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[helm.git] / matita / contribs / dama / dama / sandwich_corollary.ma

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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 "sandwich.ma".
+
+(* 3.17 *)
+lemma tends_uniq:
+  ∀R.∀ml:mlattice R.∀xn:sequence ml.∀x,y:ml. 
+     xn ⇝ x → xn ⇝ y → δ x y ≈ 0.
+intros (R ml xn x y H1 H2); unfold tends0 in H1 H2; unfold d2s in H1 H2;
+intro Axy; lapply (ap_le_to_lt ??? (ap_symmetric ??? Axy) (mpositive ? ml ??)) as ge0;
+cases (H1 (δ x y/1) (divide_preserves_lt ??? ge0)) (n1 Hn1); clear H1; 
+cases (H2 (δ x y/1) (divide_preserves_lt ??? ge0)) (n2 Hn2); clear H2;
+letin N ≝ (S (n2 + n1));
+cases (Hn1 N ?) (H1 H2); [apply (ltwr ? n2); rewrite < sym_plus; apply le_n;]
+cases (Hn2 N ?) (H3 H4); [apply (ltwl ? n1); rewrite < sym_plus; apply le_n;]
+clear H1 H3 Hn2 Hn1 N ge0 Axy; lapply (mtineq ?? x y (xn (S (n2+n1)))) as H5;
+cut ( δx (xn (S (n2+n1)))+ δ(xn (S (n2+n1))) y <   δx y/1 + δ(xn (S (n2+n1))) y) as H6;[2:
+  apply flt_plusr; apply (Lt≪ ? (msymmetric ????)); assumption]
+lapply (le_lt_transitive ???? H5 H6) as H7; clear H6;
+cut (δx y/1+ δ(xn (S (n2+n1))) y < δx y/1+  δx y/1) as H6; [2:apply flt_plusl; assumption]
+lapply (lt_transitive ???? H7 H6) as ABS; clear H6 H7 H4 H5 H2 n1 n2 xn;
+lapply (divpow ? (δ x y) 1) as D; lapply (Lt≪ ? (eq_sym ??? D) ABS) as H;
+change in H with ( δx y/1+ δx y/1< δx y/1+ δx y/1);
+apply (lt_coreflexive ?? H);
+qed.
+
+(* 3.18 *)