X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fdama%2Fintegration_algebras.ma;h=534882ff2a54b31dbeef313f1740168959f1af92;hb=08d8e4e422aafdc11e4230a87f2adee7facad809;hp=c44d3e4822fa91fe15ed8fc9c708c9527946a2b2;hpb=9da5a5054b66ee9264ecccb2af43c2fce3b35e64;p=helm.git

diff --git a/matita/dama/integration_algebras.ma b/matita/dama/integration_algebras.ma
index c44d3e482..534882ff2 100644
--- a/matita/dama/integration_algebras.ma
+++ b/matita/dama/integration_algebras.ma
@@ -14,303 +14,296 @@
 
 set "baseuri" "cic:/matita/integration_algebras/".
 
-include "higher_order_defs/functions.ma".
-include "nat/nat.ma".
-include "nat/orders.ma".
+include "reals.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.
+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)
+ }.
 
-definition left_inverse \def λC,op.λe:C.λinv:C→C. ∀x:C. op (inv x) x = e.
+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
+}.
 
-definition right_inverse \def λC,op.λe:C.λ inv: C→ C. ∀x:C. op x (inv x)=e. 
+interpretation "Vector space external product" 'times a b =
+ (cic:/matita/integration_algebras/emult.con _ _ a b).
 
-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).
+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
+ }.
 
-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).
+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_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 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_ring (C:Type) (plus:C→C→C) (mult:C→C→C) (zero:C) (opp:C→C) : Prop
-≝
- { (* abelian group properties *)
-    abelian_group:> is_abelian_group ? plus zero opp;
-    (* multiplicative semigroup properties *)
-    mult_assoc: associative ? mult;
-    (* ring properties *)
-    mult_plus_distr_left: distributive_left C mult plus;
-    mult_plus_distr_right: distributive_right C mult plus
- }.
- 
-record ring : Type \def
- { r_carrier:> Type;
-   r_plus: r_carrier → r_carrier → r_carrier;
-   r_mult: r_carrier → r_carrier → r_carrier;
-   r_zero: r_carrier;
-   r_opp: r_carrier → r_carrier;
-   r_ring_properties:> is_ring ? r_plus r_mult r_zero r_opp
+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
  }.
 
-notation "0" with precedence 89
-for @{ 'zero }.
-
-interpretation "Ring zero" 'zero =
- (cic:/matita/integration_algebras/r_zero.con _).
-
-interpretation "Ring plus" 'plus a b =
- (cic:/matita/integration_algebras/r_plus.con _ a b).
-
-interpretation "Ring mult" 'times a b =
- (cic:/matita/integration_algebras/r_mult.con _ a b).
+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
+ }.
 
-interpretation "Ring opp" 'uminus a =
- (cic:/matita/integration_algebras/r_opp.con _ a).
+definition induced_distance ≝
+ λR:real.λV:vector_space R.λnorm:V→R.
+  λf,g:V.norm (f - g).
 
-lemma eq_mult_zero_x_zero: ∀R:ring.∀x:R.0*x=0.
+theorem induced_distance_is_distance:
+ ∀R:real.∀V:vector_space R.∀norm:V→R.
+  is_norm ? ? norm → is_distance ? ? (induced_distance ? ? norm).
  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 ? ? ? ? ? R) in H1;
- generalize in match (eq_f ? ? (λy.-(0*x)+y) ? ? H1); intro; clear H1;
- rewrite < (plus_assoc ? ? ? ? R) in H;
- rewrite > (opp_inverse ? ? ? ? R) in H;
- rewrite > (zero_neutral ? ? ? ? R) in H;
- assumption.
+ 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.
 
-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 ? ? ? ? ? R) in H1;
-generalize in match (eq_f ? ? (\lambda y. (-(x*0)) +y) ? ? H1);intro;
-clear H1;
-rewrite < (plus_assoc ? ? ? ? R) in H;
-rewrite > (opp_inverse ? ? ? ? R) in H;
-rewrite > (zero_neutral ? ? ? ? R) in H;
-assumption.
-
-
-record is_field (C:Type) (plus:C→C→C) (mult:C→C→C) (zero,one:C) (opp:C→C)
- (inv:∀x:C.x ≠ zero →C)  : Prop
-≝
- {  (* ring properties *)
-    ring_properties:> is_ring ? plus mult zero opp;
-    (* multiplicative abelian properties *)
-    mult_comm: symmetric ? mult;
-    (* multiplicative monoid properties *)
-    one_neutral: left_neutral ? mult one;
-    (* multiplicative group properties *)
-    inv_inverse: ∀x.∀p: x ≠ zero. mult (inv x p) x = one;
-    (* integral domain *)
-    not_eq_zero_one: zero ≠ one
+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
  }.
 
-lemma cancellationlaw: \forall R:ring. \forall x,y,z:R. 
-(x+y=x+z) \to (y=z). 
-intros;
-generalize in match (eq_f ? ? (\lambda a. (-x +a)) ? ? H);
-intros; clear H;
-rewrite < (plus_assoc ? ? ? ? R) in H1;
-rewrite < (plus_assoc ? ? ? ? R) in H1;
-rewrite > (opp_inverse ? ? ? ? R) in H1;
-rewrite > (zero_neutral ? ? ? ? R) in H1;
-rewrite > (zero_neutral ? ? ? ? R) in H1;
-assumption.
-qed.
-
+record lattice (C:Type) : Type \def
+ { join: C → C → C;
+   meet: C → C → C;
+   l_lattice_properties: is_lattice ? join meet
+ }.
 
-lemma opp_opp: \forall R:ring. \forall x:R. (-(-x))=x.
-intros; 
-apply (cancellationlaw ? (-x) ? ?); 
-rewrite  > (opp_inverse ? ? ? ? R (x)); 
-rewrite > (plus_comm ? ? ? ? R);
-rewrite > (opp_inverse ? ? ? ? R);
-reflexivity.
-qed.
+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).
 
-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)
-  ].
+definition carrier_of_lattice ≝
+ λC:Type.λL:lattice C.C.
 
-record field : Type \def
- { f_ring:> ring;
-   one: f_ring;
-   inv: ∀x:f_ring. x ≠ 0 → f_ring;
-   field_properties:>
-    is_field ? (r_plus f_ring) (r_mult f_ring) (r_zero f_ring) one
-     (r_opp f_ring) inv
+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)
  }.
 
-definition sum_field ≝
- λF:field. sum ? (r_plus F) (r_zero F) (one F).
- 
-notation "1" with precedence 89
-for @{ 'one }.
-
-interpretation "Field one" 'one =
- (cic:/matita/integration_algebras/one.con _).
-
-record is_ordered_field_ch0 (C:Type) (plus,mult:C→C→C) (zero,one:C) (opp:C→C)
- (inv:∀x:C.x ≠ zero → C) (le:C→C→Prop) : Prop \def
- { (* field properties *)
-   of_is_field:> is_field C plus mult zero one opp inv;
-   of_mult_compat: ∀a,b. le zero a → le zero b → le zero (mult a b);
-   of_plus_compat: ∀a,b,c. le a b → le (plus a c) (plus b c);
-   of_weak_tricotomy : ∀a,b. a≠b → le a b ∨ le b a;
-   (* 0 characteristics *)
-   of_char0: ∀n. n > O → sum ? plus zero one n ≠ zero
- }.
- 
-record ordered_field_ch0 : Type \def
- { of_field:> field;
-   of_le: of_field → of_field → Prop;
-   of_ordered_field_properties:>
-    is_ordered_field_ch0 ? (r_plus of_field) (r_mult of_field) (r_zero of_field)
-     (one of_field) (r_opp of_field) (inv of_field) of_le
+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
  }.
 
-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;
-*)  
+definition absolute_value \def λK.λS:riesz_space K.λf.join ? S f (-f).   
 
-axiom not_eq_sum_field_zero: ∀F,n. n > O → sum_field F n ≠ 0.
+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) →
+     a = 0
+  }.
 
-record is_vector_space (K: field) (C:Type) (plus:C→C→C) (zero:C) (opp:C→C)
- (emult:K→C→C) : Prop
-≝
- { (* abelian group properties *)
-   vs_abelian_group: is_abelian_group ? plus zero opp;
-   (* other properties *)
-   vs_nilpotent: ∀v. emult 0 v = zero;
-   vs_neutral: ∀v. emult 1 v = v;
-   vs_distributive: ∀a,b,v. emult (a + b) v = plus (emult a v) (emult b v);
-   vs_associative: ∀a,b,v. emult (a * b) v = emult a (emult b v)
+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
  }.
 
-record vector_space : Type \def
-{vs_ :
-
+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)
+ }.
 
-}
-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
+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
  }.
 
-(* This should be a let-in field of the riesz_space!!! *)
-definition le_ \def λC.λmeet:C→C→C.λf,g. meet f g = f.
+definition induced_norm ≝
+ λR:real.λV:integration_riesz_space R.λf:V.
+  integral ? ? (absolute_value ? ? f).
 
-record is_riesz_space (K:ordered_field_ch0) (C:Type) (plus:C→C→C) (zero:C)
- (opp:C→C) (emult:K→C→C) (join,meet:C→C→C) : Prop \def
- { (* vector space properties *)
-   rs_vector_space: is_vector_space K C plus zero opp emult;
-   (* lattice properties *)
-   rs_lattice: is_lattice C join meet;
-   (* other properties *)
-   rs_compat_le_plus: ∀f,g,h. le_ ? meet f g → le_ ? meet (plus f h) (plus g h);
-   rs_compat_le_times: ∀a,f. 0≤a → le_ ? meet zero f → le_ ? meet zero (emult a f)  
- }.
- 
-definition absolute_value \def λC:Type.λopp.λjoin:C→C→C.λf.join f (opp f).   
-
-record is_archimedean_riesz_space (K:ordered_field_ch0) (C:Type) (plus:C→C→C)
- (zero:C) (opp:C→C) (emult:K→C→C) (join,meet:C→C→C)
- :Prop \def
-  { ars_riesz_space: is_riesz_space ? ? plus zero opp emult join meet;
-    ars_archimedean: ∃u.∀n,a.∀p:n > O.
-     le_ C meet (absolute_value ? opp join a)
-      (emult (inv K (sum_field K n) (not_eq_sum_field_zero K n p)) u) →
-     a = zero
-  }.
+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) (C:Type) (plus:C→C→C) (zero:C) (opp:C→C)
- (emult:K→C→C) (mult:C→C→C) : Prop
+record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
 ≝
- { (* vector space properties *)
-   a_vector_space_properties: is_vector_space ? ? plus zero opp emult;
-   (* ring properties *)
-   a_ring: is_ring ? plus mult zero opp;
+ { (* ring properties *)
+   a_ring: is_ring V mult one;
    (* algebra properties *)
-   a_associative_left: ∀a,f,g. emult a (mult f g) = mult (emult a f) g;
-   a_associative_right: ∀a,f,g. emult a (mult f g) = mult f (emult a g)
+   a_associative_left: ∀a,f,g. a * (mult f g) = mult (a * f) g;
+   a_associative_right: ∀a,f,g. a * (mult f g) = mult f (a * g)
  }.
- 
- 
-record is_f_algebra (K: ordered_field_ch0) (C:Type) (plus: C\to C \to C) 
-(zero:C) (opp: C \to C) (emult: Type_OF_ordered_field_ch0 K\to C\to C) (mult: C\to C\to C) 
-(join,meet: C\to C\to C) : Prop
+
+record algebra (K: field) (V:vector_space K) (a_one:V) : Type \def
+ { a_mult: V → V → V;
+   a_algebra_properties: is_algebra ? ? a_mult a_one
+ }.
+
+interpretation "Algebra product" 'times 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)).
+
+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 
-{ archimedean_riesz_properties:> is_archimedean_riesz_space K C
- plus zero opp emult join meet ;          
-algebra_properties:> is_algebra ? ? plus zero opp emult mult;
-compat_mult_le: \forall f,g: C. le_ ? meet zero f \to le_ ? meet zero g \to
- le_ ? meet zero (mult f g);
-compat_mult_meet: \forall f,g,h. meet f g = zero \to meet (mult h f) g = zero
+{ compat_mult_le:
+   ∀f,g:S.
+    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
+}.
+
+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 : Type \def 
-{
+(* 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).
 
-}
+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)
+ }.
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