X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fdama%2Fintegration_algebras.ma;h=cbe629dac0ca6654fa5cca095a13317f4851ef3f;hb=656cb8eb95d1fa723cbd45c8ab4069764aa539a8;hp=0339a088929b0b5b6850770930867bab35b2f009;hpb=1238984f493ee077187fcbe1f6ca416da9d43bd7;p=helm.git

diff --git a/matita/dama/integration_algebras.ma b/matita/dama/integration_algebras.ma
index 0339a0889..cbe629dac 100644
--- a/matita/dama/integration_algebras.ma
+++ b/matita/dama/integration_algebras.ma
@@ -14,347 +14,287 @@
 
 set "baseuri" "cic:/matita/integration_algebras/".
 
-include "higher_order_defs/functions.ma".
-include "nat/nat.ma".
-include "nat/orders.ma".
-
-definition left_neutral \def λC,op.λe:C. ∀x:C. op e x = x.
-
-definition right_neutral \def λC,op. λe:C. ∀x:C. op x e=x.
+include "vector_spaces.ma".
+include "lattices.ma".
+
+(**************** Riesz Spaces ********************)
+
+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_
+ }.
 
-definition left_inverse \def λC,op.λe:C.λinv:C→C. ∀x:C. op (inv x) x = e.
+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.
 
-definition right_inverse \def λC,op.λe:C.λ inv: C→ C. ∀x:C. op x (inv x)=e. 
+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.
 
-definition distributive_left ≝
- λA:Type.λf:A→A→A.λg:A→A→A.
-  ∀x,y,z. f x (g y z) = g (f x y) (f x z).
+coercion cic:/matita/integration_algebras/rs_ordered_abelian_group.con.
 
-definition distributive_right ≝
- λA:Type.λf:A→A→A.λg:A→A→A.
-  ∀x,y,z. f (g x y) z = g (f x z) (f y z).
+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
+ }.
 
-record is_abelian_group (C:Type) (plus:C→C→C) (zero:C) (opp:C→C) : Prop \def
- { (* abelian additive semigroup properties *)
-    plus_assoc_: associative ? plus;
-    plus_comm_: symmetric ? plus;
-    (* additive monoid properties *)
-    zero_neutral_: left_neutral ? plus zero;
-    (* additive group properties *)
-    opp_inverse_: left_inverse ? plus zero opp
+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 abelian_group : Type \def
- { carrier:> Type;
-   plus: carrier → carrier → carrier;
-   zero: carrier;
-   opp: carrier → carrier;
-   ag_abelian_group_properties: is_abelian_group ? plus zero opp
+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)
  }.
 
-notation "0" with precedence 89
-for @{ 'zero }.
+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)
+ }.
 
-interpretation "Ring zero" 'zero =
- (cic:/matita/integration_algebras/zero.con _).
+definition absolute_value ≝ λK.λS:riesz_space K.λf:S.f ∨ -f.   
 
-interpretation "Ring plus" 'plus a b =
- (cic:/matita/integration_algebras/plus.con _ a b).
+(**************** Normed Riesz spaces ****************************)
 
-interpretation "Ring opp" 'uminus a =
- (cic:/matita/integration_algebras/opp.con _ a).
- 
-theorem plus_assoc: ∀G:abelian_group. associative ? (plus G).
- intro;
- apply (plus_assoc_ ? ? ? ? (ag_abelian_group_properties G)).
-qed.
+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.
 
-theorem plus_comm: ∀G:abelian_group. symmetric ? (plus G).
- intro;
- apply (plus_comm_ ? ? ? ? (ag_abelian_group_properties G)).
-qed.
+record riesz_norm (R:real) (V:riesz_space R) : Type ≝
+ { rn_norm:> norm R V;
+   rn_riesz_norm_property: is_riesz_norm ? ? rn_norm
+ }.
 
-theorem zero_neutral: ∀G:abelian_group. left_neutral ? (plus G) 0.
- intro;
- apply (zero_neutral_ ? ? ? ? (ag_abelian_group_properties G)).
-qed.
+(*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).
 
-theorem opp_inverse: ∀G:abelian_group. left_inverse ? (plus G) 0 (opp G).
- intro;
- apply (opp_inverse_ ? ? ? ? (ag_abelian_group_properties G)).
-qed.
+coercion cic:/matita/integration_algebras/rn_function.con 1.
 
-lemma cancellationlaw: ∀G:abelian_group.∀x,y,z:G. x+y=x+z → y=z. 
-intros;
-generalize in match (eq_f ? ? (λa.-x +a) ? ? H);
-intros; clear H;
-rewrite < plus_assoc in H1;
-rewrite < plus_assoc in H1;
-rewrite > opp_inverse in H1;
-rewrite > zero_neutral in H1;
-rewrite > zero_neutral in H1;
-assumption.
-qed.
+(************************** 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 ***************************)
 
-(****************************** rings *********************************)
+record is_archimedean_riesz_space (K) (S:riesz_space K) : Prop
+\def
+  { 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
+  }.
 
-record is_ring (G:abelian_group) (mult:G→G→G) (one:G) : Prop
-≝
- {  (* multiplicative monoid properties *)
-    mult_assoc_: associative ? mult;
-    one_neutral_left_: left_neutral ? mult one;
-    one_neutral_right_: right_neutral ? mult one;
-    (* ring properties *)
-    mult_plus_distr_left_: distributive_left ? mult (plus G);
-    mult_plus_distr_right_: distributive_right ? mult (plus G);
-    not_eq_zero_one_: (0 ≠ one)
- }.
- 
-record ring : Type \def
- { r_abelian_group:> abelian_group;
-   mult: r_abelian_group → r_abelian_group → r_abelian_group;
-   one: r_abelian_group;
-   r_ring_properties: is_ring r_abelian_group mult one
+record archimedean_riesz_space (K:ordered_field_ch0) : Type \def
+ { ars_riesz_space:> riesz_space K;
+   ars_archimedean_property: is_archimedean_riesz_space ? ars_riesz_space
  }.
 
-theorem mult_assoc: ∀R:ring.associative ? (mult R).
- intros;
- apply (mult_assoc_ ? ? ? (r_ring_properties R)).
-qed.
+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. 0<v → 0 < (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_positive_linear: is_positive_linear ? ? integral;
+   irs_limit1:
+    ∀f:irs_archimedean_riesz_space.
+     tends_to ?
+      (λn.integral (f ∧ ((sum_field R n)*irs_unit)))
+       (integral f);
+   irs_limit2:
+    ∀f:irs_archimedean_riesz_space.
+     tends_to ?
+      (λn.
+        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;
+   irs_quotient_space1:
+    ∀f,g:irs_archimedean_riesz_space.
+     integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0 → f=g
+ }.
 
-theorem one_neutral_left: ∀R:ring.left_neutral ? (mult R) (one R).
- intros;
- apply (one_neutral_left_ ? ? ? (r_ring_properties R)).
-qed.
+definition induced_norm_fun ≝
+ λR:real.λV:integration_riesz_space R.λf:V.
+  integral ? V (absolute_value ? ? f).
 
-theorem one_neutral_right: ∀R:ring.right_neutral ? (mult R) (one R).
+lemma induced_norm_is_norm:
+ ∀R:real.∀V:integration_riesz_space R.is_norm R V (induced_norm_fun ? V).
+ elim daemon.(*
  intros;
- apply (one_neutral_right_ ? ? ? (r_ring_properties R)).
+ apply mk_is_norm;
+  [ apply mk_is_semi_norm;
+     [ unfold induced_norm_fun;
+       intros;
+       apply positive;
+       [ apply (irs_positive_linear ? 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.
 
-theorem mult_plus_distr_left: ∀R:ring.distributive_left ? (mult R) (plus R).
- intros;
- apply (mult_plus_distr_left_ ? ? ? (r_ring_properties R)).
-qed.
+definition induced_norm ≝
+ λR:real.λV:integration_riesz_space R.
+  mk_norm ? ? (induced_norm_fun ? V) (induced_norm_is_norm ? V).
 
-theorem mult_plus_distr_right: ∀R:ring.distributive_right ? (mult R) (plus R).
+lemma is_riesz_norm_induced_norm:
+ ∀R:real.∀V:integration_riesz_space R.
+  is_riesz_norm ? ? (induced_norm ? V).
  intros;
- apply (mult_plus_distr_right_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem not_eq_zero_one: ∀R:ring.0 ≠ one R.
+ unfold is_riesz_norm;
  intros;
- apply (not_eq_zero_one_ ? ? ? (r_ring_properties R)).
+ unfold induced_norm;
+ simplify;
+ unfold induced_norm_fun;
+ (* difficile *)
+ elim daemon.
 qed.
 
-interpretation "Ring mult" 'times a b =
- (cic:/matita/integration_algebras/mult.con _ a b).
+definition induced_riesz_norm ≝
+ λR:real.λV:integration_riesz_space R.
+  mk_riesz_norm ? ? (induced_norm ? V) (is_riesz_norm_induced_norm ? V).
 
-notation "1" with precedence 89
-for @{ 'one }.
-
-interpretation "Field one" 'one =
- (cic:/matita/integration_algebras/one.con _).
-
-lemma eq_mult_zero_x_zero: ∀R:ring.∀x:R.0*x=0.
- intros;
- generalize in match (zero_neutral R 0); intro;
- generalize in match (eq_f ? ? (λy.y*x) ? ? H); intro; clear H;
- rewrite > mult_plus_distr_right in H1;
- generalize in match (eq_f ? ? (λy.-(0*x)+y) ? ? H1); intro; clear H1;
- rewrite < plus_assoc in H;
- rewrite > opp_inverse in H;
- rewrite > zero_neutral in H;
- assumption.
-qed.
+definition distance_induced_by_integral ≝
+ λR:real.λV:integration_riesz_space R.
+  induced_distance ? ? (induced_norm R V).
 
-lemma eq_mult_x_zero_zero: ∀R:ring.∀x:R.x*0=0.
-intros;
-generalize in match (zero_neutral R 0);
-intro;
-generalize in match (eq_f ? ? (\lambda y.x*y) ? ? H); intro; clear H;
-rewrite > mult_plus_distr_left in H1;
-generalize in match (eq_f ? ? (\lambda y. (-(x*0)) +y) ? ? H1);intro;
-clear H1;
-rewrite < plus_assoc in H;
-rewrite > opp_inverse in H;
-rewrite > zero_neutral in H;
-assumption.
-qed.
+definition is_complete_integration_riesz_space ≝
+ λR:real.λV:integration_riesz_space R.
+  is_complete ? ? (distance_induced_by_integral ? V).
 
-record is_field (R:ring) (inv:∀x:R.x ≠ 0 → R) : Prop
-≝
- {  (* multiplicative abelian properties *)
-    mult_comm_: symmetric ? (mult R);
-    (* multiplicative group properties *)
-    inv_inverse_: ∀x.∀p: x ≠ 0. mult ? (inv x p) x = 1
+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
  }.
 
-lemma opp_opp: \forall R:ring. \forall x:R. (-(-x))=x.
-intros; 
-apply (cancellationlaw ? (-x) ? ?); 
-rewrite > (opp_inverse R x); 
-rewrite > plus_comm;
-rewrite > opp_inverse;
-reflexivity.
-qed.
-
+(* now we prove that any complete integration riesz space is an L-space *)
 
-let rec sum (C:Type) (plus:C→C→C) (zero,one:C) (n:nat) on n  ≝
- match n with
-  [ O ⇒ zero
-  | (S m) ⇒ plus one (sum C plus zero one m)
+(*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
   ].
-
-record field : Type \def
- { f_ring:> ring;
-   inv: ∀x:f_ring. x ≠ 0 → f_ring;
-   field_properties: is_field f_ring inv
- }.
- 
-theorem mult_comm: ∀F:field.symmetric ? (mult F).
- intro;
- apply (mult_comm_ ? ? (field_properties F)).
-qed.
-
-theorem inv_inverse: ∀F:field.∀x.∀p: x ≠ 0. mult ? (inv F x p) x = 1.
- intro;
- apply (inv_inverse_ ? ? (field_properties F)).
-qed.
-
-definition sum_field ≝
- λF:field. sum ? (plus F) (zero F) (one F).
- 
-record is_ordered_field_ch0 (F:field) (le:F→F→Prop) : Prop \def
- { of_mult_compat: ∀a,b. le 0 a → le 0 b → le 0 (a*b);
-   of_plus_compat: ∀a,b,c. le a b → le (a+c) (b+c);
-   of_weak_tricotomy : ∀a,b. a≠b → le a b ∨ le b a;
-   (* 0 characteristics *)
-   of_char0: ∀n. n > O → sum ? (plus F) 0 1 n ≠ 0
- }.
- 
-record ordered_field_ch0 : Type \def
- { of_field:> field;
-   of_le: of_field → of_field → Prop;
-   of_ordered_field_properties:> is_ordered_field_ch0 of_field of_le
- }.
-
-interpretation "Ordered field le" 'leq a b =
- (cic:/matita/integration_algebras/of_le.con _ a b).
- 
-definition lt \def λF:ordered_field_ch0.λa,b:F.a ≤ b ∧ a ≠ b.
-
-interpretation "Ordered field lt" 'lt a b =
- (cic:/matita/integration_algebras/lt.con _ a b).
-
-(*incompleto
-lemma le_zero_x_to_le_opp_x_zero: ∀F:ordered_field_ch0.∀x:F. 0 ≤ x → -x ≤ 0.
-intros;
- generalize in match (of_plus_compat ? ? ? ? ? ? ? ? F ? ? (-x) H); intro;
- rewrite > (zero_neutral ? ? ? ? F) in H1;
- rewrite > (plus_comm  ? ? ?  ? F) in H1;
- rewrite > (opp_inverse ? ? ? ? F) in H1;
- 
- assumption.
-qed.*)
-
-axiom le_x_zero_to_le_zero_opp_x: ∀F:ordered_field_ch0.∀x:F. x ≤ 0 → 0 ≤ -x.
-(* intros;
- generalize in match (of_plus_compat ? ? ? ? ? ? ? ? F ? ? (-x) H); intro;
- rewrite > (zero_neutral ? ? ? ? F) in H1;
- rewrite > (plus_comm ? ? ? ? F) in H1;
- rewrite > (opp_inverse ? ? ? ? F) in H1;
- assumption.
 qed.*)
 
-(*
-lemma eq_opp_x_times_opp_one_x: ∀F:ordered_field_ch0.∀x:F.-x = -1*x.
- intros;
- 
-lemma not_eq_x_zero_to_lt_zero_mult_x_x:
- ∀F:ordered_field_ch0.∀x:F. x ≠ 0 → 0 < x * x.
- intros;
- elim (of_weak_tricotomy ? ? ? ? ? ? ? ? F ? ? H);
-  [ generalize in match (le_x_zero_to_le_zero_opp_x F ? H1); intro;
-    generalize in match (of_mult_compat ? ? ? ? ? ? ? ?  F ? ? H2 H2); intro;
-*)  
-
-axiom not_eq_sum_field_zero: ∀F,n. n > O → sum_field F n ≠ 0.
-
-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 K vs_abelian_group emult
-}.
-
-interpretation "Vector space external product" 'times a b =
- (cic:/matita/integration_algebras/emult.con _ _ a b).
-
-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 lattice (C:Type) : Type \def
- { join: C → C → C;
-   meet: C → C → C;
-   l_lattice_properties: is_lattice ? join meet
- }.
-
-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 carrier_of_lattice ≝
- λC:Type.λL:lattice C.C.
-
-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)
- }.
-
-record riesz_space : Type \def
- { rs_ordered_field_ch0: ordered_field_ch0;
-   rs_vector_space:> vector_space rs_ordered_field_ch0;
-   rs_lattice:> lattice rs_vector_space;
-   rs_riesz_space_properties: is_riesz_space ? rs_vector_space rs_lattice
- }.
-
-definition absolute_value \def λS:riesz_space.λf.join ? S f (-f).   
-
-record is_archimedean_riesz_space (S:riesz_space) : Prop
-\def
-  { ars_archimedean: ∃u.∀n.∀a.∀p:n > O.
-     le ? S
-      (absolute_value S a)
-      (emult ? S
-        (inv ? (sum_field (rs_ordered_field_ch0 S) n) (not_eq_sum_field_zero ? n p))
-        u) →
-     a = 0
-  }.
-
-record archimedean_riesz_space : Type \def
- { ars_riesz_space:> riesz_space;
-   ars_archimedean_property: is_archimedean_riesz_space ars_riesz_space
- }.
+(**************************** f-ALGEBRAS ********************************)
 
 record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
 ≝
@@ -365,41 +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) : Type \def
- { a_mult: V → V → V;
-   a_one: V;
-   a_algebra_properties: is_algebra K V a_mult a_one
+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).
-
-interpretation "Field one" 'one =
- (cic:/matita/integration_algebras/a_one.con _).
+ (cic:/matita/integration_algebras/a_mult.con _ a b).
 
 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.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 (S:archimedean_riesz_space)
- (A:algebra (rs_ordered_field_ch0 (ars_riesz_space S)) S) : 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 : Type \def 
-{ fa_archimedean_riesz_space:> archimedean_riesz_space;
-  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) : 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: ∀A:f_algebra. symmetric ? (a_mult ? ? A).
+axiom symmetric_a_mult:
+ ∀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 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.