X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Flebesgue.ma;h=5957235ccc183793c5c1e1a209c9f4ca1ecc4de5;hb=6abb01e8b00db927e16aa790354d1da57af7875b;hp=5740db8de5a980f06c254679301fcc0fd522e396;hpb=e7966515539754b5a6efeed89ff8935ca137d651;p=helm.git

diff --git a/helm/software/matita/contribs/dama/dama/lebesgue.ma b/helm/software/matita/contribs/dama/dama/lebesgue.ma
index 5740db8de..5957235cc 100644
--- a/helm/software/matita/contribs/dama/dama/lebesgue.ma
+++ b/helm/software/matita/contribs/dama/dama/lebesgue.ma
@@ -16,102 +16,152 @@
 include "sandwich.ma".
 include "property_exhaustivity.ma".
 
+(* 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 ≤ (pi1exT23 ???? p) j.
+   ∀a:sequence (os_l C).∀s:segment C.∀H:∀i:nat.a i ∈ s. 
+     ∀x:C.∀p:order_converge C a x. 
+       ∀j. seg_l (os_l C) s (λl.le (os_l C) 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);
+cases (wloss_prop (os_l C))(W W);rewrite <W;
+[ intro H2; cases (SSa (seg_l_ C s) H2) (w Hw); simplify in Hw;
+  lapply (H (w+j)) as K; unfold in K;
+  whd in K:(? % ? ? ? ?); simplify in K:(%); rewrite <W in K; cases K;
+  whd in H1:(? % ? ? ? ?); simplify in H1:(%); rewrite <W in H1; 
+  simplify in H1; apply (H1 Hw);
+| intro H2; cases (SSa (seg_u_ C s) H2) (w Hw); simplify in Hw;
+  lapply (H (w+j)) as K; unfold in K;
+  whd in K:(? % ? ? ? ?);simplify in K:(%); rewrite <W in K; cases K; 
+  whd in H3:(? % ? ? ? ?);simplify in H3:(%); rewrite <W in H3;
+  simplify in H3; apply (H3 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.   
-          
+   ∀a:sequence (os_l C).∀s:segment C.∀H:∀i:nat.a i ∈ s. 
+     ∀x:C.∀p:order_converge C a x. 
+       ∀j. seg_u (os_l C) s (λu.le (os_l C) (pi2exT23 ???? p j) u).
+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;
+cases (wloss_prop (os_l C))(W W); unfold os_r; unfold dual_hos; simplify;rewrite <W;
+[ intro H2; cases (SIa (seg_u_ (os_l C) s) H2) (w Hw); simplify in Hw;
+  lapply (H (w+j)) as K; unfold in K; whd in K:(? % ? ? ? ?); simplify in K:(%); 
+  rewrite <W in K; cases K; whd in H3:(? % ? ? ? ?); simplify in H3:(%); rewrite <W in H3;
+  simplify in H3; apply (H3 Hw);
+| intro H2; cases (SIa (seg_l_ C s) H2) (w Hw); simplify in Hw;
+  lapply (H (w+j)) as K; unfold in K;  whd in K:(? % ? ? ? ?); simplify in K:(%);
+  rewrite <W in K; cases K; whd in H1:(? % ? ? ? ?); simplify in H1:(%);
+  rewrite <W in H1; simplify in H1; apply (H1 Hw);]
+qed. 
+
+alias symbol "upp" = "uppper".
+alias symbol "leq" = "Ordered set less or equal than".
+lemma cases_in_segment: 
+  ∀C:half_ordered_set.∀s:segment C.∀x. x ∈ s → seg_l C s (λl.l ≤≤ x) ∧ seg_u C s (λu.x ≤≤ u).
+intros; unfold in H; cases (wloss_prop C) (W W); rewrite<W in H; [cases H; split;assumption]
+cases H; split; assumption;
+qed. 
+
+lemma trans_under_upp:
+ ∀O:ordered_set.∀s:‡O.∀x,y:O.
+  x ≤ y → 𝕦_s (λu.y ≤ u) → 𝕦_s (λu.x ≤ u).
+intros; cases (wloss_prop (os_l O)) (W W); unfold; unfold in H1; rewrite<W in H1 ⊢ %;
+apply (le_transitive ??? H H1);
+qed.
+
+lemma trans_under_low:
+ ∀O:ordered_set.∀s:‡O.∀x,y:O.
+  y ≤ x → 𝕝_s (λl.l ≤ y) → 𝕝_s (λl.l ≤ x).
+intros; cases (wloss_prop (os_l O)) (W W); unfold; unfold in H1; rewrite<W in H1 ⊢ %;
+apply (le_transitive ??? H1 H);
+qed.
+
+lemma l2sl:
+  ∀C,s.∀x,y:half_segment_ordered_set C s. \fst x ≤≤ \fst y → x ≤≤ y.
+intros; unfold in H ⊢ %; intro; apply H; clear H; unfold in H1 ⊢ %;
+cases (wloss_prop C) (W W); whd in H1:(? (% ? ?) ? ? ? ?); simplify in H1:(%); 
+rewrite < W in H1 ⊢ %; apply H1;
+qed.
+
 (* Theorem 3.10 *)
 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].
-       uniform_converge {[l,u]} (⌊n,≪a n,H n≫⌋) ≪x,h≫.
+      x ∈ s ∧ 
+      ∀h:x ∈ s.
+       uniform_converge {[s]} (⌊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; 
+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 (trans_under_upp ?? (xi i) (a i) 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 (trans_under_low ?? (yi i) (a i) K Pl);] 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]
+[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 {[l,u]} ≪?, h≫ Xi Yi Ai); [4: 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;]]
+    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_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].
-       uniform_converge {[l,u]} (⌊n,≪a n,H n≫⌋) ≪x,h≫.
+      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 ???? 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;
+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 (trans_under_upp ?? (xi i) (a i) 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 (trans_under_low ?? (yi i) (a i) K Pl);] clear H2;
 letin Xi ≝ (⌊n,≪xi n, Hxi n≫⌋);
 letin Yi ≝ (⌊n,≪yi n, Hyi n≫⌋);
-cases (restrict_uniform_convergence_uparrow ? S ?? (H l u) Xi x Hx);
-cases (restrict_uniform_convergence_downarrow ? S ?? (H l u) Yi x Hy);
+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;
+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 {[l,u]} ≪?, h≫ Xi Yi Ai); [4: assumption;|2:apply H3;|3:apply H5]
-intro j; cases (Hxy j); cases H7; cases H8; split; [apply (H9 0);|apply (H11 0)]
+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.