after a PITA, lebergue is dualized!

[helm.git] / helm / software / matita / contribs / dama / dama / property_sigma.ma

diff --git a/helm/software/matita/contribs/dama/dama/property_sigma.ma b/helm/software/matita/contribs/dama/dama/property_sigma.ma
index ece02e9ba982f11bfbb8688428cb6e28a8ce9fed..9d9485f9c41c84eb6b595472ff6b5eaaa6f730a6 100644 (file)
--- a/helm/software/matita/contribs/dama/dama/property_sigma.ma
+++ b/helm/software/matita/contribs/dama/dama/property_sigma.ma
@@ -20,7 +20,7 @@ alias num (instance 0) = "natural number".
 definition property_sigma ≝
   λC:ordered_uniform_space.
    ∀U.us_unifbase ? U → 
-     ∃V:sequence (C square → Prop).
+     ∃V:sequence (C squareB → Prop).
        (∀i.us_unifbase ? (V i)) ∧ 
        ∀a:sequence C.∀x:C.(a ↑ x ∨ a ↓ x) → 
          (∀n.∀i,j.n ≤ i → n ≤ j → V n 〈a i,a j〉) → U 〈a 0,x〉.
@@ -57,61 +57,63 @@ qed.
 lemma max_le_r: ∀n,m,z.max n m ≤ z → m ≤ z.
 intros; rewrite > sym_max in H; apply (max_le_l ??? H); 
 qed.
-      
+  
 (* Lemma 3.6 *)   
 lemma sigma_cauchy:
   ∀C:ordered_uniform_space.property_sigma C →
     ∀a:sequence C.∀l:C.(a ↑ l ∨ a ↓ l) → a is_cauchy → a uniform_converges l.
 intros 8; cases (H ? H3) (w H5); cases H5 (H8 H9); clear H5;
+letin spec ≝ (λz,k:nat.∀i,j,l:nat.k ≤ i → k ≤ j → l ≤ z → w l 〈a i,a j〉);
 letin m ≝ (hide ? (let rec aux (i:nat) : nat ≝
   match i with
   [ O ⇒ match H2 (w i) ? with [ ex_introT k _ ⇒ k ]
   | S i' ⇒ max (match H2 (w i) ? with [ ex_introT k _ ⇒ k ]) (S (aux i'))
   ] in aux
-  : 
-  ∀z:nat.∃k:nat.∀i,j,l.k ≤ i → k ≤ j → l ≤ z → w l 〈a i, a j〉));
+  : ∀z.∃k. spec z k)); unfold spec in aux ⊢ %;
   [1,2:apply H8;
   |3: intros 3; cases (H2 (w n) (H8 n)); simplify in ⊢ (? (? % ?) ?→?);
       simplify in ⊢ (?→? (? % ?) ?→?);
       intros; lapply (H5 i j) as H14;
         [2: apply (max_le_l ??? H6);|3:apply (max_le_l ??? H7);]
       cases (le_to_or_lt_eq ?? H10); [2: destruct H11; destruct H4; assumption]
-      generalize in match H6; generalize in match H7;
-      cases (aux n1); simplify in ⊢ (? (? ? %) ?→? (? ? %) ?→?); intros;
-      apply H12; [3: apply le_S_S_to_le; assumption]
+      cases (aux n1) in H6 H7 ⊢ %; simplify in ⊢ (? (? ? %) ?→? (? ? %) ?→?); intros;
+      apply H6; [3: apply le_S_S_to_le; assumption]
       apply lt_to_le; apply (max_le_r w1); assumption;
   |4: intros; clear H6; rewrite > H4 in H5; 
       rewrite < (le_n_O_to_eq ? H11); apply H5; assumption;] 
-cut (((m : nat→nat) : sequence nat_ordered_set) is_strictly_increasing) as Hm; [2:
+cut ((⌊x,(m x:nat)⌋ : sequence nat_ordered_set) is_strictly_increasing) as Hm; [2:
   intro n; change with (S (m n) ≤ m (S n)); unfold m; 
   whd in ⊢ (? ? %); apply (le_max ? (S (m n)));]
-cut (((m : nat→nat) : sequence nat_ordered_set) is_increasing) as Hm1; [2:
+cut ((⌊x,(m x:nat)⌋ : sequence nat_ordered_set) is_increasing) as Hm1; [2:
   intro n; intro L; change in L with (m (S n) < m n);
   lapply (Hm n) as L1; change in L1 with (m n < m (S n));
   lapply (trans_lt ??? L L1) as L3; apply (not_le_Sn_n ? L3);] 
-clearbody m;
-cut ((λx:nat.a (m x))↑ l ∨ (λx:nat.a (m x)) ↓ l) as H10; [2: 
+clearbody m; unfold spec in m Hm Hm1; clear spec;
+cut (⌊x,a (m x)⌋ ↑ l ∨ ⌊x,a (m x)⌋ ↓ l) as H10; [2: 
  cases H1;
- [ left; apply (selection_uparrow ?? Hm a l H4);
- | right; apply (selection_downarrow ?? Hm a l H4);]] 
+ [ left; apply (selection_uparrow ? Hm a l H4);
+ | right; apply (selection_downarrow ? Hm a l H4);]] 
 lapply (H9 ?? H10) as H11; [
   exists [apply (m 0:nat)] intros;
   cases H1;
     [cases H5; cases H7; apply (ous_convex ?? H3 ? H11 (H12 (m O)));
     |cases H5; cases H7; cases (us_phi4 ?? H3 〈(a (m O)),l〉); 
      lapply (H14 H11) as H11'; apply (ous_convex ?? H3 〈l,(a (m O))〉 H11' (H12 (m O)));]
-  simplify; repeat split; [1,6:intro X; cases (os_coreflexive ?? X)|*: try apply H12;]
-  [1:change with (a (m O) ≤ a i);
-     apply (trans_increasing ?? H6); intro; apply (le_to_not_lt ?? H4 H14);
-  |2:change with (a i ≤ a (m O));
-     apply (trans_decreasing ?? H6); intro; apply (le_to_not_lt ?? H4 H16);]]  
+  simplify; repeat split; 
+  [1,6: apply (le_reflexive l); 
+  |2,5: apply (H12 (\fst (m 0)));
+  |3,8: apply (H12 i);
+  |4:change with (a (m O) ≤ a i);
+     apply (trans_increasing a H6 (\fst (m 0)) i); intro; apply (le_to_not_lt ?? H4 H14);
+  |7:change with (a i ≤ a (m O));
+     apply (trans_decreasing ? H6 (\fst (m 0)) i); intro; apply (le_to_not_lt ?? H4 H16);]]  
 clear H10; intros (p q r); change with (w p 〈a (m q),a (m r)〉);
 generalize in match (refl_eq nat (m p)); 
 generalize in match (m p) in ⊢ (? ? ? % → %); intro X; cases X (w1 H15); clear X; 
 intros (H16); simplify in H16:(? ? ? %); destruct H16;
 apply H15; [3: apply le_n]
-[1: lapply (trans_increasing ?? Hm1 p q) as T; [apply not_lt_to_le; apply T;]
+[1: lapply (trans_increasing ? Hm1 p q) as T; [apply not_lt_to_le; apply T;]
     apply (le_to_not_lt p q H4);
-|2: lapply (trans_increasing ?? Hm1 p r) as T; [apply not_lt_to_le; apply T;]
+|2: lapply (trans_increasing ? Hm1 p r) as T; [apply not_lt_to_le; apply T;]
     apply (le_to_not_lt p r H5);]
 qed.