(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/sequence/".
+include "excess.ma".
-include "ordered_set.ma".
+definition sequence := λO:Type.nat → O.
-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 fun_of_sequence: ∀O:Type.sequence O → nat → O ≝ λO.λx:sequence O.x.
-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.
-
-record is_sup (O:pordered_set) (a:nat→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 ≝
- { 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 ≝
- { 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 ≝
- { 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 ≝
- { 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;
- bbs_is_bounded_below:> is_bounded_below ? bbs_seq
- }.
-
-record bounded_above_sequence (O:pordered_set) : Type ≝
- { bas_seq:1> nat→O;
- bas_is_bounded_above:> is_bounded_above ? bas_seq
- }.
-
-record bounded_sequence (O:pordered_set) : Type ≝
- { bs_seq:1> nat → 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.
- 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.
- 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.
- ib_lower_bound ? b (bbs_is_bounded_below ? b).
-
-lemma lower_bound_is_lower_bound:
- ∀O:pordered_set.∀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.
- ib_upper_bound ? b (bas_is_bounded_above ? b).
-
-lemma upper_bound_is_upper_bound:
- ∀O:pordered_set.∀b:bounded_above_sequence O.
- is_upper_bound ? b (upper_bound ? b).
-intros; unfold upper_bound; apply ib_upper_bound_is_upper_bound.
-qed.
-
-definition reverse_excedence: excedence → excedence.
-intros (E); apply (mk_excedence E); [apply (λx,y.exc_relation E y x)]
-cases E (T f cRf cTf); simplify;
-[1: unfold Not; intros (x H); apply (cRf x); assumption
-|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));
-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;
-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;
-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;
-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;
-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));
-[1: apply is_lower_bound_reverse_is_upper_bound; apply inf_lower_bound; assumption
-|2: unfold reverse_pordered_set; 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));
-[1: apply is_upper_bound_reverse_is_lower_bound; apply sup_upper_bound; assumption
-|2: unfold reverse_pordered_set; 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.
-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 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.
-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 is_upper_bound_reverse_is_lower_bound; assumption;]
-qed.
+coercion cic:/matita/sequence/fun_of_sequence.con 1.
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