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).
+interpretation "Excess upper bound" 'upper_bound e s x = (upper_bound e s x).
+interpretation "Excess lower bound" 'lower_bound e s x = (upper_bound (dual_exc e) s x).
+interpretation "Excess increasing" 'increasing e s = (increasing e s).
+interpretation "Excess decreasing" 'decreasing e s = (increasing (dual_exc e) s).
+interpretation "Excess strong sup" 'strong_sup e s x = (strong_sup e s x).
+interpretation "Excess strong inf" 'strong_inf e s x = (strong_sup (dual_exc e) s x).
lemma strong_sup_is_weak:
∀O:excess.∀s:sequence O.∀x:O.strong_sup ? s x → weak_sup ? s x.