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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "infsup.ma".
+
+definition shift ≝ λT:Type.λs:sequence T.λk:nat.λn.s (n+k).
+
+(* 3.28 *)
+definition limsup ≝
+  λE:excess.λxn:sequence E.λx:E.∃alpha:sequence E. 
+    (∀k.(alpha k) is_strong_sup (shift ? xn k) in E) ∧ 
+     x is_strong_inf alpha in E.     
+
+notation < "x \nbsp 'is_limsup' \nbsp s" non associative with precedence 50 for @{'limsup $_ $s $x}.
+notation < "x \nbsp 'is_liminf' \nbsp s" non associative with precedence 50 for @{'liminf $_ $s $x}.
+notation > "x 'is_limsup' s 'in' e" non associative with precedence 50 for @{'limsup $e $s $x}.
+notation > "x 'is_liminf' s 'in' e" non associative with precedence 50 for @{'liminf $e $s $x}.
+
+interpretation "Excess limsup" 'limsup e s x = (cic:/matita/limit/limsup.con e s x).
+interpretation "Excess liminf" 'liminf e s x = (cic:/matita/limit/limsup.con (cic:/matita/excess/dual_exc.con e) s x).
+
+(* 3.29 *)  
+definition lim ≝ 
+  λE:excess.λxn:sequence E.λx:E. x is_limsup xn in E ∧ x is_liminf xn in E.
+
+notation < "x \nbsp 'is_lim' \nbsp s" non associative with precedence 50 for @{'lim $_ $s $x}.
+notation > "x 'is_lim' s 'in' e" non associative with precedence 50 for @{'lim $e $s $x}.
+interpretation "Excess lim" 'lim e s x = (cic:/matita/limit/lim.con e s x).
+
+lemma sup_decreasing:
+  ∀E:excess.∀xn:sequence E.
+   ∀alpha:sequence E. (∀k.(alpha k) is_strong_sup xn in E) → 
+     alpha is_decreasing in E.
+intros (E xn alpha H); unfold strong_sup in H; unfold upper_bound in H; unfold;
+intro r; 
+elim (H r) (H1r H2r);
+elim (H (S r)) (H1sr H2sr); clear H H2r H1sr;
+intro e; cases (H2sr (alpha r) e) (w Hw); clear e H2sr;
+cases (H1r w Hw);
+qed.
+
+include "tend.ma".
+include "metric_lattice.ma".
+  
+(* 3.30 *)   
+lemma lim_tends: 
+  ∀R.∀ml:mlattice R.∀xn:sequence ml.∀x:ml.
+    x is_lim xn in ml → xn ⇝ x.
+intros (R ml xn x Hl); unfold lim in Hl; unfold limsup in Hl;
+cases Hl (e1 e2); cases e1 (an Han); cases e2 (bn Hbn); clear Hl e1 e2;
+cases Han (SSan SIxan); cases Hbn (SSbn SIxbn); clear Han Hbn;
+cases SIxan (LBxan Hxg); cases SIxbn (UPxbn Hxl); clear SIxbn SIxan;
+change in UPxbn:(%) with (x is_lower_bound bn in ml);
+unfold upper_bound in UPxbn LBxan; change  
+intros (e He);  
+(* 2.6 OC *)