set "baseuri" "cic:/matita/integration_algebras/".
-include "reals.ma".
+include "vector_spaces.ma".
+include "lattices.ma".
-record is_vector_space (K: field) (G:abelian_group) (emult:K→G→G) : Prop
-≝
- { vs_nilpotent: ∀v. emult 0 v = 0;
- vs_neutral: ∀v. emult 1 v = v;
- vs_distributive: ∀a,b,v. emult (a + b) v = (emult a v) + (emult b v);
- vs_associative: ∀a,b,v. emult (a * b) v = emult a (emult b v)
- }.
-
-record vector_space (K:field): Type \def
-{ vs_abelian_group :> abelian_group;
- emult: K → vs_abelian_group → vs_abelian_group;
- vs_vector_space_properties :> is_vector_space ? vs_abelian_group emult
-}.
+(**************** Riesz Spaces ********************)
-interpretation "Vector space external product" 'times a b =
- (cic:/matita/integration_algebras/emult.con _ _ a b).
-
-record is_semi_norm (R:real) (V: vector_space R) (semi_norm:V→R) : Prop \def
- { sn_positive: ∀x:V. 0 ≤ semi_norm x;
- sn_omogeneous: ∀a:R.∀x:V. semi_norm (a*x) = (abs ? a) * semi_norm x;
- sn_triangle_inequality: ∀x,y:V. semi_norm (x + y) ≤ semi_norm x + semi_norm y
+record is_riesz_space (K:ordered_field_ch0) (V:vector_space K)
+ (L:lattice V)
+: Prop
+\def
+ { 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)
}.
-theorem eq_semi_norm_zero_zero:
- ∀R:real.∀V:vector_space R.∀semi_norm:V→R.
- is_semi_norm ? ? semi_norm →
- semi_norm 0 = 0.
- intros;
- (* facile *)
- elim daemon.
-qed.
-
-record is_norm (R:real) (V:vector_space R) (norm:V → R) : Prop \def
- { n_semi_norm:> is_semi_norm ? ? norm;
- n_properness: ∀x:V. norm x = 0 → x = 0
+record riesz_space (K:ordered_field_ch0) : Type \def
+ { rs_vector_space:> vector_space K;
+ rs_lattice:> lattice rs_vector_space;
+ rs_riesz_space_properties: is_riesz_space ? rs_vector_space rs_lattice
}.
-record is_semi_distance (R:real) (C:Type) (semi_d: C→C→R) : Prop \def
- { sd_positive: ∀x,y:C. 0 ≤ semi_d x y;
- sd_properness: \forall x:C. semi_d x x = 0;
- sd_triangle_inequality: ∀x,y,z:C. semi_d x z ≤ semi_d x y + semi_d z y
+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 is_distance (R:real) (C:Type) (d:C→C→R) : Prop \def
- { d_semi_distance:> is_semi_distance ? ? d;
- d_properness: ∀x,y:C. d x y = 0 → x=y
+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 induced_distance ≝
- λR:real.λV:vector_space R.λnorm:V→R.
- λf,g:V.norm (f - g).
+definition absolute_value \def λK.λS:riesz_space K.λf.join ? S f (-f).
-theorem induced_distance_is_distance:
- ∀R:real.∀V:vector_space R.∀norm:V→R.
- is_norm ? ? norm → is_distance ? ? (induced_distance ? ? norm).
- intros;
- apply mk_is_distance;
- [ apply mk_is_semi_distance;
- [ unfold induced_distance;
- intros;
- apply sn_positive;
- apply n_semi_norm;
- assumption
- | unfold induced_distance;
- intros;
- unfold minus;
- rewrite < plus_comm;
- rewrite > opp_inverse;
- apply eq_semi_norm_zero_zero;
- apply n_semi_norm;
- assumption
- | unfold induced_distance;
- intros;
- (* ??? *)
- elim daemon
- ]
- | unfold induced_distance;
- intros;
- generalize in match (n_properness ? ? ? H ? H1);
- intro;
- (* facile *)
- elim daemon
- ].
-qed.
+(**************** Normed Riesz spaces ****************************)
-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
- }.
+definition is_riesz_norm ≝
+ 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 lattice (C:Type) : Type \def
- { join: C → C → C;
- meet: C → C → C;
- l_lattice_properties: is_lattice ? join meet
+record riesz_norm (R:real) (V:riesz_space R) : Type ≝
+ { rn_norm:> norm R V;
+ rn_riesz_norm_property: is_riesz_norm ? ? rn_norm
}.
-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.
+(*CSC: non fa la chiusura delle coercion verso funclass *)
+definition rn_function ≝
+ λR:real.λV:riesz_space R.λnorm:riesz_norm ? V.
+ n_function R V (rn_norm ? ? norm).
-interpretation "Lattice lt" 'lt a b =
- (cic:/matita/integration_algebras/lt.con _ _ a b).
+coercion cic:/matita/integration_algebras/rn_function.con 1.
-definition carrier_of_lattice ≝
- λC:Type.λL:lattice C.C.
-
-record is_riesz_space (K:ordered_field_ch0) (V:vector_space K)
- (L:lattice (Type_OF_vector_space ? 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)
- }.
-
-record riesz_space (K:ordered_field_ch0) : Type \def
- { rs_vector_space:> vector_space K;
- rs_lattice:> lattice rs_vector_space;
- rs_riesz_space_properties: is_riesz_space ? rs_vector_space rs_lattice
+(************************** 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
}.
-
-definition absolute_value \def λK.λS:riesz_space K.λf.join ? S f (-f).
+*)
+(******************** 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 ?
integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0 → f=g
}.
-definition induced_norm ≝
+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 ? 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;
+ [ unfold induced_norm_fun;
intros;
- apply i_positive;
- [ apply (irs_integral_properties ? V)
+ apply positive;
+ [ apply (irs_positive_linear ? V)
| (* difficile *)
elim daemon
]
| intros;
- unfold induced_norm;
+ unfold induced_norm_fun;
(* facile *)
elim daemon
| intros;
- unfold induced_norm;
+ unfold induced_norm_fun;
(* difficile *)
elim daemon
]
| intros;
- unfold induced_norm in H;
+ unfold induced_norm_fun in H;
apply irs_quotient_space1;
unfold minus;
rewrite < plus_comm;
rewrite < eq_zero_opp_zero;
rewrite > zero_neutral;
assumption
- ].
+ ].*)
qed.
-definition distance_induced_by_integral ≝
+definition induced_norm ≝
λR:real.λV:integration_riesz_space R.
- induced_distance ? ? (induced_norm R V).
+ mk_norm ? ? (induced_norm_fun ? V) (induced_norm_is_norm ? V).
-theorem distance_induced_by_integral_is_distance:
+lemma is_riesz_norm_induced_norm:
∀R:real.∀V:integration_riesz_space R.
- is_distance ? ? (distance_induced_by_integral ? V).
+ is_riesz_norm ? ? (induced_norm ? V).
intros;
- unfold distance_induced_by_integral;
- apply induced_distance_is_distance;
- apply induced_norm_is_norm.
+ unfold is_riesz_norm;
+ intros;
+ unfold induced_norm;
+ simplify;
+ unfold induced_norm_fun;
+ (* difficile *)
+ elim daemon.
qed.
+definition induced_riesz_norm ≝
+ λR:real.λV:integration_riesz_space R.
+ mk_riesz_norm ? ? (induced_norm ? V) (is_riesz_norm_induced_norm ? V).
+
+definition distance_induced_by_integral ≝
+ λR:real.λV:integration_riesz_space R.
+ induced_distance ? ? (induced_norm R V).
+
+definition is_complete_integration_riesz_space ≝
+ λR:real.λV:integration_riesz_space R.
+ is_complete ? ? (distance_induced_by_integral ? V).
+
+record complete_integration_riesz_space (R:real) : Type ≝
+ { cirz_integration_riesz_space:> integration_riesz_space R;
+ cirz_complete_integration_riesz_space_property:
+ is_complete_integration_riesz_space ? cirz_integration_riesz_space
+ }.
+
+(* now we prove that any complete integration riesz space is an L-space *)
+
+(*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;
+ constructor 1;
+ [ apply cirz_complete_integration_riesz_space_property
+ | intros;
+ unfold induced_riesz_norm;
+ simplify;
+ unfold induced_norm;
+ simplify;
+ unfold induced_norm_fun;
+ (* difficile *)
+ elim daemon
+ ].
+qed.*)
+
+(**************************** f-ALGEBRAS ********************************)
+
record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
≝
{ (* ring properties *)
ifa_f_algebra:>
f_algebra ? ifa_integration_riesz_space
(irs_unit ? ifa_integration_riesz_space)
- }.
\ No newline at end of file
+ }.
projects / helm.git / blobdiff