record ordered_uniform_space : Type ≝ {
ous_stuff :> ordered_uniform_space_;
- ous_convex: ∀U.us_unifbase ous_stuff U → convex (os_l ous_stuff) U
+ ous_convex_l: ∀U.us_unifbase ous_stuff U → convex (os_l ous_stuff) U;
+ ous_convex_r: ∀U.us_unifbase ous_stuff U → convex (os_r ous_stuff) U
}.
definition half_ordered_set_OF_ordered_uniform_space : ordered_uniform_space → half_ordered_set.
|2: simplify; unfold convex; intros 3; cases s1; intros;
(* TODO: x2sx is for ≰, we need one for ≤ *)
cases H (u HU); cases HU (Gu HuU); clear HU H;
- lapply depth=0 (ous_convex ?? Gu 〈\fst h,\fst h1〉 ???????) as K3;
+ lapply depth=0 (ous_convex_l ?? Gu 〈\fst h,\fst h1〉 ???????) as K3;
[2: intro; apply H2; apply (x2sx (os_l O) s h h1 H);
|3: apply 〈\fst (\fst y),\fst (\snd y)〉;
|4: intro; change in H with (\fst (\fst y) ≰ \fst h1); apply H3;
|7: change with (\fst (\fst y) ≤ \fst (\snd y)); intro; apply H6;
apply (x2sx (os_l O) s (\fst y) (\snd y) H);
|8: apply (restrict O s U u y HuU K3);
- |1: apply (unrestrict O s ?? 〈h,h1〉 HuU H1);]]
+ |1: apply (unrestrict O s ?? 〈h,h1〉 HuU H1);]
+|3: simplify; unfold convex; intros 3; cases s1; intros; (* TODO *)
+ cases H (u HU); cases HU (Gu HuU); clear HU H;
+ lapply depth=0 (ous_convex_r ?? Gu 〈\fst h,\fst h1〉 ???????) as K3;
+ [2: intro; apply H2; apply (x2sx (os_r O) s h h1 H);
+ |3: apply 〈\fst (\fst y),\fst (\snd y)〉;
+ |4: intro; (*change in H with (\fst (\fst y) ≱ \fst h1);*) apply H3;
+ apply (x2sx (os_r O) s (\fst y) h1 H);
+ |5: (*change with (\fst h ≥ \fst (\fst y));*) intro; apply H4;
+ apply (x2sx (os_r O) s h (\fst y) H);
+ |6: (*change with (\fst (\snd y) ≤ \fst h1);*) intro; apply H5;
+ apply (x2sx (os_r O) s (\snd y) h1 H);
+ |7: (*change with (\fst (\fst y) ≤ \fst (\snd y));*) intro; apply H6;
+ apply (x2sx (os_r O) s (\fst y) (\snd y) H);
+ |8: apply (restrict O s U u y HuU K3);
+ |1: apply (unrestrict O s ?? 〈h,h1〉 HuU H1);]
+]
qed.
-interpretation "Ordered uniform space segment" 'segment_set a b =
- (segment_ordered_uniform_space _ a b).
+interpretation "Ordered uniform space segment" 'segset a =
+ (segment_ordered_uniform_space _ a).
(* Lemma 3.2 *)
alias symbol "pi1" = "exT \fst".
∀a:sequence {[s]}.
(⌊n, \fst (a n)⌋ : sequence O) uniform_converges (\fst x) →
a uniform_converges x.
-intros 8; cases H1; cases H2; clear H2 H1;
+intros 7; cases H1; cases H2; clear H2 H1;
cases (H ? H3) (m Hm); exists [apply m]; intros;
-apply (restrict ? l u ??? H4); apply (Hm ? H1);
+apply (restrict ? s ??? H4); apply (Hm ? H1);
qed.
definition order_continuity ≝