include "sandwich.ma".
include "property_exhaustivity.ma".
-lemma foo:
+(* NOT DUALIZED *)
+alias symbol "low" = "lower".
+alias id "le" = "cic:/matita/dama/ordered_set/le.con".
+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);
+ ∀a:sequence (os_l C).∀s:segment C.∀H:∀i:nat.a i ∈ s.
+ ∀x:C.∀p:order_converge C a x.
+ ∀j. 𝕝_s ≤ (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 𝕝_s H2) (w Hw); simplify in Hw;
+lapply (H (w+j)) as K; cases (cases_in_segment ? s ? K); apply H3; apply Hw;
qed.
-
-
+
+alias symbol "upp" = "uppper".
+alias symbol "leq" = "Ordered set less or equal than".
+lemma order_converges_smaller_upsegment:
+ ∀C:ordered_set.
+ ∀a:sequence (os_l C).∀s:segment C.∀H:∀i:nat.a i ∈ s.
+ ∀x:C.∀p:order_converge C a x.
+ ∀j. (pi2exT23 ???? p j) ≤ 𝕦_s.
+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 (SIa 𝕦_s H2) (w Hw); lapply (H (w+j)) as K;
+cases (cases_in_segment ? s ? K); apply H1; apply Hw;
+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].
+ (∀s:‡C.order_continuity {[s]}) →
+ ∀a:sequence C.∀s:‡C.∀H:∀i:nat.a i ∈ s.
∀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;
+ x ∈ s ∧
+ ∀h:x ∈ s.
+ uniform_converge {[s]} (⌊n,≪a n,H n≫⌋) ≪x,h≫.
+intros;
+generalize in match (order_converges_bigger_lowsegment ? a s H1 ? H2);
+generalize in match (order_converges_smaller_upsegment ? a s H1 ? H2);
+cases H2 (xi yi Hx Hy Hxy); clear H2; simplify in ⊢ ((?→???%) → (?→???%) → ?); intros;
+cut (∀i.xi i ∈ s) as Hxi; [2:
+ intros; apply (prove_in_segment (os_l C)); [apply (H3 i)] cases (Hxy i) (H5 _); cases H5 (H7 _);
+ lapply (H7 0) as K; cases (cases_in_segment ? s ? (H1 i)) (Pl Pu);
+ simplify in K:(? ? % ?); apply (hle_transitive (os_l C) (xi i) (a i) 𝕦_s K Pu);] clear H3;
+cut (∀i.yi i ∈ s) as Hyi; [2:
+ intros; apply (prove_in_segment (os_l C)); [2:apply (H2 i)] cases (Hxy i) (_ H5); cases H5 (H7 _);
+ lapply (H7 0) as K; cases (cases_in_segment ? s ? (H1 i)) (Pl Pu); simplify in K;
+ apply (le_transitive 𝕝_s ? ? ? K);apply Pl;] clear H2;
split;
-[2: intro h;
- cases (H l u (λn:nat.sig_in ?? (a n) (H1 n)) (sig_in ?? x h));
-
+[1: apply (uparrow_to_in_segment s ? Hxi ? Hx);
+|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 {[s]} ≪x, h≫ Xi Yi Ai); [4: assumption;]
+ [1: intro j; cases (Hxy j); cases H3; cases H4; split; clear H3 H4; simplify in H5 H7;
+ [apply (l2sl ? s (Xi j) (Ai j) (H5 0));|apply (l2sl ? s (Ai j) (Yi j) (H7 0))]
+ |2: cases (H s Xi ≪?,h≫) (Ux Uy); apply Ux; cases Hx; split; [intro i; apply (l2sl ? s (Xi i) (Xi (S i)) (H3 i));]
+ cases H4; split; [intro i; apply (l2sl ? s (Xi i) ≪x,h≫ (H5 i))]
+ intros (y Hy);cases (H6 (\fst y));[2:apply (sx2x ? s ? y Hy)]
+ exists [apply w] apply (x2sx ? s (Xi w) y H7);
+ |3: cases (H s Yi ≪?,h≫) (Ux Uy); apply Uy; cases Hy; split; [intro i; apply (l2sl ? s (Yi (S i)) (Yi i) (H3 i));]
+ cases H4; split; [intro i; apply (l2sl ? s ≪x,h≫ (Yi i) (H5 i))]
+ intros (y Hy);cases (H6 (\fst y));[2:apply (sx2x ? s y ≪x,h≫ Hy)]
+ exists [apply w] apply (x2sx ? s y (Yi w) 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].
+ (∀s:‡C.exhaustive {[s]}) →
+ ∀a:sequence C.∀s:‡C.∀H:∀i:nat.a i ∈ s.
∀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:
+ x ∈ s ∧
+ ∀h:x ∈ s.
+ uniform_converge {[s]} (⌊n,≪a n,H n≫⌋) ≪x,h≫.
+intros (C S);
+generalize in match (order_converges_bigger_lowsegment ? a s H1 ? H2);
+generalize in match (order_converges_smaller_upsegment ? a s H1 ? H2);
+cases H2 (xi yi Hx Hy Hxy); clear H2; simplify in ⊢ ((?→???%) → (?→???%) → ?); intros;
+cut (∀i.xi i ∈ s) as Hxi; [2:
+ intros; apply (prove_in_segment (os_l C)); [apply (H3 i)] cases (Hxy i) (H5 _); cases H5 (H7 _);
+ lapply (H7 0) as K; cases (cases_in_segment ? s ? (H1 i)) (Pl Pu);
+ simplify in K:(? ? % ?); apply (hle_transitive (os_l C) (xi i) (a i) 𝕦_s K Pu);] clear H3;
+cut (∀i.yi i ∈ s) as Hyi; [2:
+ intros; apply (prove_in_segment (os_l C)); [2:apply (H2 i)] cases (Hxy i) (_ H5); cases H5 (H7 _);
+ lapply (H7 0) as K; cases (cases_in_segment ? s ? (H1 i)) (Pl Pu); simplify in K;
+ apply (le_transitive 𝕝_s ? ? ? K);apply Pl;] clear H2;
+letin Xi ≝ (⌊n,≪xi n, Hxi n≫⌋);
+letin Yi ≝ (⌊n,≪yi n, Hyi n≫⌋);
+cases (restrict_uniform_convergence_uparrow ? S ? (H s) Xi x Hx);
+cases (restrict_uniform_convergence_downarrow ? S ? (H s) Yi x Hy);
+split; [1: assumption]
+intros 3;
+lapply (uparrow_upperlocated xi x Hx)as Ux;
+lapply (downarrow_lowerlocated yi x Hy)as Uy;
+letin Ai ≝ (⌊n,≪a n, H1 n≫⌋);
+apply (sandwich {[s]} ≪x, h≫ Xi Yi Ai); [4: assumption;|2:apply H3;|3:apply H5]
+intro j; cases (Hxy j); cases H7; cases H8; split;
+[apply (l2sl ? s (Xi j) (Ai j) (H9 0));|apply (l2sl ? s (Ai j) (Yi j) (H11 0))]
+qed.
+
+
-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
projects / helm.git / blobdiff