(* *)
(**************************************************************************)
-
-
include "metric_space.ma".
include "lattice.ma".
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_with: ms_carr ? ml_mspace_ = Type_OF_lattice ml_lattice
}.
lemma ml_mspace: ∀R.mlattice_ R → metric_space R.
-intros (R ml); apply (mk_metric_space R ml); unfold Type_OF_mlattice_;
-cases (ml_with_ ? ml); simplify;
+intros (R ml); apply (mk_metric_space R (Type_OF_mlattice_ ? ml));
+unfold Type_OF_mlattice_; 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));
|apply (mtineq ? (ml_mspace_ ? ml))]
coercion cic:/matita/metric_lattice/ml_mspace.con.
+alias symbol "plus" = "Abelian group plus".
+alias symbol "leq" = "Excess less or equal than".
record mlattice (R : todgroup) : Type ≝ {
ml_carr :> mlattice_ R;
ml_prop1: ∀a,b:ml_carr. 0 < δ a b → a # b;
- ml_prop2: ∀a,b,c:ml_carr. δ (a∨b) (a∨c) + δ (a∧b) (a∧c) ≤ δ b c
+ ml_prop2: ∀a,b,c:ml_carr. δ (a∨b) (a∨c) + δ (a∧b) (a∧c) ≤ (δ b c)
}.
+interpretation "Metric lattice leq" 'leq a b =
+ (cic:/matita/excess/le.con (cic:/matita/metric_lattice/excess_OF_mlattice1.con _ _) a b).
+interpretation "Metric lattice geq" 'geq a b =
+ (cic:/matita/excess/le.con (cic:/matita/metric_lattice/excess_OF_mlattice.con _ _) a b).
+
lemma eq_to_ndlt0: ∀R.∀ml:mlattice R.∀a,b:ml. a ≈ b → ¬ 0 < δ a b.
intros (R ml a b E); intro H; apply E; apply ml_prop1;
assumption;
intros; apply (eq_trans ???? (msymmetric ??y x));
apply (eq_trans ????? (msymmetric ??z y)); apply meq_l; assumption;
qed.
-
lemma dap_to_lt: ∀R.∀ml:mlattice R.∀x,y:ml. δ x y # 0 → 0 < δ x y.
intros; split [apply mpositive] apply ap_symmetric; assumption;
apply (le_transitive ????? (ml_prop2 ?? (y) ??));
cut ( δx y+ δy z ≈ δ(y∨x) (y∨z)+ δ(y∧x) (y∧z)); [
apply (le_rewr ??? (δx y+ δy z)); [assumption] apply le_reflexive]
-lapply (le_to_eqm ??? Lxy) as Dxm; lapply (le_to_eqm ??? Lyz) as Dym;
-lapply (le_to_eqj ??? Lxy) as Dxj; lapply (le_to_eqj ??? Lyz) as Dyj; clear Lxy Lyz;
+lapply (le_to_eqm y x Lxy) as Dxm; lapply (le_to_eqm z y Lyz) as Dym;
+lapply (le_to_eqj x y Lxy) as Dxj; lapply (le_to_eqj y z Lyz) as Dyj; clear Lxy Lyz;
+STOP
apply (Eq≈ (δ(x∧y) y + δy z) (meq_l ????? Dxm));
apply (Eq≈ (δ(x∧y) (y∧z) + δy z) (meq_r ????? Dym));
-apply (Eq≈ (δ(x∧y) (y∧z) + δ(x∨y) z) (meq_l ????? Dxj));
-apply (Eq≈ (δ(x∧y) (y∧z) + δ(x∨y) (y∨z))); [
- apply (feq_plusl ? (δ(x∧y) (y∧z)) ?? (meq_r ??? (x∨y) ? Dyj));]
+apply (Eq≈ (δ(x∧y) (y∧z) + δ(y∨x) z));[
+ apply feq_plusl; apply meq_l; clear Dyj Dxm Dym; assumption]
+apply (Eq≈ (δ(x∧y) (y∧z) + δ(y∨x) (z∨y))); [
+ apply (feq_plusl ? (δ(x∧y) (y∧z)) ?? (meq_r ??? (y∨x) ? Dyj));]
apply (Eq≈ ? (plus_comm ???));
-apply (Eq≈ (δ(y∨x) (y∨z)+ δ(x∧y) (y∧z)) (meq_l ????? (join_comm ?x y)));
+apply (Eq≈ (δ(y∨x) (y∨z)+ δ(x∧y) (y∧z)));[
+ apply feq_plusr; apply meq_r; apply (join_comm ??);]
apply feq_plusl;
-apply (Eq≈ (δ(y∧x) (y∧z)) (meq_l ????? (meet_comm ?x y)));
+apply (Eq≈ (δ(y∧x) (y∧z)) (meq_l ????? (meet_comm ??)));
apply eq_reflexive;
qed.
(* 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 *)