record ordered_uniform_space : Type ≝ {
ous_stuff :> ordered_uniform_space_;
- ous_convex: ∀U.us_unifbase ous_stuff U → convex 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.
coercion bs2_of_bss2 nocomposites.
-(*
-lemma xxx :
- ∀O,s,x.bs2_of_bss2 (ordered_set_OF_ordered_uniform_space O) s x
- =
- x.
-intros; reflexivity;
-*)
+
+lemma a2sa :
+ ∀O:ordered_uniform_space.∀s:‡(ordered_set_OF_ordered_uniform_space O).
+ ∀x:
+ bs_carr
+ (square_bishop_set
+ (bishop_set_of_ordered_set
+ (segment_ordered_set
+ (ordered_set_OF_ordered_uniform_space O) s))).
+ (\fst x) ≈ (\snd x) →
+ (\fst (bs2_of_bss2 (ordered_set_OF_ordered_uniform_space O) s x))
+ ≈
+ (\snd (bs2_of_bss2 (ordered_set_OF_ordered_uniform_space O) s x)).
+intros 3; cases x (a b); clear x; cases a (x Hx); cases b (y Hy); clear a b;
+simplify; intros 2 (K H); apply K; clear K; whd; whd in H; cases H;
+[left|right] apply x2sx; assumption;
+qed.
+
lemma segment_ordered_uniform_space:
∀O:ordered_uniform_space.∀s:‡O.ordered_uniform_space.
apply (mk_uniform_space (bishop_set_of_ordered_set {[s]}) f);
[1: intros (U H); intro x; simplify;
cases H (w Hw); cases Hw (Gw Hwp); clear H Hw; intro Hm;
- lapply (us_phi1 O w Gw x) as IH;[2:intro;apply Hm;cases H; clear H;
- [left;apply (x2sx ? s (\fst x) (\snd x) H1);
- |right;apply (x2sx ? s ?? H1);]
-
+ lapply (us_phi1 O w Gw x (a2sa ??? Hm)) as IH;
apply (restrict ? s ??? Hwp IH);
|2: intros (U V HU HV); cases HU (u Hu); cases HV (v Hv); clear HU HV;
cases Hu (Gu HuU); cases Hv (Gv HvV); clear Hu Hv;
cases (us_phi2 O u v Gu Gv) (w HW); cases HW (Gw Hw); clear HW;
- exists; [apply (λb:{[l,r]} squareB.w b)] split;
+ exists; [apply (λb:{[s]} squareB.w b)] split;
[1: unfold f; simplify; clearbody f;
exists; [apply w]; split; [assumption] intro b; simplify;
unfold segment_square_of_ordered_set_square;
cases b; intros; split; intros; assumption;
|2: intros 2 (x Hx); cases (Hw ? Hx); split;
- [apply (restrict O l r ??? HuU H)|apply (restrict O l r ??? HvV H1);]]
+ [apply (restrict O s ??? HuU H)|apply (restrict O s ??? HvV H1);]]
|3: intros (U Hu); cases Hu (u HU); cases HU (Gu HuU); clear Hu HU;
cases (us_phi3 O u Gu) (w HW); cases HW (Gw Hwu); clear HW;
- exists; [apply (λx:{[l,r]} squareB.w x)] split;
+ exists; [apply (λx:{[s]} squareB.w x)] split;
[1: exists;[apply w];split;[assumption] intros; simplify; intro;
unfold segment_square_of_ordered_set_square;
cases b; intros; split; intro; assumption;
- |2: intros 2 (x Hx); apply (restrict O l r ??? HuU); apply Hwu;
+ |2: intros 2 (x Hx); apply (restrict O s ??? HuU); apply Hwu;
cases Hx (m Hm); exists[apply (\fst m)] apply Hm;]
|4: intros (U HU x); cases HU (u Hu); cases Hu (Gu HuU); clear HU Hu;
cases (us_phi4 O u Gu x) (Hul Hur);
split; intros;
- [1: lapply (invert_restriction_agreement O l r ?? HuU) as Ra;
- apply (restrict O l r ?? x Ra);
- apply Hul; apply (unrestrict O l r ??? HuU H);
- |2: apply (restrict O l r ??? HuU); apply Hur;
- apply (unrestrict O l r ??? (invert_restriction_agreement O l r ?? HuU) H);]]
+ [1: lapply (invert_restriction_agreement O s ?? HuU) as Ra;
+ apply (restrict O s ?? x Ra);
+ apply Hul; apply (unrestrict O s ??? HuU H);
+ |2: apply (restrict O s ??? HuU); apply Hur;
+ apply (unrestrict O s ??? (invert_restriction_agreement O s ?? HuU) H);]]
|2: simplify; reflexivity;]
-|2: simplify; unfold convex; intros;
- cases H (u HU); cases HU (Gu HuU); clear HU H;
- lapply (ous_convex ?? Gu p ? H2 y H3) as Cu;
- [1: apply (unrestrict O l r ??? HuU); apply H1;
- |2: apply (restrict O l r ??? HuU Cu);]]
+|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_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;
+ apply (x2sx (os_l O) s (\fst y) h1 H);
+ |5: change with (\fst h ≤ \fst (\fst y)); intro; apply H4;
+ apply (x2sx (os_l O) s h (\fst y) H);
+ |6: change with (\fst (\snd y) ≤ \fst h1); intro; apply H5;
+ apply (x2sx (os_l O) s (\snd y) h1 H);
+ |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);]
+|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".
lemma restric_uniform_convergence:
- ∀O:ordered_uniform_space.∀l,u:O.
- ∀x:{[l,u]}.
- ∀a:sequence {[l,u]}.
+ ∀O:ordered_uniform_space.∀s:‡O.
+ ∀x:{[s]}.
+ ∀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 ≝
projects / helm.git / blobdiff