1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/integration_algebras/".
17 include "vector_spaces.ma".
18 include "lattices.ma".
20 (**************** Riesz Spaces ********************)
22 record is_riesz_space (K:ordered_field_ch0) (V:vector_space K)
26 { rs_compat_le_plus: ∀f,g,h:V. os_le ? L f g → os_le ? L (f+h) (g+h);
27 rs_compat_le_times: ∀a:K.∀f:V. zero K≤a → os_le ? L (zero V) f → os_le ? L (zero V) (a*f)
30 record riesz_space (K:ordered_field_ch0) : Type \def
31 { rs_vector_space:> vector_space K;
32 rs_lattice:> lattice rs_vector_space;
33 rs_riesz_space_properties: is_riesz_space ? rs_vector_space rs_lattice
36 record is_positive_linear (K) (V:riesz_space K) (T:V→K) : Prop ≝
37 { positive: ∀u:V. os_le ? V 0 u → os_le ? K 0 (T u);
38 linear1: ∀u,v:V. T (u+v) = T u + T v;
39 linear2: ∀u:V.∀k:K. T (k*u) = k*(T u)
42 record sequentially_order_continuous (K) (V:riesz_space K) (T:V→K) : Prop ≝
44 ∀a:nat→V.∀l:V.is_increasing ? ? a → is_sup ? V a l →
45 is_increasing ? K (λn.T (a n)) ∧ tends_to ? (λn.T (a n)) (T l)
48 definition absolute_value \def λK.λS:riesz_space K.λf.join ? S f (-f).
50 (**************** Normed Riesz spaces ****************************)
52 definition is_riesz_norm ≝
53 λR:real.λV:riesz_space R.λnorm:norm R V.
54 ∀f,g:V. os_le ? V (absolute_value ? V f) (absolute_value ? V g) →
55 os_le ? R (n_function R V norm f) (n_function R V norm g).
57 record riesz_norm (R:real) (V:riesz_space R) : Type ≝
59 rn_riesz_norm_property: is_riesz_norm ? ? rn_norm
62 (*CSC: non fa la chiusura delle coercion verso funclass *)
63 definition rn_function ≝
64 λR:real.λV:riesz_space R.λnorm:riesz_norm ? V.
65 n_function R V (rn_norm ? ? norm).
67 coercion cic:/matita/integration_algebras/rn_function.con 1.
69 (************************** L-SPACES *************************************)
71 record is_l_space (R:real) (V:riesz_space R) (norm:riesz_norm ? V) : Prop ≝
72 { ls_banach: is_complete ? V (induced_distance ? ? norm);
73 ls_linear: ∀f,g:V. le ? V 0 f → le ? V 0 g → norm (f+g) = norm f + norm g
76 (******************** ARCHIMEDEAN RIESZ SPACES ***************************)
78 record is_archimedean_riesz_space (K) (S:riesz_space K) : Prop
80 { ars_archimedean: ∃u:S.∀n.∀a.∀p:n > O.
82 (absolute_value ? S a)
83 ((inv K (sum_field K n) (not_eq_sum_field_zero K n p))* u) →
87 record archimedean_riesz_space (K:ordered_field_ch0) : Type \def
88 { ars_riesz_space:> riesz_space K;
89 ars_archimedean_property: is_archimedean_riesz_space ? ars_riesz_space
92 definition is_weak_unit ≝
93 (* This definition is by Spitters. He cites Fremlin 353P, but:
94 1. that theorem holds only in f-algebras (as in Spitters, but we are
95 defining it on Riesz spaces)
96 2. Fremlin proves |x|/\u=0 \to u=0. How do we remove the absolute value?
97 λR:real.λV:archimedean_riesz_space R.λunit: V.
98 ∀x:V. meet x unit = 0 → u = 0.
99 3. Fremlin proves u > 0 implies x /\ u > 0 > 0 for Archimedean spaces
100 only. We pick this definition for now.
101 *) λR:real.λV:archimedean_riesz_space R.λe:V.
102 ∀v:V. lt ? V 0 v → lt ? V 0 (meet ? V v e).
104 (* Here we are avoiding a construction (the quotient space to define
105 f=g iff I(|f-g|)=0 *)
106 record integration_riesz_space (R:real) : Type \def
107 { irs_archimedean_riesz_space:> archimedean_riesz_space R;
108 irs_unit: irs_archimedean_riesz_space;
109 irs_weak_unit: is_weak_unit ? ? irs_unit;
110 integral: irs_archimedean_riesz_space → R;
111 irs_positive_linear: is_positive_linear ? ? integral;
113 ∀f:irs_archimedean_riesz_space.
115 (λn.integral (meet ? irs_archimedean_riesz_space f
116 ((sum_field R n)*irs_unit)))
119 ∀f:irs_archimedean_riesz_space.
122 integral (meet ? irs_archimedean_riesz_space f
123 ((inv ? (sum_field R (S n))
124 (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))
127 ∀f,g:irs_archimedean_riesz_space.
128 integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0 → f=g
131 definition induced_norm_fun ≝
132 λR:real.λV:integration_riesz_space R.λf:V.
133 integral ? V (absolute_value ? ? f).
135 lemma induced_norm_is_norm:
136 ∀R:real.∀V:integration_riesz_space R.is_norm R V (induced_norm_fun ? V).
140 [ apply mk_is_semi_norm;
141 [ unfold induced_norm_fun;
144 [ apply (irs_positive_linear ? V)
149 unfold induced_norm_fun;
153 unfold induced_norm_fun;
158 unfold induced_norm_fun in H;
159 apply irs_quotient_space1;
162 rewrite < eq_zero_opp_zero;
163 rewrite > zero_neutral;
168 definition induced_norm ≝
169 λR:real.λV:integration_riesz_space R.
170 mk_norm ? ? (induced_norm_fun ? V) (induced_norm_is_norm ? V).
172 lemma is_riesz_norm_induced_norm:
173 ∀R:real.∀V:integration_riesz_space R.
174 is_riesz_norm ? ? (induced_norm ? V).
176 unfold is_riesz_norm;
180 unfold induced_norm_fun;
185 definition induced_riesz_norm ≝
186 λR:real.λV:integration_riesz_space R.
187 mk_riesz_norm ? ? (induced_norm ? V) (is_riesz_norm_induced_norm ? V).
189 definition distance_induced_by_integral ≝
190 λR:real.λV:integration_riesz_space R.
191 induced_distance ? ? (induced_norm R V).
193 definition is_complete_integration_riesz_space ≝
194 λR:real.λV:integration_riesz_space R.
195 is_complete ? ? (distance_induced_by_integral ? V).
197 record complete_integration_riesz_space (R:real) : Type ≝
198 { cirz_integration_riesz_space:> integration_riesz_space R;
199 cirz_complete_integration_riesz_space_property:
200 is_complete_integration_riesz_space ? cirz_integration_riesz_space
203 (* now we prove that any complete integration riesz space is an L-space *)
205 (*theorem is_l_space_l_space_induced_by_integral:
206 ∀R:real.∀V:complete_integration_riesz_space R.
207 is_l_space ? ? (induced_riesz_norm ? V).
210 [ apply cirz_complete_integration_riesz_space_property
212 unfold induced_riesz_norm;
216 unfold induced_norm_fun;
222 (**************************** f-ALGEBRAS ********************************)
224 record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
226 { (* ring properties *)
227 a_ring: is_ring V mult one;
228 (* algebra properties *)
229 a_associative_left: ∀a,f,g. a * (mult f g) = mult (a * f) g;
230 a_associative_right: ∀a,f,g. a * (mult f g) = mult f (a * g)
233 record algebra (K: field) (V:vector_space K) (a_one:V) : Type \def
235 a_algebra_properties: is_algebra ? ? a_mult a_one
238 interpretation "Algebra product" 'times a b =
239 (cic:/matita/integration_algebras/a_mult.con _ _ _ a b).
241 definition ring_of_algebra ≝
242 λK.λV:vector_space K.λone:V.λA:algebra ? V one.
243 mk_ring V (a_mult ? ? ? A) one
244 (a_ring ? ? ? ? (a_algebra_properties ? ? ? A)).
246 coercion cic:/matita/integration_algebras/ring_of_algebra.con.
248 record is_f_algebra (K) (S:archimedean_riesz_space K) (one: S)
249 (A:algebra ? S one) : Prop
253 le ? S 0 f → le ? S 0 g → le ? S 0 (a_mult ? ? ? A f g);
256 meet ? S f g = 0 → meet ? S (a_mult ? ? ? A h f) g = 0
259 record f_algebra (K:ordered_field_ch0) (R:archimedean_riesz_space K) (one:R) :
261 { fa_algebra:> algebra ? R one;
262 fa_f_algebra_properties: is_f_algebra ? ? ? fa_algebra
265 (* to be proved; see footnote 2 in the paper by Spitters *)
266 axiom symmetric_a_mult:
267 ∀K,R,one.∀A:f_algebra K R one. symmetric ? (a_mult ? ? ? A).
269 record integration_f_algebra (R:real) : Type \def
270 { ifa_integration_riesz_space:> integration_riesz_space R;
272 f_algebra ? ifa_integration_riesz_space
273 (irs_unit ? ifa_integration_riesz_space)