X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fdama%2Fintegration_algebras.ma;h=46bebc0e9da6745255d64691b99f5fe6a00eb7e2;hb=7e30c63fcf9f9fe1780ba7aa4d95fd0d8658548b;hp=28185a78570f29201ea49340294b8b7726617c99;hpb=d61b0ec37e7943a058dc9c6381efe49db7eb575f;p=helm.git

diff --git a/helm/software/matita/dama/integration_algebras.ma b/helm/software/matita/dama/integration_algebras.ma
index 28185a785..46bebc0e9 100644
--- a/helm/software/matita/dama/integration_algebras.ma
+++ b/helm/software/matita/dama/integration_algebras.ma
@@ -14,35 +14,7 @@
 
 set "baseuri" "cic:/matita/integration_algebras/".
 
-include "reals.ma".
-
-record is_vector_space (K: field) (G:abelian_group) (emult:K→G→G) : Prop
-≝
- { vs_nilpotent: ∀v. emult 0 v = 0;
-   vs_neutral: ∀v. emult 1 v = v;
-   vs_distributive: ∀a,b,v. emult (a + b) v = (emult a v) + (emult b v);
-   vs_associative: ∀a,b,v. emult (a * b) v = emult a (emult b v)
- }.
-
-record vector_space (K:field): Type \def
-{ vs_abelian_group :> abelian_group;
-  emult: K → vs_abelian_group → vs_abelian_group;
-  vs_vector_space_properties :> is_vector_space ? vs_abelian_group emult
-}.
-
-interpretation "Vector space external product" 'times a b =
- (cic:/matita/integration_algebras/emult.con _ _ a b).
-
-record is_semi_norm (R:real) (V: vector_space R) (semi_norm:V→R) : Prop \def
- { sn_positive: ∀x:V. 0 ≤ semi_norm x;
-   sn_omogeneous: ∀a:R.∀x:V. semi_norm (a*x) = (abs ? a) * semi_norm x;
-   sn_triangle_inequality: ∀x,y:V. semi_norm (x + y) ≤ semi_norm x + semi_norm y
- }.
-
-record is_norm (R:real) (V:vector_space R) (norm:V → R) : Prop \def
- { n_semi_norm:> is_semi_norm ? ? norm;
-   n_properness: ∀x:V. norm x = 0 → x = 0
- }.
+include "vector_spaces.ma".
 
 record is_lattice (C:Type) (join,meet:C→C→C) : Prop \def
  { (* abelian semigroup properties *)
@@ -66,6 +38,11 @@ definition le \def λC:Type.λL:lattice C.λf,g. meet ? L f g = f.
 interpretation "Lattice le" 'leq a b =
  (cic:/matita/integration_algebras/le.con _ _ a b).
 
+definition lt \def λC:Type.λL:lattice C.λf,g. le ? L f g ∧ f ≠ g.
+
+interpretation "Lattice lt" 'lt a b =
+ (cic:/matita/integration_algebras/lt.con _ _ a b).
+
 definition carrier_of_lattice ≝
  λC:Type.λL:lattice C.C.
 
@@ -85,6 +62,33 @@ record riesz_space (K:ordered_field_ch0) : Type \def
 
 definition absolute_value \def λK.λS:riesz_space K.λf.join ? S f (-f).   
 
+(*CSC: qui la notazione non fa capire!!! *)
+definition is_riesz_norm ≝
+ λR:real.λV:riesz_space R.λnorm:norm ? V.
+  ∀f,g:V. le ? V (absolute_value ? V f) (absolute_value ? V g) →
+   of_le R (norm f) (norm g).
+
+record riesz_norm (R:real) (V:riesz_space R) : Type ≝
+ { rn_norm:> norm ? V;
+   rn_riesz_norm_property: is_riesz_norm ? ? rn_norm
+ }.
+
+(*CSC: non fa la chiusura delle coercion verso funclass *)
+definition rn_function ≝
+ λR:real.λV:riesz_space R.λnorm:riesz_norm ? V.
+  n_function ? ? (rn_norm ? ? norm).
+
+coercion cic:/matita/integration_algebras/rn_function.con 1.
+
+(************************** L-SPACES *************************************)
+
+record is_l_space (R:real) (V:riesz_space R) (norm:riesz_norm ? V) : Prop ≝
+ { ls_banach: is_complete ? V (induced_distance ? ? norm);
+   ls_linear: ∀f,g:V. le ? V 0 f → le ? V 0 g → norm (f+g) = norm f + norm g
+ }.
+
+(******************** ARCHIMEDEAN RIESZ SPACES ***************************)
+
 record is_archimedean_riesz_space (K) (S:riesz_space K) : Prop
 \def
   { ars_archimedean: ∃u.∀n.∀a.∀p:n > O.
@@ -113,7 +117,10 @@ definition is_weak_unit ≝
    2. Fremlin proves |x|/\u=0 \to u=0. How do we remove the absolute value?
  λR:real.λV:archimedean_riesz_space R.λunit: V.
   ∀x:V. meet x unit = 0 → u = 0.
-*) λR:real.λV:archimedean_riesz_space R.λe:V.True.
+  3. Fremlin proves u > 0 implies x /\ u > 0  > 0 for Archimedean spaces
+   only. We pick this definition for now.
+*) λR:real.λV:archimedean_riesz_space R.λe:V.
+    ∀v:V. lt ? V 0 v → lt ? V 0 (meet ? V v e).
 
 (* Here we are avoiding a construction (the quotient space to define
    f=g iff I(|f-g|)=0 *)
@@ -139,9 +146,101 @@ record integration_riesz_space (R:real) : Type \def
          ) * irs_unit))) 0;
    irs_quotient_space1:
     ∀f,g:irs_archimedean_riesz_space.
-     f=g → integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0
+     integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0 → f=g
  }.
 
+definition induced_norm_fun ≝
+ λR:real.λV:integration_riesz_space R.λf:V.
+  integral ? ? (absolute_value ? ? f).
+
+lemma induced_norm_is_norm:
+ ∀R:real.∀V:integration_riesz_space R.is_norm ? V (induced_norm_fun ? V).
+ intros;
+ apply mk_is_norm;
+  [ apply mk_is_semi_norm;
+     [ unfold induced_norm_fun;
+       intros;
+       apply i_positive;
+       [ apply (irs_integral_properties ? V)
+       | (* difficile *)
+         elim daemon
+       ]
+     | intros;
+       unfold induced_norm_fun;
+       (* facile *)
+       elim daemon
+     | intros;
+       unfold induced_norm_fun;
+       (* difficile *)
+       elim daemon
+     ]
+  | intros;
+    unfold induced_norm_fun in H;
+    apply irs_quotient_space1;
+    unfold minus;
+    rewrite < plus_comm;
+    rewrite < eq_zero_opp_zero;
+    rewrite > zero_neutral;
+    assumption
+  ].
+qed.
+
+definition induced_norm ≝
+ λR:real.λV:integration_riesz_space R.
+  mk_norm ? ? (induced_norm_fun ? V) (induced_norm_is_norm ? V).
+
+lemma is_riesz_norm_induced_norm:
+ ∀R:real.∀V:integration_riesz_space R.
+  is_riesz_norm ? ? (induced_norm ? V).
+ intros;
+ unfold is_riesz_norm;
+ intros;
+ unfold induced_norm;
+ simplify;
+ unfold induced_norm_fun;
+ (* difficile *)
+ elim daemon.
+qed.
+
+definition induced_riesz_norm ≝
+ λR:real.λV:integration_riesz_space R.
+  mk_riesz_norm ? ? (induced_norm ? V) (is_riesz_norm_induced_norm ? V).
+
+definition distance_induced_by_integral ≝
+ λR:real.λV:integration_riesz_space R.
+  induced_distance ? ? (induced_norm R V).
+
+definition is_complete_integration_riesz_space ≝
+ λR:real.λV:integration_riesz_space R.
+  is_complete ? ? (distance_induced_by_integral ? V).
+
+record complete_integration_riesz_space (R:real) : Type ≝
+ { cirz_integration_riesz_space:> integration_riesz_space R;
+   cirz_complete_integration_riesz_space_property:
+    is_complete_integration_riesz_space ? cirz_integration_riesz_space
+ }.
+
+(* now we prove that any complete integration riesz space is an L-space *)
+
+theorem is_l_space_l_space_induced_by_integral:
+ ∀R:real.∀V:complete_integration_riesz_space R.
+  is_l_space ? ? (induced_riesz_norm ? V).
+ intros;
+ constructor 1;
+  [ apply cirz_complete_integration_riesz_space_property
+  | intros;
+    unfold induced_riesz_norm;
+    simplify;
+    unfold induced_norm;
+    simplify;
+    unfold induced_norm_fun;
+    (* difficile *)
+    elim daemon
+  ].
+qed.
+
+(**************************** f-ALGEBRAS ********************************)
+
 record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
 ≝
  { (* ring properties *)
@@ -192,4 +291,4 @@ record integration_f_algebra (R:real) : Type \def
    ifa_f_algebra:>
     f_algebra ? ifa_integration_riesz_space
      (irs_unit ? ifa_integration_riesz_space)
- }.
\ No newline at end of file
+ }.