ok

[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 a4ca3b4f4b6029d692b618aae05eee9438190230..ae55b5551dae807bd16ab72fae962914c0e70b4d 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
 }.
@@ -100,6 +100,93 @@ apply (Eq≈ (δ(y∧x) (y∧z))); [apply (meq_l ????? (meet_comm ???));]
 apply eq_reflexive;   
 qed.  
 
+include "sequence.ma".
+include "nat/plus.ma".
+
+lemma ltwl: ∀a,b,c:nat. b + a < c → a < c.
+intros 3 (x y z); elim y (H z IH H); [apply H]
+apply IH; apply lt_S_to_lt; apply H;
+qed.
+
+lemma ltwr: ∀a,b,c:nat. a + b < c → a < c.
+intros 3 (x y z); rewrite > sym_plus; apply ltwl;
+qed. 
+
+
+definition d2s ≝ 
+  λR.λml:mlattice R.λs:sequence ml.λk.λn. δ (s n) k.
+(*
+notation "s ⇝ 'Zero'" non associative with precedence 50 for @{'tends0 $s }.
+  
+interpretation "tends to" 'tends s x = 
+  (cic:/matita/sequence/tends0.con _ s).    
+*)
+
+axiom core1: ∀G:todgroup.∀e:G.0<e → e/3 + e/2 + e/2 < e.
+
+alias symbol "leq" = "ordered sets less or equal than".
+alias symbol "and" = "constructive and".
+alias symbol "exists" = "constructive exists (Type)".
+theorem carabinieri:
+  ∀R.∀ml:mlattice R.∀an,bn,xn:sequence ml.
+    (∀n. (an n ≤ xn n) ∧ (xn n ≤ bn n)) →
+    ∀x:ml. tends0 ? (d2s ? ml an x) → tends0 ? (d2s ? ml bn x) →
+    tends0 ? (d2s ? ml xn x).
+intros (R ml an bn xn H x Ha Hb); unfold tends0 in Ha Hb ⊢ %. unfold d2s in Ha Hb ⊢ %.
+intros (e He);
+alias num (instance 0) = "natural number".
+elim (Ha (e/2) (divide_preserves_lt ??? He)) (n1 H1); clear Ha; 
+elim (Hb (e/3) (divide_preserves_lt ??? He)) (n2 H2); clear Hb;
+constructor 1; [apply (n1 + n2);] intros (n3 Hn3);
+lapply (ltwr ??? Hn3) as Hn1n3; lapply (ltwl ??? Hn3) as Hn2n3;
+elim (H1 ? Hn1n3) (H3 H4); elim (H2 ? Hn2n3) (H5 H6); clear Hn1n3 Hn2n3;
+elim (H n3) (H7 H8); clear H H1 H2; 
+lapply (le_to_eqm ??? H7) as Dxm; lapply (le_to_eqj ??? H7) as Dym;
+(* the main step *)
+cut (δ (xn n3) x ≤ δ (bn n3) x + δ (an n3) x + δ (an n3) x); [2:
+  apply (le_transitive ???? (mtineq ???? (an n3)));
+  lapply (le_mtri ????? H7 H8);
+  lapply (feq_plusr ? (- δ(xn n3) (bn n3)) ?? Hletin);
+  cut ( δ(an n3) (bn n3)+- δ(xn n3) (bn n3)≈( δ(an n3) (xn n3))); [2:
+    apply (Eq≈  (0 + δ(an n3) (xn n3)) ? (zero_neutral ??));
+    apply (Eq≈  (δ(an n3) (xn n3) + 0) ? (plus_comm ???));
+    apply (Eq≈  (δ(an n3) (xn n3) +  (-δ(xn n3) (bn n3) +  δ(xn n3) (bn n3))) ? (opp_inverse ??));
+    apply (Eq≈  (δ(an n3) (xn n3) +  (δ(xn n3) (bn n3) + -δ(xn n3) (bn n3))) ? (plus_comm ?? (δ(xn n3) (bn n3))));
+    apply (Eq≈  ? ? (eq_sym ??? (plus_assoc ????))); assumption;] clear Hletin1;
+  apply (le_rewl ???  ( δ(an n3) (xn n3)+ δ(an n3) x));[
+    apply feq_plusr; apply msymmetric;]
+  apply (le_rewl ???  (δ(an n3) (bn n3)+- δ(xn n3) (bn n3)+ δ(an n3) x));[
+    apply feq_plusr; assumption;]
+  clear Hcut Hletin Dym Dxm H8 H7 H6 H5 H4 H3;
+  apply (le_rewl ??? (- δ(xn n3) (bn n3)+ δ(an n3) (bn n3)+δ(an n3) x));[
+    apply feq_plusr; apply plus_comm;]
+  apply (le_rewl ??? (- δ(xn n3) (bn n3)+ (δ(an n3) (bn n3)+δ(an n3) x)) (plus_assoc ????));
+  apply (le_rewl ??? ((δ(an n3) (bn n3)+δ(an n3) x)+- δ(xn n3) (bn n3)) (plus_comm ???));
+  apply lew_opp; [apply mpositive] apply fle_plusr;
+  apply (le_rewr ???? (plus_comm ???));
+  apply (le_rewr ??? ( δ(an n3) x+ δx (bn n3)) (msymmetric ????));
+  apply mtineq;]
+split; [
+  apply (lt_le_transitive ????? (mpositive ????));
+  split; elim He; [apply  le_zero_x_to_le_opp_x_zero; assumption;]
+  cases t1; [ 
+    left; apply exc_zero_opp_x_to_exc_x_zero;
+    apply (Ex≫ ? (eq_opp_opp_x_x ??));assumption;]
+  right;  apply exc_opp_x_zero_to_exc_zero_x;
+  apply (Ex≪ ? (eq_opp_opp_x_x ??)); assumption;]
+clear Dxm Dym H7 H8 Hn3 H5 H3 n1 n2;
+apply (le_lt_transitive ???? ? (core1 ?? He));
+apply (le_transitive ???? Hcut);
+apply (le_transitive ??  (e/3+ δ(an n3) x+ δ(an n3) x)); [
+  repeat apply fle_plusr; cases H6; assumption;]
+apply (le_transitive ??  (e/3+ e/2 + δ(an n3) x)); [
+  apply fle_plusr; apply fle_plusl; cases H4; assumption;]
+apply (le_transitive ??  (e/3+ e/2 + e/2)); [
+  apply fle_plusl; cases H4; assumption;]
+apply le_reflexive;
+qed.
+
+  
 (* 3.17 conclusione: δ x y ≈ 0 *)
 (* 3.20 conclusione: δ x y ≈ 0 *)
 (* 3.21 sup forte