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" 'segment_set a =
+ (segment_ordered_uniform_space _ a).
(* Lemma 3.2 *)
alias symbol "pi1" = "exT \fst".
lemma restric_uniform_convergence:
∀O:ordered_uniform_space.∀s:‡O.
∀x:{[s]}.
- ∀a:sequence {[s]}.
+ ∀a:sequence (segment_ordered_uniform_space O 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 ≝
(a is_increasing → a is_upper_located → a is_cauchy) ∧
(b is_decreasing → b is_lower_located → b is_cauchy).
-lemma segment_upperbound:
- ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.u is_upper_bound ⌊n,\fst (a n)⌋.
-intros 5; change with (\fst (a n) ≤ u); cases (a n); cases H; assumption;
+lemma h_segment_upperbound:
+ ∀C:half_ordered_set.
+ ∀s:segment C.
+ ∀a:sequence (half_segment_ordered_set C s).
+ (seg_u C s) (upper_bound ? ⌊n,\fst (a n)⌋).
+intros; cases (wloss_prop C); unfold; rewrite < H; simplify; intro n;
+cases (a n); simplify; unfold in H1; rewrite < H in H1; cases H1;
+simplify in H2 H3; rewrite < H in H2 H3; assumption;
qed.
-lemma segment_lowerbound:
- ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.l is_lower_bound ⌊n,\fst (a n)⌋.
-intros 5; change with (l ≤ \fst (a n)); cases (a n); cases H; assumption;
-qed.
+notation "'segment_upperbound'" non associative with precedence 90 for @{'segment_upperbound}.
+notation "'segment_lowerbound'" non associative with precedence 90 for @{'segment_lowerbound}.
-lemma segment_preserves_uparrow:
- ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.∀x,h.
- ⌊n,\fst (a n)⌋ ↑ x → a ↑ ≪x,h≫.
-intros; cases H (Ha Hx); split [apply Ha] cases Hx;
-split; [apply H1] intros;
-cases (H2 (\fst y)); [2: apply H3;] exists [apply w] assumption;
-qed.
+interpretation "segment_upperbound" 'segment_upperbound = (h_segment_upperbound (os_l _)).
+interpretation "segment_lowerbound" 'segment_lowerbound = (h_segment_upperbound (os_r _)).
-lemma segment_preserves_downarrow:
- ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.∀x,h.
- ⌊n,\fst (a n)⌋ ↓ x → a ↓ ≪x,h≫.
-intros; cases H (Ha Hx); split [apply Ha] cases Hx;
-split; [apply H1] intros;
-cases (H2 (\fst y));[2:apply H3]; exists [apply w] assumption;
+lemma h_segment_preserves_uparrow:
+ ∀C:half_ordered_set.∀s:segment C.∀a:sequence (half_segment_ordered_set C s).
+ ∀x,h. uparrow C ⌊n,\fst (a n)⌋ x → uparrow (half_segment_ordered_set C s) a ≪x,h≫.
+intros; cases H (Ha Hx); split;
+[ intro n; intro H; apply (Ha n); apply (sx2x ???? H);
+| cases Hx; split;
+ [ intro n; intro H; apply (H1 n);apply (sx2x ???? H);
+ | intros; cases (H2 (\fst y)); [2: apply (sx2x ???? H3);]
+ exists [apply w] apply (x2sx ?? (a w) y H4);]]
qed.
-
+
+notation "'segment_preserves_uparrow'" non associative with precedence 90 for @{'segment_preserves_uparrow}.
+notation "'segment_preserves_downarrow'" non associative with precedence 90 for @{'segment_preserves_downarrow}.
+
+interpretation "segment_preserves_uparrow" 'segment_preserves_uparrow = (h_segment_preserves_uparrow (os_l _)).
+interpretation "segment_preserves_downarrow" 'segment_preserves_downarrow = (h_segment_preserves_uparrow (os_r _)).
+
+lemma hint_pippo:
+ ∀C,s.
+ sequence
+ (Type_of_ordered_set
+ (segment_ordered_set
+ (ordered_set_OF_ordered_uniform_space C) s))
+ →
+ sequence (Type_OF_uniform_space (segment_ordered_uniform_space C s)). intros; assumption;
+qed.
+
+coercion hint_pippo nocomposites.
+
(* Fact 2.18 *)
lemma segment_cauchy:
- ∀C:ordered_uniform_space.∀l,u:C.∀a:sequence {[l,u]}.
+ ∀C:ordered_uniform_space.∀s:‡C.∀a:sequence {[s]}.
a is_cauchy → ⌊n,\fst (a n)⌋ is_cauchy.
-intros 7;
+intros 6;
alias symbol "pi1" (instance 3) = "pair pi1".
alias symbol "pi2" = "pair pi2".
-apply (H (λx:{[l,u]} squareB.U 〈\fst (\fst x),\fst (\snd x)〉));
+apply (H (λx:{[s]} squareB.U 〈\fst (\fst x),\fst (\snd x)〉));
(unfold segment_ordered_uniform_space; simplify);
exists [apply U] split; [assumption;]
intro; cases b; intros; simplify; split; intros; assumption;
| 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: 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));
+ 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));
definition seg_u ≝
λO:half_ordered_set.λs:segment O.λP: O → CProp.
- wloss O ? (λl,u.P u) (seg_l_ ? s) (seg_u_ ? s).
+ wloss O ? (λl,u.P u) (seg_u_ ? s) (seg_l_ ? s).
definition seg_l ≝
λO:half_ordered_set.λs:segment O.λP: O → CProp.
- wloss O ? (λl,u.P u) (seg_u_ ? s) (seg_l_ ? s).
+ wloss O ? (λl,u.P u) (seg_l_ ? s) (seg_u_ ? s).
interpretation "uppper" 'upp s P = (seg_u (os_l _) s P).
interpretation "lower" 'low s P = (seg_l (os_l _) s P).
definition in_segment ≝
λO:half_ordered_set.λs:segment O.λx:O.
- wloss O ? (λp1,p2.p1 ∧ p2) (seg_u ? s (λu.u ≤≤ x)) (seg_l ? s (λl.x ≤≤ l)).
+ wloss O ? (λp1,p2.p1 ∧ p2) (seg_l ? s (λl.l ≤≤ x)) (seg_u ? s (λu.x ≤≤ u)).
notation "‡O" non associative with precedence 90 for @{'segment $O}.
interpretation "Ordered set sergment" 'segment x = (segment x).
*)
(* Definition 2.10 *)
+
alias symbol "pi2" = "pair pi2".
alias symbol "pi1" = "pair pi1".
+(*
definition square_segment ≝
λO:half_ordered_set.λs:segment O.λx: square_half_ordered_set O.
in_segment ? s (\fst x) ∧ in_segment ? s (\snd x).
-
+*)
definition convex ≝
λO:half_ordered_set.λU:square_half_ordered_set O → Prop.
∀s.U s → le O (\fst s) (\snd s) →
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