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[helm.git] / matita / contribs / dama / dama / premetric_lattice.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 "metric_space.ma".
+
+record premetric_lattice_ (R : todgroup) : Type ≝ {
+  pml_carr:> metric_space R;
+  meet: pml_carr → pml_carr → pml_carr;
+  join: pml_carr → pml_carr → pml_carr
+}.
+
+interpretation "valued lattice meet" 'and a b =
+ (cic:/matita/premetric_lattice/meet.con _ _ a b).
+
+interpretation "valued lattice join" 'or a b =
+ (cic:/matita/premetric_lattice/join.con _ _ a b).
+ 
+record premetric_lattice_props (R : todgroup) (ml : premetric_lattice_ R) : Prop ≝ {
+  prop1a: ∀a : ml.δ (a ∧ a) a ≈ 0;
+  prop1b: ∀a : ml.δ (a ∨ a) a ≈ 0;
+  prop2a: ∀a,b: ml. δ (a ∨ b) (b ∨ a) ≈ 0;
+  prop2b: ∀a,b: ml. δ (a ∧ b) (b ∧ a) ≈ 0;
+  prop3a: ∀a,b,c: ml. δ (a ∨ (b ∨ c)) ((a ∨ b) ∨ c) ≈ 0;
+  prop3b: ∀a,b,c: ml. δ (a ∧ (b ∧ c)) ((a ∧ b) ∧ c) ≈ 0;
+  prop4a: ∀a,b: ml. δ (a ∨ (a ∧ b)) a ≈ 0;
+  prop4b: ∀a,b: ml. δ (a ∧ (a ∨ b)) a ≈ 0;
+  prop5: ∀a,b,c: ml. δ (a ∨ b) (a ∨ c) + δ (a ∧ b) (a ∧ c) ≤ δ b c
+}.
+
+record pmlattice (R : todgroup) : Type ≝ {
+  carr :> premetric_lattice_ R;
+  ispremetriclattice:> premetric_lattice_props R carr
+}.
+  
+include "lattice.ma".
+
+lemma lattice_of_pmlattice: ∀R: todgroup. pmlattice R → lattice.
+intros (R pml); apply (mk_lattice (apart_of_metric_space ? pml));
+[apply (join ? pml)|apply (meet ? pml)
+|3,4,5,6,7,8,9,10: intros (x y z); whd; intro H; whd in H; cases H (LE AP);]
+[apply (prop1b ? pml pml x);    |apply (prop1a ? pml pml x);
+|apply (prop2a ? pml pml x y);  |apply (prop2b ? pml pml x y); 
+|apply (prop3a ? pml pml x y z);|apply (prop3b ? pml pml x y z);
+|apply (prop4a ? pml pml x y);  |apply (prop4b ? pml pml x y);]
+try (apply ap_symmetric; assumption); intros 4 (x y z H); change with (0 < (δ y z));
+[ change in H with (0 < δ (x ∨ y) (x ∨ z));
+  apply (lt_le_transitive ???? H);  
+  apply (le0plus_le ???? (mpositive ? pml ??) (prop5 ? pml pml x y z));
+| change in H with (0 < δ (x ∧ y) (x ∧ z));
+  apply (lt_le_transitive ???? H);  
+  apply (le0plus_le ???? (mpositive ? pml (x∨y) (x∨z)));
+  apply (le_rewl ??? ? (plus_comm ???));
+  apply (prop5 ? pml pml);] 
+qed.
+
+coercion cic:/matita/premetric_lattice/lattice_of_pmlattice.con.
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