X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fdama%2Fmetric_lattice.ma;h=f0242da284896e612f5774aa801bc881a9606ff8;hb=3deaedbac3407f8b4602b885a6405d4e0cc3e955;hp=81eab3137e9f121d28f84a96e4f29c9b4b62b129;hpb=a1f4ef3daaeed7a3121a40afe55f321565669da8;p=helm.git

diff --git a/helm/software/matita/dama/metric_lattice.ma b/helm/software/matita/dama/metric_lattice.ma
index 81eab3137..f0242da28 100644
--- a/helm/software/matita/dama/metric_lattice.ma
+++ b/helm/software/matita/dama/metric_lattice.ma
@@ -12,20 +12,18 @@
 (*                                                                        *)
 (**************************************************************************)
 
-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)
+  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))]
@@ -33,24 +31,27 @@ qed.
 
 coercion cic:/matita/metric_lattice/ml_mspace.con.
 
-record is_mlattice (R : ogroup) (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 ≝ {
+alias symbol "plus" = "Abelian group plus".
+alias symbol "leq" = "Excess less or equal than".
+record mlattice (R : todgroup) : Type ≝ {
   ml_carr :> mlattice_ R;
-  ml_props:> is_mlattice R ml_carr
+  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)
 }.
 
+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 ?? ml);
+intros (R ml a b E); intro H; apply E; apply ml_prop1;
 assumption;
 qed.
 
 lemma eq_to_dzero: ∀R.∀ml:mlattice R.∀x,y:ml.x ≈ y → δ x y ≈ 0.
 intros (R ml x y H); intro H1; apply H; clear H; 
-apply (ml_prop1 ?? ml); split [apply mpositive] apply ap_symmetric;
+apply ml_prop1; split [apply mpositive] apply ap_symmetric;
 assumption;
 qed.
 
@@ -69,14 +70,13 @@ 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;
 qed.
 
 lemma dap_to_ap: ∀R.∀ml:mlattice R.∀x,y:ml. δ x y # 0 → x # y.
-intros (R ml x y H); apply (ml_prop1 ?? ml); split; [apply mpositive;]
+intros (R ml x y H); apply ml_prop1; split; [apply mpositive;]
 apply ap_symmetric; assumption;
 qed.
 
@@ -84,119 +84,27 @@ qed.
 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) ??)); 
+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;
-apply (Eq≈ (δ(x∧y) y + δy z)); [apply feq_plusr; apply (meq_l ????? Dxm);]
-apply (Eq≈ (δ(x∧y) (y∧z) + δy z)); [apply feq_plusr; apply (meq_r ????? Dym);]
-apply (Eq≈ (δ(x∧y) (y∧z) + δ(x∨y) z)); [apply feq_plusl; apply (meq_l ????? Dxj);]
-apply (Eq≈ (δ(x∧y) (y∧z) + δ(x∨y) (y∨z))); [apply feq_plusl; apply (meq_r ????? Dyj);]
+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(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))); [apply feq_plusr; apply (meq_l ????? (join_comm ???));]
+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))); [apply (meq_l ????? (meet_comm ???));]
+apply (Eq≈ (δ(y∧x) (y∧z)) (meq_l ????? (meet_comm ??)));
 apply eq_reflexive;   
 qed.  
 
-include "sequence.ma".
-include "nat/plus.ma".
-
-definition d2s ≝ 
-  λR.λml:mlattice R.λs:sequence (pordered_set_of_excedence 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).    
-*)
-
-lemma lew_opp : ∀O:ogroup.∀a,b,c:O.0 ≤ b → a ≤ c → a + -b ≤ c.
-intros (O a b c L0 L);
-apply (le_transitive ????? L);
-apply (plus_cancl_le ??? (-a));
-apply (le_rewr ??? 0 (opp_inverse ??));
-apply (le_rewl ??? (-a+a+-b) (plus_assoc ????));
-apply (le_rewl ??? (0+-b) (opp_inverse ??));
-apply (le_rewl ??? (-b) (zero_neutral ?(-b)));
-apply le_zero_x_to_le_opp_x_zero;
-assumption;
-qed.
-
-
-lemma ltw_opp : ∀O:ogroup.∀a,b,c:O.0 < b → a < c → a + -b < c.
-intros (O a b c P L);
-apply (lt_transitive ????? L);
-apply (plus_cancl_lt ??? (-a));
-apply (lt_rewr ??? 0 (opp_inverse ??));
-apply (lt_rewl ??? (-a+a+-b) (plus_assoc ????));
-apply (lt_rewl ??? (0+-b) (opp_inverse ??));
-apply (lt_rewl ??? ? (zero_neutral ??));
-apply lt_zero_x_to_lt_opp_x_zero;
-assumption;
-qed.
-
-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. 
-
-alias symbol "leq" = "ordered sets less or equal than".
-alias symbol "and" = "constructive and".
-alias symbol "exists" = "constructive exists (Type)".
-theorem carabinieri: (* non trova la coercion .... *)
-  ∀R.∀ml:mlattice R.∀an,bn,xn:sequence (pordered_set_of_excedence 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);
-elim (Ha ? He) (n1 H1); clear Ha; elim (Hb e 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;
 
-  
 (* 3.17 conclusione: δ x y ≈ 0 *)
 (* 3.20 conclusione: δ x y ≈ 0 *)
 (* 3.21 sup forte
@@ -206,4 +114,4 @@ clear Dxm Dym H7 H8 Hn3 H5 H3 n1 n2;
 (* 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 *)