(* *)
(**************************************************************************)
-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_ = apart_of_excess (pl_carr ml_lattice)
}.
lemma ml_mspace: ∀R.mlattice_ R → metric_space R.
-intros (R ml); apply (mk_metric_space R ml); unfold Type_OF_mlattice_;
+intros (R ml); apply (mk_metric_space R (apart_of_excess ml));
+unfold apartness_OF_mlattice_;
+[rewrite < (ml_with_ ? ml); apply (metric ? (ml_mspace_ ? ml))]
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))]
+[apply (mpositive ? (ml_mspace_ ? ml));|apply (mreflexive ? (ml_mspace_ ? ml));
+|apply (msymmetric ? (ml_mspace_ ? ml));|apply (mtineq ? (ml_mspace_ ? ml))]
qed.
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".
+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
}.
-axiom meq_joinl: ∀R.∀ml:mlattice R.∀x,y,z:ml. x≈z → δx y ≈ δz y.
+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;
+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; split [apply mpositive] apply ap_symmetric;
+assumption;
+qed.
-lemma meq_joinr: ∀R.∀ml:mlattice R.∀x,y,z:ml. x≈z → δy x ≈ δy z.
+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 (le_rewl ??? (0+δz y) (eq_to_dzero ???? H));
+ apply (le_rewl ??? (δz y) (zero_neutral ??)); apply le_reflexive;
+| apply (le_transitive ???? (mtineq ???y x));
+ apply (le_rewl ??? (0+δx y) (eq_to_dzero ??z x H));
+ apply (le_rewl ??? (δx y) (zero_neutral ??)); apply le_reflexive;]
+qed.
+
+(* 3.3 *)
+lemma meq_r: ∀R.∀ml:mlattice R.∀x,y,z:ml. x≈z → δy x ≈ δy z.
intros; apply (eq_trans ???? (msymmetric ??y x));
-apply (eq_trans ????? (msymmetric ??z y)); apply meq_joinl; assumption;
+apply (eq_trans ????? (msymmetric ??z y)); apply meq_l; assumption;
qed.
+
-(*
-lemma foo: ∀R.∀ml:mlattice R.∀x,y,z:ml. δx y ≈ δ(y∨x) (y∨z).
-intros; apply le_le_eq; [
- apply (le_rewr ???? (meq_joinl ????? (join_comm ???)));
- apply (le_rewr ???? (meq_joinr ????? (join_comm ???)));
-*)
+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; split; [apply mpositive;]
+apply ap_symmetric; assumption;
+qed.
+
+interpretation "Lattive meet le" 'leq a b =
+ (cic:/matita/excess/le.con (cic:/matita/lattice/excess_OF_lattice1.con _) a b).
+
+interpretation "Lattive join le (aka ge)" 'geq a b =
+ (cic:/matita/excess/le.con (cic:/matita/lattice/excess_OF_lattice.con _) a b).
+
+lemma le_to_ge: ∀l:lattice.∀a,b:l.a ≤ b → b ≥ a.
+intros(l a b H); apply H;
+qed.
+
+lemma ge_to_le: ∀l:lattice.∀a,b:l.b ≥ a → a ≤ b.
+intros(l a b H); apply H;
+qed.
+
+lemma eq_to_eq:∀l:lattice.∀a,b:l.
+ (eq (apart_of_excess (pl_carr (latt_jcarr l))) a b) →
+ (eq (apart_of_excess (pl_carr (latt_mcarr l))) a b).
+intros 3; unfold eq; unfold apartness_OF_lattice;
+unfold apartness_OF_lattice_1; unfold latt_jcarr; simplify;
+unfold dual_exc; simplify; intros 2 (H H1); apply H;
+cases H1;[right|left]assumption;
+qed.
+
+coercion cic:/matita/metric_lattice/eq_to_eq.con nocomposites.
(* 3.11 *)
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∧x) ??));
-(* auto type. assert failure *)
+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 ?? Lxy) as Dxm; lapply (le_to_eqm ?? Lyz) as Dym;
+lapply (le_to_eqj ?? (le_to_ge ??? Lxy)) as Dxj; lapply (le_to_eqj ?? (le_to_ge ??? Lyz)) as Dyj; clear Lxy Lyz;
+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;
+ unfold apartness_OF_mlattice1;
+ exact (eq_to_eq ??? Dxj);]
+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 y z);]
+apply feq_plusl;
+apply (Eq≈ (δ(y∧x) (y∧z)) (meq_l ????? (meet_comm x y)));
+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