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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 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/integration_algebras/".
19 record is_vector_space (K: field) (G:abelian_group) (emult:K→G→G) : Prop
21 { vs_nilpotent: ∀v. emult 0 v = 0;
22 vs_neutral: ∀v. emult 1 v = v;
23 vs_distributive: ∀a,b,v. emult (a + b) v = (emult a v) + (emult b v);
24 vs_associative: ∀a,b,v. emult (a * b) v = emult a (emult b v)
27 record vector_space (K:field): Type \def
28 { vs_abelian_group :> abelian_group;
29 emult: K → vs_abelian_group → vs_abelian_group;
30 vs_vector_space_properties :> is_vector_space K vs_abelian_group emult
33 interpretation "Vector space external product" 'times a b =
34 (cic:/matita/integration_algebras/emult.con _ _ a b).
36 record is_semi_norm (R:real) (V: vector_space R) (semi_norm:V→R) : Prop \def
37 { sn_positive: ∀x:V. 0 ≤ semi_norm x;
38 sn_omogeneous: ∀a:R.∀x:V. semi_norm (a*x) = (abs ? a) * semi_norm x;
39 sn_triangle_inequality: ∀x,y:V. semi_norm (x + y) ≤ semi_norm x + semi_norm y
42 record is_norm (R:real) (V:vector_space R) (norm:V → R)
44 { n_semi_norm:> is_semi_norm ? ? norm;
45 n_properness: ∀x:V. norm x = 0 → x = 0
48 record is_lattice (C:Type) (join,meet:C→C→C) : Prop \def
49 { (* abelian semigroup properties *)
50 l_comm_j: symmetric ? join;
51 l_associative_j: associative ? join;
52 l_comm_m: symmetric ? meet;
53 l_associative_m: associative ? meet;
54 (* other properties *)
55 l_adsorb_j_m: ∀f,g. join f (meet f g) = f;
56 l_adsorb_m_j: ∀f,g. meet f (join f g) = f
59 record lattice (C:Type) : Type \def
62 l_lattice_properties: is_lattice ? join meet
65 definition le \def λC:Type.λL:lattice C.λf,g. meet ? L f g = f.
67 interpretation "Lattice le" 'leq a b =
68 (cic:/matita/integration_algebras/le.con _ _ a b).
70 definition carrier_of_lattice ≝
71 λC:Type.λL:lattice C.C.
73 record is_riesz_space (K:ordered_field_ch0) (V:vector_space K)
74 (L:lattice (Type_OF_vector_space ? V))
77 { rs_compat_le_plus: ∀f,g,h. le ? L f g → le ? L (f+h) (g+h);
78 rs_compat_le_times: ∀a:K.∀f. of_le ? 0 a → le ? L 0 f → le ? L 0 (a*f)
81 record riesz_space (K:ordered_field_ch0) : Type \def
82 { rs_vector_space:> vector_space K;
83 rs_lattice:> lattice rs_vector_space;
84 rs_riesz_space_properties: is_riesz_space ? rs_vector_space rs_lattice
87 definition absolute_value \def λK.λS:riesz_space K.λf.join ? S f (-f).
89 record is_archimedean_riesz_space (K) (S:riesz_space K) : Prop
91 { ars_archimedean: ∃u.∀n.∀a.∀p:n > O.
93 (absolute_value ? S a)
95 (inv ? (sum_field K n) (not_eq_sum_field_zero ? n p))
100 record archimedean_riesz_space (K:ordered_field_ch0) : Type \def
101 { ars_riesz_space:> riesz_space K;
102 ars_archimedean_property: is_archimedean_riesz_space ? ars_riesz_space
105 record is_integral (K) (R:archimedean_riesz_space K) (I:R→K) : Prop
107 { i_positive: ∀f:R. le ? R 0 f → of_le K 0 (I f);
108 i_linear1: ∀f,g:R. I (f + g) = I f + I g;
109 i_linear2: ∀f:R.∀k:K. I (emult ? R k f) = k*(I f)
112 definition is_weak_unit ≝
113 (* This definition is by Spitters. He cites Fremlin 353P, but:
114 1. that theorem holds only in f-algebras (as in Spitters, but we are
115 defining it on Riesz spaces)
116 2. Fremlin proves |x|/\u=0 \to u=0. How do we remove the absolute value?
117 λR:real.λV:archimedean_riesz_space R.λunit: V.
118 ∀x:V. meet x unit = 0 → u = 0.
119 *) λR:real.λV:archimedean_riesz_space R.λe:V.True.
121 record integration_riesz_space (R:real) : Type \def
122 { irs_archimedean_riesz_space:> archimedean_riesz_space R;
123 irs_unit: irs_archimedean_riesz_space;
124 irs_weak_unit: is_weak_unit ? ? irs_unit;
125 integral: irs_archimedean_riesz_space → R;
126 irs_integral_properties: is_integral R irs_archimedean_riesz_space integral;
128 ∀f:irs_archimedean_riesz_space.
130 (λn.integral (meet ? irs_archimedean_riesz_space f
131 ((sum_field R n)*irs_unit)))
134 ∀f:irs_archimedean_riesz_space.
137 integral (meet ? irs_archimedean_riesz_space f
138 ((inv ? (sum_field R (S n))
139 (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))
142 ∀f,g:irs_archimedean_riesz_space.
143 f=g → integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0
146 record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
148 { (* ring properties *)
149 a_ring: is_ring V mult one;
150 (* algebra properties *)
151 a_associative_left: ∀a,f,g. a * (mult f g) = mult (a * f) g;
152 a_associative_right: ∀a,f,g. a * (mult f g) = mult f (a * g)
155 record algebra (K: field) (V:vector_space K) (a_one:V) : Type \def
157 a_algebra_properties: is_algebra K V a_mult a_one
160 interpretation "Algebra product" 'times a b =
161 (cic:/matita/integration_algebras/a_mult.con _ _ _ a b).
163 definition ring_of_algebra ≝
164 λK.λV:vector_space K.λone:V.λA:algebra ? V one.
165 mk_ring V (a_mult ? ? ? A) one
166 (a_ring ? ? ? ? (a_algebra_properties ? ? ? A)).
168 coercion cic:/matita/integration_algebras/ring_of_algebra.con.
170 record is_f_algebra (K) (S:archimedean_riesz_space K) (one: S)
171 (A:algebra ? S one) : Prop
175 le ? S 0 f → le ? S 0 g → le ? S 0 (a_mult ? ? ? A f g);
178 meet ? S f g = 0 → meet ? S (a_mult ? ? ? A h f) g = 0
181 record f_algebra (K:ordered_field_ch0) (R:archimedean_riesz_space K) (one:R) :
183 { fa_algebra:> algebra ? R one;
184 fa_f_algebra_properties: is_f_algebra ? ? ? fa_algebra
187 (* to be proved; see footnote 2 in the paper by Spitters *)
188 axiom symmetric_a_mult:
189 ∀K,R,one.∀A:f_algebra K R one. symmetric ? (a_mult ? ? ? A).
191 (* Here we are avoiding a construction (the quotient space to define
192 f=g iff I(|f-g|)=0 *)
193 record integration_f_algebra (R:real) : Type \def
194 { ifa_integration_riesz_space:> integration_riesz_space R;
196 f_algebra ? ifa_integration_riesz_space
197 (irs_unit ? ifa_integration_riesz_space)