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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 "sequence.ma".
+
+definition upper_bound ≝ λO:excess.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
+
+definition weak_sup ≝
+  λO:excess.λs:sequence O.λx.
+    upper_bound ? s x ∧ (∀y:O.upper_bound ? s y → x ≤ y).
+
+definition strong_sup ≝
+  λO:excess.λs:sequence O.λx.upper_bound ? s x ∧ (∀y:O.x ≰ y → ∃n.s n ≰ y).
+
+definition increasing ≝ λO:excess.λa:sequence O.∀n:nat.a n ≤ a (S n).
+
+notation < "x \nbsp 'is_upper_bound' \nbsp s" non associative with precedence 50 for @{'upper_bound $_ $s $x}.
+notation < "x \nbsp 'is_lower_bound' \nbsp s" non associative with precedence 50 for @{'lower_bound $_ $s $x}.
+notation < "s \nbsp 'is_increasing'"          non associative with precedence 50 for @{'increasing $_ $s}.
+notation < "s \nbsp 'is_decreasing'"          non associative with precedence 50 for @{'decreasing $_ $s}.
+notation < "x \nbsp 'is_strong_sup' \nbsp s"  non associative with precedence 50 for @{'strong_sup $_ $s $x}.
+notation < "x \nbsp 'is_strong_inf' \nbsp s"  non associative with precedence 50 for @{'strong_inf $_ $s $x}.
+
+notation > "x 'is_upper_bound' s 'in' e" non associative with precedence 50 for @{'upper_bound $e $s $x}.
+notation > "x 'is_lower_bound' s 'in' e" non associative with precedence 50 for @{'lower_bound $e $s $x}.
+notation > "s 'is_increasing' 'in' e"    non associative with precedence 50 for @{'increasing $e $s}.
+notation > "s 'is_decreasing' 'in' e"    non associative with precedence 50 for @{'decreasing $e $s}.
+notation > "x 'is_strong_sup' s 'in' e"  non associative with precedence 50 for @{'strong_sup $e $s $x}.
+notation > "x 'is_strong_inf' s 'in' e"  non associative with precedence 50 for @{'strong_inf $e $s $x}.
+
+interpretation "Excess upper bound" 'upper_bound e s x = (cic:/matita/infsup/upper_bound.con e s x).
+interpretation "Excess lower bound" 'lower_bound e s x = (cic:/matita/infsup/upper_bound.con (cic:/matita/excess/dual_exc.con e) s x).
+interpretation "Excess increasing"  'increasing e s    = (cic:/matita/infsup/increasing.con e s).
+interpretation "Excess decreasing"  'decreasing e s    = (cic:/matita/infsup/increasing.con (cic:/matita/excess/dual_exc.con e) s).
+interpretation "Excess strong sup"  'strong_sup e s x  = (cic:/matita/infsup/strong_sup.con e s x).
+interpretation "Excess strong inf"  'strong_inf e s x  = (cic:/matita/infsup/strong_sup.con (cic:/matita/excess/dual_exc.con e) s x).
+
+lemma strong_sup_is_weak: 
+  ∀O:excess.∀s:sequence O.∀x:O.strong_sup ? s x → weak_sup ? s x.
+intros (O s x Ssup); elim Ssup (Ubx M); clear Ssup; split; [assumption]
+intros 3 (y Uby E); cases (M ? E) (n En); unfold in Uby; cases (Uby ? En);
+qed.