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 ↓ x) →
(∀n.∀i,j.n ≤ i → n ≤ j → V n 〈a i,a j〉) → U 〈a 0,x〉.
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.
-
+
(* Lemma 3.6 *)
lemma sigma_cauchy:
∀C:ordered_uniform_space.property_sigma C →
∀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;
-alias symbol "pair" = "pair".
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
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]
- generalize in match H6; generalize in match H7;
- cases (aux n1); simplify in ⊢ (? (? ? %) ?→? (? ? %) ?→?); intros;
- apply H12; [3: apply le_S_S_to_le; 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 H6; rewrite > H4 in H5;
rewrite < (le_n_O_to_eq ? H11); apply H5; assumption;]
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);]]
+ [ 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;
- 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:intro X; cases (os_coreflexive ?? X)|*: try apply H12;]
- [1:change with (a (m O) ≤ a i);
- apply (trans_increasing ?? H6); intro; apply (le_to_not_lt ?? H4 H14);
- |2:change with (a i ≤ a (m O));
- apply (trans_decreasing ?? H6); intro; apply (le_to_not_lt ?? H4 H16);]]
+ cases H1; cases H5; cases H7; cases (us_phi4 ?? H3 〈l,a i〉);
+ apply H15; change with (U 〈a i,l〉);
+ [apply (ous_convex_l C ? H3 ? H11 (H12 (m O)));
+ |apply (ous_convex_r C ? H3 ? H11 (H12 (m O)));]
+ [1:apply (H12 i);
+ |3: apply (le_reflexive l);
+ |4: apply (H12 i);
+ |2:change with (a (m O) ≤ a i);
+ apply (trans_increasing a H6 (\fst (m 0)) i); intro; apply (le_to_not_lt ?? H4 H16);
+ |5:apply (H12 i);
+ |7:apply (ge_reflexive (l : hos_carr (os_r C)));
+ |8:apply (H12 i);
+ |6: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));
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;]
+[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;]
+|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.