X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fdama%2Fintegration_algebras.ma;h=534882ff2a54b31dbeef313f1740168959f1af92;hb=08d8e4e422aafdc11e4230a87f2adee7facad809;hp=b2fb189e9dbd35196e6c7ac7370bab0246ccca47;hpb=8b62b96fea74985e303e093d9b7ead91089c664e;p=helm.git

diff --git a/matita/dama/integration_algebras.ma b/matita/dama/integration_algebras.ma
index b2fb189e9..534882ff2 100644
--- a/matita/dama/integration_algebras.ma
+++ b/matita/dama/integration_algebras.ma
@@ -27,12 +27,80 @@ record is_vector_space (K: field) (G:abelian_group) (emult:K→G→G) : Prop
 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 K vs_abelian_group emult
+  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
+ }.
+
+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.
+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
+ }.
+
+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
+ }.
+
+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
+ }.
+
+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 is_lattice (C:Type) (join,meet:C→C→C) : Prop \def
  { (* abelian semigroup properties *)
    l_comm_j: symmetric ? join;
@@ -55,6 +123,11 @@ 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.
 
@@ -79,9 +152,7 @@ record is_archimedean_riesz_space (K) (S:riesz_space K) : Prop
   { ars_archimedean: ∃u.∀n.∀a.∀p:n > O.
      le ? S
       (absolute_value ? S a)
-      (emult ? S
-        (inv ? (sum_field K n) (not_eq_sum_field_zero ? n p))
-        u) →
+      ((inv ? (sum_field K n) (not_eq_sum_field_zero ? n p))* u) →
      a = 0
   }.
 
@@ -90,6 +161,101 @@ 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
+      defining it on Riesz spaces)
+   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.
+  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 *)
+record integration_riesz_space (R:real) : Type \def
+ { irs_archimedean_riesz_space:> archimedean_riesz_space R;
+   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_limit1:
+    ∀f:irs_archimedean_riesz_space.
+     tends_to ?
+      (λn.integral (meet ? irs_archimedean_riesz_space 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
+         ((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;
+   irs_quotient_space1:
+    ∀f,g:irs_archimedean_riesz_space.
+     integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0 → f=g
+ }.
+
+definition induced_norm ≝
+ 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 ? V).
+ intros;
+ apply mk_is_norm;
+  [ apply mk_is_semi_norm;
+     [ unfold induced_norm;
+       intros;
+       apply i_positive;
+       [ apply (irs_integral_properties ? V)
+       | (* difficile *)
+         elim daemon
+       ]
+     | intros;
+       unfold induced_norm;
+       (* facile *)
+       elim daemon
+     | intros;
+       unfold induced_norm;
+       (* difficile *)
+       elim daemon
+     ]
+  | intros;
+    unfold induced_norm 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 ≝
+ λR:real.λV:integration_riesz_space R.
+  induced_distance ? ? (induced_norm R V).
+
+theorem distance_induced_by_integral_is_distance:
+ ∀R:real.∀V:integration_riesz_space R.
+  is_distance ? ? (distance_induced_by_integral ? V).
+ intros;
+ unfold distance_induced_by_integral;
+ apply induced_distance_is_distance;
+ apply induced_norm_is_norm.
+qed.
+
 record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
 ≝
  { (* ring properties *)
@@ -99,80 +265,45 @@ 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) : Type \def
+record algebra (K: field) (V:vector_space K) (a_one:V) : Type \def
  { a_mult: V → V → V;
-   a_one: V;
-   a_algebra_properties: is_algebra K V a_mult a_one
+   a_algebra_properties: is_algebra ? ? a_mult a_one
  }.
 
 interpretation "Algebra product" 'times a b =
  (cic:/matita/integration_algebras/a_mult.con _ _ _ a b).
 
-interpretation "Algebra one" 'one =
- (cic:/matita/integration_algebras/a_one.con _ _ _).
-
 definition ring_of_algebra ≝
- λK.λV:vector_space K.λA:algebra ? V.
-  mk_ring V (a_mult ? ? A) (a_one ? ? A)
-   (a_ring ? ? ? ? (a_algebra_properties ? ? A)).
+ λK.λV:vector_space K.λone:V.λA:algebra ? V one.
+  mk_ring V (a_mult ? ? ? A) one
+   (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) (A:algebra ? S) : Prop
+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);
+    le ? S 0 f → le ? S 0 g → le ? S 0 (a_mult ? ? ? A f g);
   compat_mult_meet:
    ∀f,g,h:S.
-    meet ? S f g = 0 → meet ? S (a_mult ? ? A h f) g = 0
+    meet ? S f g = 0 → meet ? S (a_mult ? ? ? A h f) g = 0
 }.
 
-record f_algebra (K:ordered_field_ch0) : Type \def 
-{ fa_archimedean_riesz_space:> archimedean_riesz_space K;
-  fa_algebra:> algebra ? fa_archimedean_riesz_space;
-  fa_f_algebra_properties: is_f_algebra ? fa_archimedean_riesz_space fa_algebra
+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
 }.
 
 (* to be proved; see footnote 2 in the paper by Spitters *)
-axiom symmetric_a_mult: ∀K.∀A:f_algebra K. symmetric ? (a_mult ? ? A).
-
-
-definition tends_to : ∀F:ordered_field_ch0.∀f:nat→F.∀l:F.Prop.
- alias symbol "leq" = "Ordered field le".
- alias id "le" = "cic:/matita/nat/orders/le.ind#xpointer(1/1)".
- apply
-  (λF:ordered_field_ch0.λf:nat → F.λl:F.
-    ∀n:nat.∃m:nat.∀j:nat. le m j →
-     l - (inv F (sum_field F (S n)) ?) ≤ f j ∧
-     f j ≤ l + (inv F (sum_field F (S n)) ?));
- apply not_eq_sum_field_zero;
- unfold;
- auto new.
-qed.
+axiom symmetric_a_mult:
+ ∀K,R,one.∀A:f_algebra K R one. symmetric ? (a_mult ? ? ? A).
 
-record is_integral (K) (A:f_algebra K) (I:Type_OF_f_algebra ? A→K) : Prop
-\def
- { i_positive: ∀f:Type_OF_f_algebra ? A. le ? (lattice_OF_f_algebra ? A) 0 f → of_le K 0 (I f);
-   i_linear1: ∀f,g:Type_OF_f_algebra ? A. I (f + g) = I f + I g;
-   i_linear2: ∀f:A.∀k:K. I (emult ? A k f) = k*(I f)
- }.
-
-(* Here we are avoiding a construction (the quotient space to define
-   f=g iff I(|f-g|)=0 *)
-record is_integration_f_algebra (K) (A:f_algebra K) (I:Type_OF_f_algebra ? A→K) : Prop
-\def
- { ifa_integral: is_integral ? ? I;
-   ifa_limit1:
-    ∀f:A. tends_to ? (λn.I(meet ? A f ((sum_field K n)*(a_one ? ? A)))) (I f);
-   ifa_limit2:
-    ∀f:A.
-     tends_to ?
-      (λn.
-        I (meet ? A f
-         ((inv ? (sum_field K (S n))
-           (not_eq_sum_field_zero K (S n) (le_S_S O n (le_O_n n)))
-         ) * (a_one ? ? A)))) 0;
-   ifa_quotient_space1:
-    ∀f,g:A. f=g → I(absolute_value ? A (f - g)) = 0
- }.
+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