--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+
+
+include "ordered_group.ma".
+
+record vlattice (R : togroup) : Type ≝ {
+ wl_carr:> Type;
+ value: wl_carr → R;
+ join: wl_carr → wl_carr → wl_carr;
+ meet: wl_carr → wl_carr → wl_carr;
+ meet_refl: ∀x. value (meet x x) ≈ value x;
+ join_refl: ∀x. value (join x x) ≈ value x;
+ meet_comm: ∀x,y. value (meet x y) ≈ value (meet y x);
+ join_comm: ∀x,y. value (join x y) ≈ value (join y x);
+ join_assoc: ∀x,y,z. value (join x (join y z)) ≈ value (join (join x y) z);
+ 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;
+ 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 =
+ (cic:/matita/prevalued_lattice/meet.con _ _ a b).
+
+interpretation "valued lattice join" 'or a b =
+ (cic:/matita/prevalued_lattice/join.con _ _ a b).
+
+notation < "\nbsp \mu a" non associative with precedence 80 for @{ 'value2 $a}.
+interpretation "lattice value" 'value2 a = (cic:/matita/prevalued_lattice/value.con _ _ a).
+
+notation "\mu" non associative with precedence 80 for @{ 'value }.
+interpretation "lattice value" 'value = (cic:/matita/prevalued_lattice/value.con _ _).
+
+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≈ (μz + μx) (modular_mjp ????));
+apply (Eq≈ (μz + μy) H); clear H;
+apply (Eq≈ (μ(z∨y) + μ(z∧y)) (modular_mjp ??z y));
+apply (plus_cancl ??? (- μ (z ∨ y)));
+apply (Eq≈ ? (plus_assoc ????));
+apply (Eq≈ (0+ μ(z∧y)) (opp_inverse ??));
+apply (Eq≈ ? (zero_neutral ??));
+apply (Eq≈ (- μ(z∨y)+ μ(z∨y)+ μ(z∧x)) ? (plus_assoc ????));
+apply (Eq≈ (0+ μ(z∧x)) ? (opp_inverse ??));
+apply (Eq≈ (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≈ ? H1); clear H1;
+apply (Eq≈ ?? (plus_assoc ????));
+apply (Eq≈ (μy+ μz + 0) ? (opp_inverse ??));
+apply (Eq≈ ?? (plus_comm ???));
+apply (Eq≈ (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).
+(* CSC: questa è la causa per cui la hint per cercare i duplicati ci sta 1 mese *)
+(* exact modularj; *)
+intros (R L y z);
+lapply (modular_mjp ?? y z) as H1;
+apply (plus_cancl ??? (μ(y ∨ z)));
+apply (Eq≈ ? H1); clear H1;
+apply (Eq≈ ?? (plus_comm ???));
+apply (Eq≈ ?? (plus_assoc ????));
+apply (Eq≈ (μy+ μz + 0) ? (opp_inverse ??));
+apply (Eq≈ ?? (plus_comm ???));
+apply (Eq≈ (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 (modular_mjp ?? x (y ∨ z)) as H1;
+apply (Eq≈ (μ(x∨(y∨z))+ μ(x∧(y∨z)) +-μ(x∨(y∨z))) ? (feq_plusr ???? H1)); clear H1;
+apply (Eq≈ ? ? (plus_comm ???));
+apply (Eq≈ (- μ(x∨(y∨z))+ μ(x∨(y∨z))+ μ(x∧(y∨z))) ? (plus_assoc ????));
+apply (Eq≈ (0+μ(x∧(y∨z))) ? (opp_inverse ??));
+apply (Eq≈ (μ(x∧(y∨z))) ? (zero_neutral ??));
+apply eq_reflexive.
+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≈ (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≈ ? ? (plus_comm ???));
+apply (Eq≈ (- μ(x∧(y∧z))+ μ(x∧(y∧z))+ μ(x∨y∧z)) ? (plus_assoc ????));
+apply (Eq≈ (0+ μ(x∨y∧z)) ? (opp_inverse ??));
+apply eq_sym; apply 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);
+apply (Eq≈ ? (modularjm ?? x y z));
+apply (Eq≈ ( μx+ (μy+ μz+- μ(y∨z)) +- μ(x∧(y∧z)))); [
+ apply feq_plusr; apply feq_plusl; apply (modularm ?? y z);]
+apply (Eq≈ (μx+ μy+ μz+- μ(y∨z)+- μ(x∧(y∧z)))); [2:
+ apply feq_plusl; apply feq_opp;
+ apply (Eq≈ ? (meet_assoc ?????));
+ apply (Eq≈ ? (meet_comm ????));
+ apply eq_reflexive;]
+apply feq_plusr; apply (Eq≈ ? (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 (Eq≈ ? (modularmj ?? x y z));
+apply (Eq≈ ( μx+ (μy+ μz+- μ(y∧z)) +- μ(x∨(y∨z)))); [
+ apply feq_plusr; apply feq_plusl; apply (modularj ?? y z);]
+apply (Eq≈ (μx+ μy+ μz+- μ(y∧z)+- μ(x∨(y∨z)))); [2:
+ apply feq_plusl; apply feq_opp;
+ apply (Eq≈ ? (join_assoc ?????));
+ apply (Eq≈ ? (join_comm ????));
+ apply eq_reflexive;]
+apply feq_plusr; apply (Eq≈ ? (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≈ (- (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≈ (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≈ ( 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≈ (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≈ (- (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≈ (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≈ ( 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≈ ?? (plus_assoc ????));
+ apply (Eq≈ (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.
projects / helm.git / blobdiff