record mlattice_ (R : todgroup) : Type ≝ {
ml_mspace_: metric_space R;
- ml_lattice:> lattice;
- ml_with_: ms_carr ? ml_mspace_ = ap_carr (l_carr ml_lattice)
+ ml_lattice_: lattice;
+ ml_with_: ms_carr ? ml_mspace_ = l_carr ml_lattice_;
+ ml_with2_: l_carr ml_lattice_ = apart_of_metric_space ? ml_mspace_
}.
+lemma ml_lattice: ∀R.mlattice_ R → lattice.
+intros (R ml); apply (mk_lattice (apart_of_metric_space ? (ml_mspace_ ? ml))); try unfold eq;
+cases (ml_with2_ ? ml);
+[apply (join (ml_lattice_ ? ml));|apply (meet (ml_lattice_ ? ml));
+|apply (join_refl (ml_lattice_ R ml));| apply (meet_refl (ml_lattice_ ? ml));
+|apply (join_comm (ml_lattice_ ? ml));| apply (meet_comm (ml_lattice_ ? ml));
+|apply (join_assoc (ml_lattice_ ? ml));|apply (meet_assoc (ml_lattice_ ? ml));
+|apply (absorbjm (ml_lattice_ ? ml)); |apply (absorbmj (ml_lattice_ ? ml));
+|apply (strong_extj (ml_lattice_ ? ml));|apply (strong_extm (ml_lattice_ ? ml));]
+qed.
+
+coercion cic:/matita/metric_lattice/ml_lattice.con.
+
lemma ml_mspace: ∀R.mlattice_ R → metric_space R.
-intros (R ml); apply (mk_metric_space R ml); unfold Type_OF_mlattice_;
+intros (R ml); apply (mk_metric_space R ml);
cases (ml_with_ ? ml); simplify;
[apply (metric ? (ml_mspace_ ? ml));|apply (mpositive ? (ml_mspace_ ? ml));
|apply (mreflexive ? (ml_mspace_ ? ml));|apply (msymmetric ? (ml_mspace_ ? ml));
(* 3.22 sup debole (più piccolo dei maggioranti) *)
(* 3.23 conclusion: δ x sup(...) ≈ 0 *)
(* 3.25 vero nel reticolo e basta (niente δ) *)
-(* 3.36 conclusion: δ x y ≈ 0 *)
\ No newline at end of file
+(* 3.36 conclusion: δ x y ≈ 0 *)
mreflexive: ∀a. metric a a ≈ 0;
msymmetric: ∀a,b. metric a b ≈ metric b a;
mtineq: ∀a,b,c:ms_carr. metric a b ≤ metric a c + metric c b
- (* manca qualcosa per essere una metrica e non una semimetrica *)
}.
notation < "\nbsp \delta a \nbsp b" non associative with precedence 80 for @{ 'delta2 $a $b}.
qed.
coercion cic:/matita/metric_space/apart_of_metric_space.con.
+
+lemma ap2delta: ∀R.∀m:metric_space R.∀x,y:m.x#y → 0 < δ x y.
+intros 2 (R m); cases m 0; simplify; intros; assumption; qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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".
+
+include "metric_lattice.ma".
+
+(* 3.17 *)
+lemma tends_uniq:
+ ∀R.∀ml:mlattice R.∀xn:sequence ml.
+ ∀x,y:apart_of_metric_space ? ml.
+ (* BUG: it inserts a compesoed coercion instead of an hand made one,
+ what to do? prefer the human made one or allow to kill a coercion?
+ *)
+ xn ⇝ x → xn ⇝ y → x ≈ y.
+intros (R ml xn x y H1 H2); unfold tends0 in H1 H2; unfold d2s in H1 H2;
+intro Axy; lapply (ap2delta R ml x y Axy) 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.
+
+