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/".
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_lattice (C:Type) (join,meet:C→C→C) : Prop \def
37 { (* abelian semigroup properties *)
38 l_comm_j: symmetric ? join;
39 l_associative_j: associative ? join;
40 l_comm_m: symmetric ? meet;
41 l_associative_m: associative ? meet;
42 (* other properties *)
43 l_adsorb_j_m: ∀f,g. join f (meet f g) = f;
44 l_adsorb_m_j: ∀f,g. meet f (join f g) = f
47 record lattice (C:Type) : Type \def
50 l_lattice_properties: is_lattice ? join meet
53 definition le \def λC:Type.λL:lattice C.λf,g. meet ? L f g = f.
55 interpretation "Lattice le" 'leq a b =
56 (cic:/matita/integration_algebras/le.con _ _ a b).
58 definition carrier_of_lattice ≝
59 λC:Type.λL:lattice C.C.
61 record is_riesz_space (K:ordered_field_ch0) (V:vector_space K)
62 (L:lattice (Type_OF_vector_space ? V))
65 { rs_compat_le_plus: ∀f,g,h. le ? L f g → le ? L (f+h) (g+h);
66 rs_compat_le_times: ∀a:K.∀f. of_le ? 0 a → le ? L 0 f → le ? L 0 (a*f)
69 record riesz_space (K:ordered_field_ch0) : Type \def
70 { rs_vector_space:> vector_space K;
71 rs_lattice:> lattice rs_vector_space;
72 rs_riesz_space_properties: is_riesz_space ? rs_vector_space rs_lattice
75 definition absolute_value \def λK.λS:riesz_space K.λf.join ? S f (-f).
77 record is_archimedean_riesz_space (K) (S:riesz_space K) : Prop
79 { ars_archimedean: ∃u.∀n.∀a.∀p:n > O.
81 (absolute_value ? S a)
83 (inv ? (sum_field K n) (not_eq_sum_field_zero ? n p))
88 record archimedean_riesz_space (K:ordered_field_ch0) : Type \def
89 { ars_riesz_space:> riesz_space K;
90 ars_archimedean_property: is_archimedean_riesz_space ? ars_riesz_space
93 record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
95 { (* ring properties *)
96 a_ring: is_ring V mult one;
97 (* algebra properties *)
98 a_associative_left: ∀a,f,g. a * (mult f g) = mult (a * f) g;
99 a_associative_right: ∀a,f,g. a * (mult f g) = mult f (a * g)
102 record algebra (K: field) (V:vector_space K) : Type \def
105 a_algebra_properties: is_algebra K V a_mult a_one
108 interpretation "Algebra product" 'times a b =
109 (cic:/matita/integration_algebras/a_mult.con _ _ _ a b).
111 interpretation "Algebra one" 'one =
112 (cic:/matita/integration_algebras/a_one.con _ _ _).
114 definition ring_of_algebra ≝
115 λK.λV:vector_space K.λA:algebra ? V.
116 mk_ring V (a_mult ? ? A) (a_one ? ? A)
117 (a_ring ? ? ? ? (a_algebra_properties ? ? A)).
119 coercion cic:/matita/integration_algebras/ring_of_algebra.con.
121 record is_f_algebra (K) (S:archimedean_riesz_space K) (A:algebra ? S) : Prop
125 le ? S 0 f → le ? S 0 g → le ? S 0 (a_mult ? ? A f g);
128 meet ? S f g = 0 → meet ? S (a_mult ? ? A h f) g = 0
131 record f_algebra (K:ordered_field_ch0) : Type \def
132 { fa_archimedean_riesz_space:> archimedean_riesz_space K;
133 fa_algebra:> algebra ? fa_archimedean_riesz_space;
134 fa_f_algebra_properties: is_f_algebra ? fa_archimedean_riesz_space fa_algebra
137 (* to be proved; see footnote 2 in the paper by Spitters *)
138 axiom symmetric_a_mult: ∀K.∀A:f_algebra K. symmetric ? (a_mult ? ? A).
140 record is_integral (K) (A:f_algebra K) (I:Type_OF_f_algebra ? A→K) : Prop
142 { i_positive: ∀f:Type_OF_f_algebra ? A. le ? (lattice_OF_f_algebra ? A) 0 f → of_le K 0 (I f);
143 i_linear1: ∀f,g:Type_OF_f_algebra ? A. I (f + g) = I f + I g;
144 i_linear2: ∀f:A.∀k:K. I (emult ? A k f) = k*(I f)
147 (* Here we are avoiding a construction (the quotient space to define
148 f=g iff I(|f-g|)=0 *)
149 record is_integration_f_algebra (K) (A:f_algebra K) (I:Type_OF_f_algebra ? A→K) : Prop
151 { ifa_integral: is_integral ? ? I;
153 ∀f:A. tends_to ? (λn.I(meet ? A f ((sum_field K n)*(a_one ? ? A)))) (I f);
159 ((inv ? (sum_field K (S n))
160 (not_eq_sum_field_zero K (S n) (le_S_S O n (le_O_n n)))
161 ) * (a_one ? ? A)))) 0;
163 ∀f,g:A. f=g → I(absolute_value ? A (f - g)) = 0