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(* 02110-1301 USA                                                     *)
(** %\subsection*{ support :  counting\_elements.v }%*)
Set Implicit Arguments.
Unset Strict Implicit.
From Algebra Require Export Diff.
From Algebra Require Export Singleton.
Require Export has_n_elements.
From Algebra Require Export Union.
Require Export const.
(** - Once we have has_n_elements, it would seem to be easy to define has_at_least_n_elements and has_at_most_n_elements. But one has to take a little care. We could not just say, e.g. [  Definition has_at_least_n_elements := ] [    [n:nat;A:Setoid](EXT m:nat | (le n m) /\ (has_n_elements m A)).  ] since the set may be infinite, and then there is no such m to be found.  - But we may adapt the definition of has_n_elements slightly. It said that we can make a sequence [v] of length [n] with all elements distinct---[(distinct v)]; covering the whole set---[(seq_set v)='(full v)]. Now we just drop this last requirement so we only need to point out a sequence of $n$ different elements:  %\label{hasatleastnelements}% *)
Definition has_at_least_n_elements (n : Nat) (A : Setoid) :=
exists v : seq n A, distinct v.
(** - Note that, contrarily, we may not just drop the other requirement to obtain  [Definition has_at_most_n_elements_wrong := [n:nat;A:Setoid](EXT v:(seq n A) | (full A)='(seq_set v)).]  since this doesn't work out well for empty sets: it has at most n elements for any n, but with this definition we'd need to exhibit an n-length sequence, which we can't do for $n>0$.*)
Definition has_at_most_n_elements (n : Nat) (A : Setoid) :=
exists m : nat, m <= n /\ has_n_elements m A.
intros.
red in |- *.
red in H.
inversion_clear H.
rename x into Bsize.
inversion_clear H2.
exists Bsize.
split.
generalize (eq_ind _ (fun n => Bsize <= n) H _ H1).
auto.
apply has_n_elements_comp with Bsize B; auto with algebra.
Qed.
intros.
rewrite <- H1.
red in |- *.
red in H.
inversion_clear H.
rename x into Bseq.
exists (Map_to_equal_subsets H0 Bseq).
red in |- *; intros; intro; red in H2; (apply (H2 i j); auto with algebra).
simpl in |- *.
red in |- *.
apply Trans with (subtype_elt (Bseq i)); auto with algebra.
apply Trans with (subtype_elt (Bseq j)); auto with algebra.
apply Trans with (subtype_elt (Map_to_equal_subsets H0 Bseq i));
auto with algebra.
apply Trans with (subtype_elt (Map_to_equal_subsets H0 Bseq j));
auto with algebra.
Qed.
intros.
inversion_clear H.
exists x.
inversion_clear H0; auto.
Qed.
intros.
exists n.
split; auto with arith.
Qed.
intros.
red in |- *; red in H.
inversion_clear H.
exists (Map_embed x).
fold (distinct (Map_embed x)) in |- *.
fold (distinct x) in H0.
apply Map_embed_preserves_distinct; auto with algebra.
Qed.
intros.
apply subset_has_at_least_n_elements with B; auto with algebra.
apply has_n_elements_then_has_at_least_n_elements; auto with algebra.
Qed.
intros.
inversion H.
set (B' := seq_set (Map_embed x)) in *.
assert (has_n_elements n B').
red in |- *.
exists (seq_set_seq (Map_embed x)).
split.
unfold B' in |- *.
simpl in |- *.
red in |- *; simpl in |- *.
split; intro; auto.
clear H3.
unfold subtype_image_equal in |- *.
simpl in |- *.
destruct x0.
simpl in subtype_prf.
rename subtype_elt into a.
inversion_clear subtype_prf.
exists x0.
simpl in |- *.
auto.
red in |- *.
intros.
simpl in |- *.
unfold subtype_image_equal in |- *.
simpl in |- *.
intro; (apply (H2 i j); auto with algebra).
assert (exists a : A, in_part a B' /\ ~ in_part a C).
apply has_extra_element with m n; auto with algebra.
assert (included B' B).
red in |- *.
intros a Ha.
simpl in Ha.
inversion_clear Ha.
apply in_part_comp_l with (subtype_elt (x x0)); auto with algebra.
inversion_clear H4.
rename x0 into a.
exists a.
inversion_clear H6.
split; auto.
Qed.
intros.
unfold has_at_least_n_elements in H.
unfold has_at_most_n_elements in |- *.
induction n.
exists 0; split; auto.
red in |- *.
exists (empty_seq A).
split.
simpl in |- *.
red in |- *.
intro a; split; simpl in |- *; auto.
intros _.
apply False_ind; auto with algebra.
apply H.
exists (const_seq 1 a).
intros; intro.
auto with algebra.
red in |- *.
intros.
auto with algebra.
case
(classic
(exists v : seq (S n) A,
(forall i j : fin (S n), ~ i =' j in _ -> ~ v i =' v j in _))).
intro; exists (S n).
split; auto.
red in |- *.
inversion_clear H0.
rename x into aa.
exists aa.
split; auto.
simpl in |- *; red in |- *; simpl in |- *.
intro a; split; auto; intros _.
apply NNPP; intro; apply H.
exists (a;; aa).
red in |- *.
destruct i; destruct j.
destruct index as [| n0];
[ destruct index0 as [| n0] | destruct index0 as [| n1] ].
simpl in |- *.
intuition.
simpl in |- *.
intros _.
intro; apply H0.
exists (Build_finiteT (lt_S_n n0 (S n) in_range_prf0)); auto with algebra.
simpl in |- *.
intros _.
intro; apply H0.
exists (Build_finiteT (lt_S_n n0 (S n) in_range_prf)).
apply Sym; auto with algebra.
simpl in |- *.
intro.
apply
(H1 (Build_finiteT (lt_S_n n0 (S n) in_range_prf))
(Build_finiteT (lt_S_n n1 (S n) in_range_prf0)));
auto with algebra.
intro.
generalize (IHn H0); intro p; inversion_clear p.
exists x.
inversion_clear H1; split; auto.
Qed.
intros.
red in H, H0.
inversion_clear H; inversion_clear H0.
inversion_clear H.
rename x0 into m.
red in |- *; red in H2.
inversion_clear H2.
inversion_clear H.
rename x into aas.
rename x0 into Aseq.
exists aas.
split; auto.
apply Trans with (seq_set Aseq); auto with algebra.
simpl in |- *; red in |- *; simpl in |- *.
intro a; split; simpl in |- *.
2: simpl in H2; red in H2; simpl in H2.
2: elim (H2 a); auto.
intro.
inversion_clear H.
rename x into i.
cut (exists j : fin n, aas j =' Aseq i in _).
intros.
inversion_clear H.
exists x.
apply Trans with (Aseq i); auto with algebra.
clear H4 a.
generalize (exists_difference H0).
intro; inversion_clear H.
rename x into d.
cut (exists j : fin (m + d), cast_seq aas H4 j =' Aseq i in _).
intro.
inversion_clear H.
rename x into j.
exists (cast_fin j (sym_eq H4)).
apply Trans with (cast_seq aas H4 j); auto with algebra.
set (aas' := cast_seq aas H4) in *.
assert (distinct aas').
unfold aas' in |- *.
apply cast_seq_preserves_distinct; auto with algebra.
assert (included (seq_set aas') (seq_set Aseq)).
simpl in H2; red in H2; simpl in H2.
red in |- *.
simpl in |- *; intros.
inversion_clear H5.
elim (H2 x); auto with algebra.
clearbody aas'.
clear H4 H2 H0 H1 aas n.
move d after m.
fold (distinct Aseq) in H3.
induction m.
apply False_ind; auto with algebra.
case (classic (head aas' =' Aseq i in _)).
intros.
exists (Build_finiteT (lt_O_Sn (m + d))).
apply Trans with (head aas'); auto with algebra.
intros.
elim (H5 (head aas')).
2: unfold head in |- *; simpl in |- *.
2: fold (m + d) in |- *.
2: exists (Build_finiteT (lt_O_Sn (m + d))).
2: apply Refl.
intros j Hj.
assert (distinct (omit Aseq j)).
apply omit_preserves_distinct; auto with algebra.
assert (~ j =' i in fin _).
assert (~ Aseq i =' Aseq j in _).
intro.
apply H0; auto with algebra.
apply Trans with (Aseq j); auto with algebra.
intro; apply H2.
apply (Map_compatible_prf Aseq); auto with algebra.
elim (omit_removes' Aseq H2).
intros i' HAseqi'.
assert (distinct (Seqtl aas')).
apply Seqtl_preserves_distinct; auto with algebra.
generalize (IHm (omit Aseq j) H1 i' (Seqtl aas') H4).
intros.
assert (included (seq_set (Seqtl aas')) (seq_set (omit Aseq j))).
2: specialize (H6 H7).
2: inversion_clear H6.
2: destruct x.
2: exists (Build_finiteT (lt_n_S _ _ in_range_prf)).
2: apply Trans with (Seqtl aas' (Build_finiteT in_range_prf));
auto with algebra.
2: apply Trans with (omit Aseq j i'); auto with algebra.
clear H6.
clear IHm HAseqi' i' H2 H0 i.
cut
(forall p : fin (m + d),
exists q : fin m, Seqtl aas' p =' omit Aseq j q in _).
intro.
red in |- *.
simpl in |- *.
intros a Ha.
inversion_clear Ha.
generalize (H0 x); intro p; inversion_clear p.
exists x0.
apply Trans with (Seqtl aas' x); auto with algebra.
intro.
simpl in |- *.
destruct p.
set (p' := Build_finiteT (lt_n_S index (m + d) in_range_prf)) in *.
assert (~ p' =' Build_finiteT (lt_O_Sn (m + d)) in fin _).
unfold p' in |- *; simpl in |- *.
auto with arith.
assert
(forall k : fin (S m + d), exists l : fin (S m), aas' k =' Aseq l in _).
red in H5.
simpl in H5.
intro.
generalize (H5 (aas' k)); intro.
assert (exists i : finiteT (S (m + d)), aas' k =' aas' i in _).
exists k.
apply Refl.
generalize (H2 H6).
intro.
inversion_clear H7.
exists x.
auto.
generalize (H2 p'); intro.
inversion_clear H6.
rename x into q.
cut (exists q0 : finiteT m, Aseq q =' omit Aseq j q0 in _).
intro.
inversion_clear H6.
exists x.
apply Trans with (Aseq q); auto with algebra.
cut (~ j =' q in _).
intro.
elim (omit_removes' Aseq H6).
intros; exists x; auto.
assert (~ aas' p' =' head aas' in _).
unfold head in |- *.
fold (m + d) in |- *.
apply H; auto with algebra.
assert (~ Aseq q =' head aas' in _).
intro; (apply H6; auto with algebra).
apply Trans with (Aseq q); auto with algebra.
assert (~ Aseq q =' Aseq j in _).
intro; (apply H8; auto with algebra).
apply Trans with (Aseq j); auto with algebra.
intro; (apply H9; auto with algebra).
Qed.
intros; red in |- *.
red in H, H0.
inversion_clear H.
inversion_clear H1.
assert (forall i : fin n, in_part (Map_embed x i) C).
intro.
simpl in |- *.
apply H0; auto with algebra.
exists (cast_map_to_subset H1).
red in |- *; intros.
red in H2; (apply (H2 i j); auto with algebra).
Qed.
red in |- *; intros.
assert (has_at_most_n_elements m A).
exists n.
split; auto with arith.
generalize (has_n_elements_by_at_least_at_most H1 H2).
intro.
assert (n =' m in _).
assert (has_n_elements n (full A)).
apply full_preserves_has_n_elements; auto with algebra.
assert (has_n_elements m (full A)).
apply full_preserves_has_n_elements; auto with algebra.
apply (has_n_elements_inj H4 H5); auto with algebra.
rewrite H4 in H0.
generalize lt_irrefl; intro p; red in p.
apply p with m; auto with algebra.
Qed.

Lemma union_has_at_most_n_plus_m_elements :
forall (A : Setoid) (B C : part_set A) (n m : Nat),
has_n_elements n B ->
has_n_elements m C -> has_at_most_n_elements (n + m) (union B C).