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 pre_riesz_space (K:ordered_field_ch0) : Type \def
23 { rs_vector_space:> vector_space K;
25 rs_with: os_carrier rs_lattice_ = rs_vector_space
28 lemma rs_lattice: ∀K:ordered_field_ch0.pre_riesz_space K → lattice.
32 | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? (rs_with ? V));
34 | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? (rs_with ? V));
38 (λa:Type.λH:os_carrier (rs_lattice_ ? V)=a.
40 (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C)
41 (l_join (rs_lattice_ K V)) a H)
42 (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C)
43 (l_meet (rs_lattice_ K V)) a H))
46 apply l_lattice_properties
50 coercion cic:/matita/integration_algebras/rs_lattice.con.
52 record is_riesz_space (K:ordered_field_ch0) (V:pre_riesz_space K) : Prop ≝
53 { rs_compat_le_plus: ∀f,g,h:V. f≤g → f+h≤g+h;
54 rs_compat_le_times: ∀a:K.∀f:V. zero K≤a → zero V≤f → zero V≤a*f
57 record riesz_space (K:ordered_field_ch0) : Type \def
58 { rs_pre_riesz_space:> pre_riesz_space K;
59 rs_riesz_space_properties: is_riesz_space ? rs_pre_riesz_space
62 record is_positive_linear (K) (V:riesz_space K) (T:V→K) : Prop ≝
63 { positive: ∀u:V. (0:carrier V)≤u → (0:carrier K)≤T u;
64 linear1: ∀u,v:V. T (u+v) = T u + T v;
65 linear2: ∀u:V.∀k:K. T (k*u) = k*(T u)
68 record sequentially_order_continuous (K) (V:riesz_space K) (T:V→K) : Prop ≝
70 ∀a:nat→V.∀l:V.is_increasing ? a → is_sup V a l →
71 is_increasing K (λn.T (a n)) ∧ tends_to ? (λn.T (a n)) (T l)
74 definition absolute_value \def λK.λS:riesz_space K.λf.l_join S f (-f).
76 (**************** Normed Riesz spaces ****************************)
78 definition is_riesz_norm ≝
79 λR:real.λV:riesz_space R.λnorm:norm R V.
80 ∀f,g:V. absolute_value ? V f ≤ absolute_value ? V g →
81 n_function R V norm f ≤ n_function R V norm g.
83 record riesz_norm (R:real) (V:riesz_space R) : Type ≝
85 rn_riesz_norm_property: is_riesz_norm ? ? rn_norm
88 (*CSC: non fa la chiusura delle coercion verso funclass *)
89 definition rn_function ≝
90 λR:real.λV:riesz_space R.λnorm:riesz_norm ? V.
91 n_function R V (rn_norm ? ? norm).
93 coercion cic:/matita/integration_algebras/rn_function.con 1.
95 (************************** L-SPACES *************************************)
97 record is_l_space (R:real) (V:riesz_space R) (norm:riesz_norm ? V) : Prop ≝
98 { ls_banach: is_complete ? V (induced_distance ? ? norm);
99 ls_linear: ∀f,g:V. le ? V 0 f → le ? V 0 g → norm (f+g) = norm f + norm g
102 (******************** ARCHIMEDEAN RIESZ SPACES ***************************)
104 record is_archimedean_riesz_space (K) (S:riesz_space K) : Prop
106 { ars_archimedean: ∃u:S.∀n.∀a.∀p:n > O.
107 absolute_value ? S a ≤
108 (inv K (sum_field K n) (not_eq_sum_field_zero K n p))* u →
112 record archimedean_riesz_space (K:ordered_field_ch0) : Type \def
113 { ars_riesz_space:> riesz_space K;
114 ars_archimedean_property: is_archimedean_riesz_space ? ars_riesz_space
117 definition is_weak_unit ≝
118 (* This definition is by Spitters. He cites Fremlin 353P, but:
119 1. that theorem holds only in f-algebras (as in Spitters, but we are
120 defining it on Riesz spaces)
121 2. Fremlin proves |x|/\u=0 \to u=0. How do we remove the absolute value?
122 λR:real.λV:archimedean_riesz_space R.λunit: V.
123 ∀x:V. meet x unit = 0 → u = 0.
124 3. Fremlin proves u > 0 implies x /\ u > 0 > 0 for Archimedean spaces
125 only. We pick this definition for now.
126 *) λR:real.λV:archimedean_riesz_space R.λe:V.
127 ∀v:V. lt V 0 v → lt V 0 (l_meet V v e).
129 (* Here we are avoiding a construction (the quotient space to define
130 f=g iff I(|f-g|)=0 *)
131 record integration_riesz_space (R:real) : Type \def
132 { irs_archimedean_riesz_space:> archimedean_riesz_space R;
133 irs_unit: irs_archimedean_riesz_space;
134 irs_weak_unit: is_weak_unit ? ? irs_unit;
135 integral: irs_archimedean_riesz_space → R;
136 irs_positive_linear: is_positive_linear ? ? integral;
138 ∀f:irs_archimedean_riesz_space.
140 (λn.integral (l_meet irs_archimedean_riesz_space f
141 ((sum_field R n)*irs_unit)))
144 ∀f:irs_archimedean_riesz_space.
147 integral (l_meet irs_archimedean_riesz_space f
148 ((inv ? (sum_field R (S n))
149 (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))
152 ∀f,g:irs_archimedean_riesz_space.
153 integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0 → f=g
156 definition induced_norm_fun ≝
157 λR:real.λV:integration_riesz_space R.λf:V.
158 integral ? V (absolute_value ? ? f).
160 lemma induced_norm_is_norm:
161 ∀R:real.∀V:integration_riesz_space R.is_norm R V (induced_norm_fun ? V).
165 [ apply mk_is_semi_norm;
166 [ unfold induced_norm_fun;
169 [ apply (irs_positive_linear ? V)
174 unfold induced_norm_fun;
178 unfold induced_norm_fun;
183 unfold induced_norm_fun in H;
184 apply irs_quotient_space1;
187 rewrite < eq_zero_opp_zero;
188 rewrite > zero_neutral;
193 definition induced_norm ≝
194 λR:real.λV:integration_riesz_space R.
195 mk_norm ? ? (induced_norm_fun ? V) (induced_norm_is_norm ? V).
197 lemma is_riesz_norm_induced_norm:
198 ∀R:real.∀V:integration_riesz_space R.
199 is_riesz_norm ? ? (induced_norm ? V).
201 unfold is_riesz_norm;
205 unfold induced_norm_fun;
210 definition induced_riesz_norm ≝
211 λR:real.λV:integration_riesz_space R.
212 mk_riesz_norm ? ? (induced_norm ? V) (is_riesz_norm_induced_norm ? V).
214 definition distance_induced_by_integral ≝
215 λR:real.λV:integration_riesz_space R.
216 induced_distance ? ? (induced_norm R V).
218 definition is_complete_integration_riesz_space ≝
219 λR:real.λV:integration_riesz_space R.
220 is_complete ? ? (distance_induced_by_integral ? V).
222 record complete_integration_riesz_space (R:real) : Type ≝
223 { cirz_integration_riesz_space:> integration_riesz_space R;
224 cirz_complete_integration_riesz_space_property:
225 is_complete_integration_riesz_space ? cirz_integration_riesz_space
228 (* now we prove that any complete integration riesz space is an L-space *)
230 (*theorem is_l_space_l_space_induced_by_integral:
231 ∀R:real.∀V:complete_integration_riesz_space R.
232 is_l_space ? ? (induced_riesz_norm ? V).
235 [ apply cirz_complete_integration_riesz_space_property
237 unfold induced_riesz_norm;
241 unfold induced_norm_fun;
247 (**************************** f-ALGEBRAS ********************************)
249 record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
251 { (* ring properties *)
252 a_ring: is_ring V mult one;
253 (* algebra properties *)
254 a_associative_left: ∀a,f,g. a * (mult f g) = mult (a * f) g;
255 a_associative_right: ∀a,f,g. a * (mult f g) = mult f (a * g)
258 record algebra (K: field) : Type \def
259 { a_vector_space:> vector_space K;
260 a_one: a_vector_space;
261 a_mult: a_vector_space → a_vector_space → a_vector_space;
262 a_algebra_properties: is_algebra ? ? a_mult a_one
265 interpretation "Algebra product" 'times a b =
266 (cic:/matita/integration_algebras/a_mult.con _ a b).
268 definition ring_of_algebra ≝
270 mk_ring A (a_mult ? A) (a_one ? A)
271 (a_ring ? ? ? ? (a_algebra_properties ? A)).
273 coercion cic:/matita/integration_algebras/ring_of_algebra.con.
275 record pre_f_algebra (K:ordered_field_ch0) : Type ≝
276 { fa_archimedean_riesz_space:> archimedean_riesz_space K;
277 fa_algebra_:> algebra K;
278 fa_with: a_vector_space ? fa_algebra_ = rs_vector_space ? fa_archimedean_riesz_space
281 lemma fa_algebra: ∀K:ordered_field_ch0.pre_f_algebra K → algebra K.
284 [ apply (rs_vector_space ? A)
291 coercion cic:/matita/integration_algebras/fa_algebra.con.
293 record is_f_algebra (K) (A:pre_f_algebra K) : Prop ≝
296 zero A ≤ f → zero A ≤ g → zero A ≤ a_mult ? A f g;
299 l_meet A f g = (zero A) → l_meet A (a_mult ? A h f) g = (zero A)
302 record f_algebra (K:ordered_field_ch0) : Type ≝
303 { fa_pre_f_algebra:> pre_f_algebra K;
304 fa_f_algebra_properties: is_f_algebra ? fa_pre_f_algebra
307 (* to be proved; see footnote 2 in the paper by Spitters *)
308 axiom symmetric_a_mult:
309 ∀K.∀A:f_algebra K. symmetric ? (a_mult ? A).
311 record integration_f_algebra (R:real) : Type \def
312 { ifa_integration_riesz_space:> integration_riesz_space R;
313 ifa_f_algebra_: f_algebra R;
315 fa_archimedean_riesz_space ? ifa_f_algebra_ =
316 irs_archimedean_riesz_space ? ifa_integration_riesz_space
319 axiom ifa_f_algebra: ∀R:real.integration_f_algebra R → f_algebra R.
321 coercion cic:/matita/integration_algebras/ifa_f_algebra.con.