sandwitch theorem done

[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 917e2a199037d55157ccae646a30bd5da2c66eac..cca6430983486656f7863db6224c02d4c7d22ed3 100644 (file)
--- a/helm/software/matita/dama/metric_lattice.ma
+++ b/helm/software/matita/dama/metric_lattice.ma
@@ -17,7 +17,7 @@ set "baseuri" "cic:/matita/metric_lattice/".
 include "metric_space.ma".
 include "lattice.ma".
 
-record mlattice_ (R : ogroup) : Type ≝ {
+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)
@@ -33,12 +33,12 @@ qed.
 
 coercion cic:/matita/metric_lattice/ml_mspace.con.
 
-record is_mlattice (R : ogroup) (ml: mlattice_ R) : Type ≝ {
+record is_mlattice (R : todgroup) (ml: mlattice_ R) : Type ≝ {
   ml_prop1: ∀a,b:ml. 0 < δ a b → a # b;
   ml_prop2: ∀a,b,c:ml. δ (a∨b) (a∨c) + δ (a∧b) (a∧c) ≤ δ b c
 }.
 
-record mlattice (R : ogroup) : Type ≝ {
+record mlattice (R : todgroup) : Type ≝ {
   ml_carr :> mlattice_ R;
   ml_props:> is_mlattice R ml_carr
 }.
@@ -54,13 +54,6 @@ apply (ml_prop1 ?? ml); split [apply mpositive] apply ap_symmetric;
 assumption;
 qed.
 
-(*
-lemma lt_to_dpos: ∀R.∀ml:mlattice R.∀x,y:ml.x < y → 0 < δ x y.
-intros 4; repeat (unfold in ⊢ (? % ? ?→?)); simplify; unfold excl;
-intro H; elim H (H1 H2); elim H2 (H3 H3); [cases (H1 H3)]
-split; [apply mpositive]
-*)
-
 lemma meq_l: ∀R.∀ml:mlattice R.∀x,y,z:ml. x≈z → δx y ≈ δz y.
 intros (R ml x y z); apply le_le_eq;
 [ apply (le_transitive ???? (mtineq ???y z)); 
@@ -76,10 +69,7 @@ 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 ap_le_to_lt: ∀O:ogroup.∀a,c:O.c # a → c ≤ a → c < a.
-intros (R a c A L); split; 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;
@@ -95,22 +85,29 @@ lemma le_mtri:
   ∀R.∀ml:mlattice R.∀x,y,z:ml. x ≤ y → y ≤ z → δ x z ≈ δ x y + δ y z.
 intros (R ml x y z Lxy Lyz); apply le_le_eq; [apply mtineq]
 apply (le_transitive ????? (ml_prop2 ?? ml (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;
-  apply (eq_trans ?? (δ(x∧y) y + δy z)); [apply feq_plusr; apply (meq_l ????? Dxm);]
-  apply (eq_trans ?? (δ(x∧y) (y∧z) + δy z)); [apply feq_plusr; apply (meq_r ????? Dym);]
-  apply (eq_trans ?? (δ(x∧y) (y∧z) + δ(x∨y) z)); [apply feq_plusl; apply (meq_l ????? Dxj);]
-  apply (eq_trans ?? (δ(x∧y) (y∧z) + δ(x∨y) (y∨z))); [apply feq_plusl; apply (meq_r ????? Dyj);]
-  apply (eq_trans ?? ? ? (plus_comm ???));
-  apply (eq_trans ?? (δ(y∨x) (y∨z)+ δ(x∧y) (y∧z))); [apply feq_plusr; apply (meq_l ????? (join_comm ???));]
-  apply feq_plusl;
-  apply (eq_trans ?? (δ(y∧x) (y∧z))); [apply (meq_l ????? (meet_comm ???));]
-  apply eq_reflexive;   
+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;
+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)) (meq_r ????? Dyj));
+apply (Eq≈ ? (plus_comm ???));
+apply (Eq≈ (δ(y∨x) (y∨z)+ δ(x∧y) (y∧z)) (meq_l ????? (join_comm ?x y)));
+apply feq_plusl;
+apply (Eq≈ (δ(y∧x) (y∧z)) (meq_l ????? (meet_comm ?x y)));
+apply eq_reflexive;   
 qed.  
 
 
-
+(* 3.17 conclusione: δ x y ≈ 0 *)
+(* 3.20 conclusione: δ x y ≈ 0 *)
+(* 3.21 sup forte
+   strong_sup x ≝ ∀n. s n ≤ x ∧ ∀y x ≰ y → ∃n. s n ≰ y
+   strong_sup_zoli x ≝  ∀n. s n ≤ x ∧ ∄y. y#x ∧ y ≤ x
+*)
+(* 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