+++ /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 *)
-(* *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/valued_lattice/".
-
-include "ordered_groups.ma".
-
-record vlattice (R : ogroup) : Type ≝ {
- vl_carr:> Type;
- value: vl_carr → R;
- join: vl_carr → vl_carr → vl_carr;
- meet: vl_carr → vl_carr → vl_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/valued_lattice/meet.con _ _ a b).
-
-interpretation "valued lattice join" 'or a b =
- (cic:/matita/valued_lattice/join.con _ _ a b).
-
-notation < "\nbsp \mu a" non associative with precedence 80 for @{ 'value2 $a}.
-interpretation "lattice value" 'value2 a = (cic:/matita/valued_lattice/value.con _ _ a).
-
-notation "\mu" non associative with precedence 80 for @{ 'value }.
-interpretation "lattice value" 'value = (cic:/matita/valued_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_trans ?? (μz + μx) ? (modular_mjp ????));
-apply (eq_trans ?? (μz + μy) ? H); clear H;
-apply (eq_trans ?? (μ(z∨y) + μ(z∧y)) ? (modular_mjp ??z y));
-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) ?? (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).
-(* 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_trans ?? ? ? H1); clear H1;
-apply (eq_trans ????? (plus_comm ???));
-apply (eq_trans ?? ? ?? (plus_assoc ????));
-apply (eq_trans ?? (μy+ μz + 0) ?? (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 (modular_mjp ?? x (y ∨ z)) as H1;
-apply (eq_trans ?? (μ(x∨(y∨z))+ μ(x∧(y∨z)) +-μ(x∨(y∨z))) ?? (feq_plusr ???? H1)); clear H1;
-apply (eq_trans ?? ? ?? (plus_comm ???));
-apply (eq_trans ?? (- μ(x∨(y∨z))+ μ(x∨(y∨z))+ μ(x∧(y∨z))) ?? (plus_assoc ????));
-apply (eq_trans ?? (0+μ(x∧(y∨z))) ?? (opp_inverse ??));
-apply (eq_trans ?? (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_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)) ?? (plus_assoc ????));
-apply (eq_trans ?? (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_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 (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.
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