-lemma inf_respects_le:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- ∀a:bounded_below_sequence ? O.∀m:O.
- is_upper_bound ? ? a m → inf ? ? a ≤ m.
- intros (C O');
- apply (or_transitive ? ? O' ? (sup ? ? (mk_bounded_sequence ? ? a ? ?)));
- [ apply (bbs_is_bounded_below ? ? a)
- | apply (mk_is_bounded_above ? ? ? m H)
- | apply inf_le_sup
- | apply
- (sup_least_upper_bound ? ? ? ?
- (sup_is_sup ? ? (mk_bounded_sequence C O' a a
- (mk_is_bounded_above C O' a m H))));
- assumption
- ].
-qed.
-
-definition is_sequentially_monotone ≝
- λC.λO:ordered_set C.λf:O→O.
- ∀a:nat→O.∀p:is_increasing ? ? a.
- is_increasing ? ? (λi.f (a i)).
-
-record is_order_continuous (C)
- (O:dedekind_sigma_complete_ordered_set C) (f:O→O) : Prop
-≝
- { ioc_is_sequentially_monotone: is_sequentially_monotone ? ? f;
- ioc_is_upper_bound_f_sup:
- ∀a:bounded_above_sequence ? O.
- is_upper_bound ? ? (λi.f (a i)) (f (sup ? ? a));
- ioc_respects_sup:
- ∀a:bounded_above_sequence ? O.
- is_increasing ? ? a →
- f (sup ? ? a) =
- sup ? ? (mk_bounded_above_sequence ? ? (λi.f (a i))
- (mk_is_bounded_above ? ? ? (f (sup ? ? a))
- (ioc_is_upper_bound_f_sup a)));
- ioc_is_lower_bound_f_inf:
- ∀a:bounded_below_sequence ? O.
- is_lower_bound ? ? (λi.f (a i)) (f (inf ? ? a));
- ioc_respects_inf:
- ∀a:bounded_below_sequence ? O.
- is_decreasing ? ? a →
- f (inf ? ? a) =
- inf ? ? (mk_bounded_below_sequence ? ? (λi.f (a i))
- (mk_is_bounded_below ? ? ? (f (inf ? ? a))
- (ioc_is_lower_bound_f_inf a)))
- }.
-
-theorem tail_inf_increasing:
- ∀C.∀O:dedekind_sigma_complete_ordered_set C.
- ∀a:bounded_below_sequence ? O.
- let y ≝ λi.mk_bounded_below_sequence ? ? (λj.a (i+j)) ? in
- let x ≝ λi.inf ? ? (y i) in
- is_increasing ? ? x.
- [ apply (mk_is_bounded_below ? ? ? (ib_lower_bound ? ? a a));
- simplify;
- intro;
- apply (ib_lower_bound_is_lower_bound ? ? a a)
- | intros;
- unfold is_increasing;
- intro;
- unfold x in ⊢ (? ? ? ? %);
- apply (inf_greatest_lower_bound ? ? ? ? (inf_is_inf ? ? (y (S n))));
- change with (is_lower_bound ? ? (y (S n)) (inf ? ? (y n)));
- unfold is_lower_bound;
- intro;
- generalize in match (inf_lower_bound ? ? ? ? (inf_is_inf ? ? (y n)) (S n1));
- (*CSC: coercion per FunClass inserita a mano*)
- suppose (inf ? ? (y n) ≤ bbs_seq ? ? (y n) (S n1)) (H);
- cut (bbs_seq ? ? (y n) (S n1) = bbs_seq ? ? (y (S n)) n1);
- [ rewrite < Hcut;
- assumption
- | unfold y;
- simplify;
- auto paramodulation library
- ]
- ].
-qed.
-
-definition is_liminf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- bounded_below_sequence ? O → O → Prop.
- intros;
- apply
- (is_sup ? ? (λi.inf ? ? (mk_bounded_below_sequence ? ? (λj.b (i+j)) ?)) t);
- apply (mk_is_bounded_below ? ? ? (ib_lower_bound ? ? b b));
- simplify;
- intros;
- apply (ib_lower_bound_is_lower_bound ? ? b b).
-qed.
-
-definition liminf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- bounded_sequence ? O → O.
- intros;
- apply
- (sup ? ?
- (mk_bounded_above_sequence ? ?
- (λi.inf ? ?
- (mk_bounded_below_sequence ? ?
- (λj.b (i+j)) ?)) ?));
- [ apply (mk_is_bounded_below ? ? ? (ib_lower_bound ? ? b b));
- simplify;
- intros;
- apply (ib_lower_bound_is_lower_bound ? ? b b)
- | apply (mk_is_bounded_above ? ? ? (ib_upper_bound ? ? b b));
- unfold is_upper_bound;
- intro;
- change with
- (inf C O
- (mk_bounded_below_sequence C O (\lambda j:nat.b (n+j))
- (mk_is_bounded_below C O (\lambda j:nat.b (n+j)) (ib_lower_bound C O b b)
- (\lambda j:nat.ib_lower_bound_is_lower_bound C O b b (n+j))))
-\leq ib_upper_bound C O b b);
- apply (inf_respects_le ? O);
- simplify;
- intro;
- apply (ib_upper_bound_is_upper_bound ? ? b b)
- ].