(**************************************************************************)
include "ordered_uniform.ma".
+include "property_sigma.ma".
(* Definition 3.7 *)
-definition exhaustivity ≝
+definition exhaustive ≝
λC:ordered_uniform_space.
∀a,b:sequence C.
(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;
+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.
+
+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.
+
+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;
+qed.
+
+(* Fact 2.18 *)
+lemma segment_cauchy:
+ ∀C:ordered_uniform_space.∀l,u:C.∀a:sequence {[l,u]}.
+ a is_cauchy → ⌊n,\fst (a n)⌋ is_cauchy.
+intros 7;
+alias symbol "pi1" (instance 3) = "pair pi1".
+alias symbol "pi2" = "pair pi2".
+apply (H (λx:{[l,u]} square.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;
+qed.
+
(* Lemma 3.8 *)
+lemma restrict_uniform_convergence_uparrow:
+ ∀C:ordered_uniform_space.property_sigma C →
+ ∀l,u:C.exhaustive {[l,u]} →
+ ∀a:sequence {[l,u]}.∀x:C. ⌊n,\fst (a n)⌋ ↑ x →
+ x∈[l,u] ∧ ∀h:x ∈ [l,u].a uniform_converges ≪x,h≫.
+intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
+[1: split;
+ [1: apply (supremum_is_upper_bound C ?? Hx u);
+ apply (segment_upperbound ? l);
+ |2: apply (le_transitive ? ??? ? (H2 O));
+ apply (segment_lowerbound ?l u);]
+|2: intros;
+ lapply (uparrow_upperlocated ? a ≪x,h≫) as Ha1;
+ [2: apply segment_preserves_uparrow;split; assumption;]
+ lapply (segment_preserves_supremum ? l u a ≪?,h≫) as Ha2;
+ [2:split; assumption]; cases Ha2; clear Ha2;
+ cases (H1 a a); lapply (H6 H4 Ha1) as HaC;
+ lapply (segment_cauchy ? l u ? HaC) as Ha;
+ lapply (sigma_cauchy ? H ? x ? Ha); [left; split; assumption]
+ apply restric_uniform_convergence; assumption;]
+qed.
+
+lemma restrict_uniform_convergence_downarrow:
+ ∀C:ordered_uniform_space.property_sigma C →
+ ∀l,u:C.exhaustive {[l,u]} →
+ ∀a:sequence {[l,u]}.∀x:C. ⌊n,\fst (a n)⌋ ↓ x →
+ x∈[l,u] ∧ ∀h:x ∈ [l,u].a uniform_converges ≪x,h≫.
+intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
+[1: split;
+ [2: apply (infimum_is_lower_bound C ?? Hx l);
+ apply (segment_lowerbound ? l u);
+ |1: apply (le_transitive ???? (H2 O));
+ apply (segment_upperbound ? l u);]
+|2: intros;
+ lapply (downarrow_lowerlocated ? a ≪x,h≫) as Ha1;
+ [2: apply segment_preserves_downarrow;split; assumption;]
+ lapply (segment_preserves_infimum ?l u a ≪?,h≫) as Ha2;
+ [2:split; assumption]; cases Ha2; clear Ha2;
+ cases (H1 a a); lapply (H7 H4 Ha1) as HaC;
+ lapply (segment_cauchy ? l u ? HaC) as Ha;
+ lapply (sigma_cauchy ? H ? x ? Ha); [right; split; assumption]
+ apply restric_uniform_convergence; assumption;]
+qed.
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