-record is_dedekind_sigma_complete (O:ordered_set) : Type ≝
- { dsc_inf: ∀a:nat→O.∀m:O. is_lower_bound ? a m → ex ? (λs:O.is_inf O a s);
- dsc_inf_proof_irrelevant:
- ∀a:nat→O.∀m,m':O.∀p:is_lower_bound ? a m.∀p':is_lower_bound ? a m'.
- (match dsc_inf a m p with [ ex_intro s _ ⇒ s ]) =
- (match dsc_inf a m' p' with [ ex_intro s' _ ⇒ s' ]);
- dsc_sup: ∀a:nat→O.∀m:O. is_upper_bound ? a m → ex ? (λs:O.is_sup O a s);
- dsc_sup_proof_irrelevant:
- ∀a:nat→O.∀m,m':O.∀p:is_upper_bound ? a m.∀p':is_upper_bound ? a m'.
- (match dsc_sup a m p with [ ex_intro s _ ⇒ s ]) =
- (match dsc_sup a m' p' with [ ex_intro s' _ ⇒ s' ])
- }.
-
-record dedekind_sigma_complete_ordered_set : Type ≝
- { dscos_ordered_set:> ordered_set;
- dscos_dedekind_sigma_complete_properties:>
- is_dedekind_sigma_complete dscos_ordered_set
- }.
-
-definition inf:
- ∀O:dedekind_sigma_complete_ordered_set.
- bounded_below_sequence O → O.
- intros;
- elim
- (dsc_inf O (dscos_dedekind_sigma_complete_properties O) b);
- [ apply a
- | apply (lower_bound ? b)
- | apply lower_bound_is_lower_bound
- ]
-qed.
-
-lemma inf_is_inf:
- ∀O:dedekind_sigma_complete_ordered_set.
- ∀a:bounded_below_sequence O.
- is_inf ? a (inf ? a).
- intros;
- unfold inf;
- simplify;
- elim (dsc_inf O (dscos_dedekind_sigma_complete_properties O) a
-(lower_bound O a) (lower_bound_is_lower_bound O a));
- simplify;
- assumption.
-qed.
-
-lemma inf_proof_irrelevant:
- ∀O:dedekind_sigma_complete_ordered_set.
- ∀a,a':bounded_below_sequence O.
- bbs_seq ? a = bbs_seq ? a' →
- inf ? a = inf ? a'.
- intros 3;
- elim a 0;
- elim a';
- simplify in H;
- generalize in match i1;
- clear i1;
- rewrite > H;
- intro;
- simplify;
- rewrite < (dsc_inf_proof_irrelevant O O f (ib_lower_bound ? f i)
- (ib_lower_bound ? f i2) (ib_lower_bound_is_lower_bound ? f i)
- (ib_lower_bound_is_lower_bound ? f i2));
- reflexivity.
-qed.
-
-definition sup:
- ∀O:dedekind_sigma_complete_ordered_set.
- bounded_above_sequence O → O.
- intros;
- elim
- (dsc_sup O (dscos_dedekind_sigma_complete_properties O) b);
- [ apply a
- | apply (upper_bound ? b)
- | apply upper_bound_is_upper_bound
- ].
-qed.
-
-lemma sup_is_sup:
- ∀O:dedekind_sigma_complete_ordered_set.
- ∀a:bounded_above_sequence O.
- is_sup ? a (sup ? a).
- intros;
- unfold sup;
- simplify;
- elim (dsc_sup O (dscos_dedekind_sigma_complete_properties O) a
-(upper_bound O a) (upper_bound_is_upper_bound O a));
- simplify;
- assumption.
-qed.
-
-lemma sup_proof_irrelevant:
- ∀O:dedekind_sigma_complete_ordered_set.
- ∀a,a':bounded_above_sequence O.
- bas_seq ? a = bas_seq ? a' →
- sup ? a = sup ? a'.
- intros 3;
- elim a 0;
- elim a';
- simplify in H;
- generalize in match i1;
- clear i1;
- rewrite > H;
- intro;
- simplify;
- rewrite < (dsc_sup_proof_irrelevant O O f (ib_upper_bound ? f i2)
- (ib_upper_bound ? f i) (ib_upper_bound_is_upper_bound ? f i2)
- (ib_upper_bound_is_upper_bound ? f i));
- reflexivity.
-qed.
-
-axiom daemon: False.
-
-theorem inf_le_sup:
- ∀O:dedekind_sigma_complete_ordered_set.
- ∀a:bounded_sequence O. inf ? a ≤ sup ? a.
- intros (O');
- apply (or_transitive ? ? O' ? (a O));
- [ elim daemon (*apply (inf_lower_bound ? ? ? ? (inf_is_inf ? ? a))*)
- | elim daemon (*apply (sup_upper_bound ? ? ? ? (sup_is_sup ? ? a))*)
- ].
-qed.
-
-lemma inf_respects_le:
- ∀O:dedekind_sigma_complete_ordered_set.
- ∀a:bounded_below_sequence O.∀m:O.
- is_upper_bound ? a m → inf ? a ≤ m.
- intros (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 O' a a
- (mk_is_bounded_above O' a m H))));
- assumption
- ].
-qed.
-
-definition is_sequentially_monotone ≝
- λO:ordered_set.λf:O→O.
- ∀a:nat→O.∀p:is_increasing ? a.
- is_increasing ? (λi.f (a i)).
-
-record is_order_continuous
- (O:dedekind_sigma_complete_ordered_set) (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:
- ∀O:dedekind_sigma_complete_ordered_set.
- ∀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:
- ∀O:dedekind_sigma_complete_ordered_set.
- 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.