Bertrand's conjecture (weak), some work in progress

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

diff --git a/helm/software/matita/dama/premetric_lattice.ma b/helm/software/matita/dama/premetric_lattice.ma
index 61c4b826d7453e9f3785b14a2e9ba139775fac1b..bfba3710afb881c65bfae772f9977346500ace86 100644 (file)
--- a/helm/software/matita/dama/premetric_lattice.ma
+++ b/helm/software/matita/dama/premetric_lattice.ma
@@ -12,11 +12,11 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/premetric_lattice/".
+
 
 include "metric_space.ma".
 
-record premetric_lattice_ (R : ogroup) : Type ≝ {
+record premetric_lattice_ (R : todgroup) : Type ≝ {
   pml_carr:> metric_space R;
   meet: pml_carr → pml_carr → pml_carr;
   join: pml_carr → pml_carr → pml_carr
@@ -28,7 +28,7 @@ interpretation "valued lattice meet" 'and a b =
 interpretation "valued lattice join" 'or a b =
  (cic:/matita/premetric_lattice/join.con _ _ a b).
  
-record premetric_lattice_props (R : ogroup) (ml : premetric_lattice_ R) : Prop ≝ {
+record premetric_lattice_props (R : todgroup) (ml : premetric_lattice_ R) : Prop ≝ {
   prop1a: ∀a : ml.δ (a ∧ a) a ≈ 0;
   prop1b: ∀a : ml.δ (a ∨ a) a ≈ 0;
   prop2a: ∀a,b: ml. δ (a ∨ b) (b ∨ a) ≈ 0;
@@ -40,21 +40,30 @@ record premetric_lattice_props (R : ogroup) (ml : premetric_lattice_ R) : Prop 
   prop5: ∀a,b,c: ml. δ (a ∨ b) (a ∨ c) + δ (a ∧ b) (a ∧ c) ≤ δ b c
 }.
 
-record pmlattice (R : ogroup) : Type ≝ {
+record pmlattice (R : todgroup) : Type ≝ {
   carr :> premetric_lattice_ R;
   ispremetriclattice:> premetric_lattice_props R carr
 }.
   
 include "lattice.ma".
 
-lemma lattice_of_pmlattice: ∀R: ogroup. pmlattice R → lattice.
+lemma lattice_of_pmlattice: ∀R: todgroup. pmlattice R → lattice.
 intros (R pml); apply (mk_lattice (apart_of_metric_space ? pml));
-[apply (join ? pml)|apply (meet ? pml)]
-intros (x y z); whd; intro H; whd in H; cases H (LE AP);
-[apply (prop2a ? pml pml x y);  |apply (prop2b ? pml pml x y); 
+[apply (join ? pml)|apply (meet ? pml)
+|3,4,5,6,7,8,9,10: intros (x y z); whd; intro H; whd in H; cases H (LE AP);]
+[apply (prop1b ? pml pml x);    |apply (prop1a ? pml pml x);
+|apply (prop2a ? pml pml x y);  |apply (prop2b ? pml pml x y); 
 |apply (prop3a ? pml pml x y z);|apply (prop3b ? pml pml x y z);
 |apply (prop4a ? pml pml x y);  |apply (prop4b ? pml pml x y);]
-apply ap_symmetric; assumption;
+try (apply ap_symmetric; assumption); intros 4 (x y z H); change with (0 < (δ y z));
+[ change in H with (0 < δ (x ∨ y) (x ∨ z));
+  apply (lt_le_transitive ???? H);  
+  apply (le0plus_le ???? (mpositive ? pml ??) (prop5 ? pml pml x y z));
+| change in H with (0 < δ (x ∧ y) (x ∧ z));
+  apply (lt_le_transitive ???? H);  
+  apply (le0plus_le ???? (mpositive ? pml (x∨y) (x∨z)));
+  apply (le_rewl ??? ? (plus_comm ???));
+  apply (prop5 ? pml pml);] 
 qed.
 
 coercion cic:/matita/premetric_lattice/lattice_of_pmlattice.con.
\ No newline at end of file