(**************************************************************************)
include "ordered_uniform.ma".
-
+include "russell_support.ma".
(* Definition 3.5 *)
alias num (instance 0) = "natural number".
definition property_sigma ≝
λC:ordered_uniform_space.
∀U.us_unifbase ? U →
- ∃V:sequence (C square → Prop).
+ ∃V:sequence (C squareB → Prop).
(∀i.us_unifbase ? (V i)) ∧
- ∀a:sequence C.∀x:C.a ↑ x →
+ ∀a:sequence C.∀x:C.(a ↑ x ∨ a ↓ x) →
(∀n.∀i,j.n ≤ i → n ≤ j → V n 〈a i,a j〉) → U 〈a 0,x〉.
definition max ≝
lemma max_le_r: ∀n,m,z.max n m ≤ z → m ≤ z.
intros; rewrite > sym_max in H; apply (max_le_l ??? H);
qed.
-
-
-definition hide ≝ λT:Type.λx:T.x.
-
-notation < "\blacksquare" non associative with precedence 50 for @{'hide}.
-interpretation "hide" 'hide = (hide _ _).
-interpretation "hide2" 'hide = (hide _ _ _).
-
-definition inject ≝ λP.λa:nat.λp:P a. ex_introT ? P ? p.
-coercion cic:/matita/dama/property_sigma/inject.con 0 1.
-definition eject ≝ λP.λc: ∃n:nat.P n. match c with [ ex_introT w _ ⇒ w].
-coercion cic:/matita/dama/property_sigma/eject.con.
-
+
(* Lemma 3.6 *)
lemma sigma_cauchy:
∀C:ordered_uniform_space.property_sigma C →
- ∀a:sequence C.∀l:C.a ↑ l → a is_cauchy → a uniform_converges l.
-intros 8; cases H1; cases H5; clear H5;
-cases (H ? H3); cases H5; clear H5;
+ ∀a:sequence C.∀l:C.(a ↑ l ∨ a ↓ l) → a is_cauchy → a uniform_converges l.
+intros 8; cases (H ? H3) (w H5); cases H5 (H8 H9); clear H5;
+letin spec ≝ (λz,k:nat.∀i,j,l:nat.k ≤ i → k ≤ j → l ≤ z → w l 〈a i,a j〉);
letin m ≝ (hide ? (let rec aux (i:nat) : nat ≝
match i with
[ O ⇒ match H2 (w i) ? with [ ex_introT k _ ⇒ k ]
| S i' ⇒ max (match H2 (w i) ? with [ ex_introT k _ ⇒ k ]) (S (aux i'))
] in aux
- :
- ∀z:nat.∃k:nat.∀i,j,l.k ≤ i → k ≤ j → l ≤ z → w l 〈a i, a j〉));
+ : ∀z.∃k. spec z k)); unfold spec in aux ⊢ %;
[1,2:apply H8;
|3: intros 3; cases (H2 (w n) (H8 n)); simplify in ⊢ (? (? % ?) ?→?);
simplify in ⊢ (?→? (? % ?) ?→?);
- intros; lapply (H10 i j) as H14;
- [2: apply (max_le_l ??? H11);|3:apply (max_le_l ??? H12);]
- cases (le_to_or_lt_eq ?? H13); [2: destruct H15; destruct H5; assumption]
- generalize in match H11; generalize in match H12;
- cases (aux n1); simplify in ⊢ (? (? ? %) ?→? (? ? %) ?→?); intros;
- apply H16; [3: apply le_S_S_to_le; assumption]
+ intros; lapply (H5 i j) as H14;
+ [2: apply (max_le_l ??? H6);|3:apply (max_le_l ??? H7);]
+ cases (le_to_or_lt_eq ?? H10); [2: destruct H11; destruct H4; assumption]
+ cases (aux n1) in H6 H7 ⊢ %; simplify in ⊢ (? (? ? %) ?→? (? ? %) ?→?); intros;
+ apply H6; [3: apply le_S_S_to_le; assumption]
apply lt_to_le; apply (max_le_r w1); assumption;
- |4: intros; clear H11; rewrite > H5 in H10;
- rewrite < (le_n_O_to_eq ? H14); apply H10; assumption;]
-cut (((m : nat→nat) : sequence nat_ordered_set) is_strictly_increasing) as Hm; [2:
+ |4: intros; clear H6; rewrite > H4 in H5;
+ rewrite < (le_n_O_to_eq ? H11); apply H5; assumption;]
+cut ((⌊x,(m x:nat)⌋ : sequence nat_ordered_set) is_strictly_increasing) as Hm; [2:
intro n; change with (S (m n) ≤ m (S n)); unfold m;
whd in ⊢ (? ? %); apply (le_max ? (S (m n)));]
-cut (((m : nat→nat) : sequence nat_ordered_set) is_increasing) as Hm1; [2:
+cut ((⌊x,(m x:nat)⌋ : sequence nat_ordered_set) is_increasing) as Hm1; [2:
intro n; intro L; change in L with (m (S n) < m n);
lapply (Hm n) as L1; change in L1 with (m n < m (S n));
lapply (trans_lt ??? L L1) as L3; apply (not_le_Sn_n ? L3);]
-clearbody m;
-lapply (selection ?? Hm a l H1) as H10;
+clearbody m; unfold spec in m Hm Hm1; clear spec;
+cut (⌊x,a (m x)⌋ ↑ l ∨ ⌊x,a (m x)⌋ ↓ l) as H10; [2:
+ cases H1;
+ [ left; apply (selection_uparrow ? Hm a l H4);
+ | right; apply (selection_downarrow ? Hm a l H4);]]
lapply (H9 ?? H10) as H11; [
exists [apply (m 0:nat)] intros;
- apply (ous_convex ?? H3 ? H11 (H6 (m 0)));
- simplify; repeat split; [intro X; cases (os_coreflexive ?? X)|2,3:apply H6;]
- change with (a (m O) ≤ a i);
- apply (trans_increasing ?? H4); intro; whd in H12;
- apply (not_le_Sn_n i); apply (transitive_le ??? H12 H5)]
+ cases H1;
+ [cases H5; cases H7; apply (ous_convex ?? H3 ? H11 (H12 (m O)));
+ |cases H5; cases H7; cases (us_phi4 ?? H3 〈(a (m O)),l〉);
+ lapply (H14 H11) as H11'; apply (ous_convex ?? H3 〈l,(a (m O))〉 H11' (H12 (m O)));]
+ simplify; repeat split;
+ [1,6: apply (le_reflexive l);
+ |2,5: apply (H12 (\fst (m 0)));
+ |3,8: apply (H12 i);
+ |4:change with (a (m O) ≤ a i);
+ apply (trans_increasing a H6 (\fst (m 0)) i); intro; apply (le_to_not_lt ?? H4 H14);
+ |7:change with (a i ≤ a (m O));
+ apply (trans_decreasing ? H6 (\fst (m 0)) i); intro; apply (le_to_not_lt ?? H4 H16);]]
clear H10; intros (p q r); change with (w p 〈a (m q),a (m r)〉);
generalize in match (refl_eq nat (m p));
-(*
-intro E; cases p in E : (? ? ? ?);
-*)
generalize in match (m p) in ⊢ (? ? ? % → %); intro X; cases X (w1 H15); clear X;
intros (H16); simplify in H16:(? ? ? %); destruct H16;
apply H15; [3: apply le_n]
-[1: lapply (trans_increasing ?? Hm1 p q) as T; [apply not_lt_to_le; apply T;]
- apply (le_to_not_lt p q H5);
-|2: lapply (trans_increasing ?? Hm1 p r) as T; [apply not_lt_to_le; apply T;]
- apply (le_to_not_lt p r H10);]
+[1: lapply (trans_increasing ? Hm1 p q) as T; [apply not_lt_to_le; apply T;]
+ apply (le_to_not_lt p q H4);
+|2: lapply (trans_increasing ? Hm1 p r) as T; [apply not_lt_to_le; apply T;]
+ apply (le_to_not_lt p r H5);]
qed.
projects / helm.git / blobdiff