From 0e93f77172427eed198b974e32c7f3e80d2c0251 Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Fri, 30 Nov 2007 12:40:56 +0000
Subject: [PATCH] 3.11 !!!

---
 helm/software/matita/dama/lattice.ma        | 31 ++++++++++-
 helm/software/matita/dama/metric_lattice.ma | 59 ++++++++++++++++++---
 helm/software/matita/dama/ordered_group.ma  | 10 ++++
 helm/software/matita/dama/sequence.ma       |  2 +
 4 files changed, 93 insertions(+), 9 deletions(-)

diff --git a/helm/software/matita/dama/lattice.ma b/helm/software/matita/dama/lattice.ma
index 46662b588..9e91376d1 100644
--- a/helm/software/matita/dama/lattice.ma
+++ b/helm/software/matita/dama/lattice.ma
@@ -55,4 +55,33 @@ intro l; apply (mk_excedence l (excl l));
   assumption]
 qed.    
 
-coercion cic:/matita/lattice/excedence_of_lattice.con.
\ No newline at end of file
+coercion cic:/matita/lattice/excedence_of_lattice.con.
+
+lemma feq_ml: ∀ml:lattice.∀a,b,c:ml. a ≈ b → (c ∧ a) ≈ (c ∧ b).
+intros (l a b c H); unfold eq in H ⊢ %; unfold Not in H ⊢ %;
+intro H1; apply H; clear H; apply (strong_extm ???? H1);
+qed.
+
+lemma feq_jl: ∀ml:lattice.∀a,b,c:ml. a ≈ b → (c ∨ a) ≈ (c ∨ b).
+intros (l a b c H); unfold eq in H ⊢ %; unfold Not in H ⊢ %;
+intro H1; apply H; clear H; apply (strong_extj ???? H1);
+qed.
+
+lemma le_to_eqm: ∀ml:lattice.∀a,b:ml. a ≤ b → a ≈ (a ∧ b).
+intros (l a b H); 
+  unfold le in H; unfold excedence_of_lattice in H;
+  unfold excl in H; simplify in H; 
+unfold eq; assumption;
+qed.
+
+lemma le_to_eqj: ∀ml:lattice.∀a,b:ml. a ≤ b → b ≈ (a ∨ b).
+intros (l a b H); lapply (le_to_eqm ??? H) as H1;
+lapply (feq_jl ??? b H1) as H2;
+apply (eq_trans ????? (join_comm ???));
+apply (eq_trans ?? (b∨a∧b) ?? H2); clear H1 H2 H;
+apply (eq_trans ?? (b∨(b∧a)) ?? (feq_jl ???? (meet_comm ???)));
+apply eq_sym; apply absorbjm;
+qed.
+
+
+
diff --git a/helm/software/matita/dama/metric_lattice.ma b/helm/software/matita/dama/metric_lattice.ma
index 6028ada59..917e2a199 100644
--- 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.  
+
+
+
diff --git a/helm/software/matita/dama/ordered_group.ma b/helm/software/matita/dama/ordered_group.ma
index 439a6e262..5b0f0aa9b 100644
--- a/helm/software/matita/dama/ordered_group.ma
+++ b/helm/software/matita/dama/ordered_group.ma
@@ -185,6 +185,16 @@ apply (le_rewr ??? b (zero_neutral ??));
 assumption;
 qed.
 
+lemma le_le0plus: 
+  ∀G:ogroup.∀a,b:G. 0 ≤ a → 0 ≤ b → 0 ≤ a + b.
+intros (G a b L1 L2); apply (le_transitive ???? L1);
+apply (plus_cancl_le ??? (-a));
+apply (le_rewl ??? 0 (opp_inverse ??));
+apply (le_rewr ??? (-a + a + b) (plus_assoc ????));
+apply (le_rewr ??? (0 + b) (opp_inverse ??));
+apply (le_rewr ??? b (zero_neutral ??));
+assumption;
+qed.
 
 
   
diff --git a/helm/software/matita/dama/sequence.ma b/helm/software/matita/dama/sequence.ma
index 1350545ae..52baef839 100644
--- a/helm/software/matita/dama/sequence.ma
+++ b/helm/software/matita/dama/sequence.ma
@@ -72,6 +72,8 @@ definition decreasing ≝
   λO:pordered_set.λa:sequence O.∀n:nat.a (S n) ≤ a n.
 
 
+
+
 (*
 
 definition is_upper_bound ≝ λO:pordered_set.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
-- 
2.39.2