- ∀O:ordered_set.∀a:bounded_above_sequence O.∀l:O.
- is_inf (reverse_ordered_set O) a l → is_sup O a l.
- intros;
- apply mk_is_sup;
- [ apply reverse_is_lower_bound_is_upper_bound;
- change in l with (os_carrier (reverse_ordered_set O));
- apply (inf_lower_bound ? ? ? H)
- | intros;
- change in l with (os_carrier (reverse_ordered_set O));
- change in v with (os_carrier (reverse_ordered_set O));
- change with (os_le (reverse_ordered_set O) v l);
- apply (inf_greatest_lower_bound ? ? ? H);
- change in v with (os_carrier O);
- apply is_upper_bound_reverse_is_lower_bound;
- assumption
- ].
-qed.
-
-
-definition reverse_dedekind_sigma_complete_ordered_set:
- dedekind_sigma_complete_ordered_set → dedekind_sigma_complete_ordered_set.
- intros;
- apply mk_dedekind_sigma_complete_ordered_set;
- [ apply (reverse_ordered_set d)
- | elim daemon
- (*apply mk_is_dedekind_sigma_complete;
- [ intros;
- elim (dsc_sup ? ? d a m) 0;
- [ generalize in match H; clear H;
- generalize in match m; clear m;
- elim d;
- simplify in a1;
- simplify;
- change in a1 with (Type_OF_ordered_set ? (reverse_ordered_set ? o));
- apply (ex_intro ? ? a1);
- simplify in H1;
- change in m with (Type_OF_ordered_set ? o);
- apply (is_sup_to_reverse_is_inf ? ? ? ? H1)
- | generalize in match H; clear H;
- generalize in match m; clear m;
- elim d;
- simplify;
- change in t with (Type_OF_ordered_set ? o);
- simplify in t;
- apply reverse_is_lower_bound_is_upper_bound;
- assumption
- ]
- | apply is_sup_reverse_is_inf;
- | apply m
- |
- ]*)
- ].
-qed.
-
-definition reverse_bounded_sequence:
- ∀O:dedekind_sigma_complete_ordered_set.
- bounded_sequence O →
- bounded_sequence (reverse_dedekind_sigma_complete_ordered_set O).
- intros;
- apply mk_bounded_sequence;
- [ apply bs_seq;
- unfold reverse_dedekind_sigma_complete_ordered_set;
- simplify;
- elim daemon
- | elim daemon
- | elim daemon
- ].
-qed.
-
-definition limsup ≝
- λO:dedekind_sigma_complete_ordered_set.
- λa:bounded_sequence O.
- liminf (reverse_dedekind_sigma_complete_ordered_set O)
- (reverse_bounded_sequence O a).
-
-notation "hvbox(〈a〉)"
- non associative with precedence 45
-for @{ 'hide_everything_but $a }.
-
-interpretation "mk_bounded_above_sequence" 'hide_everything_but a
-= (cic:/matita/ordered_sets/bounded_above_sequence.ind#xpointer(1/1/1) _ _ a _).
-
-interpretation "mk_bounded_below_sequence" 'hide_everything_but a
-= (cic:/matita/ordered_sets/bounded_below_sequence.ind#xpointer(1/1/1) _ _ a _).
-
-theorem eq_f_sup_sup_f:
- ∀O':dedekind_sigma_complete_ordered_set.
- ∀f:O'→O'. ∀H:is_order_continuous ? f.
- ∀a:bounded_above_sequence O'.
- ∀p:is_increasing ? a.
- f (sup ? a) = sup ? (mk_bounded_above_sequence ? (λi.f (a i)) ?).
- [ apply (mk_is_bounded_above ? ? (f (sup ? a)));
- apply ioc_is_upper_bound_f_sup;
- assumption
- | intros;
- apply ioc_respects_sup;
- assumption
- ].
-qed.
-
-theorem eq_f_sup_sup_f':
- ∀O':dedekind_sigma_complete_ordered_set.
- ∀f:O'→O'. ∀H:is_order_continuous ? f.
- ∀a:bounded_above_sequence O'.
- ∀p:is_increasing ? a.
- ∀p':is_bounded_above ? (λi.f (a i)).
- f (sup ? a) = sup ? (mk_bounded_above_sequence ? (λi.f (a i)) p').
- intros;
- rewrite > (eq_f_sup_sup_f ? f H a H1);
- apply sup_proof_irrelevant;
- reflexivity.
-qed.
-
-theorem eq_f_liminf_sup_f_inf:
- ∀O':dedekind_sigma_complete_ordered_set.
- ∀f:O'→O'. ∀H:is_order_continuous ? f.
- ∀a:bounded_sequence O'.
- let p1 := ? in
- f (liminf ? a) =
- sup ?
- (mk_bounded_above_sequence ?
- (λi.f (inf ?
- (mk_bounded_below_sequence ?
- (λj.a (i+j))
- ?)))
- p1).
- [ apply (mk_is_bounded_below ? ? (ib_lower_bound ? a a));
- simplify;
- intro;
- apply (ib_lower_bound_is_lower_bound ? a a)
- | apply (mk_is_bounded_above ? ? (f (sup ? a)));
- unfold is_upper_bound;
- intro;
- apply (or_transitive ? ? O' ? (f (a n)));
- [ generalize in match (ioc_is_lower_bound_f_inf ? ? H);
- intro H1;
- simplify in H1;
- rewrite > (plus_n_O n) in ⊢ (? ? ? (? (? ? ? %)));
- apply (H1 (mk_bounded_below_sequence O' (\lambda j:nat.a (n+j))
-(mk_is_bounded_below O' (\lambda j:nat.a (n+j)) (ib_lower_bound O' a a)
- (\lambda j:nat.ib_lower_bound_is_lower_bound O' a a (n+j)))) O);
- | elim daemon (*apply (ioc_is_upper_bound_f_sup ? ? ? H)*)
- ]
- | intros;
- unfold liminf;
- clearbody p1;
- generalize in match (\lambda n:nat
-.inf_respects_le O'
- (mk_bounded_below_sequence O' (\lambda j:nat.a (plus n j))
- (mk_is_bounded_below O' (\lambda j:nat.a (plus n j))
- (ib_lower_bound O' a a)
- (\lambda j:nat.ib_lower_bound_is_lower_bound O' a a (plus n j))))
- (ib_upper_bound O' a a)
- (\lambda n1:nat.ib_upper_bound_is_upper_bound O' a a (plus n n1)));
- intro p2;
- apply (eq_f_sup_sup_f' ? f H (mk_bounded_above_sequence O'
-(\lambda i:nat
- .inf O'
- (mk_bounded_below_sequence O' (\lambda j:nat.a (plus i j))
- (mk_is_bounded_below O' (\lambda j:nat.a (plus i j))
- (ib_lower_bound O' a a)
- (\lambda n:nat.ib_lower_bound_is_lower_bound O' a a (plus i n)))))
-(mk_is_bounded_above O'
- (\lambda i:nat
- .inf O'
- (mk_bounded_below_sequence O' (\lambda j:nat.a (plus i j))
- (mk_is_bounded_below O' (\lambda j:nat.a (plus i j))
- (ib_lower_bound O' a a)
- (\lambda n:nat.ib_lower_bound_is_lower_bound O' a a (plus i n)))))
- (ib_upper_bound O' a a) p2)));
- unfold bas_seq;
- change with
- (is_increasing ? (\lambda i:nat
-.inf O'
- (mk_bounded_below_sequence O' (\lambda j:nat.a (plus i j))
- (mk_is_bounded_below O' (\lambda j:nat.a (plus i j))
- (ib_lower_bound O' a a)
- (\lambda n:nat.ib_lower_bound_is_lower_bound O' a a (plus i n))))));
- apply tail_inf_increasing
- ].
-qed.
-
-
-
-
-definition lt ≝ λO:ordered_set.λa,b:O.a ≤ b ∧ a ≠ b.
-
-interpretation "Ordered set lt" 'lt a b =
- (cic:/matita/ordered_sets/lt.con _ a b).