 (xs ys : list positive) :
     length xs = length ys →
     p1 ++ positives_flatten xs = p2 ++ positives_flatten ys →

theories/listset.v

@@ -29,7 +29,7 @@ Proof.
   destruct X as [l]; split; [|by intros; simplify_eq/=].
   rewrite elem_of_equiv_empty; intros Hl.
   destruct l as [|x l]; [done|]. feed inversion (Hl x). left.
-Qed. 
+Qed.
 Global Instance listset_empty_dec (X : listset A) : Decision (X ≡ ∅).
 Proof.
  refine (cast_if (decide (listset_car X = [])));

theories/numbers.v

@@ -213,7 +213,7 @@ Proof.
   - by rewrite Preverse_xO, Preverse_app, IH.
   - reflexivity.
 Qed.
-    
+
 Instance Preverse_inj : Inj (=) (=) Preverse.
 Proof.
   intros p q eq.
@@ -571,7 +571,7 @@ Proof.
   by apply Qcplus_le_mono_l.
 Qed.
 Lemma Qcplus_nonneg_pos (x y : Qc) : 0 ≤ x → 0 < y → 0 < x + y.
-Proof. rewrite (Qcplus_comm x). auto using Qcplus_pos_nonneg. Qed. 
+Proof. rewrite (Qcplus_comm x). auto using Qcplus_pos_nonneg. Qed.
 Lemma Qcplus_pos_pos (x y : Qc) : 0 < x → 0 < y → 0 < x + y.
 Proof. auto using Qcplus_pos_nonneg, Qclt_le_weak. Qed.
 Lemma Qcplus_nonneg_nonneg (x y : Qc) : 0 ≤ x → 0 ≤ y → 0 ≤ x + y.

theories/sets.v

@@ -564,7 +564,7 @@ Section semi_set.
     Lemma union_list_reverse_L Xs : ⋃ (reverse Xs) = ⋃ Xs.
     Proof. unfold_leibniz. apply union_list_reverse. Qed.
     Lemma empty_union_list_L Xs : ⋃ Xs = ∅ ↔ Forall (.= ∅) Xs.
-    Proof. unfold_leibniz. by rewrite empty_union_list. Qed. 
+    Proof. unfold_leibniz. by rewrite empty_union_list. Qed.
   End leibniz.
 
   Lemma not_elem_of_iff `{!RelDecision (∈@{C})} X Y x :
@@ -604,7 +604,7 @@ Section set.
 
   (** Intersection *)
   Lemma subseteq_intersection X Y : X ⊆ Y ↔ X ∩ Y ≡ X.
-  Proof. set_solver. Qed. 
+  Proof. set_solver. Qed.
   Lemma subseteq_intersection_1 X Y : X ⊆ Y → X ∩ Y ≡ X.
   Proof. apply subseteq_intersection. Qed.
   Lemma subseteq_intersection_2 X Y : X ∩ Y ≡ X → X ⊆ Y.

theories/tactics.v

@@ -520,7 +520,7 @@ Ltac find_pat pat tac :=
       tryif tac x then idtac else fail 2
   end.
 
-(** Coq's [firstorder] tactic fails or loops on rather small goals already. In 
+(** Coq's [firstorder] tactic fails or loops on rather small goals already. In
 particular, on those generated by the tactic [unfold_elem_ofs] which is used
 to solve propositions on sets. The [naive_solver] tactic implements an
 ad-hoc and incomplete [firstorder]-like solver using Ltac's backtracking