X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Flebesgue.ma;h=251352d1f6240f8a97275b7ffc3c9e6a4fe4597d;hb=bb7af347df386afcd3ea2adea8e7e982e3a5a253;hp=1b34801d2949e2516096675c462066eeed18d7d1;hpb=c00f22f7afa508881c8d116928e1c460600ba0ac;p=helm.git

diff --git a/helm/software/matita/contribs/dama/dama/lebesgue.ma b/helm/software/matita/contribs/dama/dama/lebesgue.ma
index 1b34801d2..251352d1f 100644
--- a/helm/software/matita/contribs/dama/dama/lebesgue.ma
+++ b/helm/software/matita/contribs/dama/dama/lebesgue.ma
@@ -16,67 +16,110 @@
 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