set "baseuri" "cic:/matita/integration_algebras/".
include "vector_spaces.ma".
+include "lattices.ma".
-record is_lattice (C:Type) (join,meet:C→C→C) : Prop \def
- { (* abelian semigroup properties *)
- l_comm_j: symmetric ? join;
- l_associative_j: associative ? join;
- l_comm_m: symmetric ? meet;
- l_associative_m: associative ? meet;
- (* other properties *)
- l_adsorb_j_m: ∀f,g. join f (meet f g) = f;
- l_adsorb_m_j: ∀f,g. meet f (join f g) = f
- }.
-
-record lattice (C:Type) : Type \def
- { join: C → C → C;
- meet: C → C → C;
- l_lattice_properties: is_lattice ? join meet
- }.
-
-definition le \def λC:Type.λL:lattice C.λf,g. meet ? L f g = f.
-
-interpretation "Lattice le" 'leq a b =
- (cic:/matita/integration_algebras/le.con _ _ a b).
-
-definition lt \def λC:Type.λL:lattice C.λf,g. le ? L f g ∧ f ≠ g.
-
-interpretation "Lattice lt" 'lt a b =
- (cic:/matita/integration_algebras/lt.con _ _ a b).
-
-definition carrier_of_lattice ≝
- λC:Type.λL:lattice C.C.
+(**************** Riesz Spaces ********************)
record is_riesz_space (K:ordered_field_ch0) (V:vector_space K)
- (L:lattice (Type_OF_vector_space ? V))
+ (L:lattice V)
: Prop
\def
- { rs_compat_le_plus: ∀f,g,h. le ? L f g → le ? L (f+h) (g+h);
- rs_compat_le_times: ∀a:K.∀f. of_le ? 0 a → le ? L 0 f → le ? L 0 (a*f)
+ { rs_compat_le_plus: ∀f,g,h:V. os_le ? L f g → os_le ? L (f+h) (g+h);
+ rs_compat_le_times: ∀a:K.∀f:V. zero K≤a → os_le ? L (zero V) f → os_le ? L (zero V) (a*f)
}.
record riesz_space (K:ordered_field_ch0) : Type \def
rs_riesz_space_properties: is_riesz_space ? rs_vector_space rs_lattice
}.
+record is_positive_linear (K) (V:riesz_space K) (T:V→K) : Prop ≝
+ { positive: ∀u:V. os_le ? V 0 u → os_le ? K 0 (T u);
+ linear1: ∀u,v:V. T (u+v) = T u + T v;
+ linear2: ∀u:V.∀k:K. T (k*u) = k*(T u)
+ }.
+
+record sequentially_order_continuous (K) (V:riesz_space K) (T:V→K) : Prop ≝
+ { soc_incr:
+ ∀a:nat→V.∀l:V.is_increasing ? ? a → is_sup ? V a l →
+ is_increasing ? K (λn.T (a n)) ∧ tends_to ? (λn.T (a n)) (T l)
+ }.
+
definition absolute_value \def λK.λS:riesz_space K.λf.join ? S f (-f).
-(*CSC: qui la notazione non fa capire!!! *)
+(**************** Normed Riesz spaces ****************************)
+
definition is_riesz_norm ≝
- λR:real.λV:riesz_space R.λnorm:norm ? V.
- ∀f,g:V. le ? V (absolute_value ? V f) (absolute_value ? V g) →
- of_le R (norm f) (norm g).
+ λR:real.λV:riesz_space R.λnorm:norm R V.
+ ∀f,g:V. os_le ? V (absolute_value ? V f) (absolute_value ? V g) →
+ os_le ? R (n_function R V norm f) (n_function R V norm g).
record riesz_norm (R:real) (V:riesz_space R) : Type ≝
- { rn_norm:> norm ? V;
+ { rn_norm:> norm R V;
rn_riesz_norm_property: is_riesz_norm ? ? rn_norm
}.
(*CSC: non fa la chiusura delle coercion verso funclass *)
definition rn_function ≝
λR:real.λV:riesz_space R.λnorm:riesz_norm ? V.
- n_function ? ? (rn_norm ? ? norm).
+ n_function R V (rn_norm ? ? norm).
coercion cic:/matita/integration_algebras/rn_function.con 1.
(************************** L-SPACES *************************************)
-
+(*
record is_l_space (R:real) (V:riesz_space R) (norm:riesz_norm ? V) : Prop ≝
{ ls_banach: is_complete ? V (induced_distance ? ? norm);
ls_linear: ∀f,g:V. le ? V 0 f → le ? V 0 g → norm (f+g) = norm f + norm g
}.
-
+*)
(******************** ARCHIMEDEAN RIESZ SPACES ***************************)
record is_archimedean_riesz_space (K) (S:riesz_space K) : Prop
\def
- { ars_archimedean: ∃u.∀n.∀a.∀p:n > O.
- le ? S
+ { ars_archimedean: ∃u:S.∀n.∀a.∀p:n > O.
+ os_le ? S
(absolute_value ? S a)
- ((inv ? (sum_field K n) (not_eq_sum_field_zero ? n p))* u) →
+ ((inv K (sum_field K n) (not_eq_sum_field_zero K n p))* u) →
a = 0
}.
ars_archimedean_property: is_archimedean_riesz_space ? ars_riesz_space
}.
-record is_integral (K) (R:archimedean_riesz_space K) (I:R→K) : Prop
-\def
- { i_positive: ∀f:R. le ? R 0 f → of_le K 0 (I f);
- i_linear1: ∀f,g:R. I (f + g) = I f + I g;
- i_linear2: ∀f:R.∀k:K. I (k*f) = k*(I f)
- }.
-
definition is_weak_unit ≝
(* This definition is by Spitters. He cites Fremlin 353P, but:
1. that theorem holds only in f-algebras (as in Spitters, but we are
irs_unit: irs_archimedean_riesz_space;
irs_weak_unit: is_weak_unit ? ? irs_unit;
integral: irs_archimedean_riesz_space → R;
- irs_integral_properties: is_integral ? ? integral;
+ irs_positive_linear: is_positive_linear ? ? integral;
irs_limit1:
∀f:irs_archimedean_riesz_space.
tends_to ?
definition induced_norm_fun ≝
λR:real.λV:integration_riesz_space R.λf:V.
- integral ? ? (absolute_value ? ? f).
+ integral ? V (absolute_value ? ? f).
lemma induced_norm_is_norm:
- ∀R:real.∀V:integration_riesz_space R.is_norm ? V (induced_norm_fun ? V).
+ ∀R:real.∀V:integration_riesz_space R.is_norm R V (induced_norm_fun ? V).
+ elim daemon.(*
intros;
apply mk_is_norm;
[ apply mk_is_semi_norm;
[ unfold induced_norm_fun;
intros;
- apply i_positive;
- [ apply (irs_integral_properties ? V)
+ apply positive;
+ [ apply (irs_positive_linear ? V)
| (* difficile *)
elim daemon
]
rewrite < eq_zero_opp_zero;
rewrite > zero_neutral;
assumption
- ].
+ ].*)
qed.
definition induced_norm ≝
(* now we prove that any complete integration riesz space is an L-space *)
-theorem is_l_space_l_space_induced_by_integral:
+(*theorem is_l_space_l_space_induced_by_integral:
∀R:real.∀V:complete_integration_riesz_space R.
is_l_space ? ? (induced_riesz_norm ? V).
intros;
(* difficile *)
elim daemon
].
-qed.
+qed.*)
(**************************** f-ALGEBRAS ********************************)
projects / helm.git / blobdiff