X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fdama%2Fintegration_algebras.ma;h=cbe629dac0ca6654fa5cca095a13317f4851ef3f;hb=f79567e3b0abcb508c94b66d69d967c4df83082a;hp=322e59a56f89f19fae55d4f4d8b9022f7f14cb66;hpb=87da9f97fb41575b986b9567265abef4e67d07af;p=helm.git

diff --git a/matita/dama/integration_algebras.ma b/matita/dama/integration_algebras.ma
index 322e59a56..cbe629dac 100644
--- a/matita/dama/integration_algebras.ma
+++ b/matita/dama/integration_algebras.ma
@@ -22,36 +22,87 @@ include "lattices.ma".
 record pre_riesz_space (K:ordered_field_ch0) : Type \def
  { rs_vector_space:> vector_space K;
    rs_lattice_: lattice;
-   rs_with: os_carrier rs_lattice_ = rs_vector_space
+   rs_ordered_abelian_group_: ordered_abelian_group;
+   rs_with1:
+    og_abelian_group rs_ordered_abelian_group_ = vs_abelian_group ? rs_vector_space;
+   rs_with2:
+    og_ordered_set rs_ordered_abelian_group_ = ordered_set_of_lattice rs_lattice_
  }.
 
-lemma rs_lattice: ∀K:ordered_field_ch0.pre_riesz_space K → lattice.
+lemma rs_lattice: ∀K.pre_riesz_space K → lattice.
  intros (K V);
- apply mk_lattice;
-  [ apply (carrier V) 
-  | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? (rs_with ? V));
-    apply l_join
-  | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? (rs_with ? V));
-    apply l_meet
-  | apply 
-     (eq_rect' ? ?
-      (λa:Type.λH:os_carrier (rs_lattice_ ? V)=a.
-       is_lattice a
-        (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C)
-          (l_join (rs_lattice_ K V)) a H)
-        (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C)
-          (l_meet (rs_lattice_ K V)) a H))
-      ? ? (rs_with ? V));
-    simplify;
-    apply l_lattice_properties
+ cut (os_carrier (rs_lattice_ ? V) = V);
+  [ apply mk_lattice;
+     [ apply (carrier V) 
+     | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? Hcut);
+       apply l_join
+     | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? Hcut);
+       apply l_meet
+     | apply 
+        (eq_rect' ? ?
+         (λa:Type.λH:os_carrier (rs_lattice_ ? V)=a.
+          is_lattice a
+           (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C)
+             (l_join (rs_lattice_ K V)) a H)
+           (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C)
+             (l_meet (rs_lattice_ K V)) a H))
+         ? ? Hcut);
+       simplify;
+       apply l_lattice_properties
+     ]
+  | transitivity (os_carrier (rs_ordered_abelian_group_ ? V));
+    [ apply (eq_f ? ? os_carrier);
+      symmetry;
+      apply rs_with2
+    | apply (eq_f ? ? carrier);
+      apply rs_with1
+    ]
   ].
 qed.
 
 coercion cic:/matita/integration_algebras/rs_lattice.con.
+ 
+lemma rs_ordered_abelian_group: ∀K.pre_riesz_space K → ordered_abelian_group.
+ intros (K V);
+ apply mk_ordered_abelian_group;
+  [ apply mk_pre_ordered_abelian_group;
+     [ apply (vs_abelian_group ? (rs_vector_space ? V))
+     | apply (ordered_set_of_lattice (rs_lattice ? V))
+     | reflexivity
+     ]
+  | simplify;
+    generalize in match
+     (og_ordered_abelian_group_properties (rs_ordered_abelian_group_ ? V));
+    intro P;
+    unfold in P;
+    elim daemon(*
+    apply
+     (eq_rect ? ?
+      (λO:ordered_set.
+        ∀f,g,h.
+         os_le O f g →
+          os_le O
+           (plus (abelian_group_OF_pre_riesz_space K V) f h)
+           (plus (abelian_group_OF_pre_riesz_space K V) g h))
+      ? ? (rs_with2 ? V));
+    apply
+     (eq_rect ? ?
+      (λG:abelian_group.
+        ∀f,g,h.
+         os_le (ordered_set_OF_pre_riesz_space K V) f g →
+          os_le (ordered_set_OF_pre_riesz_space K V)
+           (plus (abelian_group_OF_pre_riesz_space K V) f h)
+           (plus (abelian_group_OF_pre_riesz_space K V) g h))
+      ? ? (rs_with1 ? V));
+    simplify;
+    apply og_ordered_abelian_group_properties*)
+  ]
+qed.
+
+coercion cic:/matita/integration_algebras/rs_ordered_abelian_group.con.
 
 record is_riesz_space (K:ordered_field_ch0) (V:pre_riesz_space K) : Prop ≝
- { rs_compat_le_plus: ∀f,g,h:V. f≤g → f+h≤g+h;
-   rs_compat_le_times: ∀a:K.∀f:V. zero K≤a → zero V≤f → zero V≤a*f
+ { rs_compat_le_times: ∀a:K.∀f:V. 0≤a → 0≤f → 0≤a*f
  }.
 
 record riesz_space (K:ordered_field_ch0) : Type \def
@@ -60,7 +111,7 @@ record riesz_space (K:ordered_field_ch0) : Type \def
  }.
 
 record is_positive_linear (K) (V:riesz_space K) (T:V→K) : Prop ≝
- { positive: ∀u:V. os_le V 0 u → os_le K 0 (T u);
+ { positive: ∀u:V. 0≤u → 0≤T u;
    linear1: ∀u,v:V. T (u+v) = T u + T v;
    linear2: ∀u:V.∀k:K. T (k*u) = k*(T u)
  }.
@@ -71,14 +122,14 @@ record sequentially_order_continuous (K) (V:riesz_space K) (T:V→K) : Prop ≝
      is_increasing K (λn.T (a n)) ∧ tends_to ? (λn.T (a n)) (T l)
  }.
 
-definition absolute_value \def λK.λS:riesz_space K.λf.l_join S f (-f).   
+definition absolute_value ≝ λK.λS:riesz_space K.λf:S.f ∨ -f.   
 
 (**************** Normed Riesz spaces ****************************)
 
 definition is_riesz_norm ≝
  λR:real.λV:riesz_space R.λnorm:norm R V.
-  ∀f,g:V. os_le V (absolute_value ? V f) (absolute_value ? V g) →
-   os_le R (n_function R V norm f) (n_function R V norm g).
+  ∀f,g:V. absolute_value ? V f ≤ absolute_value ? V g →
+   n_function R V norm f ≤ n_function R V norm g.
 
 record riesz_norm (R:real) (V:riesz_space R) : Type ≝
  { rn_norm:> norm R V;
@@ -104,9 +155,8 @@ record is_l_space (R:real) (V:riesz_space R) (norm:riesz_norm ? V) : Prop ≝
 record is_archimedean_riesz_space (K) (S:riesz_space K) : Prop
 \def
   { ars_archimedean: ∃u:S.∀n.∀a.∀p:n > O.
-     os_le S
-      (absolute_value ? S a)
-      ((inv K (sum_field K n) (not_eq_sum_field_zero K n p))* u) →
+     absolute_value ? S a ≤
+      (inv K (sum_field K n) (not_eq_sum_field_zero K n p))* u →
      a = 0
   }.
 
@@ -125,7 +175,7 @@ definition is_weak_unit ≝
   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 (l_meet V v e).
+    ∀v:V. 0<v → 0 < (v ∧ e).
 
 (* Here we are avoiding a construction (the quotient space to define
    f=g iff I(|f-g|)=0 *)
@@ -138,14 +188,13 @@ record integration_riesz_space (R:real) : Type \def
    irs_limit1:
     ∀f:irs_archimedean_riesz_space.
      tends_to ?
-      (λn.integral (l_meet irs_archimedean_riesz_space f
-       ((sum_field R n)*irs_unit)))
+      (λn.integral (f ∧ ((sum_field R n)*irs_unit)))
        (integral f);
    irs_limit2:
     ∀f:irs_archimedean_riesz_space.
      tends_to ?
       (λn.
-        integral (l_meet irs_archimedean_riesz_space f
+        integral (f ∧
          ((inv ? (sum_field R (S n))
            (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))
          ) * irs_unit))) 0;
@@ -275,7 +324,7 @@ coercion cic:/matita/integration_algebras/ring_of_algebra.con.
 
 record pre_f_algebra (K:ordered_field_ch0) : Type ≝
  { fa_archimedean_riesz_space:> archimedean_riesz_space K;
-   fa_algebra_:> algebra K;
+   fa_algebra_: algebra K;
    fa_with: a_vector_space ? fa_algebra_ = rs_vector_space ? fa_archimedean_riesz_space
  }.
 
@@ -292,12 +341,9 @@ lemma fa_algebra: ∀K:ordered_field_ch0.pre_f_algebra K → algebra K.
 coercion cic:/matita/integration_algebras/fa_algebra.con.
 
 record is_f_algebra (K) (A:pre_f_algebra K) : Prop ≝ 
-{ compat_mult_le:
-   ∀f,g:A.
-    os_le A (zero A) f → os_le A (zero A) g → os_le A (zero A) (a_mult ? A f g);
+{ compat_mult_le: ∀f,g:A.0 ≤ f → 0 ≤ g → 0 ≤ f*g;
   compat_mult_meet:
-   ∀f,g,h:A.
-    l_meet A f g = (zero A) → l_meet A (a_mult ? A h f) g = (zero A)
+   ∀f,g,h:A.(f ∧ g) = 0 → ((h*f) ∧ g) = 0
 }.
 
 record f_algebra (K:ordered_field_ch0) : Type ≝ 
@@ -319,4 +365,4 @@ record integration_f_algebra (R:real) : Type \def
 
 axiom ifa_f_algebra: ∀R:real.integration_f_algebra R → f_algebra R.
 
-coercion cic:/matita/integration_algebras/ifa_f_algebra.con.
\ No newline at end of file
+coercion cic:/matita/integration_algebras/ifa_f_algebra.con.