X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fdama%2Fintegration_algebras.ma;h=1ebfad8344ec62f45b73933bad58ce4fd1810a04;hb=d69878e5f8e69bc0fb71b10a14938803f7f87369;hp=0339a088929b0b5b6850770930867bab35b2f009;hpb=40c2f7eaa04e0baae6933f87260abaa7c2f78dd1;p=helm.git

diff --git a/helm/software/matita/dama/integration_algebras.ma b/helm/software/matita/dama/integration_algebras.ma
index 0339a0889..1ebfad834 100644
--- a/helm/software/matita/dama/integration_algebras.ma
+++ b/helm/software/matita/dama/integration_algebras.ma
@@ -14,272 +14,7 @@
 
 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.
-
-definition left_inverse \def λC,op.λe:C.λinv:C→C. ∀x:C. op (inv x) x = e.
-
-definition right_inverse \def λC,op.λe:C.λ inv: C→ C. ∀x:C. op x (inv x)=e. 
-
-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).
-
-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_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 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
- }.
-
-notation "0" with precedence 89
-for @{ 'zero }.
-
-interpretation "Ring zero" 'zero =
- (cic:/matita/integration_algebras/zero.con _).
-
-interpretation "Ring plus" 'plus a b =
- (cic:/matita/integration_algebras/plus.con _ a b).
-
-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.
-
-theorem plus_comm: ∀G:abelian_group. symmetric ? (plus G).
- intro;
- apply (plus_comm_ ? ? ? ? (ag_abelian_group_properties G)).
-qed.
-
-theorem zero_neutral: ∀G:abelian_group. left_neutral ? (plus G) 0.
- intro;
- apply (zero_neutral_ ? ? ? ? (ag_abelian_group_properties G)).
-qed.
-
-theorem opp_inverse: ∀G:abelian_group. left_inverse ? (plus G) 0 (opp G).
- intro;
- apply (opp_inverse_ ? ? ? ? (ag_abelian_group_properties G)).
-qed.
-
-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.
-
-(****************************** rings *********************************)
-
-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
- }.
-
-theorem mult_assoc: ∀R:ring.associative ? (mult R).
- intros;
- apply (mult_assoc_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem one_neutral_left: ∀R:ring.left_neutral ? (mult R) (one R).
- intros;
- apply (one_neutral_left_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem one_neutral_right: ∀R:ring.right_neutral ? (mult R) (one R).
- intros;
- apply (one_neutral_right_ ? ? ? (r_ring_properties R)).
-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.
-
-theorem mult_plus_distr_right: ∀R:ring.distributive_right ? (mult R) (plus R).
- intros;
- apply (mult_plus_distr_right_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem not_eq_zero_one: ∀R:ring.0 ≠ one R.
- intros;
- apply (not_eq_zero_one_ ? ? ? (r_ring_properties R)).
-qed.
-
-interpretation "Ring mult" 'times a b =
- (cic:/matita/integration_algebras/mult.con _ a b).
-
-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.
-
-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.
-
-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
- }.
-
-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.
-
-
-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)
-  ].
-
-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.
+include "reals.ma".
 
 record is_vector_space (K: field) (G:abelian_group) (emult:K→G→G) : Prop
 ≝
@@ -298,6 +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;
@@ -331,29 +80,69 @@ record is_riesz_space (K:ordered_field_ch0) (V:vector_space K)
    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;
+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
  }.
 
-definition absolute_value \def λS:riesz_space.λf.join ? S f (-f).   
+definition absolute_value \def λK.λS:riesz_space K.λf.join ? S f (-f).   
 
-record is_archimedean_riesz_space (S:riesz_space) : Prop
+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)
+      (absolute_value ? S a)
       (emult ? S
-        (inv ? (sum_field (rs_ordered_field_ch0 S) n) (not_eq_sum_field_zero ? n p))
+        (inv ? (sum_field K 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
+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 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
@@ -365,41 +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 "Field 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 (S:archimedean_riesz_space)
- (A:algebra (rs_ordered_field_ch0 (ars_riesz_space S)) 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 : 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) (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: ∀A:f_algebra. symmetric ? (a_mult ? ? A).
+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 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