From 6abb01e8b00db927e16aa790354d1da57af7875b Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Tue, 28 Oct 2008 15:35:05 +0000
Subject: [PATCH] lebesge works

---
 .../matita/contribs/dama/dama/lebesgue.ma     | 176 +++++++++++-------
 .../matita/contribs/dama/dama/ordered_set.ma  |   4 +-
 .../dama/dama/property_exhaustivity.ma        | 152 ++++++---------
 .../matita/contribs/dama/dama/sandwich.ma     |   9 +-
 4 files changed, 180 insertions(+), 161 deletions(-)

diff --git a/helm/software/matita/contribs/dama/dama/lebesgue.ma b/helm/software/matita/contribs/dama/dama/lebesgue.ma
index 2e6b0a1e4..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 l 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 (ge_transitive u ??? (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 l ? (yi i) ? (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 (ge_transitive u ?? ? (H8 0)); cases (Hyi 0); assumption
-    |2: apply (le_transitive l ? x ? (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 (ge_transitive u ?? ? (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 l ? (yi i) ? (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;
 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. 
 
 
diff --git a/helm/software/matita/contribs/dama/dama/ordered_set.ma b/helm/software/matita/contribs/dama/dama/ordered_set.ma
index a4b40be7a..2a1089fec 100644
--- a/helm/software/matita/contribs/dama/dama/ordered_set.ma
+++ b/helm/software/matita/contribs/dama/dama/ordered_set.ma
@@ -36,6 +36,7 @@ record half_ordered_set: Type ≝ {
 
 definition hos_excess ≝ λO:half_ordered_set.wloss O ? (hos_excess_ O). 
 
+(*
 lemma find_leq : half_ordered_set → half_ordered_set.
 intro O; constructor 1;
 [1: apply (hos_carr O);
@@ -49,6 +50,7 @@ intro O; constructor 1;
     rewrite > (H1 ? (hos_excess_ O)); lapply (hos_cotransitive O ?? z H);
     [assumption] cases Hletin;[right|left]assumption;]
 qed.
+*)
 
 definition dual_hos : half_ordered_set → half_ordered_set.
 intro; constructor 1;
@@ -75,8 +77,6 @@ intro hos;
 constructor 1; [apply hos; | apply (dual_hos hos); | reflexivity] 
 qed.
 
-(* coercion half2full. *)
-  
 definition Type_of_ordered_set : ordered_set → Type.
 intro o; apply (hos_carr (os_l o)); qed.
 
diff --git a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
index 0e0f013e9..48551f85e 100644
--- a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
+++ b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
@@ -69,22 +69,37 @@ definition exhaustive ≝
      (b is_decreasing → b is_lower_located → b is_cauchy).
 
 lemma prove_in_segment: 
- ∀O:ordered_set.∀s:segment (os_l O).∀x:O.
-   𝕝_s (λl.l ≤ x) → 𝕦_s (λu.x ≤ u) → x ∈ s.
-intros; unfold; cases (wloss_prop (os_l O)); rewrite < H2;
+ ∀O:half_ordered_set.∀s:segment O.∀x:O.
+   seg_l O s (λl.l ≤≤ x) → seg_u O s (λu.x ≤≤ u) → x ∈ s.
+intros; unfold; cases (wloss_prop O); rewrite < H2;
 split; assumption;
 qed.
 
-lemma under_wloss_upperbound: 
- ∀C:half_ordered_set.∀s:segment C.∀a:sequence C.
-  seg_u C s (upper_bound C a) → 
-    ∀i.seg_u C s (λu.a i ≤≤ u).
-intros; unfold in H; unfold;
-cases (wloss_prop C); rewrite <H1 in H ⊢ %;
-apply (H i);
-qed. 
+lemma h_uparrow_to_in_segment:
+  ∀C:half_ordered_set.
+   ∀s:segment C.
+     ∀a:sequence C.
+      (∀i.a i ∈ s) →
+       ∀x:C. uparrow C a x → 
+         in_segment C s x.
+intros (C H a H1 x H2); unfold in H2; cases H2; clear H2;unfold in H3 H4; cases H4; clear H4; unfold in H2;
+cases (wloss_prop C) (W W); apply prove_in_segment; unfold; rewrite <W;simplify;
+[ apply (hle_transitive ??? x ? (H2 O)); lapply (H1 O) as K; unfold in K; rewrite <W in K;
+  cases K; unfold in H4 H6; rewrite <W in H6 H4; simplify in H4 H6; assumption;
+| intro; cases (H5 ? H4); clear H5 H4;lapply(H1 w) as K; unfold in K; rewrite<W in K;
+  cases K; unfold in H5 H4; rewrite<W in H4 H5; simplify in H4 H5; apply (H5 H6);    
+| apply (hle_transitive ??? x ?  (H2 O)); lapply (H1 0) as K; unfold in K; rewrite <W in K;
+  cases K; unfold in H4 H6; rewrite <W in H4 H6; simplify in H4 H6; assumption;
+| intro; cases (H5 ? H4); clear H5 H4;lapply(H1 w) as K; unfold in K; rewrite<W in K;
+  cases K; unfold in H5 H4; rewrite<W in H4 H5; simplify in H4 H5; apply (H4 H6);]    
+qed.
 
+notation "'uparrow_to_in_segment'" non associative with precedence 90 for @{'uparrow_to_in_segment}.
+notation "'downarrow_to_in_segment'" non associative with precedence 90 for @{'downarrow_to_in_segment}.
 
+interpretation "uparrow_to_in_segment" 'uparrow_to_in_segment = (h_uparrow_to_in_segment (os_l _)).
+interpretation "downarrow_to_in_segment" 'downarrow_to_in_segment = (h_uparrow_to_in_segment (os_r _)).
+ 
 (* Lemma 3.8 NON DUALIZZATO *)
 lemma restrict_uniform_convergence_uparrow:
   ∀C:ordered_uniform_space.property_sigma C →
@@ -93,72 +108,17 @@ lemma restrict_uniform_convergence_uparrow:
       ∀x:C. ⌊n,\fst (a n)⌋ ↑ x → 
        in_segment (os_l C) s x ∧ ∀h:x ∈ s.a uniform_converges ≪x,h≫.
 intros; split;
-[1: unfold in H2; cases H2; clear H2;unfold in H3 H4; cases H4; clear H4; unfold in H2;
-    cases (wloss_prop (os_l C)) (W W); apply prove_in_segment; unfold; rewrite <W;
-    simplify;
-    [ apply (le_transitive ?? x ? (H2 O));  
-      lapply (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a) O) as K;
-      unfold in K; whd in K:(?%????); simplify in K; rewrite <W in K; apply K;
-    | intro; cases (H5 ? H4); clear H5 H4;    
-      lapply (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) w) as K;
-      unfold in K; whd in K:(?%????); simplify in K; rewrite <W in K;
-      apply K; apply H6;
-    | intro; unfold in H4; rewrite <W in H4;
-      lapply depth=0 (H5 (seg_u_ (os_l C) s)) as k; unfold in k:(%???→?);
-      simplify in k; rewrite <W in k; lapply (k
-     simplify;intro; cases (H5 ? H4); clear H5 H4;    
-      lapply (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) w) as K;
-      unfold in K; whd in K:(?%????); simplify in K; rewrite <W in K;
-      apply K; apply H6;    
-      
-      
-      
-    cases H2 (Ha Hx); clear H2; cases Hx; split; 
-    lapply depth=0 (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) O) as W1;
-    lapply depth=0 (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a) O) as W2;
-    lapply (H2 O); simplify in Hletin; simplify in W2 W1;
-    cases a in Hletin W2 W1; simplify; cases (f O); simplify; intros;
-    whd in H6:(? % ? ? ? ?);
-    simplify in H6:(%);
-    cases (wloss_prop (os_l C)); rewrite <H8 in H5 H6 ⊢ %;
-    [ change in H6 with (le (os_l C) (seg_l_ (os_l C) s) w);
-      apply (le_transitive ??? H6 H7);
-    | apply (le_transitive (seg_u_ (os_l C) s) w x H6 H7);
-    |  
-      lapply depth=0 (supremum_is_upper_bound ? x Hx (seg_u_ (os_l C) s)) as K;    
-      lapply depth=0 (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a));
-      apply K; intro; lapply (Hletin n); unfold; unfold in Hletin1;
-      rewrite < H8 in Hletin1; intro; apply Hletin1; clear Hletin1; apply H9;
-    | lapply depth=0 (h_supremum_is_upper_bound (os_r C) ⌊n,\fst (a n)⌋ x Hx (seg_l_ (os_r C) s)) as K;    
-      lapply depth=0 (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a));
-      apply K; intro; lapply (Hletin n); unfold; unfold in Hletin1;
-whd in Hletin1:(? % ? ? ? ?);
-simplify in Hletin1:(%);
-      rewrite < H8 in Hletin1; intro; apply Hletin1; clear Hletin1; apply H9;
-  
-      
-        apply (segment_upperbound ? l);
-    generalize in match (H2 O); generalize in match Hx; unfold supremum;
-    unfold upper_bound; whd in ⊢ (?→%→?); rewrite < H4;
-    split; unfold; rewrite < H4; simplify;
-      [1: lapply (infimum_is_lower_bound ? ? Hx u); 
-
-
-
-split;
-    [1: apply (supremum_is_upper_bound ? x Hx u); 
-        apply (segment_upperbound ? l);
-    |2: apply (le_transitive l ? x ? (H2 O));
-        apply (segment_lowerbound ? l u a 0);]
+[1: apply (uparrow_to_in_segment s ⌊n,\fst (a \sub n)⌋ ? x H2); 
+    simplify; intros; cases (a i); assumption;
 |2: intros;
     lapply (uparrow_upperlocated a ≪x,h≫) as Ha1;
-      [2: apply (segment_preserves_uparrow C l u);split; assumption;] 
-    lapply (segment_preserves_supremum C 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;]
+      [2: apply (segment_preserves_uparrow s); assumption;] 
+    lapply (segment_preserves_supremum s a ≪?,h≫ H2) as Ha2; 
+    cases Ha2; clear Ha2;
+    cases (H1 a a); lapply (H5 H3 Ha1) as HaC;
+    lapply (segment_cauchy C s ? HaC) as Ha;
+    lapply (sigma_cauchy ? H  ? x ? Ha); [left; assumption]
+    apply (restric_uniform_convergence C s ≪x,h≫ a Hletin)]
 qed.
 
 lemma hint_mah1:
@@ -185,24 +145,34 @@ lemma hint_mah4:
   
 coercion hint_mah4 nocomposites.
 
+lemma hint_mah5:
+  ∀C. segment (hos_carr (os_r C)) → segment (hos_carr (os_l C)).
+  intros; assumption; qed.
+  
+coercion hint_mah5 nocomposites.
+
+lemma hint_mah6:
+  ∀C. segment (hos_carr (os_l C)) → segment (hos_carr (os_r C)).
+  intros; assumption; qed.
+
+coercion hint_mah6 nocomposites.
+
 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 ? x Hx l); 
-        apply (segment_lowerbound ? l u);
-    |1: lapply (ge_transitive ? ? x ? (H2 O)); [apply u||assumption]
-        apply (segment_upperbound ? l u a 0);]
+    ∀s:segment (os_l C).exhaustive (segment_ordered_uniform_space C s) →
+     ∀a:sequence (segment_ordered_uniform_space C s).
+      ∀x:C. ⌊n,\fst (a n)⌋ ↓ x → 
+       in_segment (os_l C) s x ∧ ∀h:x ∈ s.a uniform_converges ≪x,h≫.
+intros; split;       
+[1: apply (downarrow_to_in_segment s ⌊n,\fst (a n)⌋ ? x); [2: apply H2]; 
+    simplify; intros; cases (a i); assumption;
 |2: intros;
     lapply (downarrow_lowerlocated a ≪x,h≫) as Ha1;
-      [2: apply (segment_preserves_downarrow ? l u);split; assumption;]
-          lapply (segment_preserves_infimum C l u a ≪x,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;]
+      [2: apply (segment_preserves_downarrow s a x h H2);] 
+    lapply (segment_preserves_infimum s a ≪?,h≫ H2) as Ha2; 
+    cases Ha2; clear Ha2;
+    cases (H1 a a); lapply (H6 H3 Ha1) as HaC;
+    lapply (segment_cauchy C s ? HaC) as Ha;
+    lapply (sigma_cauchy ? H  ? x ? Ha); [right; assumption]
+    apply (restric_uniform_convergence C s ≪x,h≫ a Hletin)]
 qed.
diff --git a/helm/software/matita/contribs/dama/dama/sandwich.ma b/helm/software/matita/contribs/dama/dama/sandwich.ma
index e55b6d3b3..68bb4453c 100644
--- a/helm/software/matita/contribs/dama/dama/sandwich.ma
+++ b/helm/software/matita/contribs/dama/dama/sandwich.ma
@@ -33,15 +33,14 @@ cases (us_phi3 ? ? Gv) (W GW); cases GW (Gw HW); clear GW;
 cases (H1 ? Gw) (ma Hma); cases (H2 ? Gw) (mb Hmb); clear H1 H2;
 exists [apply (ma + mb)] intros; apply (HV 〈l,(x i)〉); 
 unfold; simplify; exists [apply (a i)] split;
-[2: apply (ous_convex ?? Gv 〈a i,b i〉); cases (H i) (Lax Lxb); clear H;
+[2: apply (ous_convex_l ?? Gv 〈a i,b i〉); cases (H i) (Lax Lxb); clear H;
     [1: apply HW; exists [apply l]simplify; split; 
         [1: cases (us_phi4 ?? Gw 〈(a i),l〉); apply H2; clear H2 H;
             apply (Hma i); rewrite > sym_plus in H1; apply (le_w_plus mb); assumption;
         |2: apply Hmb; apply (le_w_plus ma); assumption] 
-    |2: simplify; apply (le_transitive (a i) ?? Lax Lxb);
-    |3: simplify; repeat split; try assumption; 
-        [1: apply (le_transitive ?? (b i) Lax Lxb);
-        |2: intro X; cases (exc_coreflexive (a i) X)]]
+    |2,3: simplify; apply (le_transitive (a i) ?? Lax Lxb);
+    |4: apply (le_reflexive);
+    |5,6: assumption;]
 |1: apply HW; exists[apply l] simplify; split;
     [1: apply (us_phi1 ?? Gw); unfold; apply eq_reflexive;
     |2: apply Hma;  rewrite > sym_plus in H1; apply (le_w_plus mb); assumption;]]
-- 
2.39.2