snapshot with more duality, almost where we left without duality

[helm.git] / helm / software / matita / dama / metric_lattice.ma

diff --git a/helm/software/matita/dama/metric_lattice.ma b/helm/software/matita/dama/metric_lattice.ma
index 968ae8f3b65515e9e1411fac2530981e949a52a9..9f5811b6d710c0c896f3b74c04bb8d29ab757fa1 100644 (file)
--- a/helm/software/matita/dama/metric_lattice.ma
+++ b/helm/software/matita/dama/metric_lattice.ma
@@ -18,25 +18,25 @@ include "lattice.ma".
 record mlattice_ (R : todgroup) : Type ≝ {
   ml_mspace_: metric_space R;
   ml_lattice:> lattice;
-  ml_with_: ms_carr ? ml_mspace_ = apart_of_excess (pl_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 (apart_of_excess ml)); 
-unfold apartness_OF_mlattice_; 
-[rewrite < (ml_with_ ? ml); apply (metric ? (ml_mspace_ ? ml))]
-cases (ml_with_ ? ml); simplify;
-[apply (mpositive ? (ml_mspace_ ? ml));|apply (mreflexive ? (ml_mspace_ ? ml));
-|apply (msymmetric ? (ml_mspace_ ? ml));|apply (mtineq ? (ml_mspace_ ? ml))]
+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))]
 qed.
 
 coercion cic:/matita/metric_lattice/ml_mspace.con.
 
 alias symbol "plus" = "Abelian group plus".
+alias symbol "leq" = "ordered sets 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)
 }.
 
 lemma eq_to_ndlt0: ∀R.∀ml:mlattice R.∀a,b:ml. a ≈ b → ¬ 0 < δ a b.
@@ -65,7 +65,6 @@ lemma meq_r: ∀R.∀ml:mlattice R.∀x,y,z:ml. x≈z → δy x ≈ δy z.
 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;
@@ -76,31 +75,6 @@ intros (R ml x y H); apply ml_prop1; split; [apply mpositive;]
 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.
@@ -109,22 +83,18 @@ 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 ?? (le_to_ge ??? Lxy)) as Dxj; lapply (le_to_eqj ?? (le_to_ge ??? Lyz)) as Dyj; clear Lxy Lyz;
+lapply (le_to_eqj ?? Lxy) as Dxj; lapply (le_to_eqj ?? 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(y∨x) z));[
-  apply feq_plusl; apply meq_l; clear Dyj Dxm Dym;
-  unfold apartness_OF_mlattice1;
-  exact (eq_to_eq ??? Dxj);]
+  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)));[
-  apply feq_plusr;
-  apply meq_r; 
-  apply (join_comm 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.