include "sandwich.ma".
include "property_exhaustivity.ma".
-lemma foo:
+lemma order_converges_bigger_lowsegment:
∀C:ordered_set.
∀a:sequence C.∀l,u:C.∀H:∀i:nat.a i ∈ [l,u].
∀x:C.∀p:a order_converges x.
- ∀j.l ≤ (match p with [ex_introT xi _ ⇒ xi] j).
-intros; cases p; simplify; cases H1; clear H1; cases H2; clear H2;
-cases (H3 j); cases H1; clear H3 H1; clear H4 H6; cases H5; clear H5;
-cases H2; clear H2; intro; cases (H5 ? H2);
-cases (H (w2+j)); apply (H8 H6);
+ ∀j.l ≤ (pi1exT23 ???? p) j.
+intros; cases p (xi yi Ux Dy Hxy); clear p; simplify;
+cases Ux (Ixi Sxi); clear Ux; cases Dy (Dyi Iyi); clear Dy;
+cases (Hxy j) (Ia Sa); clear Hxy; cases Ia (Da SSa); cases Sa (Inca SIa); clear Ia Sa;
+intro H2; cases (SSa ? H2) (w Hw); simplify in Hw;
+cases (H (w+j)) (Hal Hau); apply (Hau Hw);
qed.
-
-
+
+lemma order_converges_smaller_upsegment:
+ ∀C:ordered_set.
+ ∀a:sequence C.∀l,u:C.∀H:∀i:nat.a i ∈ [l,u].
+ ∀x:C.∀p:a order_converges x.
+ ∀j.(pi2exT23 ???? p) j ≤ u.
+intros; cases p; clear p; simplify; cases H1; clear H1; cases H2; clear H2;
+cases (H3 j); clear H3; cases H2; cases H7; clear H2 H7;
+intro H2; cases (H10 ? H2);
+cases (H (w1+j)); apply (H11 H7);
+qed.
+
(* Theorem 3.10 *)
-theorem lebesgue:
+theorem lebesgue_oc:
∀C:ordered_uniform_space.
(∀l,u:C.order_continuity {[l,u]}) →
∀a:sequence C.∀l,u:C.∀H:∀i:nat.a i ∈ [l,u].
∀x:C.a order_converges x →
x ∈ [l,u] ∧
- ∀h:x ∈ [l,u]. (* manca il pullback? *)
- uniform_converge
- (uniform_space_OF_ordered_uniform_space
- (segment_ordered_uniform_space C l u))
- (λn.sig_in C (λx.x∈[l,u]) (a n) (H n))
- (sig_in ?? x h).
-intros; cases H2 (xi H4); cases H4 (yi H5); cases H5; clear H4 H5;
-cases H3; cases H5; cases H4; clear H3 H4 H5 H2;
+ ∀h:x ∈ [l,u].
+ uniform_converge {[l,u]} (⌊n,≪a n,H n≫⌋) ≪x,h≫.
+intros;
+generalize in match (order_converges_bigger_lowsegment ???? H1 ? H2);
+generalize in match (order_converges_smaller_upsegment ???? H1 ? H2);
+cases H2 (xi yi Hx Hy Hxy); clear H2; simplify in ⊢ (% → % → ?); intros;
+cut (∀i.xi i ∈ [l,u]) as Hxi; [2:
+ intros; split; [2:apply H3] cases (Hxy i) (H5 _); cases H5 (H7 _);
+ apply (le_transitive ???? (H7 0)); simplify;
+ cases (H1 i); assumption;] clear H3;
+cut (∀i.yi i ∈ [l,u]) as Hyi; [2:
+ intros; split; [apply H2] cases (Hxy i) (_ H5); cases H5 (H7 _);
+ apply (le_transitive ????? (H7 0)); simplify;
+ cases (H1 i); assumption;] clear H2;
split;
-[2: intro h;
- cases (H l u (λn:nat.sig_in ?? (a n) (H1 n)) (sig_in ?? x h));
-
+[1: cases Hx; cases H3; cases Hy; cases H7; split;
+ [1: apply (le_transitive ???? (H8 0)); cases (Hyi 0); assumption
+ |2: apply (le_transitive ????? (H4 0)); cases (Hxi 0); assumption]
+|2: intros 3 (h);
+ letin Xi ≝ (⌊n,≪xi n, Hxi n≫⌋);
+ letin Yi ≝ (⌊n,≪yi n, Hyi n≫⌋);
+ letin Ai ≝ (⌊n,≪a n, H1 n≫⌋);
+ apply (sandwich {[l,u]} ≪?, h≫ Xi Yi Ai); try assumption;
+ [1: intro j; cases (Hxy j); cases H3; cases H4; split;
+ [apply (H5 0);|apply (H7 0)]
+ |2: cases (H l u Xi ≪?,h≫) (Ux Uy); apply Ux; cases Hx; split; [apply H3;]
+ cases H4; split; [apply H5] intros (y Hy);cases (H6 (\fst y));[2:apply Hy];
+ exists [apply w] apply H7;
+ |3: cases (H l u Yi ≪?,h≫) (Ux Uy); apply Uy; cases Hy; split; [apply H3;]
+ cases H4; split; [apply H5] intros (y Hy);cases (H6 (\fst y));[2:apply Hy];
+ exists [apply w] apply H7;]]
+qed.
+
(* Theorem 3.9 *)
-theorem lebesgue:
+theorem lebesgue_se:
∀C:ordered_uniform_space.property_sigma C →
(∀l,u:C.exhaustive {[l,u]}) →
∀a:sequence C.∀l,u:C.∀H:∀i:nat.a i ∈ [l,u].
∀x:C.a order_converges x →
x ∈ [l,u] ∧
- ∀h:x ∈ [l,u]. (* manca il pullback? *)
- uniform_converge
- (uniform_space_OF_ordered_uniform_space
- (segment_ordered_uniform_space C l u))
- (λn.sig_in C (λx.x∈[l,u]) (a n) (H n))
- (sig_in ?? x h).
-intros; cases H3 (xi H4); cases H4 (yi H5); cases H5; cases H6; cases H8;
-cases H9; cases H10; cases H11; clear H3 H4 H5 H6 H8 H9 H10 H11 H15 H16;
-lapply (uparrow_upperlocated ? xi x)as Ux;[2: split; assumption]
-lapply (downarrow_lowerlocated ? yi x)as Uy;[2: split; assumption]
-cases (restrict_uniform_convergence ? H ?? (H1 l u) (λn:nat.sig_in ?? (a n) (H2 n)) x);
-[ split; assumption]
-split; simplify;
- [1: intro; cases (H7 n); cases H3;
-
-
- lapply (sandwich ? x xi yi a );
- [2: intro; cases (H7 i); cases H3; cases H4; split[apply (H5 0)|apply (H8 0)]
- |3: intros 2;
- cases (restrict_uniform_convergence ? H ?? (H1 l u) ? x);
- [1:
-
-lapply (restrict_uniform_convergence ? H ?? (H1 l u)
- (λn:nat.sig_in ?? (a n) (H2 n)) x);
- [2:split; assumption]
\ No newline at end of file
+ ∀h:x ∈ [l,u].
+ uniform_converge {[l,u]} (⌊n,≪a n,H n≫⌋) ≪x,h≫.
+intros (C S);
+generalize in match (order_converges_bigger_lowsegment ???? H1 ? H2);
+generalize in match (order_converges_smaller_upsegment ???? H1 ? H2);
+cases H2 (xi yi Hx Hy Hxy); clear H2; simplify in ⊢ (% → % → ?); intros;
+cut (∀i.xi i ∈ [l,u]) as Hxi; [2:
+ intros; split; [2:apply H3] cases (Hxy i) (H5 _); cases H5 (H7 _);
+ apply (le_transitive ???? (H7 0)); simplify;
+ cases (H1 i); assumption;] clear H3;
+cut (∀i.yi i ∈ [l,u]) as Hyi; [2:
+ intros; split; [apply H2] cases (Hxy i) (_ H5); cases H5 (H7 _);
+ apply (le_transitive ????? (H7 0)); simplify;
+ cases (H1 i); assumption;] clear H2;
+split;
+[1: cases Hx; cases H3; cases Hy; cases H7; split;
+ [1: apply (le_transitive ???? (H8 0)); cases (Hyi 0); assumption
+ |2: apply (le_transitive ????? (H4 0)); cases (Hxi 0); assumption]
+|2: intros 3;
+ lapply (uparrow_upperlocated ? xi x Hx)as Ux;
+ lapply (downarrow_lowerlocated ? yi x Hy)as Uy;
+ letin Xi ≝ (⌊n,≪xi n, Hxi n≫⌋);
+ letin Yi ≝ (⌊n,≪yi n, Hyi n≫⌋);
+ letin Ai ≝ (⌊n,≪a n, H1 n≫⌋);
+ apply (sandwich {[l,u]} ≪?, h≫ Xi Yi Ai); try assumption;
+ [1: intro j; cases (Hxy j); cases H3; cases H4; split;
+ [apply (H5 0);|apply (H7 0)]
+ |2: cases (restrict_uniform_convergence_uparrow ? S ?? (H l u) Xi x Hx);
+ apply (H4 h);
+ |3: cases (restrict_uniform_convergence_downarrow ? S ?? (H l u) Yi x Hy);
+ apply (H4 h);]]
+qed.
projects / helm.git / blobdiff