From 4a073df57b56bf6d89ec0f806d5471388940deda Mon Sep 17 00:00:00 2001
From: Enrico Zoli <??>
Date: Fri, 3 Nov 2006 16:58:51 +0000
Subject: [PATCH] Integration f_algebras declassed. We suppose to prove
 everything on integration_riesz_spaces. However, the definition or weak order
 unit is complex because of the sup on sequences.

---
 matita/dama/integration_algebras.ma | 115 ++++++++++++++++++----------
 1 file changed, 76 insertions(+), 39 deletions(-)

diff --git a/matita/dama/integration_algebras.ma b/matita/dama/integration_algebras.ma
index 2a2aae51d..1ebfad834 100644
--- a/matita/dama/integration_algebras.ma
+++ b/matita/dama/integration_algebras.ma
@@ -33,6 +33,20 @@ record vector_space (K:field): Type \def
 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:Type_OF_vector_space ? 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:Type_OF_vector_space ? V → R)
+ : Prop \def
+ { n_semi_norm:> is_semi_norm ? ? norm;
+   n_properness: ∀x:V. norm x = 0 → x = 0
+ }.
+
 record is_lattice (C:Type) (join,meet:C→C→C) : Prop \def
  { (* abelian semigroup properties *)
    l_comm_j: symmetric ? join;
@@ -90,6 +104,47 @@ 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:Type_OF_archimedean_riesz_space ? 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 (emult ? R 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.
+*) λR:real.λV:archimedean_riesz_space R.λe:V.True.
+
+record integration_riesz_space (R:real) : Type \def
+ { irs_archimedean_riesz_space:> archimedean_riesz_space R;
+   irs_unit: Type_OF_archimedean_riesz_space ? irs_archimedean_riesz_space;
+   irs_weak_unit: is_weak_unit ? ? irs_unit;
+   integral: Type_OF_archimedean_riesz_space ? irs_archimedean_riesz_space → R;
+   irs_integral_properties: is_integral R irs_archimedean_riesz_space 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.
+     f=g → integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0
+ }.
+
 record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
 ≝
  { (* ring properties *)
@@ -99,66 +154,48 @@ 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
  }.
 
 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:Type_OF_vector_space ? 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:Type_OF_archimedean_riesz_space ? 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).
-
-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)
- }.
+axiom symmetric_a_mult:
+ ∀K,R,one.∀A:f_algebra K R one. symmetric ? (a_mult ? ? ? A).
 
 (* 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
-- 
2.39.2