(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/metric_lattice/".
-
include "metric_space.ma".
include "lattice.ma".
-record mlattice_ (R : ogroup) : Type ≝ {
+record mlattice_ (R : todgroup) : Type ≝ {
ml_mspace_: metric_space R;
ml_lattice:> lattice;
- ml_with_: ms_carr ? ml_mspace_ = ap_carr (l_carr ml_lattice)
+ ml_with: ms_carr ? ml_mspace_ = Type_OF_lattice ml_lattice
}.
lemma ml_mspace: ∀R.mlattice_ R → metric_space R.
-intros (R ml); apply (mk_metric_space R ml); unfold Type_OF_mlattice_;
-cases (ml_with_ ? ml); simplify;
+intros (R ml); apply (mk_metric_space R (Type_OF_mlattice_ ? ml));
+unfold Type_OF_mlattice_; cases (ml_with ? ml); simplify;
[apply (metric ? (ml_mspace_ ? ml));|apply (mpositive ? (ml_mspace_ ? ml));
|apply (mreflexive ? (ml_mspace_ ? ml));|apply (msymmetric ? (ml_mspace_ ? ml));
|apply (mtineq ? (ml_mspace_ ? ml))]
coercion cic:/matita/metric_lattice/ml_mspace.con.
-record is_mlattice (R : ogroup) (ml: mlattice_ R) : Type ≝ {
- ml_prop1: ∀a,b:ml. 0 < δ a b → a # b;
- ml_prop2: ∀a,b,c:ml. δ (a∨b) (a∨c) + δ (a∧b) (a∧c) ≤ δ b c
-}.
-
-record mlattice (R : ogroup) : Type ≝ {
+alias symbol "plus" = "Abelian group plus".
+alias symbol "leq" = "Excess less or equal than".
+record mlattice (R : todgroup) : Type ≝ {
ml_carr :> mlattice_ R;
- ml_props:> is_mlattice R ml_carr
+ ml_prop1: ∀a,b:ml_carr. 0 < δ a b → a # b;
+ ml_prop2: ∀a,b,c:ml_carr. δ (a∨b) (a∨c) + δ (a∧b) (a∧c) ≤ (δ b c)
}.
+interpretation "Metric lattice leq" 'leq a b =
+ (cic:/matita/excess/le.con (cic:/matita/metric_lattice/excess_OF_mlattice1.con _ _) a b).
+interpretation "Metric lattice geq" 'geq a b =
+ (cic:/matita/excess/le.con (cic:/matita/metric_lattice/excess_OF_mlattice.con _ _) a b).
+
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);
+intros (R ml a b E); intro H; apply E; apply ml_prop1;
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;
+apply ml_prop1; split [apply mpositive] apply ap_symmetric;
assumption;
qed.
-(*
-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 (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;]
+intros (R ml x y H); apply ml_prop1; split; [apply mpositive;]
apply ap_symmetric; assumption;
qed.
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) ??));
- 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;
+apply (le_transitive ????? (ml_prop2 ?? (y) ??));
+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 y x Lxy) as Dxm; lapply (le_to_eqm z y Lyz) as Dym;
+lapply (le_to_eqj x y Lxy) as Dxj; lapply (le_to_eqj y z Lyz) as Dyj; clear Lxy Lyz;
+STOP
+apply (Eq≈ (δ(x∧y) y + δy z) (meq_l ????? Dxm));
+apply (Eq≈ (δ(x∧y) (y∧z) + δy z) (meq_r ????? Dym));
+apply (Eq≈ (δ(x∧y) (y∧z) + δ(y∨x) z));[
+ apply feq_plusl; apply meq_l; clear Dyj Dxm Dym; assumption]
+apply (Eq≈ (δ(x∧y) (y∧z) + δ(y∨x) (z∨y))); [
+ apply (feq_plusl ? (δ(x∧y) (y∧z)) ?? (meq_r ??? (y∨x) ? Dyj));]
+apply (Eq≈ ? (plus_comm ???));
+apply (Eq≈ (δ(y∨x) (y∨z)+ δ(x∧y) (y∧z)));[
+ apply feq_plusr; apply meq_r; apply (join_comm ??);]
+apply feq_plusl;
+apply (Eq≈ (δ(y∧x) (y∧z)) (meq_l ????? (meet_comm ??)));
+apply eq_reflexive;
qed.
-
+(* 3.17 conclusione: δ x y ≈ 0 *)
+(* 3.20 conclusione: δ x y ≈ 0 *)
+(* 3.21 sup forte
+ strong_sup x ≝ ∀n. s n ≤ x ∧ ∀y x ≰ y → ∃n. s n ≰ y
+ strong_sup_zoli x ≝ ∀n. s n ≤ x ∧ ∄y. y#x ∧ y ≤ x
+*)
+(* 3.22 sup debole (più piccolo dei maggioranti) *)
+(* 3.23 conclusion: δ x sup(...) ≈ 0 *)
+(* 3.25 vero nel reticolo e basta (niente δ) *)
+(* 3.36 conclusion: δ x y ≈ 0 *)
projects / helm.git / blobdiff