From d80bd0bc5253538759bfb13c98e86d07592a9359 Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Thu, 8 Nov 2007 17:03:23 +0000 Subject: [PATCH] xxx --- .../dama/constructive_pointfree/lebesgue.ma | 75 +++++++++++++++++++ 1 file changed, 75 insertions(+) create mode 100644 matita/dama/constructive_pointfree/lebesgue.ma diff --git a/matita/dama/constructive_pointfree/lebesgue.ma b/matita/dama/constructive_pointfree/lebesgue.ma new file mode 100644 index 000000000..583dc638c --- /dev/null +++ b/matita/dama/constructive_pointfree/lebesgue.ma @@ -0,0 +1,75 @@ +(**************************************************************************) +(* ___ *) +(* ||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/lebesgue/". + +include "reals.ma". +include "lattices.ma". + +record measured_set (R : real) : Type ≝ { + ms_carr :> Type; + ms_measure: ms_carr → ms_carr → R +}. + +notation "\delta" non associative with precedence 90 for @{ 'delta }. +interpretation "measure" 'delta = (cic:/matita/lebesgue/ms_measure.con _ _). + +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 +}. + +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. + +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; -- 2.39.2