set "baseuri" "cic:/matita/sequence/".
-include "ordered_set.ma".
+include "excedence.ma".
-definition is_increasing ≝ λO:pordered_set.λa:nat→O.∀n:nat.a n ≤ a (S n).
-definition is_decreasing ≝ λO:pordered_set.λa:nat→O.∀n:nat.a (S n) ≤ a n.
+definition sequence := λO:excedence.nat → O.
-definition is_upper_bound ≝ λO:pordered_set.λa:nat→O.λu:O.∀n:nat.a n ≤ u.
-definition is_lower_bound ≝ λO:pordered_set.λa:nat→O.λu:O.∀n:nat.u ≤ a n.
+definition fun_of_sequence: ∀O:excedence.sequence O → nat → O.
+intros; apply s; assumption;
+qed.
+
+coercion cic:/matita/sequence/fun_of_sequence.con 1.
+
+definition upper_bound ≝
+ λO:excedence.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
+
+definition lower_bound ≝
+ λO:excedence.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
+
+definition strong_sup ≝
+ λO:excedence.λs:sequence O.λx.
+ upper_bound ? s x ∧ (∀y:O.x ≰ y → ∃n.s n ≰ y).
+
+definition strong_inf ≝
+ λO:excedence.λs:sequence O.λx.
+ lower_bound ? s x ∧ (∀y:O.y ≰ x → ∃n.y ≰ s n).
+
+definition weak_sup ≝
+ λO:excedence.λs:sequence O.λx.
+ upper_bound ? s x ∧ (∀y:O.upper_bound ? s y → x ≤ y).
+
+definition weak_inf ≝
+ λO:excedence.λs:sequence O.λx.
+ lower_bound ? s x ∧ (∀y:O.lower_bound ? s y → y ≤ x).
+
+lemma strong_sup_is_weak:
+ ∀O:excedence.∀s:sequence O.∀x:O.strong_sup ? s x → weak_sup ? s x.
+intros (O s x Ssup); elim Ssup (Ubx M); clear Ssup; split; [assumption]
+intros 3 (y Uby E); cases (M ? E) (n En); unfold in Uby; cases (Uby ? En);
+qed.
+
+lemma strong_inf_is_weak:
+ ∀O:excedence.∀s:sequence O.∀x:O.strong_inf ? s x → weak_inf ? s x.
+intros (O s x Ssup); elim Ssup (Ubx M); clear Ssup; split; [assumption]
+intros 3 (y Uby E); cases (M ? E) (n En); unfold in Uby; cases (Uby ? En);
+qed.
+
+include "ordered_group.ma".
+include "nat/orders.ma".
+
+definition tends0 ≝
+ λO:pogroup.λs:sequence O.
+ ∀e:O.0 < e → ∃N.∀n.N < n → -e < s n ∧ s n < e.
+
+definition increasing ≝
+ λO:excedence.λa:sequence O.∀n:nat.a n ≤ a (S n).
+
+definition decreasing ≝
+ λO:excedence.λa:sequence O.∀n:nat.a (S n) ≤ a n.
-record is_sup (O:pordered_set) (a:nat→O) (u:O) : Prop ≝
+
+
+
+(*
+
+definition is_upper_bound ≝ λO:excedence.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
+definition is_lower_bound ≝ λO:excedence.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
+
+record is_sup (O:excedence) (a:sequence O) (u:O) : Prop ≝
{ sup_upper_bound: is_upper_bound O a u;
sup_least_upper_bound: ∀v:O. is_upper_bound O a v → u≤v
}.
-record is_inf (O:pordered_set) (a:nat→O) (u:O) : Prop ≝
+record is_inf (O:excedence) (a:sequence O) (u:O) : Prop ≝
{ inf_lower_bound: is_lower_bound O a u;
inf_greatest_lower_bound: ∀v:O. is_lower_bound O a v → v≤u
}.
-record is_bounded_below (O:pordered_set) (a:nat→O) : Type ≝
+record is_bounded_below (O:excedence) (a:sequence O) : Type ≝
{ ib_lower_bound: O;
ib_lower_bound_is_lower_bound: is_lower_bound ? a ib_lower_bound
}.
-record is_bounded_above (O:pordered_set) (a:nat→O) : Type ≝
+record is_bounded_above (O:excedence) (a:sequence O) : Type ≝
{ ib_upper_bound: O;
ib_upper_bound_is_upper_bound: is_upper_bound ? a ib_upper_bound
}.
-record is_bounded (O:pordered_set) (a:nat→O) : Type ≝
+record is_bounded (O:excedence) (a:sequence O) : Type ≝
{ ib_bounded_below:> is_bounded_below ? a;
ib_bounded_above:> is_bounded_above ? a
}.
-record bounded_below_sequence (O:pordered_set) : Type ≝
- { bbs_seq:1> nat→O;
+record bounded_below_sequence (O:excedence) : Type ≝
+ { bbs_seq:> sequence O;
bbs_is_bounded_below:> is_bounded_below ? bbs_seq
}.
-record bounded_above_sequence (O:pordered_set) : Type ≝
- { bas_seq:1> nat→O;
+record bounded_above_sequence (O:excedence) : Type ≝
+ { bas_seq:> sequence O;
bas_is_bounded_above:> is_bounded_above ? bas_seq
}.
-record bounded_sequence (O:pordered_set) : Type ≝
- { bs_seq:1> nat → O;
+record bounded_sequence (O:excedence) : Type ≝
+ { bs_seq:> sequence O;
bs_is_bounded_below: is_bounded_below ? bs_seq;
bs_is_bounded_above: is_bounded_above ? bs_seq
}.
definition bounded_below_sequence_of_bounded_sequence ≝
- λO:pordered_set.λb:bounded_sequence O.
+ λO:excedence.λb:bounded_sequence O.
mk_bounded_below_sequence ? b (bs_is_bounded_below ? b).
coercion cic:/matita/sequence/bounded_below_sequence_of_bounded_sequence.con.
definition bounded_above_sequence_of_bounded_sequence ≝
- λO:pordered_set.λb:bounded_sequence O.
+ λO:excedence.λb:bounded_sequence O.
mk_bounded_above_sequence ? b (bs_is_bounded_above ? b).
coercion cic:/matita/sequence/bounded_above_sequence_of_bounded_sequence.con.
definition lower_bound ≝
- λO:pordered_set.λb:bounded_below_sequence O.
+ λO:excedence.λb:bounded_below_sequence O.
ib_lower_bound ? b (bbs_is_bounded_below ? b).
lemma lower_bound_is_lower_bound:
- ∀O:pordered_set.∀b:bounded_below_sequence O.
+ ∀O:excedence.∀b:bounded_below_sequence O.
is_lower_bound ? b (lower_bound ? b).
intros; unfold lower_bound; apply ib_lower_bound_is_lower_bound.
qed.
definition upper_bound ≝
- λO:pordered_set.λb:bounded_above_sequence O.
+ λO:excedence.λb:bounded_above_sequence O.
ib_upper_bound ? b (bas_is_bounded_above ? b).
lemma upper_bound_is_upper_bound:
- ∀O:pordered_set.∀b:bounded_above_sequence O.
+ ∀O:excedence.∀b:bounded_above_sequence O.
is_upper_bound ? b (upper_bound ? b).
intros; unfold upper_bound; apply ib_upper_bound_is_upper_bound.
qed.
|2: intros (x y z); apply Or_symmetric; apply cTf; assumption;]
qed.
-definition reverse_pordered_set: pordered_set → pordered_set.
-intros (p); apply (mk_pordered_set (reverse_excedence p));
+definition reverse_excedence: excedence → excedence.
+intros (p); apply (mk_excedence (reverse_excedence p));
generalize in match (reverse_excedence p); intros (E);
apply mk_is_porder_relation;
[apply le_reflexive|apply le_transitive|apply le_antisymmetric]
qed.
lemma is_lower_bound_reverse_is_upper_bound:
- ∀O:pordered_set.∀a:nat→O.∀l:O.
- is_lower_bound O a l → is_upper_bound (reverse_pordered_set O) a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set;
+ ∀O:excedence.∀a:sequence O.∀l:O.
+ is_lower_bound O a l → is_upper_bound (reverse_excedence O) a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excedence;
unfold reverse_excedence; simplify; fold unfold le (le ? l (a n)); apply H;
qed.
lemma is_upper_bound_reverse_is_lower_bound:
- ∀O:pordered_set.∀a:nat→O.∀l:O.
- is_upper_bound O a l → is_lower_bound (reverse_pordered_set O) a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set;
+ ∀O:excedence.∀a:sequence O.∀l:O.
+ is_upper_bound O a l → is_lower_bound (reverse_excedence O) a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excedence;
unfold reverse_excedence; simplify; fold unfold le (le ? (a n) l); apply H;
qed.
lemma reverse_is_lower_bound_is_upper_bound:
- ∀O:pordered_set.∀a:nat→O.∀l:O.
- is_lower_bound (reverse_pordered_set O) a l → is_upper_bound O a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set in H;
+ ∀O:excedence.∀a:sequence O.∀l:O.
+ is_lower_bound (reverse_excedence O) a l → is_upper_bound O a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excedence in H;
unfold reverse_excedence in H; simplify in H; apply H;
qed.
lemma reverse_is_upper_bound_is_lower_bound:
- ∀O:pordered_set.∀a:nat→O.∀l:O.
- is_upper_bound (reverse_pordered_set O) a l → is_lower_bound O a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set in H;
+ ∀O:excedence.∀a:sequence O.∀l:O.
+ is_upper_bound (reverse_excedence O) a l → is_lower_bound O a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excedence in H;
unfold reverse_excedence in H; simplify in H; apply H;
qed.
lemma is_inf_to_reverse_is_sup:
- ∀O:pordered_set.∀a:bounded_below_sequence O.∀l:O.
- is_inf O a l → is_sup (reverse_pordered_set O) a l.
-intros (O a l H); apply (mk_is_sup (reverse_pordered_set O));
+ ∀O:excedence.∀a:bounded_below_sequence O.∀l:O.
+ is_inf O a l → is_sup (reverse_excedence O) a l.
+intros (O a l H); apply (mk_is_sup (reverse_excedence O));
[1: apply is_lower_bound_reverse_is_upper_bound; apply inf_lower_bound; assumption
-|2: unfold reverse_pordered_set; simplify; unfold reverse_excedence; simplify;
+|2: unfold reverse_excedence; simplify; unfold reverse_excedence; simplify;
intros (m H1); apply (inf_greatest_lower_bound ? ? ? H); apply H1;]
qed.
lemma is_sup_to_reverse_is_inf:
- ∀O:pordered_set.∀a:bounded_above_sequence O.∀l:O.
- is_sup O a l → is_inf (reverse_pordered_set O) a l.
-intros (O a l H); apply (mk_is_inf (reverse_pordered_set O));
+ ∀O:excedence.∀a:bounded_above_sequence O.∀l:O.
+ is_sup O a l → is_inf (reverse_excedence O) a l.
+intros (O a l H); apply (mk_is_inf (reverse_excedence O));
[1: apply is_upper_bound_reverse_is_lower_bound; apply sup_upper_bound; assumption
-|2: unfold reverse_pordered_set; simplify; unfold reverse_excedence; simplify;
+|2: unfold reverse_excedence; simplify; unfold reverse_excedence; simplify;
intros (m H1); apply (sup_least_upper_bound ? ? ? H); apply H1;]
qed.
lemma reverse_is_sup_to_is_inf:
- ∀O:pordered_set.∀a:bounded_above_sequence O.∀l:O.
- is_sup (reverse_pordered_set O) a l → is_inf O a l.
+ ∀O:excedence.∀a:bounded_above_sequence O.∀l:O.
+ is_sup (reverse_excedence O) a l → is_inf O a l.
intros (O a l H); apply mk_is_inf;
[1: apply reverse_is_upper_bound_is_lower_bound;
- apply (sup_upper_bound (reverse_pordered_set O)); assumption
-|2: intros (v H1); apply (sup_least_upper_bound (reverse_pordered_set O) a l H v);
+ apply (sup_upper_bound (reverse_excedence O)); assumption
+|2: intros (v H1); apply (sup_least_upper_bound (reverse_excedence O) a l H v);
apply is_lower_bound_reverse_is_upper_bound; assumption;]
qed.
lemma reverse_is_inf_to_is_sup:
- ∀O:pordered_set.∀a:bounded_above_sequence O.∀l:O.
- is_inf (reverse_pordered_set O) a l → is_sup O a l.
+ ∀O:excedence.∀a:bounded_above_sequence O.∀l:O.
+ is_inf (reverse_excedence O) a l → is_sup O a l.
intros (O a l H); apply mk_is_sup;
[1: apply reverse_is_lower_bound_is_upper_bound;
- apply (inf_lower_bound (reverse_pordered_set O)); assumption
-|2: intros (v H1); apply (inf_greatest_lower_bound (reverse_pordered_set O) a l H v);
+ apply (inf_lower_bound (reverse_excedence O)); assumption
+|2: intros (v H1); apply (inf_greatest_lower_bound (reverse_excedence O) a l H v);
apply is_upper_bound_reverse_is_lower_bound; assumption;]
qed.
+
+*)
\ No newline at end of file