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[helm.git] / helm / software / matita / dama / constructive_pointfree / lebesgue.ma

diff --git a/helm/software/matita/dama/constructive_pointfree/lebesgue.ma b/helm/software/matita/dama/constructive_pointfree/lebesgue.ma
index a739ed71526b4224b9224642ee60010c5f2dcc29..824202a78e4a3ef832c08ac4f8ba93c2ef2d5cae 100644 (file)
--- a/helm/software/matita/dama/constructive_pointfree/lebesgue.ma
+++ b/helm/software/matita/dama/constructive_pointfree/lebesgue.ma
@@ -14,95 +14,14 @@
 
 set "baseuri" "cic:/matita/lebesgue/".
 
-include "reals.ma".
-include "lattices.ma".
+include "metric_lattice.ma".
+include "sequence.ma".
 
-axiom ltT : ∀R:real.∀a,b:R.Type.  
-  
-   
-alias symbol "or" = "constructive or".
-definition rapart ≝ λR:real.λa,b:R.a < b ∨ b < a. 
+(* Section 3.2 *)
 
-notation "a # b" non associative with precedence 50 for
- @{ 'apart $a $b}.
-notation "a \approx b" non associative with precedence 50 for
- @{ 'napart $a $b}.    
-interpretation "real apartness" 'apart a b =
- (cic:/matita/lebesgue/rapart.con _ a b). 
-interpretation "real non apartness" 'napart a b =
- (cic:/matita/constructive_connectives/Not.con
-   (cic:/matita/lebesgue/rapart.con _ a b)). 
+(* 3.17 *)
 
-record is_semi_metric (R : real) (T : Type) (d : T → T → R): Prop ≝ {
- sm_positive: ∀a,b:T. d a b ≤ 0; 
- sm_reflexive: ∀a. d a a ≈ 0;
- sm_symmetric: ∀a,b. d a b ≈ d b a;
- sm_tri_ineq: ∀a,b,c:T. d a b ≤ d a c + d c b
-}.
 
-record semimetric_set (R : real) : Type ≝ {
-  ms_carr :> Type;
-  ms_semimetric: ms_carr → ms_carr → R;
-  ms_properties:> is_semi_metric R ms_carr ms_semimetric
-}.
-
-notation < "\nbsp \delta a \nbsp b" non associative with precedence 80 for @{ 'delta2 $a $b}.
-interpretation "semimetric" 'delta2 a b = (cic:/matita/lebesgue/ms_semimetric.con _ _ a b).
-notation "\delta" non associative with precedence 80 for @{ 'delta }.
-interpretation "semimetric" 'delta = (cic:/matita/lebesgue/ms_semimetric.con _ _).
-
-record pre_semimetric_lattice (R : real) : Type ≝ {
-  ml_lattice :> lattice;
-  ml_semimetric_set_ : semimetric_set R;
-  with_ml_lattice_eq_ml_semimetric_set_: ms_carr ? ml_semimetric_set_ = ml_lattice
-}.
-
-lemma ml_semimetric_set : ∀R.∀ms:pre_semimetric_lattice R. semimetric_set R.
-intros (R ml); constructor 1; [apply (ml : Type);]
-cases ml (l ms with_); clear ml; simplify;
-[1: rewrite < with_; apply (ms_semimetric ? ms);  
-|2: cases with_; simplify; apply (ms_properties ? ms);]
-qed.
-
-coercion cic:/matita/lebesgue/ml_semimetric_set.con.
-
-record is_semimetric_lattice (R : real) (ml : pre_semimetric_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 semimetric_lattice (R : real) : Type ≝ {
-  ml_pre_semimetric_lattice:> pre_semimetric_lattice R;
-  ml_semimetric_lattice_properties: is_semimetric_lattice R ml_pre_semimetric_lattice
-}.
-
-definition apart:= 
- λR: real. λml: semimetric_lattice R. λa,b: ml. 
- (* 0 < δ a b *) ltT ? 0 (δ a b).
- (* < scazzato, ma CSC dice che poi si cambia dopo  *)    
-
-interpretation "semimetric lattice apartness" 'apart a b =
- (cic:/matita/lebesgue/apart.con _ _ a b). 
-interpretation "semimetric lattice non apartness" 'napart a b =
- (cic:/matita/constructive_connectives/Not.con
-   (cic:/matita/lebesgue/apart.con _ _ a b)).
-  
-
-lemma triangular_inequality : ∀R: real. ∀ms: semimetric_set R.∀a,b,c:ms.
- δ a b ≤ δ a c + δ c b.
-intros (R ms a b c); exact (sm_tri_ineq R ms (ms_semimetric R ms) ms a b c);
-qed. 
-
-lemma foo : ∀R: real. ∀ml: semimetric_lattice R.∀a,b,a1,b1: ml. 
-  a ≈ a1 → b ≈ b1 → (δ a b ≈ δ a1 b1).  
-intros;
-lapply (triangular_inequality ? ? a b a1) as H1;
-lapply (triangular_inequality ? ? a1 b b1) as H2;
-lapply (og_ordered_abelian_group_properties R ? ? (δ a a1) H2) as H3;
+(* 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