From 33808928db3c4b40e4739233eacb7ff554e8679e Mon Sep 17 00:00:00 2001
From: Dan Frumin <dfrumin@cs.ru.nl>
Date: Wed, 9 Aug 2017 18:05:58 +0200
Subject: [PATCH] Clean up trailing whitespaces and an unused definition.

---
 FiniteSets/disjunction.v                | 14 ++++----
 FiniteSets/fsets/extensionality.v       | 10 +++---
 FiniteSets/fsets/isomorphism.v          |  8 ++---
 FiniteSets/fsets/operations.v           | 27 ++++-----------
 FiniteSets/fsets/properties.v           | 18 +++++-----
 FiniteSets/fsets/properties_cons_repr.v | 24 +++++++-------
 FiniteSets/fsets/properties_decidable.v | 12 +++----
 FiniteSets/implementations/interface.v  | 10 +++---
 FiniteSets/implementations/lists.v      |  6 ++--
 FiniteSets/lattice.v                    | 20 +++++------
 FiniteSets/representations/T.v          | 44 ++++++++++++-------------
 FiniteSets/representations/cons_repr.v  | 22 ++++++-------
 FiniteSets/representations/definition.v | 36 ++++++++++----------
 FiniteSets/variations/enumerated.v      | 36 ++++++++++----------
 FiniteSets/variations/k_finite.v        |  2 +-
 15 files changed, 137 insertions(+), 152 deletions(-)

diff --git a/FiniteSets/disjunction.v b/FiniteSets/disjunction.v
index 9830b54..9aacaa5 100644
--- a/FiniteSets/disjunction.v
+++ b/FiniteSets/disjunction.v
@@ -12,7 +12,7 @@ Open Scope logic_scope.
 Section lor_props.
   Context `{Univalence}.
   Variable X Y Z : hProp.
-  
+
   Lemma lor_assoc : (X ∨ Y) ∨ Z = X ∨ Y ∨ Z.
   Proof.
     apply path_iff_hprop ; cbn.
@@ -20,15 +20,15 @@ Section lor_props.
       intros [xy | z] ; cbn.
       + simple refine (Trunc_ind _ _ xy).
         intros [x | y].
-        ++ apply (tr (inl x)). 
+        ++ apply (tr (inl x)).
         ++ apply (tr (inr (tr (inl y)))).
       + apply (tr (inr (tr (inr z)))).
-    * simple refine (Trunc_ind _ _).    
+    * simple refine (Trunc_ind _ _).
       intros [x | yz] ; cbn.
       + apply (tr (inl (tr (inl x)))).
       + simple refine (Trunc_ind _ _ yz).
         intros [y | z].
-        ++ apply (tr (inl (tr (inr y)))). 
+        ++ apply (tr (inl (tr (inr y)))).
         ++ apply (tr (inr z)).
   Defined.
 
@@ -131,7 +131,7 @@ Section hPropLattice.
     - intros [a b] ; apply a.
     - intros a ; apply (pair a a).
   Defined.
-  
+
   Instance lor_neutrall : NeutralL lor lfalse.
   Proof.
     unfold NeutralL.
@@ -169,7 +169,7 @@ Section hPropLattice.
       * assumption.
       * apply (tr (inl X)).
   Defined.
-  
+
   Global Instance lattice_hprop : Lattice hProp :=
     { commutative_min := _ ;
       commutative_max := _ ;
@@ -182,5 +182,5 @@ Section hPropLattice.
       absorption_min_max := _ ;
       absorption_max_min := _
   }.
-  
+
 End hPropLattice.
diff --git a/FiniteSets/fsets/extensionality.v b/FiniteSets/fsets/extensionality.v
index 7034a2c..6f2004e 100644
--- a/FiniteSets/fsets/extensionality.v
+++ b/FiniteSets/fsets/extensionality.v
@@ -6,7 +6,7 @@ Section ext.
   Context {A : Type}.
   Context `{Univalence}.
 
-  Lemma subset_union (X Y : FSet A) : 
+  Lemma subset_union (X Y : FSet A) :
     X ⊆ Y -> X ∪ Y = Y.
   Proof.
     hinduction X ; try (intros; apply path_forall; intro; apply set_path2).
@@ -17,7 +17,7 @@ Section ext.
         contradiction.
       + intro a0.
         simple refine (Trunc_ind _ _).
-        intro p ; simpl. 
+        intro p ; simpl.
         rewrite p; apply idem.
       + intros X1 X2 IH1 IH2.
         simple refine (Trunc_ind _ _).
@@ -112,8 +112,8 @@ Section ext.
     unshelve esplit.
     { intros [H1 H2]. etransitivity. apply H1^.
       rewrite comm. apply H2. }
-    intro; apply path_prod; apply set_path2. 
-    all: intro; apply set_path2.  
+    intro; apply path_prod; apply set_path2.
+    all: intro; apply set_path2.
   Defined.
 
   Lemma eq_subset (X Y : FSet A) :
@@ -149,4 +149,4 @@ Section ext.
         split ; apply subset_isIn.
   Defined.
 
-End ext.
\ No newline at end of file
+End ext.
diff --git a/FiniteSets/fsets/isomorphism.v b/FiniteSets/fsets/isomorphism.v
index c6c2105..a4fa178 100644
--- a/FiniteSets/fsets/isomorphism.v
+++ b/FiniteSets/fsets/isomorphism.v
@@ -14,7 +14,7 @@ Section Iso.
     - apply E.
     - intros a x.
       apply ({|a|} ∪ x).
-    - intros. cbn.  
+    - intros. cbn.
       etransitivity. apply assoc.
       apply (ap (∪ x)).
       apply idem.
@@ -22,7 +22,7 @@ Section Iso.
       etransitivity. apply assoc.
       etransitivity. refine (ap (∪ x) _ ).
       apply FSet.comm.
-      symmetry. 
+      symmetry.
       apply assoc.
   Defined.
 
@@ -39,10 +39,10 @@ Section Iso.
     - apply singleton_idem.
   Defined.
 
-  Lemma append_union: forall (x y: FSetC A), 
+  Lemma append_union: forall (x y: FSetC A),
       FSetC_to_FSet (x ∪ y) = (FSetC_to_FSet x) ∪ (FSetC_to_FSet y).
   Proof.
-    intros x. 
+    intros x.
     hrecursion x; try (intros; apply path_forall; intro; apply set_path2).
     - intros. symmetry. apply nl.
     - intros a x HR y. unfold union, fsetc_union in *. rewrite HR. apply assoc.
diff --git a/FiniteSets/fsets/operations.v b/FiniteSets/fsets/operations.v
index 506cdfa..6484e78 100644
--- a/FiniteSets/fsets/operations.v
+++ b/FiniteSets/fsets/operations.v
@@ -34,7 +34,7 @@ Section operations.
     - apply comm.
     - apply nl.
     - apply nr.
-    - intros; simpl. 
+    - intros; simpl.
       destruct (P x).
       + apply idem.
       + apply nl.
@@ -61,7 +61,7 @@ Section operations.
     - intros ; apply nr.
     - intros ; apply idem.
   Defined.
-        
+
   Definition product {A B : Type} : FSet A -> FSet B -> FSet (A * B).
   Proof.
     intros X Y.
@@ -76,7 +76,7 @@ Section operations.
     - intros ; apply nr.
     - intros ; apply union_idem.
   Defined.
-   
+
   Global Instance fset_subset : forall A, hasSubset (FSet A).
   Proof.
     intros A X Y.
@@ -103,21 +103,6 @@ Section operations.
         split ; apply p.
   Defined.
 
-  Definition map (A B : Type) (f : A -> B) : FSet A -> FSet B.
-  Proof.
-    hrecursion.
-    - apply ∅.
-    - intro a.
-      apply {|f a|}.
-    - apply (∪).
-    - apply assoc.
-    - apply comm.
-    - apply nl.
-    - apply nr.
-    - intros.
-      apply idem.
-  Defined.
-
   Local Ltac remove_transport
     := intros ; apply path_forall ; intro Z ; rewrite transport_arrow
        ; rewrite transport_const ; simpl.
@@ -155,6 +140,6 @@ Section operations.
     - remove_transport.
       rewrite <- (idem (Z (x; tr 1%path))).
       pointwise.
-  Defined.  
-      
-End operations.
\ No newline at end of file
+  Defined.
+
+End operations.
diff --git a/FiniteSets/fsets/properties.v b/FiniteSets/fsets/properties.v
index a14e1cb..14032e4 100644
--- a/FiniteSets/fsets/properties.v
+++ b/FiniteSets/fsets/properties.v
@@ -10,7 +10,7 @@ Section characterize_isIn.
 
   (** isIn properties *)
   Definition empty_isIn (a: A) : a ∈ ∅ -> Empty := idmap.
-  
+
   Definition singleton_isIn (a b: A) : a ∈ {|b|} -> Trunc (-1) (a = b) := idmap.
 
   Definition union_isIn (X Y : FSet A) (a : A)
@@ -20,7 +20,7 @@ Section characterize_isIn.
       {|X ∪ Y & ϕ|} = {|X & ϕ|} ∪ {|Y & ϕ|}.
   Proof.
     reflexivity.
-  Defined.  
+  Defined.
 
   Lemma comprehension_isIn (ϕ : A -> Bool) (a : A) : forall X : FSet A,
       a ∈ {|X & ϕ|} = if ϕ a then a ∈ X else False_hp.
@@ -37,7 +37,7 @@ Section characterize_isIn.
         destruct c ; destruct d ; rewrite Hc, Hd ; try reflexivity
         ; apply path_iff_hprop ; try contradiction ; intros ; strip_truncations
         ; apply (false_ne_true).
-        * apply (Hd^ @ ap ϕ X^ @ Hc). 
+        * apply (Hd^ @ ap ϕ X^ @ Hc).
         * apply (Hc^ @ ap ϕ X @ Hd).
       }
       apply (X (ϕ a) (ϕ b) idpath idpath).
@@ -57,7 +57,7 @@ End characterize_isIn.
 Section product_isIn.
   Context {A B : Type}.
   Context `{Univalence}.
-  
+
   Lemma isIn_singleproduct (a : A) (b : B) (c : A) : forall (Y : FSet B),
       (a, b) ∈ (single_product c Y) = land (BuildhProp (Trunc (-1) (a = c))) (b ∈ Y).
   Proof.
@@ -65,7 +65,7 @@ Section product_isIn.
     - apply path_hprop ; symmetry ; apply prod_empty_r.
     - intros d.
       apply path_iff_hprop.
-      * intros. 
+      * intros.
         strip_truncations.
         split ; apply tr ; try (apply (ap fst X)) ; try (apply (ap snd X)).
       * intros [Z1 Z2].
@@ -93,7 +93,7 @@ Section product_isIn.
         ** right.
            split ; try (apply (tr H1)) ; try (apply Hb2).
   Defined.
-  
+
   Definition isIn_product (a : A) (b : B) (X : FSet A) (Y : FSet B) :
     (a,b) ∈ (product X Y) = land (a ∈ X) (b ∈ Y).
   Proof.
@@ -147,7 +147,7 @@ Section properties.
   Global Instance joinsemilattice_fset : JoinSemiLattice (FSet A).
   Proof.
     split ; toHProp.
-  Defined.  
+  Defined.
 
   (** comprehension properties *)
   Lemma comprehension_false : forall (X : FSet A), {|X & fun _ => false|} = ∅.
@@ -176,7 +176,7 @@ Section properties.
   Proof.
     toHProp.
   Defined.
-  
+
   Lemma merely_choice : forall X : FSet A, hor (X = ∅) (hexists (fun a => a ∈ X)).
   Proof.
     hinduction; try (intros; apply equiv_hprop_allpath ; apply _).
@@ -295,5 +295,5 @@ Section properties.
            repeat f_ap.
            apply path_ishprop.
   Defined.
-  
+
 End properties.
diff --git a/FiniteSets/fsets/properties_cons_repr.v b/FiniteSets/fsets/properties_cons_repr.v
index 1a5ca87..dbb02be 100644
--- a/FiniteSets/fsets/properties_cons_repr.v
+++ b/FiniteSets/fsets/properties_cons_repr.v
@@ -7,7 +7,7 @@ Section properties.
   Context {A : Type}.
 
   Definition append_nl : forall (x: FSetC A), ∅ ∪ x = x
-    := fun _ => idpath. 
+    := fun _ => idpath.
 
   Lemma append_nr : forall (x: FSetC A), x ∪ ∅ = x.
   Proof.
@@ -15,20 +15,20 @@ Section properties.
     -  reflexivity.
     -  intros. apply (ap (fun y => a;;y) X).
   Defined.
-  
-  Lemma append_assoc {H: Funext}:  
+
+  Lemma append_assoc {H: Funext}:
     forall (x y z: FSetC A), x ∪ (y ∪ z) = (x ∪ y) ∪ z.
   Proof.
     hinduction
     ; try (intros ; apply path_forall ; intro
            ; apply path_forall ; intro ;  apply set_path2).
     - reflexivity.
-    - intros a x HR y z. 
+    - intros a x HR y z.
       specialize (HR y z).
-      apply (ap (fun y => a;;y) HR). 
+      apply (ap (fun y => a;;y) HR).
   Defined.
-  
-  Lemma append_singleton: forall (a: A) (x: FSetC A), 
+
+  Lemma append_singleton: forall (a: A) (x: FSetC A),
       a ;; x = x ∪ (a ;; ∅).
   Proof.
     intro a. hinduction; try (intros; apply set_path2).
@@ -38,22 +38,22 @@ Section properties.
       + apply (ap (fun y => b ;; y) HR).
   Defined.
 
-  Lemma append_comm {H: Funext}: 
+  Lemma append_comm {H: Funext}:
     forall (x1 x2: FSetC A), x1 ∪ x2 = x2 ∪ x1.
   Proof.
     hinduction ;  try (intros ; apply path_forall ; intro ; apply set_path2).
-    - intros. symmetry. apply append_nr. 
+    - intros. symmetry. apply append_nr.
     - intros a x1 HR x2.
       etransitivity.
-      apply (ap (fun y => a;;y) (HR x2)).  
+      apply (ap (fun y => a;;y) (HR x2)).
       transitivity  ((x2 ∪ x1) ∪ (a;;∅)).
-      + apply append_singleton. 
+      + apply append_singleton.
       + etransitivity.
     	* symmetry. apply append_assoc.
     	* simple refine (ap (x2 ∪) (append_singleton _ _)^).
   Defined.
 
-  Lemma singleton_idem: forall (a: A), 
+  Lemma singleton_idem: forall (a: A),
       {|a|} ∪ {|a|} = {|a|}.
   Proof.
     unfold singleton.
diff --git a/FiniteSets/fsets/properties_decidable.v b/FiniteSets/fsets/properties_decidable.v
index 83656e0..e4b3518 100644
--- a/FiniteSets/fsets/properties_decidable.v
+++ b/FiniteSets/fsets/properties_decidable.v
@@ -60,7 +60,7 @@ Section operations_isIn.
       contradiction.
     - reflexivity.
   Defined.
-  
+
   (* Union and membership *)
   Lemma union_isIn_b (X Y : FSet A) (a : A) :
     a ∈_d (X ∪ Y) = orb (a ∈_d X) (a ∈_d Y).
@@ -111,24 +111,24 @@ Section SetLattice.
     intros x y.
     apply (x ∪ y).
   Defined.
-      
+
   Instance fset_min : minimum (FSet A).
   Proof.
     intros x y.
     apply (x ∩ y).
   Defined.
-  
+
   Instance fset_bot : bottom (FSet A).
   Proof.
     unfold bottom.
     apply ∅.
   Defined.
-    
+
   Instance lattice_fset : Lattice (FSet A).
   Proof.
     split; toBool.
   Defined.
-  
+
 End SetLattice.
 
 (* With extensionality we can prove decidable equality *)
@@ -142,4 +142,4 @@ Section dec_eq.
     apply decidable_prod.
   Defined.
 
-End dec_eq.
\ No newline at end of file
+End dec_eq.
diff --git a/FiniteSets/implementations/interface.v b/FiniteSets/implementations/interface.v
index ba9443c..ed9216d 100644
--- a/FiniteSets/implementations/interface.v
+++ b/FiniteSets/implementations/interface.v
@@ -87,12 +87,12 @@ Section properties.
 
   Definition well_defined_filter : forall (A : Type) (ϕ : A -> Bool) (X Y : T A),
     set_eq A X Y -> set_eq A (filter ϕ X) (filter ϕ Y).
-  Proof.    
+  Proof.
     intros A ϕ X Y HXY.
     simplify.
     by rewrite HXY.
   Defined.
-  
+
   Lemma union_comm : forall A (X Y : T A),
     set_eq A (X ∪ Y) (Y ∪ X).
   Proof.
@@ -151,7 +151,7 @@ Proof. intros a b c Hab Hbc. apply (Hab @ Hbc). Defined.
 
 Instance View_recursion A : HitRecursion (View A) :=
   {
-    indTy := _; recTy := forall (P : Type) (HP: IsHSet P) (u : T A -> P), (forall x y : T A, set_eq T (@f) A x y -> u x = u y) -> View A -> P; 
+    indTy := _; recTy := forall (P : Type) (HP: IsHSet P) (u : T A -> P), (forall x y : T A, set_eq T (@f) A x y -> u x = u y) -> View A -> P;
     H_inductor := quotient_ind (R A); H_recursor := @quotient_rec _ (R A) _
   }.
 
@@ -169,7 +169,7 @@ assert (resp1 : forall x y y', set_eq (@f) y y' -> u x y = u x y').
 assert (resp2 : forall x x' y, set_eq (@f) x x' -> u x y = u x' y).
 { intros x x' y Hxx'.
   apply Hresp. apply Hxx'.
-  reflexivity. }  
+  reflexivity. }
 hrecursion.
 - intros a.
   hrecursion.
@@ -193,7 +193,7 @@ simple refine (View_rec2 _ _ _ _).
   apply related_classes_eq.
   unfold R in *.
   eapply well_defined_union; eauto.
-Defined.  
+Defined.
 
 Ltac reduce :=
   intros ;
diff --git a/FiniteSets/implementations/lists.v b/FiniteSets/implementations/lists.v
index 7228a95..05cdbc0 100644
--- a/FiniteSets/implementations/lists.v
+++ b/FiniteSets/implementations/lists.v
@@ -8,7 +8,7 @@ Section Operations.
   Global Instance list_empty A : hasEmpty (list A) := nil.
 
   Global Instance list_single A: hasSingleton (list A) A := fun a => cons a nil.
-  
+
   Global Instance list_union A : hasUnion (list A).
   Proof.
     intros l1 l2.
@@ -58,7 +58,7 @@ Section ListToSet.
       * strip_truncations ; apply (tr (inl z1)).
       * apply (tr (inr z2)).
   Defined.
-  
+
   Definition empty_empty : list_to_set A ∅ = ∅ := idpath.
 
   Lemma filter_comprehension (ϕ : A -> Bool) (l : list A)  :
@@ -72,7 +72,7 @@ Section ListToSet.
       * rewrite nl.
         apply IHl.
   Defined.
-  
+
   Definition singleton_single (a : A) : list_to_set A (singleton a) = {|a|} :=
     nr {|a|}.
 
diff --git a/FiniteSets/lattice.v b/FiniteSets/lattice.v
index e16c653..fe55e25 100644
--- a/FiniteSets/lattice.v
+++ b/FiniteSets/lattice.v
@@ -4,7 +4,7 @@ Require Import notation.
 
 Section binary_operation.
   Variable A : Type.
-  
+
   Class maximum :=
     max_L : operation A.
 
@@ -75,7 +75,7 @@ Section BoolLattice.
 
   Ltac solve_bool :=
     let x := fresh in
-    repeat (intro x ; destruct x) 
+    repeat (intro x ; destruct x)
     ; compute
     ; auto
     ; try contradiction.
@@ -83,12 +83,12 @@ Section BoolLattice.
   Instance maximum_bool : maximum Bool := orb.
   Instance minimum_bool : minimum Bool := andb.
   Instance bottom_bool : bottom Bool := false.
-  
+
   Global Instance lattice_bool : Lattice Bool.
   Proof.
     split ; solve_bool.
   Defined.
-  
+
   Definition and_true : forall b, andb b true = b.
   Proof.
     solve_bool.
@@ -116,7 +116,7 @@ Section BoolLattice.
   Proof.
     solve_bool.
   Defined.
-  
+
 End BoolLattice.
 
 Section fun_lattice.
@@ -141,7 +141,7 @@ Section fun_lattice.
   Proof.
     split ; solve_fun.
   Defined.
-  
+
 End fun_lattice.
 
 Section sub_lattice.
@@ -152,7 +152,7 @@ Section sub_lattice.
   Context {Hbot : P empty_L}.
 
   Definition AP : Type := sig P.
-  
+
   Instance botAP : bottom AP := (empty_L ; Hbot).
 
   Instance maxAP : maximum AP :=
@@ -174,17 +174,17 @@ Section sub_lattice.
 
   Ltac solve_sub :=
     let x := fresh in
-    repeat (intro x ; destruct x) 
+    repeat (intro x ; destruct x)
     ; simple refine (path_sigma _ _ _ _ _)
     ; simpl
     ; try (apply hprop_sub)
     ; eauto 3 with lattice_hints typeclass_instances.
-  
+
   Global Instance lattice_sub : Lattice AP.
   Proof.
     split ; solve_sub.
   Defined.
-  
+
 End sub_lattice.
 
 Create HintDb bool_lattice_hints.
diff --git a/FiniteSets/representations/T.v b/FiniteSets/representations/T.v
index f99ef53..2e435e6 100644
--- a/FiniteSets/representations/T.v
+++ b/FiniteSets/representations/T.v
@@ -7,7 +7,7 @@ Module Export T.
 
     Private Inductive T (B : Type) : Type :=
     | b : T B
-    | c : T B.    
+    | c : T B.
 
     Axiom p : A -> b A = c A.
     Axiom setT : IsHSet (T A).
@@ -23,7 +23,7 @@ Module Export T.
     Variable (bP : P (b A)).
     Variable (cP : P (c A)).
     Variable (pP : forall a : A, (p a) # bP = cP).
-    
+
     (* Induction principle *)
     Fixpoint T_ind
              (x : T A)
@@ -31,7 +31,7 @@ Module Export T.
       : P x
       := (match x return _ -> _ -> P x with
           | b => fun _ _ => bP
-          | c => fun _ _ => cP                              
+          | c => fun _ _ => cP
           end) pP H.
 
     Axiom T_ind_beta_p : forall (a : A),
@@ -68,7 +68,7 @@ Module Export T.
   End T_recursion.
 
   Instance T_recursion A : HitRecursion (T A)
-    := {indTy := _; recTy := _; 
+    := {indTy := _; recTy := _;
         H_inductor := T_ind A; H_recursor := T_rec A }.
 
 End T.
@@ -119,44 +119,44 @@ Section merely_dec_lem.
 
   Local Ltac f_prop := apply path_forall ; intro ; apply path_ishprop.
 
-  Lemma transport_code_b_x (a : A) : 
+  Lemma transport_code_b_x (a : A) :
     transport code_b (p a) = fun _ => a.
   Proof.
     f_prop.
   Defined.
 
-  Lemma transport_code_c_x (a : A) : 
+  Lemma transport_code_c_x (a : A) :
     transport code_c (p a) = fun _ => tt.
   Proof.
-    f_prop.    
+    f_prop.
   Defined.
 
-  Lemma transport_code_c_x_V (a : A) : 
+  Lemma transport_code_c_x_V (a : A) :
     transport code_c (p a)^ = fun _ => a.
-  Proof. 
-    f_prop.    
+  Proof.
+    f_prop.
   Defined.
 
-  Lemma transport_code_x_b (a : A) : 
+  Lemma transport_code_x_b (a : A) :
     transport (fun x => code x (b A)) (p a) = fun _ => a.
   Proof.
     f_prop.
   Defined.
 
-  Lemma transport_code_x_c (a : A) : 
+  Lemma transport_code_x_c (a : A) :
     transport (fun x => code x (c A)) (p a) = fun _ => tt.
   Proof.
     f_prop.
   Defined.
 
-  Lemma transport_code_x_c_V (a : A) : 
+  Lemma transport_code_x_c_V (a : A) :
     transport (fun x => code x (c A)) (p a)^ = fun _ => a.
   Proof.
     f_prop.
   Defined.
 
   Lemma ap_diag {B : Type} {x y : B} (p : x = y) :
-    ap (fun x : B => (x, x)) p = path_prod' p p.   
+    ap (fun x : B => (x, x)) p = path_prod' p p.
   Proof.
       by path_induction.
   Defined.
@@ -217,7 +217,7 @@ Section merely_dec_lem.
       refine (transport_arrow _ _ _ @ _).
       refine (transport_paths_FlFr _ _ @ _).
       rewrite transport_code_c_x_V.
-      hott_simpl.      
+      hott_simpl.
   Defined.
 
   Lemma transport_paths_FlFr_trunc :
@@ -229,7 +229,7 @@ Section merely_dec_lem.
     refine (ap tr _).
     exact ((concat_1p r)^ @ (concat_p1 (1 @ r))^).
   Defined.
-  
+
   Definition decode : forall (x y : T A), code x y -> x = y.
   Proof.
     simple refine (T_ind _ _ _ _ _ _); simpl.
@@ -248,7 +248,7 @@ Section merely_dec_lem.
         f_ap.
         refine (ap (fun x => (p x)) _).
         apply path_ishprop.
-      + intro.        
+      + intro.
         rewrite transport_code_x_c_V.
         hott_simpl.
       + intro b.
@@ -264,7 +264,7 @@ Section merely_dec_lem.
     intros p. induction p.
     simpl. revert u. simple refine (T_ind _ _ _ _ _ _).
     + simpl. reflexivity.
-    + simpl. reflexivity.    
+    + simpl. reflexivity.
     + intro a.
       apply set_path2.
   Defined.
@@ -278,12 +278,12 @@ Section merely_dec_lem.
       + simpl. intro a. apply path_ishprop.
       + intro a. apply path_forall; intros ?. apply set_path2.
     - simple refine (T_ind _ _ _ _ _ _).
-      + simpl. intro a. apply path_ishprop. 
-      + simpl. apply path_ishprop. 
+      + simpl. intro a. apply path_ishprop.
+      + simpl. apply path_ishprop.
       + intro a. apply path_forall; intros ?. apply set_path2.
     - intro a. repeat (apply path_forall; intros ?). apply set_path2.
   Defined.
-  
+
 
   Instance T_hprop (a : A) : IsHProp (b A = c A).
   Proof.
@@ -307,7 +307,7 @@ Section merely_dec_lem.
     rewrite ?decode_encode in H1.
     apply H1.
   Defined.
-  
+
   Lemma equiv_pathspace_T : (b A = c A) = A.
   Proof.
     apply path_iff_ishprop.
diff --git a/FiniteSets/representations/cons_repr.v b/FiniteSets/representations/cons_repr.v
index fef3f84..93d4c84 100644
--- a/FiniteSets/representations/cons_repr.v
+++ b/FiniteSets/representations/cons_repr.v
@@ -3,7 +3,7 @@ Require Import HoTT HitTactics.
 Require Export notation.
 
 Module Export FSetC.
-  
+
   Section FSetC.
     Private Inductive FSetC (A : Type) : Type :=
     | Nil : FSetC A
@@ -14,9 +14,9 @@ Module Export FSetC.
     Variable A : Type.
     Arguments Cns {_} _ _.
     Infix ";;" := Cns (at level 8, right associativity).
-    
+
     Axiom dupl : forall (a : A) (x : FSetC A),
-        a ;; a ;; x = a ;; x. 
+        a ;; a ;; x = a ;; x.
 
     Axiom comm : forall (a b : A) (x : FSetC A),
         a ;; b ;; x = b ;; a ;; x.
@@ -41,9 +41,9 @@ Module Export FSetC.
     Variable (duplP : forall (a: A) (x: FSetC A) (px : P x),
 	         dupl a x # cnsP a (a;;x) (cnsP a x px) = cnsP a x px).
     Variable (commP : forall (a b: A) (x: FSetC A) (px: P x),
-		 comm a b x # cnsP a (b;;x) (cnsP b x px) = 
+		 comm a b x # cnsP a (b;;x) (cnsP b x px) =
 		 cnsP b (a;;x) (cnsP a x px)).
-    
+
     (* Induction principle *)
     Fixpoint FSetC_ind
              (x : FSetC A)
@@ -76,18 +76,18 @@ Module Export FSetC.
     (* Recursion principle *)
     Definition FSetC_rec : FSetC A -> P.
     Proof.
-      simple refine (FSetC_ind _ _ _ _ _  _ _ ); 
+      simple refine (FSetC_ind _ _ _ _ _  _ _ );
         try (intros; simple refine ((transport_const _ _) @ _ ));  cbn.
       - apply nil.
-      - apply 	(fun a => fun _ => cns a). 
+      - apply 	(fun a => fun _ => cns a).
       - apply duplP.
-      - apply commP. 
+      - apply commP.
     Defined.
 
     Definition FSetC_rec_beta_dupl : forall (a: A) (x : FSetC A),
         ap FSetC_rec (dupl a x) = duplP a (FSetC_rec x).
     Proof.
-      intros. 
+      intros.
       eapply (cancelL (transport_const (dupl a x) _)).
       simple refine ((apD_const _ _)^ @ _).
       apply FSetC_ind_beta_dupl.
@@ -96,7 +96,7 @@ Module Export FSetC.
     Definition FSetC_rec_beta_comm : forall (a b: A) (x : FSetC A),
         ap FSetC_rec (comm a b x) = commP a b (FSetC_rec x).
     Proof.
-      intros. 
+      intros.
       eapply (cancelL (transport_const (comm a b x) _)).
       simple refine ((apD_const _ _)^ @ _).
       apply FSetC_ind_beta_comm.
@@ -107,7 +107,7 @@ Module Export FSetC.
 
   Instance FSetC_recursion A : HitRecursion (FSetC A) :=
     {
-      indTy := _; recTy := _; 
+      indTy := _; recTy := _;
       H_inductor := FSetC_ind A; H_recursor := FSetC_rec A
     }.
 
diff --git a/FiniteSets/representations/definition.v b/FiniteSets/representations/definition.v
index 6b2de43..2bc8fb9 100644
--- a/FiniteSets/representations/definition.v
+++ b/FiniteSets/representations/definition.v
@@ -14,7 +14,7 @@ Module Export FSet.
     Global Instance fset_union : forall A, hasUnion (FSet A) := U.
 
     Variable A : Type.
-    
+
     Axiom assoc : forall (x y z : FSet A),
         x ∪ (y ∪ z) = (x ∪ y) ∪ z.
 
@@ -38,7 +38,7 @@ Module Export FSet.
   Arguments comm {_} _ _.
   Arguments nl {_} _.
   Arguments nr {_} _.
-  Arguments idem {_} _.  
+  Arguments idem {_} _.
 
   Section FSet_induction.
     Variable A: Type.
@@ -47,22 +47,22 @@ Module Export FSet.
     Variable  (eP : P ∅).
     Variable  (lP : forall a: A, P {|a|}).
     Variable  (uP : forall (x y: FSet A), P x -> P y -> P (x ∪ y)).
-    Variable  (assocP : forall (x y z : FSet A) 
+    Variable  (assocP : forall (x y z : FSet A)
                                (px: P x) (py: P y) (pz: P z),
                   assoc x y z #
-                        (uP x (y ∪ z) px (uP y z py pz)) 
-                  = 
+                        (uP x (y ∪ z) px (uP y z py pz))
+                  =
                   (uP (x ∪ y) z (uP x y px py) pz)).
     Variable  (commP : forall (x y: FSet A) (px: P x) (py: P y),
                   comm x y #
                        uP x y px py = uP y x py px).
-    Variable  (nlP : forall (x : FSet A) (px: P x), 
+    Variable  (nlP : forall (x : FSet A) (px: P x),
                   nl x # uP ∅ x eP px = px).
-    Variable  (nrP : forall (x : FSet A) (px: P x), 
+    Variable  (nrP : forall (x : FSet A) (px: P x),
                   nr x # uP x ∅ px eP = px).
-    Variable  (idemP : forall (x : A), 
+    Variable  (idemP : forall (x : A),
                   idem x # uP {|x|} {|x|} (lP x) (lP x) = lP x).
-    
+
     (* Induction principle *)
     Fixpoint FSet_ind
              (x : FSet A)
@@ -119,7 +119,7 @@ Module Export FSet.
     Defined.
 
     Definition FSet_rec_beta_assoc : forall (x y z : FSet A),
-        ap FSet_rec (assoc x y z) 
+        ap FSet_rec (assoc x y z)
         =
         assocP (FSet_rec x) (FSet_rec y) (FSet_rec z).
     Proof.
@@ -131,7 +131,7 @@ Module Export FSet.
     Defined.
 
     Definition FSet_rec_beta_comm : forall (x y : FSet A),
-        ap FSet_rec (comm x y) 
+        ap FSet_rec (comm x y)
         =
         commP (FSet_rec x) (FSet_rec y).
     Proof.
@@ -143,7 +143,7 @@ Module Export FSet.
     Defined.
 
     Definition FSet_rec_beta_nl : forall (x : FSet A),
-        ap FSet_rec (nl x) 
+        ap FSet_rec (nl x)
         =
         nlP (FSet_rec x).
     Proof.
@@ -155,7 +155,7 @@ Module Export FSet.
     Defined.
 
     Definition FSet_rec_beta_nr : forall (x : FSet A),
-        ap FSet_rec (nr x) 
+        ap FSet_rec (nr x)
         =
         nrP (FSet_rec x).
     Proof.
@@ -167,7 +167,7 @@ Module Export FSet.
     Defined.
 
     Definition FSet_rec_beta_idem : forall (a : A),
-        ap FSet_rec (idem a) 
+        ap FSet_rec (idem a)
         =
         idemP a.
     Proof.
@@ -177,12 +177,12 @@ Module Export FSet.
       simple refine ((apD_const _ _)^ @ _).
       apply FSet_ind_beta_idem.
     Defined.
-    
+
   End FSet_recursion.
 
   Instance FSet_recursion A : HitRecursion (FSet A) :=
     {
-      indTy := _; recTy := _; 
+      indTy := _; recTy := _;
       H_inductor := FSet_ind A; H_recursor := FSet_rec A
     }.
 
@@ -200,5 +200,5 @@ Proof.
     rewrite P.
     rewrite (comm y x).
     rewrite <- (assoc x y y).
-    f_ap. 
-Defined.
\ No newline at end of file
+    f_ap.
+Defined.
diff --git a/FiniteSets/variations/enumerated.v b/FiniteSets/variations/enumerated.v
index 14b5286..8b73de8 100644
--- a/FiniteSets/variations/enumerated.v
+++ b/FiniteSets/variations/enumerated.v
@@ -47,7 +47,7 @@ Proof.
   destruct (if P a as b return ((b = true) + (b = false))
      then inl 1%path
      else inr 1%path) as [Pa' | Pa'].
-  - rewrite Pa' in Pa. contradiction (true_ne_false Pa). 
+  - rewrite Pa' in Pa. contradiction (true_ne_false Pa).
   - reflexivity.
 Defined.
 
@@ -104,7 +104,7 @@ Defined.
 Lemma enumerated_surj (A B : Type) (f : A -> B) :
   IsSurjection f -> enumerated A -> enumerated B.
 Proof.
-  intros Hsurj HeA. strip_truncations; apply tr. 
+  intros Hsurj HeA. strip_truncations; apply tr.
   destruct HeA as [eA HeA].
   exists (map f eA).
   intros x. specialize (Hsurj x).
@@ -157,7 +157,7 @@ destruct ys as [|y ys].
 Defined.
 
 Fixpoint listProd {A B} (xs : list A) (ys : list B) : list (A * B).
-Proof.  
+Proof.
 destruct xs as [|x xs].
 - exact nil.
 - refine (app _ _).
@@ -165,7 +165,7 @@ destruct xs as [|x xs].
   + exact (listProd _ _ xs ys).
 Defined.
 
-Lemma listExt_prod_sing {A B} (x : A) (y : B) (ys : list B) : 
+Lemma listExt_prod_sing {A B} (x : A) (y : B) (ys : list B) :
   listExt ys y -> listExt (listProd_sing x ys) (x, y).
 Proof.
 induction ys; simpl.
@@ -193,11 +193,11 @@ induction xs as [| x' xs]; intros x y.
       rewrite <- Hyy' in IHxs.
       apply listExt_app_l. apply IHxs. assumption.
       simpl. apply tr. left. apply tr. reflexivity.
-    * right. 
+    * right.
       apply listExt_app_l.
       apply IHxs. assumption.
       simpl. apply tr. right. assumption.
-Defined.      
+Defined.
 
 (** Properties of enumerated sets: closed under products *)
 Lemma enumerated_prod (A B : Type) `{Funext} :
@@ -221,7 +221,7 @@ Section enumerated_fset.
     | nil => ∅
     | cons x xs => {|x|} ∪ (list_to_fset xs)
     end.
-  
+
   Lemma list_to_fset_ext (ls : list A) (a : A):
     listExt ls a -> a ∈ (list_to_fset ls).
   Proof.
@@ -250,8 +250,8 @@ End enumerated_fset.
 Section fset_dec_enumerated.
   Variable A : Type.
   Context `{Univalence}.
- 
-  Definition Kf_fsetc : 
+
+  Definition Kf_fsetc :
     Kf A -> exists (X : FSetC A), forall (a : A), k_finite.map (FSetC_to_FSet X) a.
   Proof.
     intros [X HX].
@@ -260,7 +260,7 @@ Section fset_dec_enumerated.
     by rewrite <- HX.
   Defined.
 
-  Definition merely_enumeration_FSetC : 
+  Definition merely_enumeration_FSetC :
     forall (X : FSetC A),
     hexists (fun (ls : list A) => forall a, a ∈ (FSetC_to_FSet X) = listExt ls a).
   Proof.
@@ -274,13 +274,13 @@ Section fset_dec_enumerated.
     - intros. apply path_ishprop.
     - intros. apply path_ishprop.
   Defined.
-  
+
   Definition Kf_enumerated : Kf A -> enumerated A.
   Proof.
     intros HKf. apply Kf_fsetc in HKf.
     destruct HKf as [X HX].
-    pose (ls' := (merely_enumeration_FSetC X)).    
-    simple refine (@Trunc_rec _ _ _ _ _ ls'). clear ls'.    
+    pose (ls' := (merely_enumeration_FSetC X)).
+    simple refine (@Trunc_rec _ _ _ _ _ ls'). clear ls'.
     intros [ls Hls].
     apply tr. exists ls.
     intros a. rewrite <- Hls. apply (HX a).
@@ -293,7 +293,7 @@ Section subobjects.
 
   Definition enumeratedS (P : Sub A) : hProp :=
     enumerated (sigT P).
-  
+
   Lemma enumeratedS_empty : closedEmpty enumeratedS.
   Proof.
     unfold enumeratedS.
@@ -319,7 +319,7 @@ Section subobjects.
     - apply (cons (x; tr (inr Hx))).
       apply (weaken_list_r _ _ ls).
   Defined.
- 
+
   Lemma listExt_weaken (P Q : Sub A) (ls : list (sigT Q)) (x : A) (Hx : Q x) :
     listExt ls (x; Hx) -> listExt (weaken_list_r P Q ls) (x; tr (inr Hx)).
   Proof.
@@ -333,7 +333,7 @@ Section subobjects.
         exists (Hxy..1). apply path_ishprop.
       + right. apply IHls. assumption.
   Defined.
-  
+
   Fixpoint concatD {P Q : Sub A}
     (ls : list (sigT P)) (ls' : list (sigT Q)) : list (sigT (max_L P Q)).
   Proof.
@@ -382,9 +382,9 @@ Section subobjects.
   Defined.
 
   Opaque enumeratedS.
-  Definition FSet_to_enumeratedS : 
+  Definition FSet_to_enumeratedS :
     forall (X : FSet A), enumeratedS (k_finite.map X).
-  Proof.  
+  Proof.
     hinduction.
     - apply enumeratedS_empty.
     - intros a. apply enumeratedS_singleton.
diff --git a/FiniteSets/variations/k_finite.v b/FiniteSets/variations/k_finite.v
index c042744..4457bdd 100644
--- a/FiniteSets/variations/k_finite.v
+++ b/FiniteSets/variations/k_finite.v
@@ -100,7 +100,7 @@ Section structure_k_finite.
     exists {|a|}.
     cbn.
     apply path_forall.
-    intro z. 
+    intro z.
     reflexivity.
   Defined.