lemma finisced

[helm.git] / matita / dama / valued_lattice.ma

diff --git a/matita/dama/valued_lattice.ma b/matita/dama/valued_lattice.ma
index 66eb715c98f0dbf38c9ac0f35ef74af4e2591455..53855975d1f277ef24ef8fa973ee006febd6a2d7 100644 (file)
--- a/matita/dama/valued_lattice.ma
+++ b/matita/dama/valued_lattice.ma
@@ -29,9 +29,9 @@ 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;
-  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)  
+  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 y) ≤ value (meet x (join y z)) 
 }. 
 
 interpretation "valued lattice meet" 'and a b =
@@ -46,32 +46,198 @@ 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.
-  μ(x ∧ (y ∨ z)) ≈ (μ x) + (μ y) + μ z + -μ (y ∧ z) + -μ (z ∨ (x ∨ y)).
+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 modularj: ∀R.∀L:vlattice R.∀y,z:L. μ(y∨z) ≈ μy + μz + -μ (y ∧ z).
+intros (R L y z);
+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.
+qed.
+
+lemma modularm: ∀R.∀L:vlattice R.∀y,z:L. μ(y∧z) ≈ μy + μz + -μ (y ∨ z).
+intros (R L y z);
+lapply (modular_mjp ?? y z) as H1;
+apply (plus_cancl ??? (μ(y ∨ z)));
+apply (eq_trans ?? ? ? H1); clear H1;
+apply (eq_trans ????? (plus_comm ???));
+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.
+qed.
+
+lemma modularmj: ∀R.∀L:vlattice R.∀x,y,z:L.μ(x∧(y∨z))≈(μx + μ(y ∨ z) + - μ(x∨(y∨z))).
 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 :-)
- *)
+lapply (modular_mjp ?? x (y ∨ z)) as H1;
+apply (eq_trans ?? (μ(x∨(y∨z))+ μ(x∧(y∨z)) +-μ(x∨(y∨z)))); [2: apply feq_plusr; apply 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.
+qed.
 
+lemma modularjm: ∀R.∀L:vlattice R.∀x,y,z:L.μ(x∨(y∧z))≈(μx + μ(y ∧ z) + - μ(x∧(y∧z))).
+intros (R L x y z);
+lapply (modular_mjp ?? x (y ∧ z)) as H1;
+apply (eq_trans ?? (μ(x∧(y∧z))+ μ(x∨(y∧z)) +-μ(x∧(y∧z)))); [2: apply feq_plusr; apply (eq_trans ???? (plus_comm ???)); apply H1] clear H1;
+apply (eq_trans ?? ? ?? (plus_comm ???));
+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_sym; apply zero_neutral;
+qed.
 
-lemma meet_join_le1: ∀R.∀L:vlattice R.∀x,y,z:L.μ (x ∧ z) ≤ μ (x ∧ (y ∨ z)). 
+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);
-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;]
-lapply (meet_join_le ?? z x y);
-cut (- μ (z ∨ x ∨ y) ≈ - μ (z ∨ x) - μ y + μ (y ∧ (z ∨ x)));
- [2: 
- 
- 
-lemma join_meet_le1: ∀R.∀L:vlattice R.∀x,y,z:L.μ (x ∨ (y ∧ z)) ≤ μ (x ∨ z).   
-(* hint per duplicati? *)
+apply (eq_trans ?? ? ? (modularjm ?? x y z));
+apply (eq_trans ?? ( μx+ (μy+ μz+- μ(y∨z)) +- μ(x∧(y∧z))) ?); [
+  apply feq_plusr; apply feq_plusl; apply (modularm ?? y z);]
+apply (eq_trans ?? (μx+ μy+ μz+- μ(y∨z)+- μ(x∧(y∧z)))); [2:
+  apply feq_plusl; apply feq_opp;
+  apply (eq_trans ?? ? ? (meet_assoc ?????));
+  apply (eq_trans ?? ? ? (meet_comm ????));
+  apply eq_reflexive;]
+apply feq_plusr; apply (eq_trans ?? ? ? (plus_assoc ????));
+apply feq_plusr; apply plus_assoc;
+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);
-apply (le_rewr ??? (0 + μ (x ∨ z)) (zero_neutral ??));
-apply (le_rewr ??? (μ (x ∨ z) + 0) (plus_comm ???));
-apply (le_rewr ??? (μ (x ∨ z) + (-μ(y ∨ z) + μ(y ∨ z))) (opp_inverse ? ?));
+apply (eq_trans ?? ? ? (modularmj ?? x y z));
+apply (eq_trans ?? ( μx+ (μy+ μz+- μ(y∧z)) +- μ(x∨(y∨z))) ?); [
+  apply feq_plusr; apply feq_plusl; apply (modularj ?? y z);]
+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 3.57 *) 
 
+lemma join_meet_le_join: ∀R.∀L:vlattice R.∀x,y,z:L.μ (x ∨ (y ∧ z)) ≤ μ (x ∨ z). 
+intros (R L x y z);
+apply (le_rewl ??? ? (eq_sym ??? (step1_3_57' ?????)));
+apply (le_rewl ??? (μx+ μy+ μz+- μ(y∨z)+ -μ(z∧x∧y))); [
+  apply feq_plusl; apply feq_opp; apply (eq_trans ?? ? ?? (eq_sym ??? (meet_assoc ?????))); apply eq_reflexive;]
+apply (le_rewl ??? (μx+ μy+ μz+- μ(y∨z)+ (- ( μ(z∧x)+ μy+- μ((z∧x)∨y))))); [
+  apply feq_plusl; apply feq_opp; apply eq_sym; apply modularm]
+apply (le_rewl ??? (μx+ μy+ μz+- μ(y∨z)+ (- μ(z∧x)+ -μy+-- μ((z∧x)∨y)))); [
+  apply feq_plusl; apply (eq_trans ?? (- (μ(z∧x)+ μy) + -- μ((z∧x)∨y))); [
+    apply feq_plusr; apply eq_sym; apply eq_opp_plus_plus_opp_opp;]
+  apply eq_sym; apply eq_opp_plus_plus_opp_opp;]
+apply (le_rewl ??? (μx+ μy+ μz+- μ(y∨z)+(- μ(z∧x)+- μy+ μ(y∨(z∧x))))); [
+  repeat apply feq_plusl; apply eq_sym; apply (eq_trans ?? (μ((z∧x)∨y)) ? (eq_opp_opp_x_x ??));
+  apply join_comm;]
+apply (le_rewl ??? (μx+ μy+ μz+- μ(y∨z)+(- μ(z∧x)+- μy)+ μ(y∨(z∧x)))); [
+  apply eq_sym; apply plus_assoc;]
+apply (le_rewl ??? (μx+ μy+ μz+- μ(y∨z)+(- μy + - μ(z∧x))+ μ(y∨(z∧x)))); [
+  repeat apply feq_plusr; repeat apply feq_plusl; apply plus_comm;]
+apply (le_rewl ??? (μx+ μy+ μz+- μ(y∨z)+- μy + - μ(z∧x)+ μ(y∨(z∧x)))); [
+  repeat apply feq_plusr; apply eq_sym; apply plus_assoc;]
+apply (le_rewl ??? (μx+ μy+ μz+- μy + - μ(y∨z)+- μ(z∧x)+ μ(y∨(z∧x)))); [
+  repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_assoc ????));
+  apply (eq_trans ?? ( μx+ μy+ μz+(- μy+- μ(y∨z))) ? (eq_sym ??? (plus_assoc ????)));
+  apply feq_plusl; apply plus_comm;]
+apply (le_rewl ??? (μx+ μy+ -μy+ μz + - μ(y∨z)+- μ(z∧x)+ μ(y∨(z∧x)))); [
+  repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_assoc ????));
+  apply (eq_trans ?? (μx+ μy+( -μy+ μz)) ? (eq_sym ??? (plus_assoc ????)));
+  apply feq_plusl; apply plus_comm;]
+apply (le_rewl ??? (μx+ 0 + μz + - μ(y∨z)+- μ(z∧x)+ μ(y∨(z∧x)))); [
+  repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_assoc ????));
+  apply feq_plusl; apply eq_sym; apply (eq_trans ?? ? ? (plus_comm ???));
+  apply opp_inverse; apply eq_reflexive;] 
+apply (le_rewl ??? (μx+ μz + - μ(y∨z)+- μ(z∧x)+ μ(y∨(z∧x)))); [
+  repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_comm ???));
+  apply eq_sym; apply zero_neutral;]
+apply (le_rewl ??? (μz+ μx + - μ(y∨z)+- μ(z∧x)+ μ(y∨(z∧x)))); [
+  repeat apply feq_plusr; apply plus_comm;]
+apply (le_rewl ??? (μz+ μx +- μ(z∧x)+ - μ(y∨z)+ μ(y∨(z∧x)))); [
+  repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_assoc ????));
+  apply (eq_trans ?? ? ? (eq_sym ??? (plus_assoc ????))); apply feq_plusl;
+  apply plus_comm;]
+apply (le_rewl ??? (μ(z∨x)+ - μ(y∨z)+ μ(y∨(z∧x)))); [
+  repeat apply feq_plusr; apply modularj;]
+apply (le_rewl ??? (μ(z∨x)+ (- μ(y∨z)+ μ(y∨(z∧x)))) (plus_assoc ????));
+apply (le_rewr ??? (μ(x∨z) + 0)); [apply (eq_trans ?? ? ? (plus_comm ???)); apply zero_neutral]
+apply (le_rewr ??? (μ(x∨z) + (-μ(y∨z) + μ(y∨z)))); [ apply feq_plusl; apply opp_inverse]
+apply (le_rewr ??? (μ(z∨x) + (-μ(y∨z) + μ(y∨z)))); [ apply feq_plusr; apply join_comm;]
+repeat apply fle_plusl; apply join_meet_le;
+qed.
 
+lemma meet_le_meet_join: ∀R.∀L:vlattice R.∀x,y,z:L.μ (x ∧ z) ≤ μ (x ∧ (y ∨ z)). 
+intros (R L x y z);
+apply (le_rewr ??? ? (eq_sym ??? (step1_3_57 ?????)));
+apply (le_rewr ??? (μx+ μy+ μz+- μ(y∧z)+ -μ(z∨x∨y))); [
+  apply feq_plusl; apply feq_opp; apply (eq_trans ?? ? ?? (eq_sym ??? (join_assoc ?????))); apply eq_reflexive;]
+apply (le_rewr ??? (μx+ μy+ μz+- μ(y∧z)+ (- ( μ(z∨x)+ μy+- μ((z∨x)∧y))))); [
+  apply feq_plusl; apply feq_opp; apply eq_sym; apply modularj]
+apply (le_rewr ??? (μx+ μy+ μz+- μ(y∧z)+ (- μ(z∨x)+ -μy+-- μ((z∨x)∧y)))); [
+  apply feq_plusl; apply (eq_trans ?? (- (μ(z∨x)+ μy) + -- μ((z∨x)∧y))); [
+    apply feq_plusr; apply eq_sym; apply eq_opp_plus_plus_opp_opp;]
+  apply eq_sym; apply eq_opp_plus_plus_opp_opp;]
+apply (le_rewr ??? (μx+ μy+ μz+- μ(y∧z)+(- μ(z∨x)+- μy+ μ(y∧(z∨x))))); [
+  repeat apply feq_plusl; apply eq_sym; apply (eq_trans ?? (μ((z∨x)∧y)) ? (eq_opp_opp_x_x ??));
+  apply meet_comm;]
+apply (le_rewr ??? (μx+ μy+ μz+- μ(y∧z)+(- μ(z∨x)+- μy)+ μ(y∧(z∨x)))); [
+  apply eq_sym; apply plus_assoc;]
+apply (le_rewr ??? (μx+ μy+ μz+- μ(y∧z)+(- μy + - μ(z∨x))+ μ(y∧(z∨x)))); [
+  repeat apply feq_plusr; repeat apply feq_plusl; apply plus_comm;]
+apply (le_rewr ??? (μx+ μy+ μz+- μ(y∧z)+- μy + - μ(z∨x)+ μ(y∧(z∨x)))); [
+  repeat apply feq_plusr; apply eq_sym; apply plus_assoc;]
+apply (le_rewr ??? (μx+ μy+ μz+- μy + - μ(y∧z)+- μ(z∨x)+ μ(y∧(z∨x)))); [
+  repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_assoc ????));
+  apply (eq_trans ?? ( μx+ μy+ μz+(- μy+- μ(y∧z))) ? (eq_sym ??? (plus_assoc ????)));
+  apply feq_plusl; apply plus_comm;]
+apply (le_rewr ??? (μx+ μy+ -μy+ μz + - μ(y∧z)+- μ(z∨x)+ μ(y∧(z∨x)))); [
+  repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_assoc ????));
+  apply (eq_trans ?? (μx+ μy+( -μy+ μz)) ? (eq_sym ??? (plus_assoc ????)));
+  apply feq_plusl; apply plus_comm;]
+apply (le_rewr ??? (μx+ 0 + μz + - μ(y∧z)+- μ(z∨x)+ μ(y∧(z∨x)))); [
+  repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_assoc ????));
+  apply feq_plusl; apply eq_sym; apply (eq_trans ?? ? ? (plus_comm ???));
+  apply opp_inverse; apply eq_reflexive;] 
+apply (le_rewr ??? (μx+ μz + - μ(y∧z)+- μ(z∨x)+ μ(y∧(z∨x)))); [
+  repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_comm ???));
+  apply eq_sym; apply zero_neutral;]
+apply (le_rewr ??? (μz+ μx + - μ(y∧z)+- μ(z∨x)+ μ(y∧(z∨x)))); [
+  repeat apply feq_plusr; apply plus_comm;]
+apply (le_rewr ??? (μz+ μx +- μ(z∨x)+ - μ(y∧z)+ μ(y∧(z∨x)))); [
+  repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_assoc ????));
+  apply (eq_trans ?? ? ? (eq_sym ??? (plus_assoc ????))); apply feq_plusl;
+  apply plus_comm;]
+apply (le_rewr ??? (μ(z∧x)+ - μ(y∧z)+ μ(y∧(z∨x)))); [
+  repeat apply feq_plusr; apply modularm;]
+apply (le_rewr ??? (μ(z∧x)+ (- μ(y∧z)+ μ(y∧(z∨x)))) (plus_assoc ????));
+apply (le_rewl ??? (μ(x∧z) + 0)); [apply (eq_trans ?? ? ? (plus_comm ???)); apply zero_neutral]
+apply (le_rewl ??? (μ(x∧z) + (-μ(y∧z) + μ(y∧z)))); [ apply feq_plusl; apply opp_inverse]
+apply (le_rewl ??? (μ(z∧x) + (-μ(y∧z) + μ(y∧z)))); [ apply feq_plusr; apply meet_comm;]
+repeat apply fle_plusl; apply meet_join_le;
+qed.