From 20d600f225a0994c607b23226578078eb6b79bbe Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Thu, 22 Nov 2007 13:29:20 +0000
Subject: [PATCH] snapshot

---
 helm/software/matita/dama/groups.ma         | 10 ++--
 helm/software/matita/dama/ordered_groups.ma | 35 ++++++++-----
 helm/software/matita/dama/valued_lattice.ma | 56 ++++++++++++++++++---
 3 files changed, 75 insertions(+), 26 deletions(-)

diff --git a/helm/software/matita/dama/groups.ma b/helm/software/matita/dama/groups.ma
index 0df357af5..f2b4bd01c 100644
--- a/helm/software/matita/dama/groups.ma
+++ b/helm/software/matita/dama/groups.ma
@@ -182,26 +182,26 @@ intros (G x y z H1 H2); apply eq_sym; apply (feq_oppr ??y);
 [2:apply eq_sym] assumption;
 qed.
 
-lemma eq_opp: ∀G:abelian_group.∀x,y:G. x ≈ y → -x ≈ -y.
+lemma feq_opp: ∀G:abelian_group.∀x,y:G. x ≈ y → -x ≈ -y.
 intros (G x y H); apply (feq_oppl ??y ? H); apply eq_reflexive;
 qed.
 
-coercion cic:/matita/groups/eq_opp.con nocomposites.
+coercion cic:/matita/groups/feq_opp.con nocomposites.
 
 lemma eq_opp_sym: ∀G:abelian_group.∀x,y:G. y ≈ x → -x ≈ -y.
-compose eq_opp with eq_sym (H); apply H; assumption;
+compose feq_opp with eq_sym (H); apply H; assumption;
 qed.
 
 coercion cic:/matita/groups/eq_opp_sym.con nocomposites.
 
 lemma eq_opp_plusr: ∀G:abelian_group.∀x,y,z:G. x ≈ y → -(x + z) ≈ -(y + z).
-compose feq_plusr with eq_opp(H); apply H; assumption;
+compose feq_plusr with feq_opp(H); apply H; assumption;
 qed.
 
 coercion cic:/matita/groups/eq_opp_plusr.con nocomposites.
 
 lemma eq_opp_plusl: ∀G:abelian_group.∀x,y,z:G. x ≈ y → -(z + x) ≈ -(z + y).
-compose feq_plusl with eq_opp(H); apply H; assumption;
+compose feq_plusl with feq_opp(H); apply H; assumption;
 qed.
 
 coercion cic:/matita/groups/eq_opp_plusl.con nocomposites.
diff --git a/helm/software/matita/dama/ordered_groups.ma b/helm/software/matita/dama/ordered_groups.ma
index d9be248d0..0d13eda04 100644
--- a/helm/software/matita/dama/ordered_groups.ma
+++ b/helm/software/matita/dama/ordered_groups.ma
@@ -32,27 +32,36 @@ qed.
 
 coercion cic:/matita/ordered_groups/og_abelian_group.con.
 
-(* CSC: NO! Cosi' e' nel frammento negativo. Devi scriverlo con l'eccedenza!
-   Tutto il resto del file e' da rigirare nel frammento positivo.
-*)
 record ogroup : Type ≝ { 
   og_carr:> pre_ogroup;
   exc_canc_plusr: ∀f,g,h:og_carr. f+h ≰ g+h → f ≰ g
 }.
 
+notation > "'Ex'≪" non associative with precedence 50 for
+ @{'excedencerewritel}.
+ 
+interpretation "exc_rewl" 'excedencerewritel = 
+ (cic:/matita/excedence/exc_rewl.con _ _ _).
+
+notation > "'Ex'≫" non associative with precedence 50 for
+ @{'excedencerewriter}.
+ 
+interpretation "exc_rewr" 'excedencerewriter = 
+ (cic:/matita/excedence/exc_rewr.con _ _ _).
+
 lemma fexc_plusr: 
   ∀G:ogroup.∀x,y,z:G. x ≰ y → x+z ≰ y + z.
 intros 5 (G x y z L); apply (exc_canc_plusr ??? (-z));
-apply (exc_rewl ??? (x + (z + -z)) (plus_assoc ????));
-apply (exc_rewl ??? (x + (-z + z)) (plus_comm ??z));
-apply (exc_rewl ??? (x+0) (opp_inverse ??));
-apply (exc_rewl ??? (0+x) (plus_comm ???));
-apply (exc_rewl ??? x (zero_neutral ??));
-apply (exc_rewr ??? (y + (z + -z)) (plus_assoc ????));
-apply (exc_rewr ??? (y + (-z + z)) (plus_comm ??z));
-apply (exc_rewr ??? (y+0) (opp_inverse ??));
-apply (exc_rewr ??? (0+y) (plus_comm ???));
-apply (exc_rewr ??? y (zero_neutral ??) L);
+apply (Ex≪  (x + (z + -z)) (plus_assoc ????));
+apply (Ex≪  (x + (-z + z)) (plus_comm ??z));
+apply (Ex≪  (x+0) (opp_inverse ??));
+apply (Ex≪  (0+x) (plus_comm ???));
+apply (Ex≪  x (zero_neutral ??));
+apply (Ex≫ (y + (z + -z)) (plus_assoc ????));
+apply (Ex≫  (y + (-z + z)) (plus_comm ??z));
+apply (Ex≫  (y+0) (opp_inverse ??));
+apply (Ex≫  (0+y) (plus_comm ???));
+apply (Ex≫  y (zero_neutral ??) L);
 qed.
 
 coercion cic:/matita/ordered_groups/fexc_plusr.con nocomposites.
diff --git a/helm/software/matita/dama/valued_lattice.ma b/helm/software/matita/dama/valued_lattice.ma
index 66eb715c9..781b34c07 100644
--- a/helm/software/matita/dama/valued_lattice.ma
+++ b/helm/software/matita/dama/valued_lattice.ma
@@ -29,7 +29,7 @@ record vlattice (R : ogroup) : Type ≝ {
   meet_assoc: ∀x,y,z. value (meet x (meet y z)) ≈ value (meet (meet x y) z);   
   meet_wins1: ∀x,y. value (join x (meet x y)) ≈ value x;
   meet_wins2: ∀x,y. value (meet x (join x y)) ≈ value x;
-  meet_join_plus: ∀x,y. value (join x y) + value (meet x y) ≈ value x + value y;
+  modular_mjp: ∀x,y. value (join x y) + value (meet x y) ≈ value x + value y;
   join_meet_le: ∀x,y,z. value (join x (meet y z)) ≤ value (join x y);
   meet_join_le: ∀x,y,z. value (meet x (join y z)) ≤ value (meet x y)  
 }. 
@@ -46,18 +46,58 @@ interpretation "lattice value" 'value2 a = (cic:/matita/valued_lattice/value.con
 notation "\mu" non associative with precedence 80 for @{ 'value }.
 interpretation "lattice value" 'value = (cic:/matita/valued_lattice/value.con _ _).
 
-lemma foo: ∀R.∀L:vlattice R.∀x,y,z:L.
+lemma feq_joinr: ∀R.∀L:vlattice R.∀x,y,z:L. 
+  μ x ≈ μ y → μ (z ∧ x) ≈ μ (z ∧ y) → μ (z ∨ x) ≈ μ (z ∨ y).
+intros (R L x y z H H1);
+apply (plus_cancr ??? (μ(z∧x)));
+apply (eq_trans ?? (μz + μx) ? (modular_mjp ????));
+apply (eq_trans ?? (μz + μy) ? H); clear H;
+apply (eq_trans ?? (μ(z∨y) + μ(z∧y))); [1: apply eq_sym; apply modular_mjp]
+apply (plus_cancl ??? (- μ (z ∨ y)));
+apply (eq_trans ?? ? ? (plus_assoc ????));
+apply (eq_trans ?? (0+ μ(z∧y)) ? (opp_inverse ??));
+apply (eq_trans ?? ? ? (zero_neutral ??));
+apply (eq_trans ?? (- μ(z∨y)+ μ(z∨y)+ μ(z∧x)) ?? (plus_assoc ????));
+apply (eq_trans ?? (0+ μ(z∧x)) ?? (opp_inverse ??)); 
+apply (eq_trans ?? (μ (z ∧ x)) ?H1 (zero_neutral ??));
+qed.
+
+lemma step1_3_57: ∀R.∀L:vlattice R.∀x,y,z:L.
   μ(x ∧ (y ∨ z)) ≈ (μ x) + (μ y) + μ z + -μ (y ∧ z) + -μ (z ∨ (x ∨ y)).
 intros (R L x y z);
-lapply (meet_join_plus ? ? x (y ∨ z)) as H;
-lapply (meet_join_plus ? ? y z) as H1;
- (*CSC: A questo punto ti servono dei lemmi sui gruppi che non sono ancora
-   stati dimostrati. E una valanga di passi di riscrittura :-)
- *)
-
+cut (μ(x∧(y∨z))≈(μx + μ(y ∨ z) + - μ(x∨(y∨z)))); [2:
+  lapply (modular_mjp ?? x (y ∨ z)) as H1;
+  apply (eq_trans ?? (μ(x∨(y∨z))+ μ(x∧(y∨z)) +-μ(x∨(y∨z))) ?? H1); clear H1;
+  apply (eq_trans ?? ? ?? (plus_comm ???));
+  (* apply (eq_trans ?? (0+μ(x∧(y∧z))) ?? (opp_inverse ??)); ASSERT FALSE *)
+  apply (eq_trans ?? (- μ(x∨(y∨z))+ μ(x∨(y∨z))+ μ(x∧(y∨z)))); [2: apply eq_sym; apply plus_assoc;]
+  apply (eq_trans ?? (0+μ(x∧(y∨z)))); [2: apply feq_plusr; apply eq_sym; apply opp_inverse;]
+  (* apply (eq_trans ?? ? ? (eq_refl ??) (zero_neutral ??)); ASSERT FALSE *)
+  apply (eq_trans ?? (μ(x∧(y∨z)))); [apply eq_reflexive| apply eq_sym; apply zero_neutral]]
+apply (eq_trans ?? ? ? Hcut); clear Hcut;
+cut ( μ(y∨z) ≈ μy + μz + -μ (y ∧ z)); [2:
+  lapply (modular_mjp ?? y z) as H1;
+  apply (plus_cancr ??? (μ(y ∧ z)));
+  apply (eq_trans ?? ? ? H1); clear H1;
+  apply (eq_trans ?? ? ?? (plus_assoc ????));   
+  apply (eq_trans ?? (μy+ μz + 0)); [2: apply feq_plusl; apply eq_sym; apply opp_inverse]   
+  apply (eq_trans ?? ? ?? (plus_comm ???));
+  apply (eq_trans ?? (μy + μz) ?? (eq_sym ??? (zero_neutral ??)));
+  apply eq_reflexive;]
+apply (eq_trans ?? ( μx+ (μy+ μz+- μ(y∧z)) +- μ(x∨(y∨z))) ?); [
+  apply feq_plusr; apply feq_plusl; apply Hcut] clear Hcut;
+apply (eq_trans ?? (μx+ μy+ μz+- μ(y∧z)+- μ(x∨(y∨z)))); [2:
+  apply feq_plusl; apply feq_opp;
+  apply (eq_trans ?? ? ? (join_assoc ?????));
+  apply (eq_trans ?? ? ? (join_comm ????));
+  apply eq_reflexive;]
+apply feq_plusr; apply (eq_trans ?? ? ? (plus_assoc ????));
+apply feq_plusr; apply plus_assoc;
+qed.
 
 lemma meet_join_le1: ∀R.∀L:vlattice R.∀x,y,z:L.μ (x ∧ z) ≤ μ (x ∧ (y ∨ z)). 
 intros (R L x y z);
+apply (le_rewr ??? ? (step1_3_57 ?????));
 apply (le_rewr ??? (μ x + μ y + μ z + -μ (y ∧ z) + -μ(z ∨ (x ∨ y))) (foo ?????));
 apply (le_rewr ??? (μ x + μ y + μ z + -μ (y ∧ z) + -μ((z ∨ x) ∨ y))); 
   [ apply feq_plusl; apply eq_opp_sym; apply join_assoc;]
-- 
2.39.2