(* *)
(**************************************************************************)
-
-
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_ = apart_of_excess (pl_carr ml_lattice)
}.
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 (apart_of_excess ml));
+unfold apartness_OF_mlattice_;
+[rewrite < (ml_with_ ? ml); apply (metric ? (ml_mspace_ ? 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));
-|apply (mtineq ? (ml_mspace_ ? ml))]
+[apply (mpositive ? (ml_mspace_ ? ml));|apply (mreflexive ? (ml_mspace_ ? ml));
+|apply (msymmetric ? (ml_mspace_ ? ml));|apply (mtineq ? (ml_mspace_ ? ml))]
qed.
coercion cic:/matita/metric_lattice/ml_mspace.con.
+alias symbol "plus" = "Abelian group plus".
record mlattice (R : todgroup) : Type ≝ {
ml_carr :> mlattice_ R;
ml_prop1: ∀a,b:ml_carr. 0 < δ a b → a # b;
apply ap_symmetric; assumption;
qed.
+interpretation "Lattive meet le" 'leq a b =
+ (cic:/matita/excess/le.con (cic:/matita/lattice/excess_OF_lattice1.con _) a b).
+
+interpretation "Lattive join le (aka ge)" 'geq a b =
+ (cic:/matita/excess/le.con (cic:/matita/lattice/excess_OF_lattice.con _) a b).
+
+lemma le_to_ge: ∀l:lattice.∀a,b:l.a ≤ b → b ≥ a.
+intros(l a b H); apply H;
+qed.
+
+lemma ge_to_le: ∀l:lattice.∀a,b:l.b ≥ a → a ≤ b.
+intros(l a b H); apply H;
+qed.
+
+lemma eq_to_eq:∀l:lattice.∀a,b:l.
+ (eq (apart_of_excess (pl_carr (latt_jcarr l))) a b) →
+ (eq (apart_of_excess (pl_carr (latt_mcarr l))) a b).
+intros 3; unfold eq; unfold apartness_OF_lattice;
+unfold apartness_OF_lattice_1; unfold latt_jcarr; simplify;
+unfold dual_exc; simplify; intros 2 (H H1); apply H;
+cases H1;[right|left]assumption;
+qed.
+
+coercion cic:/matita/metric_lattice/eq_to_eq.con nocomposites.
+
(* 3.11 *)
lemma le_mtri:
∀R.∀ml:mlattice R.∀x,y,z:ml. x ≤ y → y ≤ z → δ x z ≈ δ x y + δ y z.
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 ?? Lxy) as Dxm; lapply (le_to_eqm ?? Lyz) as Dym;
+lapply (le_to_eqj ?? (le_to_ge ??? Lxy)) as Dxj; lapply (le_to_eqj ?? (le_to_ge ??? Lyz)) as Dyj; clear Lxy Lyz;
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;
+ unfold apartness_OF_mlattice1;
+ exact (eq_to_eq ??? Dxj);]
+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 y z);]
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 x y)));
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 *)