X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fdama%2Fintegration_algebras.ma;h=cbe629dac0ca6654fa5cca095a13317f4851ef3f;hb=f2e7cb9757a151382ed3a70d1a610e8f729a6597;hp=534882ff2a54b31dbeef313f1740168959f1af92;hpb=08d8e4e422aafdc11e4230a87f2adee7facad809;p=helm.git

diff --git a/matita/dama/integration_algebras.ma b/matita/dama/integration_algebras.ma
index 534882ff2..cbe629dac 100644
--- a/matita/dama/integration_algebras.ma
+++ b/matita/dama/integration_algebras.ma
@@ -14,145 +14,149 @@
 
 set "baseuri" "cic:/matita/integration_algebras/".
 
-include "reals.ma".
+include "vector_spaces.ma".
+include "lattices.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
-}.
+(**************** Riesz Spaces ********************)
 
-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 pre_riesz_space (K:ordered_field_ch0) : Type \def
+ { rs_vector_space:> vector_space K;
+   rs_lattice_: lattice;
+   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_
  }.
 
-theorem eq_semi_norm_zero_zero:
- ∀R:real.∀V:vector_space R.∀semi_norm:V→R.
-  is_semi_norm ? ? semi_norm →
-   semi_norm 0 = 0.
- intros;
- (* facile *)
- elim daemon.
+lemma rs_lattice: ∀K.pre_riesz_space K → lattice.
+ intros (K V);
+ 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.
 
-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
- }.
+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.
 
-record is_semi_distance (R:real) (C:Type) (semi_d: C→C→R) : Prop \def
- { sd_positive: ∀x,y:C. 0 ≤ semi_d x y;
-   sd_properness: \forall x:C. semi_d x x = 0; 
-   sd_triangle_inequality: ∀x,y,z:C. semi_d x z ≤ semi_d x y + semi_d z y
- }.
+coercion cic:/matita/integration_algebras/rs_ordered_abelian_group.con.
 
-record is_distance (R:real) (C:Type) (d:C→C→R) : Prop \def
- { d_semi_distance:> is_semi_distance ? ? d;
-   d_properness: ∀x,y:C. d x y = 0 → x=y
+record is_riesz_space (K:ordered_field_ch0) (V:pre_riesz_space K) : Prop ≝
+ { rs_compat_le_times: ∀a:K.∀f:V. 0≤a → 0≤f → 0≤a*f
  }.
 
-definition induced_distance ≝
- λR:real.λV:vector_space R.λnorm:V→R.
-  λf,g:V.norm (f - g).
-
-theorem induced_distance_is_distance:
- ∀R:real.∀V:vector_space R.∀norm:V→R.
-  is_norm ? ? norm → is_distance ? ? (induced_distance ? ? norm).
- intros;
- apply mk_is_distance;
-  [ apply mk_is_semi_distance;
-    [ unfold induced_distance;
-      intros;
-      apply sn_positive;
-      apply n_semi_norm;
-      assumption
-    | unfold induced_distance;
-      intros;
-      unfold minus;
-      rewrite < plus_comm;
-      rewrite > opp_inverse;
-      apply eq_semi_norm_zero_zero;
-      apply n_semi_norm;
-      assumption
-    | unfold induced_distance;
-      intros;
-      (* ??? *)
-      elim daemon
-    ]
-  | unfold induced_distance;
-    intros;
-    generalize in match (n_properness ? ? ? H ? H1);
-    intro;
-    (* facile *)
-    elim daemon
-  ].
-qed.
+record riesz_space (K:ordered_field_ch0) : Type \def
+ { rs_pre_riesz_space:> pre_riesz_space K;
+   rs_riesz_space_properties: is_riesz_space ? rs_pre_riesz_space
+ }.
 
-record is_lattice (C:Type) (join,meet:C→C→C) : Prop \def
- { (* abelian semigroup properties *)
-   l_comm_j: symmetric ? join;
-   l_associative_j: associative ? join;
-   l_comm_m: symmetric ? meet;
-   l_associative_m: associative ? meet;
-   (* other properties *)
-   l_adsorb_j_m: ∀f,g. join f (meet f g) = f;
-   l_adsorb_m_j: ∀f,g. meet f (join f g) = f
+record is_positive_linear (K) (V:riesz_space K) (T:V→K) : Prop ≝
+ { 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)
  }.
 
-record lattice (C:Type) : Type \def
- { join: C → C → C;
-   meet: C → C → C;
-   l_lattice_properties: is_lattice ? join meet
+record sequentially_order_continuous (K) (V:riesz_space K) (T:V→K) : Prop ≝
+ { soc_incr:
+    ∀a:nat→V.∀l:V.is_increasing ? a → is_sup V a l →
+     is_increasing K (λn.T (a n)) ∧ tends_to ? (λn.T (a n)) (T l)
  }.
 
-definition le \def λC:Type.λL:lattice C.λf,g. meet ? L f g = f.
+definition absolute_value ≝ λK.λS:riesz_space K.λf:S.f ∨ -f.   
 
-interpretation "Lattice le" 'leq a b =
- (cic:/matita/integration_algebras/le.con _ _ a b).
+(**************** Normed Riesz spaces ****************************)
 
-definition lt \def λC:Type.λL:lattice C.λf,g. le ? L f g ∧ f ≠ g.
+definition is_riesz_norm ≝
+ λR:real.λV:riesz_space R.λnorm:norm R V.
+  ∀f,g:V. absolute_value ? V f ≤ absolute_value ? V g →
+   n_function R V norm f ≤ n_function R V norm g.
 
-interpretation "Lattice lt" 'lt a b =
- (cic:/matita/integration_algebras/lt.con _ _ a b).
+record riesz_norm (R:real) (V:riesz_space R) : Type ≝
+ { rn_norm:> norm R V;
+   rn_riesz_norm_property: is_riesz_norm ? ? rn_norm
+ }.
 
-definition carrier_of_lattice ≝
- λC:Type.λL:lattice C.C.
+(*CSC: non fa la chiusura delle coercion verso funclass *)
+definition rn_function ≝
+ λR:real.λV:riesz_space R.λnorm:riesz_norm ? V.
+  n_function R V (rn_norm ? ? norm).
 
-record is_riesz_space (K:ordered_field_ch0) (V:vector_space K)
- (L:lattice (Type_OF_vector_space ? V))
-: Prop
-\def
- { rs_compat_le_plus: ∀f,g,h. le ? L f g → le ? L (f+h) (g+h);
-   rs_compat_le_times: ∀a:K.∀f. of_le ? 0 a → le ? L 0 f → le ? L 0 (a*f)
- }.
+coercion cic:/matita/integration_algebras/rn_function.con 1.
 
-record riesz_space (K:ordered_field_ch0) : Type \def
- { rs_vector_space:> vector_space K;
-   rs_lattice:> lattice rs_vector_space;
-   rs_riesz_space_properties: is_riesz_space ? rs_vector_space rs_lattice
+(************************** 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
  }.
-
-definition absolute_value \def λK.λS:riesz_space K.λf.join ? S f (-f).   
+*)
+(******************** ARCHIMEDEAN RIESZ SPACES ***************************)
 
 record is_archimedean_riesz_space (K) (S:riesz_space K) : Prop
 \def
-  { ars_archimedean: ∃u.∀n.∀a.∀p:n > O.
-     le ? S
-      (absolute_value ? S a)
-      ((inv ? (sum_field K n) (not_eq_sum_field_zero ? n p))* u) →
+  { ars_archimedean: ∃u:S.∀n.∀a.∀p:n > O.
+     absolute_value ? S a ≤
+      (inv K (sum_field K n) (not_eq_sum_field_zero K n p))* u →
      a = 0
   }.
 
@@ -161,13 +165,6 @@ record archimedean_riesz_space (K:ordered_field_ch0) : Type \def
    ars_archimedean_property: is_archimedean_riesz_space ? ars_riesz_space
  }.
 
-record is_integral (K) (R:archimedean_riesz_space K) (I:R→K) : Prop
-\def
- { i_positive: ∀f:R. le ? R 0 f → of_le K 0 (I f);
-   i_linear1: ∀f,g:R. I (f + g) = I f + I g;
-   i_linear2: ∀f:R.∀k:K. I (k*f) = k*(I f)
- }.
-
 definition is_weak_unit ≝
 (* This definition is by Spitters. He cites Fremlin 353P, but:
    1. that theorem holds only in f-algebras (as in Spitters, but we are
@@ -178,7 +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 (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 *)
@@ -187,18 +184,17 @@ record integration_riesz_space (R:real) : Type \def
    irs_unit: irs_archimedean_riesz_space;
    irs_weak_unit: is_weak_unit ? ? irs_unit;
    integral: irs_archimedean_riesz_space → R;
-   irs_integral_properties: is_integral ? ? integral;
+   irs_positive_linear: is_positive_linear ? ? integral;
    irs_limit1:
     ∀f:irs_archimedean_riesz_space.
      tends_to ?
-      (λn.integral (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 (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;
@@ -207,55 +203,99 @@ record integration_riesz_space (R:real) : Type \def
      integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0 → f=g
  }.
 
-definition induced_norm ≝
+definition induced_norm_fun ≝
  λR:real.λV:integration_riesz_space R.λf:V.
-  integral ? ? (absolute_value ? ? f).
+  integral ? V (absolute_value ? ? f).
 
 lemma induced_norm_is_norm:
- ∀R:real.∀V:integration_riesz_space R.is_norm ? V (induced_norm ? V).
+ ∀R:real.∀V:integration_riesz_space R.is_norm R V (induced_norm_fun ? V).
+ elim daemon.(*
  intros;
  apply mk_is_norm;
   [ apply mk_is_semi_norm;
-     [ unfold induced_norm;
+     [ unfold induced_norm_fun;
        intros;
-       apply i_positive;
-       [ apply (irs_integral_properties ? V)
+       apply positive;
+       [ apply (irs_positive_linear ? V)
        | (* difficile *)
          elim daemon
        ]
      | intros;
-       unfold induced_norm;
+       unfold induced_norm_fun;
        (* facile *)
        elim daemon
      | intros;
-       unfold induced_norm;
+       unfold induced_norm_fun;
        (* difficile *)
        elim daemon
      ]
   | intros;
-    unfold induced_norm in H;
+    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 distance_induced_by_integral ≝
+definition induced_norm ≝
  λR:real.λV:integration_riesz_space R.
-  induced_distance ? ? (induced_norm R V).
+  mk_norm ? ? (induced_norm_fun ? V) (induced_norm_is_norm ? V).
 
-theorem distance_induced_by_integral_is_distance:
+lemma is_riesz_norm_induced_norm:
  ∀R:real.∀V:integration_riesz_space R.
-  is_distance ? ? (distance_induced_by_integral ? V).
+  is_riesz_norm ? ? (induced_norm ? V).
+ intros;
+ unfold is_riesz_norm;
  intros;
- unfold distance_induced_by_integral;
- apply induced_distance_is_distance;
- apply induced_norm_is_norm.
+ 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 *)
@@ -265,45 +305,64 @@ record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
    a_associative_right: ∀a,f,g. a * (mult f g) = mult f (a * g)
  }.
 
-record algebra (K: field) (V:vector_space K) (a_one:V) : Type \def
- { a_mult: V → V → V;
+record algebra (K: field) : Type \def
+ { a_vector_space:> vector_space K;
+   a_one: a_vector_space;
+   a_mult: a_vector_space → a_vector_space → a_vector_space;
    a_algebra_properties: is_algebra ? ? a_mult a_one
  }.
 
 interpretation "Algebra product" 'times a b =
- (cic:/matita/integration_algebras/a_mult.con _ _ _ a b).
+ (cic:/matita/integration_algebras/a_mult.con _ a b).
 
 definition ring_of_algebra ≝
- λK.λV:vector_space K.λone:V.λA:algebra ? V one.
-  mk_ring V (a_mult ? ? ? A) one
-   (a_ring ? ? ? ? (a_algebra_properties ? ? ? A)).
+ λK.λA:algebra K.
+  mk_ring A (a_mult ? A) (a_one ? A)
+   (a_ring ? ? ? ? (a_algebra_properties ? A)).
 
 coercion cic:/matita/integration_algebras/ring_of_algebra.con.
 
-record is_f_algebra (K) (S:archimedean_riesz_space K) (one: S)
- (A:algebra ? S one) : Prop
-\def 
-{ compat_mult_le:
-   ∀f,g:S.
-    le ? S 0 f → le ? S 0 g → le ? S 0 (a_mult ? ? ? A f g);
+record pre_f_algebra (K:ordered_field_ch0) : Type ≝
+ { fa_archimedean_riesz_space:> archimedean_riesz_space K;
+   fa_algebra_: algebra K;
+   fa_with: a_vector_space ? fa_algebra_ = rs_vector_space ? fa_archimedean_riesz_space
+ }.
+
+lemma fa_algebra: ∀K:ordered_field_ch0.pre_f_algebra K → algebra K.
+ intros (K A);
+ apply mk_algebra;
+  [ apply (rs_vector_space ? A)
+  | elim daemon
+  | elim daemon
+  | elim daemon
+  ]
+ qed.
+
+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.0 ≤ f → 0 ≤ g → 0 ≤ f*g;
   compat_mult_meet:
-   ∀f,g,h:S.
-    meet ? S f g = 0 → meet ? S (a_mult ? ? ? A h f) g = 0
+   ∀f,g,h:A.(f ∧ g) = 0 → ((h*f) ∧ g) = 0
 }.
 
-record f_algebra (K:ordered_field_ch0) (R:archimedean_riesz_space K) (one:R) :
-Type \def 
-{ fa_algebra:> algebra ? R one;
-  fa_f_algebra_properties: is_f_algebra ? ? ? fa_algebra
+record f_algebra (K:ordered_field_ch0) : Type ≝ 
+{ fa_pre_f_algebra:> pre_f_algebra K;
+  fa_f_algebra_properties: is_f_algebra ? fa_pre_f_algebra
 }.
 
 (* to be proved; see footnote 2 in the paper by Spitters *)
 axiom symmetric_a_mult:
- ∀K,R,one.∀A:f_algebra K R one. symmetric ? (a_mult ? ? ? A).
+ ∀K.∀A:f_algebra K. symmetric ? (a_mult ? A).
 
 record integration_f_algebra (R:real) : Type \def
  { ifa_integration_riesz_space:> integration_riesz_space R;
-   ifa_f_algebra:>
-    f_algebra ? ifa_integration_riesz_space
-     (irs_unit ? ifa_integration_riesz_space)
- }.
\ No newline at end of file
+   ifa_f_algebra_: f_algebra R;
+   ifa_with:
+    fa_archimedean_riesz_space ? ifa_f_algebra_ =
+    irs_archimedean_riesz_space ? ifa_integration_riesz_space
+ }.
+
+axiom ifa_f_algebra: ∀R:real.integration_f_algebra R → f_algebra R.
+
+coercion cic:/matita/integration_algebras/ifa_f_algebra.con.