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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 *)
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15 set "baseuri" "cic:/matita/integration_algebras/".
17 include "vector_spaces.ma".
19 record is_lattice (C:Type) (join,meet:C→C→C) : Prop \def
20 { (* abelian semigroup properties *)
21 l_comm_j: symmetric ? join;
22 l_associative_j: associative ? join;
23 l_comm_m: symmetric ? meet;
24 l_associative_m: associative ? meet;
25 (* other properties *)
26 l_adsorb_j_m: ∀f,g. join f (meet f g) = f;
27 l_adsorb_m_j: ∀f,g. meet f (join f g) = f
30 record lattice (C:Type) : Type \def
33 l_lattice_properties: is_lattice ? join meet
36 definition le \def λC:Type.λL:lattice C.λf,g. meet ? L f g = f.
38 interpretation "Lattice le" 'leq a b =
39 (cic:/matita/integration_algebras/le.con _ _ a b).
41 definition lt \def λC:Type.λL:lattice C.λf,g. le ? L f g ∧ f ≠ g.
43 interpretation "Lattice lt" 'lt a b =
44 (cic:/matita/integration_algebras/lt.con _ _ a b).
46 definition carrier_of_lattice ≝
47 λC:Type.λL:lattice C.C.
49 record is_riesz_space (K:ordered_field_ch0) (V:vector_space K)
50 (L:lattice (Type_OF_vector_space ? V))
53 { rs_compat_le_plus: ∀f,g,h. le ? L f g → le ? L (f+h) (g+h);
54 rs_compat_le_times: ∀a:K.∀f. of_le ? 0 a → le ? L 0 f → le ? L 0 (a*f)
57 record riesz_space (K:ordered_field_ch0) : Type \def
58 { rs_vector_space:> vector_space K;
59 rs_lattice:> lattice rs_vector_space;
60 rs_riesz_space_properties: is_riesz_space ? rs_vector_space rs_lattice
63 definition absolute_value \def λK.λS:riesz_space K.λf.join ? S f (-f).
65 (*CSC: qui la notazione non fa capire!!! *)
66 definition is_riesz_norm ≝
67 λR:real.λV:riesz_space R.λnorm:norm ? V.
68 ∀f,g:V. le ? V (absolute_value ? V f) (absolute_value ? V g) →
69 of_le R (norm f) (norm g).
71 record riesz_norm (R:real) (V:riesz_space R) : Type ≝
73 rn_riesz_norm_property: is_riesz_norm ? ? rn_norm
76 (*CSC: non fa la chiusura delle coercion verso funclass *)
77 definition rn_function ≝
78 λR:real.λV:riesz_space R.λnorm:riesz_norm ? V.
79 n_function ? ? (rn_norm ? ? norm).
81 coercion cic:/matita/integration_algebras/rn_function.con 1.
83 (************************** L-SPACES *************************************)
85 record is_l_space (R:real) (V:riesz_space R) (norm:riesz_norm ? V) : Prop ≝
86 { ls_banach: is_complete ? V (induced_distance ? ? norm);
87 ls_linear: ∀f,g:V. le ? V 0 f → le ? V 0 g → norm (f+g) = norm f + norm g
90 (******************** ARCHIMEDEAN RIESZ SPACES ***************************)
92 record is_archimedean_riesz_space (K) (S:riesz_space K) : Prop
94 { ars_archimedean: ∃u.∀n.∀a.∀p:n > O.
96 (absolute_value ? S a)
97 ((inv ? (sum_field K n) (not_eq_sum_field_zero ? n p))* u) →
101 record archimedean_riesz_space (K:ordered_field_ch0) : Type \def
102 { ars_riesz_space:> riesz_space K;
103 ars_archimedean_property: is_archimedean_riesz_space ? ars_riesz_space
106 record is_integral (K) (R:archimedean_riesz_space K) (I:R→K) : Prop
108 { i_positive: ∀f:R. le ? R 0 f → of_le K 0 (I f);
109 i_linear1: ∀f,g:R. I (f + g) = I f + I g;
110 i_linear2: ∀f:R.∀k:K. I (k*f) = k*(I f)
113 definition is_weak_unit ≝
114 (* This definition is by Spitters. He cites Fremlin 353P, but:
115 1. that theorem holds only in f-algebras (as in Spitters, but we are
116 defining it on Riesz spaces)
117 2. Fremlin proves |x|/\u=0 \to u=0. How do we remove the absolute value?
118 λR:real.λV:archimedean_riesz_space R.λunit: V.
119 ∀x:V. meet x unit = 0 → u = 0.
120 3. Fremlin proves u > 0 implies x /\ u > 0 > 0 for Archimedean spaces
121 only. We pick this definition for now.
122 *) λR:real.λV:archimedean_riesz_space R.λe:V.
123 ∀v:V. lt ? V 0 v → lt ? V 0 (meet ? V v e).
125 (* Here we are avoiding a construction (the quotient space to define
126 f=g iff I(|f-g|)=0 *)
127 record integration_riesz_space (R:real) : Type \def
128 { irs_archimedean_riesz_space:> archimedean_riesz_space R;
129 irs_unit: irs_archimedean_riesz_space;
130 irs_weak_unit: is_weak_unit ? ? irs_unit;
131 integral: irs_archimedean_riesz_space → R;
132 irs_integral_properties: is_integral ? ? integral;
134 ∀f:irs_archimedean_riesz_space.
136 (λn.integral (meet ? irs_archimedean_riesz_space f
137 ((sum_field R n)*irs_unit)))
140 ∀f:irs_archimedean_riesz_space.
143 integral (meet ? irs_archimedean_riesz_space f
144 ((inv ? (sum_field R (S n))
145 (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))
148 ∀f,g:irs_archimedean_riesz_space.
149 integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0 → f=g
152 definition induced_norm_fun ≝
153 λR:real.λV:integration_riesz_space R.λf:V.
154 integral ? ? (absolute_value ? ? f).
156 lemma induced_norm_is_norm:
157 ∀R:real.∀V:integration_riesz_space R.is_norm ? V (induced_norm_fun ? V).
160 [ apply mk_is_semi_norm;
161 [ unfold induced_norm_fun;
164 [ apply (irs_integral_properties ? V)
169 unfold induced_norm_fun;
173 unfold induced_norm_fun;
178 unfold induced_norm_fun in H;
179 apply irs_quotient_space1;
182 rewrite < eq_zero_opp_zero;
183 rewrite > zero_neutral;
188 definition induced_norm ≝
189 λR:real.λV:integration_riesz_space R.
190 mk_norm ? ? (induced_norm_fun ? V) (induced_norm_is_norm ? V).
192 lemma is_riesz_norm_induced_norm:
193 ∀R:real.∀V:integration_riesz_space R.
194 is_riesz_norm ? ? (induced_norm ? V).
196 unfold is_riesz_norm;
200 unfold induced_norm_fun;
205 definition induced_riesz_norm ≝
206 λR:real.λV:integration_riesz_space R.
207 mk_riesz_norm ? ? (induced_norm ? V) (is_riesz_norm_induced_norm ? V).
209 definition distance_induced_by_integral ≝
210 λR:real.λV:integration_riesz_space R.
211 induced_distance ? ? (induced_norm R V).
213 definition is_complete_integration_riesz_space ≝
214 λR:real.λV:integration_riesz_space R.
215 is_complete ? ? (distance_induced_by_integral ? V).
217 record complete_integration_riesz_space (R:real) : Type ≝
218 { cirz_integration_riesz_space:> integration_riesz_space R;
219 cirz_complete_integration_riesz_space_property:
220 is_complete_integration_riesz_space ? cirz_integration_riesz_space
223 (* now we prove that any complete integration riesz space is an L-space *)
225 theorem is_l_space_l_space_induced_by_integral:
226 ∀R:real.∀V:complete_integration_riesz_space R.
227 is_l_space ? ? (induced_riesz_norm ? V).
230 [ apply cirz_complete_integration_riesz_space_property
232 unfold induced_riesz_norm;
236 unfold induced_norm_fun;
242 (**************************** f-ALGEBRAS ********************************)
244 record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
246 { (* ring properties *)
247 a_ring: is_ring V mult one;
248 (* algebra properties *)
249 a_associative_left: ∀a,f,g. a * (mult f g) = mult (a * f) g;
250 a_associative_right: ∀a,f,g. a * (mult f g) = mult f (a * g)
253 record algebra (K: field) (V:vector_space K) (a_one:V) : Type \def
255 a_algebra_properties: is_algebra ? ? a_mult a_one
258 interpretation "Algebra product" 'times a b =
259 (cic:/matita/integration_algebras/a_mult.con _ _ _ a b).
261 definition ring_of_algebra ≝
262 λK.λV:vector_space K.λone:V.λA:algebra ? V one.
263 mk_ring V (a_mult ? ? ? A) one
264 (a_ring ? ? ? ? (a_algebra_properties ? ? ? A)).
266 coercion cic:/matita/integration_algebras/ring_of_algebra.con.
268 record is_f_algebra (K) (S:archimedean_riesz_space K) (one: S)
269 (A:algebra ? S one) : Prop
273 le ? S 0 f → le ? S 0 g → le ? S 0 (a_mult ? ? ? A f g);
276 meet ? S f g = 0 → meet ? S (a_mult ? ? ? A h f) g = 0
279 record f_algebra (K:ordered_field_ch0) (R:archimedean_riesz_space K) (one:R) :
281 { fa_algebra:> algebra ? R one;
282 fa_f_algebra_properties: is_f_algebra ? ? ? fa_algebra
285 (* to be proved; see footnote 2 in the paper by Spitters *)
286 axiom symmetric_a_mult:
287 ∀K,R,one.∀A:f_algebra K R one. symmetric ? (a_mult ? ? ? A).
289 record integration_f_algebra (R:real) : Type \def
290 { ifa_integration_riesz_space:> integration_riesz_space R;
292 f_algebra ? ifa_integration_riesz_space
293 (irs_unit ? ifa_integration_riesz_space)