(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/lebesgue/".
-include "reals.ma".
-include "lattices.ma".
-record measured_set (R : real) : Type ≝ {
- ms_carr :> Type;
- ms_measure: ms_carr → ms_carr → R
-}.
+include "metric_lattice.ma".
+include "sequence.ma".
+include "constructive_connectives.ma".
-notation "\delta" non associative with precedence 90 for @{ 'delta }.
-interpretation "measure" 'delta = (cic:/matita/lebesgue/ms_measure.con _ _).
+(* Section 3.2 *)
-record pre_measured_lattice (R : real) : Type ≝ {
- ml_lattice :> lattice;
- ml_measured_set_ : measured_set R;
- with_ml_lattice_eq_ml_measured_set_: ms_carr ? ml_measured_set_ = ml_lattice
-}.
+(* 3.21 *)
-lemma ml_measured_set : ∀R.∀ms:pre_measured_lattice R. measured_set R.
-intros (R ml); constructor 1; [1:apply (ml : Type);] cases ml;
-rewrite < H; clear H; cases ml_measured_set_; simplify; exact f;
-qed.
-coercion cic:/matita/lebesgue/ml_measured_set.con.
+(* 3.17 *)
-record is_measured_lattice (R : real) (ml : pre_measured_lattice R) : Prop ≝ {
- prop1a: ∀a : ml.δ (a ∧ a) a = 0;
- prop1b: ∀a : ml.δ (a ∨ a) a = 0;
- prop2a: ∀a,b: ml. δ (a ∨ b) (b ∨ a) = 0;
- prop2b: ∀a,b: ml. δ (a ∧ b) (b ∧ a) = 0;
- prop3a: ∀a,b,c: ml. δ (a ∨ (b ∨ c)) ((a ∨ b) ∨ c) = 0;
- prop3b: ∀a,b,c: ml. δ (a ∧ (b ∧ c)) ((a ∧ b) ∧ c) = 0;
- prop4a: ∀a,b: ml. δ (a ∨ (a ∧ b)) a = 0;
- prop4b: ∀a,b: ml. δ (a ∧ (a ∨ b)) a = 0;
- prop5: ∀a,b,c: ml. δ (a ∨ b) (a ∨ c) + δ (a ∧ b) (a ∧ c) ≤ δ b c
-}.
-
-record measured_lattice (R : real) : Type ≝ {
- ml_pre_measured_lattice:> pre_measured_lattice R;
- ml_measured_lattice_properties: is_measured_lattice R ml_pre_measured_lattice
-}.
-
-definition apart:=
- λR: real. λml: measured_lattice R. λa,b: ml. 0 < δ a b.
- (* < scazzato, ma CSC dice che poi si cambia dopo *)
-notation "a # b" non associative with precedence 50 for
- @{ 'apart $a $b}.
-interpretation "measured lattice apartness" 'apart a b =
- (cic:/matita/lebesgue/apart.con _ _ a b).
-notation "a \approx b" non associative with precedence 50 for
- @{ 'napart $a $b}.
-interpretation "measured lattice non apartness" 'napart a b =
- (cic:/matita/logic/connectives/Not.con
- (cic:/matita/lebesgue/apart.con _ _ a b)).
-
-lemma foo : ∀R: real. ∀ml: measured_lattice R.∀a,b,a1,b1: ml.
- a ≈ a1 → b ≈ b1 → δ a b = δ a1 b1.
- (* =R scazzato *)
-intros;
+(* 3.20 *)
+lemma uniq_sup: ∀O:ogroup.∀s:sequence O.∀x,y:O. is_sup ? s x → is_sup ? s y → x ≈ y.
+intros;
\ No newline at end of file