``` ```

Finite sets library

``` ```
This functor derives additional properties from `FSetInterface.S`. Contrary to the functor in `FSetEqProperties` it uses predicates over sets instead of sets operations, i.e. `In x s` instead of `mem x s=true`, `Equal s s'` instead of `equal s s'=true`, etc.
``` Require Export FSetInterface. Require Import FSetFacts. Set Implicit Arguments. Unset Strict Implicit. Hint Unfold transpose compat_op compat_nat. Hint Extern 1 (Setoid_Theory _ _) => constructor; congruence. Module Properties (M: S).   Module ME:=OrderedTypeFacts(M.E).   Import ME.   Import M.E.   Import M.   Import Logic.   Import Peano. ```
```   Module FM := Facts M.   Import FM.   Definition Add (x : elt) (s s' : t) :=     forall y : elt, In y s' <-> E.eq x y \/ In y s.   Lemma In_dec : forall x s, {In x s} + {~ In x s}.   Proof.   intros; generalize (mem_iff s x); case (mem x s); intuition.   Qed.   Section BasicProperties. ```
properties of `Equal`
```   Lemma equal_refl : forall s, s[=]s.   Proof.   unfold Equal; intuition.   Qed.   Lemma equal_sym : forall s s', s[=]s' -> s'[=]s.   Proof.   unfold Equal; intros.   rewrite H; intuition.   Qed.   Lemma equal_trans : forall s1 s2 s3, s1[=]s2 -> s2[=]s3 -> s1[=]s3.   Proof.   unfold Equal; intros.   rewrite H; exact (H0 a).   Qed.   Variable s s' s'' s1 s2 s3 : t.   Variable x x' : elt. ```
properties of `Subset`
```   Lemma subset_refl : s[<=]s.   Proof.   unfold Subset; intuition.   Qed.   Lemma subset_antisym : s[<=]s' -> s'[<=]s -> s[=]s'.   Proof.   unfold Subset, Equal; intuition.   Qed.   Lemma subset_trans : s1[<=]s2 -> s2[<=]s3 -> s1[<=]s3.   Proof.   unfold Subset; intuition.   Qed.   Lemma subset_equal : s[=]s' -> s[<=]s'.   Proof.   unfold Subset, Equal; firstorder.   Qed.   Lemma subset_empty : empty[<=]s.   Proof.   unfold Subset; intros a; set_iff; intuition.   Qed.   Lemma subset_remove_3 : s1[<=]s2 -> remove x s1 [<=] s2.   Proof.   unfold Subset; intros H a; set_iff; intuition.   Qed.   Lemma subset_diff : s1[<=]s3 -> diff s1 s2 [<=] s3.   Proof.   unfold Subset; intros H a; set_iff; intuition.   Qed.   Lemma subset_add_3 : In x s2 -> s1[<=]s2 -> add x s1 [<=] s2.   Proof.   unfold Subset; intros H H0 a; set_iff; intuition.   rewrite <- H2; auto.   Qed.   Lemma subset_add_2 : s1[<=]s2 -> s1[<=] add x s2.   Proof.   unfold Subset; intuition.   Qed.   Lemma in_subset : In x s1 -> s1[<=]s2 -> In x s2.   Proof.   unfold Subset; intuition.   Qed.   Lemma double_inclusion : s1[=]s2 <-> s1[<=]s2 /\ s2[<=]s1.   Proof.   unfold Subset, Equal; split; intros; intuition; generalize (H a); intuition.   Qed. ```
properties of `empty`
```   Lemma empty_is_empty_1 : Empty s -> s[=]empty.   Proof.   unfold Empty, Equal; intros; generalize (H a); set_iff; tauto.   Qed.   Lemma empty_is_empty_2 : s[=]empty -> Empty s.   Proof.   unfold Empty, Equal; intros; generalize (H a); set_iff; tauto.   Qed. ```
properties of `add`
```   Lemma add_equal : In x s -> add x s [=] s.   Proof.   unfold Equal; intros; set_iff; intuition.   rewrite <- H1; auto.   Qed.   Lemma add_add : add x (add x' s) [=] add x' (add x s).   Proof.   unfold Equal; intros; set_iff; tauto.   Qed. ```
properties of `remove`
```   Lemma remove_equal : ~ In x s -> remove x s [=] s.   Proof.   unfold Equal; intros; set_iff; intuition.   rewrite H1 in H; auto.   Qed.   Lemma Equal_remove : s[=]s' -> remove x s [=] remove x s'.   Proof.   intros; rewrite H; apply equal_refl.   Qed. ```
properties of `add` and `remove`
```   Lemma add_remove : In x s -> add x (remove x s) [=] s.   Proof.   unfold Equal; intros; set_iff; elim (eq_dec x a); intuition.   rewrite <- H1; auto.   Qed.   Lemma remove_add : ~In x s -> remove x (add x s) [=] s.   Proof.   unfold Equal; intros; set_iff; elim (eq_dec x a); intuition.   rewrite H1 in H; auto.   Qed. ```
properties of `singleton`
```   Lemma singleton_equal_add : singleton x [=] add x empty.   Proof.   unfold Equal; intros; set_iff; intuition.   Qed. ```
properties of `union`
```   Lemma union_sym : union s s' [=] union s' s.   Proof.   unfold Equal; intros; set_iff; tauto.   Qed.   Lemma union_subset_equal : s[<=]s' -> union s s' [=] s'.   Proof.   unfold Subset, Equal; intros; set_iff; intuition.   Qed.   Lemma union_equal_1 : s[=]s' -> union s s'' [=] union s' s''.   Proof.   intros; rewrite H; apply equal_refl.   Qed.   Lemma union_equal_2 : s'[=]s'' -> union s s' [=] union s s''.   Proof.   intros; rewrite H; apply equal_refl.   Qed.   Lemma union_assoc : union (union s s') s'' [=] union s (union s' s'').   Proof.   unfold Equal; intros; set_iff; tauto.   Qed.   Lemma add_union_singleton : add x s [=] union (singleton x) s.   Proof.   unfold Equal; intros; set_iff; tauto.   Qed.   Lemma union_add : union (add x s) s' [=] add x (union s s').   Proof.   unfold Equal; intros; set_iff; tauto.   Qed.   Lemma union_subset_1 : s [<=] union s s'.   Proof.   unfold Subset; intuition.   Qed.   Lemma union_subset_2 : s' [<=] union s s'.   Proof.   unfold Subset; intuition.   Qed.   Lemma union_subset_3 : s[<=]s'' -> s'[<=]s'' -> union s s' [<=] s''.   Proof.   unfold Subset; intros H H0 a; set_iff; intuition.   Qed.   Lemma union_subset_4 : s[<=]s' -> union s s'' [<=] union s' s''.   Proof.   unfold Subset; intros H a; set_iff; intuition.   Qed.   Lemma union_subset_5 : s[<=]s' -> union s'' s [<=] union s'' s'.   Proof.   unfold Subset; intros H a; set_iff; intuition.   Qed.   Lemma empty_union_1 : Empty s -> union s s' [=] s'.   Proof.   unfold Equal, Empty; intros; set_iff; firstorder.   Qed.   Lemma empty_union_2 : Empty s -> union s' s [=] s'.   Proof.   unfold Equal, Empty; intros; set_iff; firstorder.   Qed.   Lemma not_in_union : ~In x s -> ~In x s' -> ~In x (union s s').   Proof.   intros; set_iff; intuition.   Qed. ```
properties of `inter`
```   Lemma inter_sym : inter s s' [=] inter s' s.   Proof.   unfold Equal; intros; set_iff; tauto.   Qed.   Lemma inter_subset_equal : s[<=]s' -> inter s s' [=] s.   Proof.   unfold Equal; intros; set_iff; intuition.   Qed.   Lemma inter_equal_1 : s[=]s' -> inter s s'' [=] inter s' s''.   Proof.   intros; rewrite H; apply equal_refl.   Qed.   Lemma inter_equal_2 : s'[=]s'' -> inter s s' [=] inter s s''.   Proof.   intros; rewrite H; apply equal_refl.   Qed.   Lemma inter_assoc : inter (inter s s') s'' [=] inter s (inter s' s'').   Proof.   unfold Equal; intros; set_iff; tauto.   Qed.   Lemma union_inter_1 : inter (union s s') s'' [=] union (inter s s'') (inter s' s'').   Proof.   unfold Equal; intros; set_iff; tauto.   Qed.   Lemma union_inter_2 : union (inter s s') s'' [=] inter (union s s'') (union s' s'').   Proof.   unfold Equal; intros; set_iff; tauto.   Qed.   Lemma inter_add_1 : In x s' -> inter (add x s) s' [=] add x (inter s s').   Proof.   unfold Equal; intros; set_iff; intuition.   rewrite <- H1; auto.   Qed.   Lemma inter_add_2 : ~ In x s' -> inter (add x s) s' [=] inter s s'.   Proof.   unfold Equal; intros; set_iff; intuition.   destruct H; rewrite H0; auto.   Qed.   Lemma empty_inter_1 : Empty s -> Empty (inter s s').   Proof.   unfold Empty; intros; set_iff; firstorder.   Qed.   Lemma empty_inter_2 : Empty s' -> Empty (inter s s').   Proof.   unfold Empty; intros; set_iff; firstorder.   Qed.   Lemma inter_subset_1 : inter s s' [<=] s.   Proof.   unfold Subset; intro a; set_iff; tauto.   Qed.   Lemma inter_subset_2 : inter s s' [<=] s'.   Proof.   unfold Subset; intro a; set_iff; tauto.   Qed.   Lemma inter_subset_3 :    s''[<=]s -> s''[<=]s' -> s''[<=] inter s s'.   Proof.   unfold Subset; intros H H' a; set_iff; intuition.   Qed. ```
properties of `diff`
```   Lemma empty_diff_1 : Empty s -> Empty (diff s s').   Proof.   unfold Empty, Equal; intros; set_iff; firstorder.   Qed.   Lemma empty_diff_2 : Empty s -> diff s' s [=] s'.   Proof.   unfold Empty, Equal; intros; set_iff; firstorder.   Qed.   Lemma diff_subset : diff s s' [<=] s.   Proof.   unfold Subset; intros a; set_iff; tauto.   Qed.   Lemma diff_subset_equal : s[<=]s' -> diff s s' [=] empty.   Proof.   unfold Subset, Equal; intros; set_iff; intuition; absurd (In a empty); auto.   Qed.   Lemma remove_diff_singleton :    remove x s [=] diff s (singleton x).   Proof.   unfold Equal; intros; set_iff; intuition.   Qed.   Lemma diff_inter_empty : inter (diff s s') (inter s s') [=] empty.   Proof.   unfold Equal; intros; set_iff; intuition; absurd (In a empty); auto.   Qed.   Lemma diff_inter_all : union (diff s s') (inter s s') [=] s.   Proof.   unfold Equal; intros; set_iff; intuition.   elim (In_dec a s'); auto.   Qed. ```
properties of `Add`
```   Lemma Add_add : Add x s (add x s).   Proof.    unfold Add; intros; set_iff; intuition.   Qed.   Lemma Add_remove : In x s -> Add x (remove x s) s.   Proof.     unfold Add; intros; set_iff; intuition.     elim (eq_dec x y); auto.     rewrite <- H1; auto.   Qed.   Lemma union_Add : Add x s s' -> Add x (union s s'') (union s' s'').   Proof.   unfold Add; intros; set_iff; rewrite H; tauto.   Qed.   Lemma inter_Add :    In x s'' -> Add x s s' -> Add x (inter s s'') (inter s' s'').   Proof.   unfold Add; intros; set_iff; rewrite H0; intuition.   rewrite <- H2; auto.   Qed.   Lemma union_Equal :    In x s'' -> Add x s s' -> union s s'' [=] union s' s''.   Proof.   unfold Add, Equal; intros; set_iff; rewrite H0; intuition.   rewrite <- H1; auto.   Qed.   Lemma inter_Add_2 :    ~In x s'' -> Add x s s' -> inter s s'' [=] inter s' s''.   Proof.   unfold Add, Equal; intros; set_iff; rewrite H0; intuition.   destruct H; rewrite H1; auto.   Qed.   End BasicProperties.   Hint Immediate equal_sym: set.   Hint Resolve equal_refl equal_trans : set.   Hint Immediate add_remove remove_add union_sym inter_sym: set.   Hint Resolve subset_refl subset_equal subset_antisym     subset_trans subset_empty subset_remove_3 subset_diff subset_add_3     subset_add_2 in_subset empty_is_empty_1 empty_is_empty_2 add_equal     remove_equal singleton_equal_add union_subset_equal union_equal_1     union_equal_2 union_assoc add_union_singleton union_add union_subset_1     union_subset_2 union_subset_3 inter_subset_equal inter_equal_1 inter_equal_2     inter_assoc union_inter_1 union_inter_2 inter_add_1 inter_add_2     empty_inter_1 empty_inter_2 empty_union_1 empty_union_2 empty_diff_1     empty_diff_2 union_Add inter_Add union_Equal inter_Add_2 not_in_union     inter_subset_1 inter_subset_2 inter_subset_3 diff_subset diff_subset_equal     remove_diff_singleton diff_inter_empty diff_inter_all Add_add Add_remove     Equal_remove add_add : set. ```

Alternative (weaker) specifications for `fold`

```   Section Old_Spec_Now_Properties.   Notation NoDup := (NoDupA E.eq). ```
When `FSets` was first designed, the order in which Ocaml's `Set.fold` takes the set elements was unspecified. This specification reflects this fact:
```   Lemma fold_0 :       forall s (A : Set) (i : A) (f : elt -> A -> A),       exists l : list elt,         NoDup l /\         (forall x : elt, In x s <-> InA E.eq x l) /\         fold f s i = fold_right f i l.   Proof.   intros; exists (rev (elements s)); split.   apply NoDupA_rev; auto.   exact E.eq_trans.   split; intros.   rewrite elements_iff; do 2 rewrite InA_alt.   split; destruct 1; generalize (In_rev (elements s) x0); exists x0; intuition.   rewrite fold_left_rev_right.   apply fold_1.   Qed. ```
An alternate (and previous) specification for `fold` was based on the recursive structure of a set. It is now lemmas `fold_1` and `fold_2`.
```   Lemma fold_1 :    forall s (A : Set) (eqA : A -> A -> Prop)      (st : Setoid_Theory A eqA) (i : A) (f : elt -> A -> A),    Empty s -> eqA (fold f s i) i.   Proof.   unfold Empty; intros; destruct (fold_0 s i f) as (l,(H1, (H2, H3))).   rewrite H3; clear H3.   generalize H H2; clear H H2; case l; simpl; intros.   refl_st.   elim (H e).   elim (H2 e); intuition.   Qed.   Lemma fold_2 :    forall s s' x (A : Set) (eqA : A -> A -> Prop)      (st : Setoid_Theory A eqA) (i : A) (f : elt -> A -> A),    compat_op E.eq eqA f ->    transpose eqA f ->    ~ In x s -> Add x s s' -> eqA (fold f s' i) (f x (fold f s i)).   Proof.   intros; destruct (fold_0 s i f) as (l,(Hl, (Hl1, Hl2)));     destruct (fold_0 s' i f) as (l',(Hl', (Hl'1, Hl'2))).   rewrite Hl2; rewrite Hl'2; clear Hl2 Hl'2.   apply fold_right_add with (eqA:=E.eq)(eqB:=eqA); auto.   eauto.   exact eq_dec.   rewrite <- Hl1; auto.   intros; rewrite <- Hl1; rewrite <- Hl'1; auto.   Qed. ```
Similar specifications for `cardinal`.
```   Lemma cardinal_fold : forall s, cardinal s = fold (fun _ => S) s 0.   Proof.   intros; rewrite cardinal_1; rewrite M.fold_1.   symmetry; apply fold_left_length; auto.   Qed.   Lemma cardinal_0 :      forall s, exists l : list elt,         NoDupA E.eq l /\         (forall x : elt, In x s <-> InA E.eq x l) /\         cardinal s = length l.   Proof.   intros; exists (elements s); intuition; apply cardinal_1.   Qed.   Lemma cardinal_1 : forall s, Empty s -> cardinal s = 0.   Proof.   intros; rewrite cardinal_fold; apply fold_1; auto.   Qed.   Lemma cardinal_2 :     forall s s' x, ~ In x s -> Add x s s' -> cardinal s' = S (cardinal s).   Proof.   intros; do 2 rewrite cardinal_fold.   change S with ((fun _ => S) x).   apply fold_2; auto.   Qed.   End Old_Spec_Now_Properties. ```

Induction principle over sets

```   Lemma cardinal_inv_1 : forall s, cardinal s = 0 -> Empty s.   Proof.     intros s; rewrite M.cardinal_1; intros H a; red.     rewrite elements_iff.     destruct (elements s); simpl in *; discriminate || inversion 1.   Qed.   Hint Resolve cardinal_inv_1.   Lemma cardinal_inv_2 :    forall s n, cardinal s = S n -> { x : elt | In x s }.   Proof.     intros; rewrite M.cardinal_1 in H.     generalize (elements_2 (s:=s)).     destruct (elements s); try discriminate.     exists e; auto.   Qed.   Lemma Equal_cardinal_aux :    forall n s s', cardinal s = n -> s[=]s' -> cardinal s = cardinal s'.   Proof.      simple induction n; intros.      rewrite H; symmetry .      apply cardinal_1.      rewrite <- H0; auto.      destruct (cardinal_inv_2 H0) as (x,H2).      revert H0.      rewrite (cardinal_2 (s:=remove x s) (s':=s) (x:=x)); auto with set.      rewrite (cardinal_2 (s:=remove x s') (s':=s') (x:=x)); auto with set.      rewrite H1 in H2; auto with set.   Qed.   Lemma Equal_cardinal : forall s s', s[=]s' -> cardinal s = cardinal s'.   Proof.     intros; apply Equal_cardinal_aux with (cardinal s); auto.   Qed.   Add Morphism cardinal : cardinal_m.   Proof.   exact Equal_cardinal.   Qed.   Hint Resolve Add_add Add_remove Equal_remove cardinal_inv_1 Equal_cardinal.   Lemma cardinal_induction :    forall P : t -> Type,    (forall s, Empty s -> P s) ->    (forall s s', P s -> forall x, ~In x s -> Add x s s' -> P s') ->    forall n s, cardinal s = n -> P s.   Proof.   simple induction n; intros; auto.   destruct (cardinal_inv_2 H) as (x,H0).   apply X0 with (remove x s) x; auto.   apply X1; auto.   rewrite (cardinal_2 (x:=x)(s:=remove x s)(s':=s)) in H; auto.   Qed.   Lemma set_induction :    forall P : t -> Type,    (forall s : t, Empty s -> P s) ->    (forall s s' : t, P s -> forall x : elt, ~In x s -> Add x s s' -> P s') ->    forall s : t, P s.   Proof.   intros; apply cardinal_induction with (cardinal s); auto.   Qed. ```
Other properties of `fold`.
```   Section Fold.   Variables (A:Set)(eqA:A->A->Prop)(st:Setoid_Theory _ eqA).   Variables (f:elt->A->A)(Comp:compat_op E.eq eqA f)(Ass:transpose eqA f).   Section Fold_1.   Variable i i':A.   Lemma fold_empty : eqA (fold f empty i) i.   Proof.   apply fold_1; auto.   Qed.   Lemma fold_equal :    forall s s', s[=]s' -> eqA (fold f s i) (fold f s' i).   Proof.   intros s; pattern s; apply set_induction; clear s; intros.   trans_st i.   apply fold_1; auto.   sym_st; apply fold_1; auto.   rewrite <- H0; auto.   trans_st (f x (fold f s i)).   apply fold_2 with (eqA := eqA); auto.   sym_st; apply fold_2 with (eqA := eqA); auto.   unfold Add in *; intros.   rewrite <- H2; auto.   Qed.   Lemma fold_add : forall s x, ~In x s ->    eqA (fold f (add x s) i) (f x (fold f s i)).   Proof.   intros; apply fold_2 with (eqA := eqA); auto.   Qed.   Lemma add_fold : forall s x, In x s ->    eqA (fold f (add x s) i) (fold f s i).   Proof.   intros; apply fold_equal; auto with set.   Qed.   Lemma remove_fold_1: forall s x, In x s ->    eqA (f x (fold f (remove x s) i)) (fold f s i).   Proof.   intros.   sym_st.   apply fold_2 with (eqA:=eqA); auto.   Qed.   Lemma remove_fold_2: forall s x, ~In x s ->    eqA (fold f (remove x s) i) (fold f s i).   Proof.   intros.   apply fold_equal; auto with set.   Qed.   Lemma fold_commutes : forall s x,    eqA (fold f s (f x i)) (f x (fold f s i)).   Proof.   intros; pattern s; apply set_induction; clear s; intros.   trans_st (f x i).   apply fold_1; auto.   sym_st.   apply Comp; auto.   apply fold_1; auto.   trans_st (f x0 (fold f s (f x i))).   apply fold_2 with (eqA:=eqA); auto.   trans_st (f x0 (f x (fold f s i))).   trans_st (f x (f x0 (fold f s i))).   apply Comp; auto.   sym_st.   apply fold_2 with (eqA:=eqA); auto.   Qed.   Lemma fold_init : forall s, eqA i i' ->    eqA (fold f s i) (fold f s i').   Proof.   intros; pattern s; apply set_induction; clear s; intros.   trans_st i.   apply fold_1; auto.   trans_st i'.   sym_st; apply fold_1; auto.   trans_st (f x (fold f s i)).   apply fold_2 with (eqA:=eqA); auto.   trans_st (f x (fold f s i')).   sym_st; apply fold_2 with (eqA:=eqA); auto.   Qed.   End Fold_1.   Section Fold_2.   Variable i:A.   Lemma fold_union_inter : forall s s',    eqA (fold f (union s s') (fold f (inter s s') i))        (fold f s (fold f s' i)).   Proof.   intros; pattern s; apply set_induction; clear s; intros.   trans_st (fold f s' (fold f (inter s s') i)).   apply fold_equal; auto with set.   trans_st (fold f s' i).   apply fold_init; auto.   apply fold_1; auto with set.   sym_st; apply fold_1; auto.   rename s'0 into s''.   destruct (In_dec x s').   trans_st (fold f (union s'' s') (f x (fold f (inter s s') i))); auto with set.   apply fold_init; auto.   apply fold_2 with (eqA:=eqA); auto with set.   rewrite inter_iff; intuition.   trans_st (f x (fold f s (fold f s' i))).   trans_st (fold f (union s s') (f x (fold f (inter s s') i))).   apply fold_equal; auto.   apply equal_sym; apply union_Equal with x; auto with set.   trans_st (f x (fold f (union s s') (fold f (inter s s') i))).   apply fold_commutes; auto.   sym_st; apply fold_2 with (eqA:=eqA); auto.   trans_st (f x (fold f (union s s') (fold f (inter s'' s') i))).   apply fold_2 with (eqA:=eqA); auto with set.   trans_st (f x (fold f (union s s') (fold f (inter s s') i))).   apply Comp;auto.   apply fold_init;auto.   apply fold_equal;auto.   apply equal_sym; apply inter_Add_2 with x; auto with set.   trans_st (f x (fold f s (fold f s' i))).   sym_st; apply fold_2 with (eqA:=eqA); auto.   Qed.   End Fold_2.   Section Fold_3.   Variable i:A.   Lemma fold_diff_inter : forall s s',    eqA (fold f (diff s s') (fold f (inter s s') i)) (fold f s i).   Proof.   intros.   trans_st (fold f (union (diff s s') (inter s s'))               (fold f (inter (diff s s') (inter s s')) i)).   sym_st; apply fold_union_inter; auto.   trans_st (fold f s (fold f (inter (diff s s') (inter s s')) i)).   apply fold_equal; auto with set.   apply fold_init; auto.   apply fold_1; auto with set.   Qed.   Lemma fold_union: forall s s', (forall x, ~In x s\/~In x s') ->    eqA (fold f (union s s') i) (fold f s (fold f s' i)).   Proof.   intros.   trans_st (fold f (union s s') (fold f (inter s s') i)).   apply fold_init; auto.   sym_st; apply fold_1; auto with set.   unfold Empty; intro a; generalize (H a); set_iff; tauto.   apply fold_union_inter; auto.   Qed.   End Fold_3.   End Fold.   Lemma fold_plus :    forall s p, fold (fun _ => S) s p = fold (fun _ => S) s 0 + p.   Proof.   assert (st := gen_st nat).   assert (fe : compat_op E.eq (@eq _) (fun _ => S)) by (unfold compat_op; auto).   assert (fp : transpose (@eq _) (fun _:elt => S)) by (unfold transpose; auto).   intros s p; pattern s; apply set_induction; clear s; intros.   rewrite (fold_1 st p (fun _ => S) H).   rewrite (fold_1 st 0 (fun _ => S) H); trivial.   assert (forall p s', Add x s s' -> fold (fun _ => S) s' p = S (fold (fun _ => S) s p)).    change S with ((fun _ => S) x).    intros; apply fold_2; auto.   rewrite H2; auto.   rewrite (H2 0); auto.   rewrite H.   simpl; auto.   Qed. ```
properties of `cardinal`
```   Lemma empty_cardinal : cardinal empty = 0.   Proof.   rewrite cardinal_fold; apply fold_1; auto.   Qed.   Hint Immediate empty_cardinal cardinal_1 : set.   Lemma singleton_cardinal : forall x, cardinal (singleton x) = 1.   Proof.   intros.   rewrite (singleton_equal_add x).   replace 0 with (cardinal empty); auto with set.   apply cardinal_2 with x; auto with set.   Qed.   Hint Resolve singleton_cardinal: set.   Lemma diff_inter_cardinal :    forall s s', cardinal (diff s s') + cardinal (inter s s') = cardinal s .   Proof.   intros; do 3 rewrite cardinal_fold.   rewrite <- fold_plus.   apply fold_diff_inter with (eqA:=@eq nat); auto.   Qed.   Lemma union_cardinal:    forall s s', (forall x, ~In x s\/~In x s') ->    cardinal (union s s')=cardinal s+cardinal s'.   Proof.   intros; do 3 rewrite cardinal_fold.   rewrite <- fold_plus.   apply fold_union; auto.   Qed.   Lemma subset_cardinal :    forall s s', s[<=]s' -> cardinal s <= cardinal s' .   Proof.   intros.   rewrite <- (diff_inter_cardinal s' s).   rewrite (inter_sym s' s).   rewrite (inter_subset_equal H); auto with arith.   Qed.   Lemma subset_cardinal_lt :    forall s s' x, s[<=]s' -> In x s' -> ~In x s -> cardinal s < cardinal s'.   Proof.   intros.   rewrite <- (diff_inter_cardinal s' s).   rewrite (inter_sym s' s).   rewrite (inter_subset_equal H).   generalize (@cardinal_inv_1 (diff s' s)).   destruct (cardinal (diff s' s)).   intro H2; destruct (H2 (refl_equal _) x).   set_iff; auto.   intros _.   change (0 + cardinal s < S n + cardinal s).   apply Plus.plus_lt_le_compat; auto with arith.   Qed.   Theorem union_inter_cardinal :    forall s s', cardinal (union s s') + cardinal (inter s s') = cardinal s + cardinal s' .   Proof.   intros.   do 4 rewrite cardinal_fold.   do 2 rewrite <- fold_plus.   apply fold_union_inter with (eqA:=@eq nat); auto.   Qed.   Lemma union_cardinal_inter :    forall s s', cardinal (union s s') = cardinal s + cardinal s' - cardinal (inter s s').   Proof.   intros.   rewrite <- union_inter_cardinal.   rewrite Plus.plus_comm.   auto with arith.   Qed.   Lemma union_cardinal_le :    forall s s', cardinal (union s s') <= cardinal s + cardinal s'.   Proof.    intros; generalize (union_inter_cardinal s s').    intros; rewrite <- H; auto with arith.   Qed.   Lemma add_cardinal_1 :    forall s x, In x s -> cardinal (add x s) = cardinal s.   Proof.   auto with set.   Qed.   Lemma add_cardinal_2 :    forall s x, ~In x s -> cardinal (add x s) = S (cardinal s).   Proof.   intros.   do 2 rewrite cardinal_fold.   change S with ((fun _ => S) x);    apply fold_add with (eqA:=@eq nat); auto.   Qed.   Lemma remove_cardinal_1 :    forall s x, In x s -> S (cardinal (remove x s)) = cardinal s.   Proof.   intros.   do 2 rewrite cardinal_fold.   change S with ((fun _ =>S) x).   apply remove_fold_1 with (eqA:=@eq nat); auto.   Qed.   Lemma remove_cardinal_2 :    forall s x, ~In x s -> cardinal (remove x s) = cardinal s.   Proof.   auto with set.   Qed.   Hint Resolve subset_cardinal union_cardinal add_cardinal_1 add_cardinal_2. End Properties. ```