3.11 !!!

[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 6028ada59748c3659533fb48b5fd7c02ec8b694a..917e2a199037d55157ccae646a30bd5da2c66eac 100644 (file)
--- a/helm/software/matita/dama/metric_lattice.ma
+++ b/helm/software/matita/dama/metric_lattice.ma
@@ -43,31 +43,74 @@ record mlattice (R : ogroup) : Type ≝ {
   ml_props:> is_mlattice R ml_carr
 }.
 
-lemma eq_to_zero: ∀R.∀ml:mlattice R.∀x,y:ml.x ≈ y → δ x y ≈ 0.
+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);
+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;
 assumption;
 qed.
 
-lemma meq_joinl: ∀R.∀ml:mlattice R.∀x,y,z:ml. x≈z → δx y ≈ δz y.
+(*
+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)); 
-  apply (le_rewl ??? (0+δz y) (eq_to_zero ???? H));
+  apply (le_rewl ??? (0+δz y) (eq_to_dzero ???? H));
   apply (le_rewl ??? (δz y) (zero_neutral ??)); apply le_reflexive;
 | apply (le_transitive ???? (mtineq ???y x));
-  apply (le_rewl ??? (0+δx y) (eq_to_zero ??z x H));
+  apply (le_rewl ??? (0+δx y) (eq_to_dzero ??z x H));
   apply (le_rewl ??? (δx y) (zero_neutral ??)); apply le_reflexive;]
 qed.
 
-lemma meq_joinr: ∀R.∀ml:mlattice R.∀x,y,z:ml. x≈z → δy x ≈ δy z.
+(* 3.3 *)
+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_joinl; assumption;
+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;
+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;]
+apply ap_symmetric; assumption;
+qed.
+
 (* 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.
 intros (R ml x y z Lxy Lyz); apply le_le_eq; [apply mtineq]
 apply (le_transitive ????? (ml_prop2 ?? ml (y) ??)); 
-(* auto type. assert failure *)
-whd;
+  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;   
+qed.  
+
+
+