helm/software/matita/dama/infsup.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "sandwich_corollary.ma".
  16
  17 (* 3.19 *)
  18 definition supremum ≝
  19   λR.λml:mlattice R.λxn:sequence ml.λx:ml.
  20     increasing ? xn → upper_bound ? xn x ∧ xn ⇝ x.
  21
  22 definition infimum ≝
  23   λR.λml:mlattice R.λxn:sequence ml.λx:ml.
  24     decreasing ? xn → lower_bound ? xn x ∧ xn ⇝ x.
  25
  26 (* 3.20 *)
  27 lemma supremum_uniq:
  28   ∀R.∀ml:mlattice R.∀xn:sequence ml.increasing ? xn → (* BUG again the wrong coercion is chosen *)
  29    ∀x,y:apart_of_metric_space ? ml.supremum ?? xn x → supremum ?? xn y → x ≈ y.
  30 intros (R ml xn Hxn x y Sx Sy);
  31 elim (Sx Hxn) (_ Hx); elim (Sy Hxn) (_ Hy);
  32 apply (tends_uniq ?? xn ?? Hx Hy);
  33 qed.
  34
  35 (* 3.21 *)