X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fdama%2Fintegration_algebras.ma;h=cbe629dac0ca6654fa5cca095a13317f4851ef3f;hb=8097af927e3e0f80c6dd2cb219b38f18e81df2a6;hp=94bc002b377d4d019d8463675c67c7f073a87005;hpb=aa863e3c45c682cd47445748275b04f91f35ef75;p=helm.git

diff --git a/matita/dama/integration_algebras.ma b/matita/dama/integration_algebras.ma
index 94bc002b3..cbe629dac 100644
--- a/matita/dama/integration_algebras.ma
+++ b/matita/dama/integration_algebras.ma
@@ -14,199 +14,355 @@
 
 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 left_inverse \def λC,op.λe:C.λinv:C→C. ∀x:C. op (inv x) x = e.
-
-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
- }.
+include "vector_spaces.ma".
+include "lattices.ma".
 
-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
-≝
- {  (* abelian group properties *)
-    abelian_group:> is_abelian_group ? plus zero opp;
-    (* abelian multiplicative semigroup properties *)
-    mult_assoc: associative ? mult;
-    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;
-    (* ring properties *)
-    mult_plus_distr: distributive ? mult plus;
-    (* integral domain *)
-    not_eq_zero_one: zero ≠ one
+(**************** 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_
  }.
 
-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)
+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.
+
+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.
+
+coercion cic:/matita/integration_algebras/rs_ordered_abelian_group.con.
 
-record field : Type \def
- { carrier:> Type;
-   plus: carrier → carrier → carrier;
-   mult: carrier → carrier → carrier;
-   zero: carrier;
-   one: carrier;
-   opp: carrier → carrier;
-   inv: ∀x:carrier. x ≠ zero → carrier;
-   field_properties: is_field ? plus mult zero one opp inv
+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 sum_field ≝
- λF:field. sum ? (plus F) (zero F) (one F).
- 
-notation "0" with precedence 89
-for @{ 'zero }.
+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
+ }.
 
-interpretation "Field zero" 'zero =
- (cic:/matita/integration_algebras/zero.con _).
+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 "1" with precedence 89
-for @{ 'one }.
+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 "Field one" 'one =
- (cic:/matita/integration_algebras/one.con _).
+definition absolute_value ≝ λK.λS:riesz_space K.λf:S.f ∨ -f.   
 
-interpretation "Field plus" 'plus a b =
- (cic:/matita/integration_algebras/plus.con _ a b).
+(**************** Normed Riesz spaces ****************************)
 
-interpretation "Field mult" 'times a b =
- (cic:/matita/integration_algebras/mult.con _ a b).
+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 "Field opp" 'uminus a =
- (cic:/matita/integration_algebras/opp.con _ a).
- 
-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 riesz_norm (R:real) (V:riesz_space R) : Type ≝
+ { rn_norm:> norm R V;
+   rn_riesz_norm_property: is_riesz_norm ? ? rn_norm
  }.
- 
-record ordered_field_ch0 : Type \def
- { of_field:> field;
-   of_le: of_field → of_field → Prop;
-   of_ordered_field_properties:>
-    is_ordered_field_ch0 ? (plus of_field) (mult of_field) (zero of_field)
-     (one of_field) (opp of_field) (inv of_field) of_le
+
+(*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).
+
+coercion cic:/matita/integration_algebras/rn_function.con 1.
+
+(************************** 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 ***************************)
 
-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.
+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
+  }.
 
-interpretation "Ordered field lt" 'lt a b =
- (cic:/matita/integration_algebras/lt.con _ a b).
+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
+ }.
 
-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.
+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
+ }.
 
-lemma le_x_zero_to_le_zero_opp_x: ∀F:ordered_field_ch0.∀x:F. x ≤ 0 → 0 ≤ -x.
+definition induced_norm_fun ≝
+ λR:real.λV:integration_riesz_space R.λf:V.
+  integral ? V (absolute_value ? ? f).
+
+lemma induced_norm_is_norm:
+ ∀R:real.∀V:integration_riesz_space R.is_norm R V (induced_norm_fun ? V).
+ elim daemon.(*
  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.
+ 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.
 
-(* To be proved for rings only *)
-lemma eq_mult_zero_x_zero: ∀F:ordered_field_ch0.∀x:F.0*x=0.
+definition induced_norm ≝
+ λR:real.λV:integration_riesz_space R.
+  mk_norm ? ? (induced_norm_fun ? V) (induced_norm_is_norm ? V).
+
+lemma is_riesz_norm_induced_norm:
+ ∀R:real.∀V:integration_riesz_space R.
+  is_riesz_norm ? ? (induced_norm ? V).
+ intros;
+ unfold is_riesz_norm;
  intros;
- generalize in match (zero_neutral ? ? ? ? F 0); intro;
- generalize in match (eq_f ? ? (λy.x*y) ? ? H); intro; clear H;
- rewrite > (mult_plus_distr ? ? ? ? ? ? ? F) in H1;
- generalize in match (eq_f ? ? (λy.-(x*0)+y) ? ? H1); intro; clear H1;
- rewrite < (plus_assoc ? ? ? ? F) in H;
- rewrite > (opp_inverse ? ? ? ? F) in H;
- rewrite > (zero_neutral ? ? ? ? F) in H;
- rewrite > (mult_comm ? ? ? ? ? ? ? F) in H;
- assumption.
+ unfold induced_norm;
+ simplify;
+ unfold induced_norm_fun;
+ (* difficile *)
+ elim daemon.
 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.
+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;
- 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;
-*)  
+ 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.*)
 
-axiom not_eq_sum_field_zero: ∀F,n. n > O → sum_field F n ≠ 0.
+(**************************** f-ALGEBRAS ********************************)
 
-record is_vector_space (K: field) (C:Type) (plus:C→C→C) (zero:C) (opp:C→C)
- (mult:K→C→C) : Prop
+record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
 ≝
- { (* abelian group properties *)
-   vs_abelian_group: is_abelian_group ? plus zero opp;
-   (* other properties *)
-   vs_nilpotent: ∀v. mult 0 v = zero;
-   vs_neutral: ∀v. mult 1 v = v;
-   vs_distributive: ∀a,b,v. mult (a + b) v = plus (mult a v) (mult b v);
-   vs_associative: ∀a,b,v. mult (a * b) v = mult a (mult b v)
+ { (* ring properties *)
+   a_ring: is_ring V mult one;
+   (* algebra properties *)
+   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 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
  }.
 
-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
+interpretation "Algebra product" 'times a b =
+ (cic:/matita/integration_algebras/a_mult.con _ a b).
+
+definition ring_of_algebra ≝
+ 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 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
  }.
 
-definition le \def λC.λmeet:C→C→C.λf,g. meet f g = f.
-
-record is_riesz_space (K:ordered_field_ch0) (C:Type) (plus:C→C→C) (zero:C)
- (opp:C→C) (mult:K→C→C) (join,meet:C→C→C) : Prop \def
- { (* vector space properties *)
-   rs_vector_space: is_vector_space K C plus zero opp mult;
-   (* 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 (mult a f)  
+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:A.(f ∧ g) = 0 → ((h*f) ∧ g) = 0
+}.
+
+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.∀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
  }.
 
-definition absolute_value \def λC:Type.λopp.λjoin:C→C→C.λf.join f (opp f).   
+axiom ifa_f_algebra: ∀R:real.integration_f_algebra R → f_algebra R.
 
-record is_archimedean_riesz_space (K:ordered_field_ch0) (C:Type) (plus:C→C→C)
- (zero:C) (opp:C→C) (mult:Type_OF_ordered_field_ch0 K→C→C) (join,meet:C→C→C) : Prop \def
-  { ars_riesz_space: is_riesz_space ? ? plus zero opp mult join meet;
-    ars_archimedean: ∃u.∀n,a.∀p:n > O.
-     le C meet (absolute_value ? opp join a)
-      (mult (inv K (sum_field K n) (not_eq_sum_field_zero K n p)) u) →
-     a = zero
-  }.
\ No newline at end of file
+coercion cic:/matita/integration_algebras/ifa_f_algebra.con.