set "baseuri" "cic:/matita/integration_algebras/".
-include "reals.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
-}.
-
-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
- }.
-
-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 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_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
- }.
-
-definition induced_distance ≝
- λR:real.λV:vector_space R.λnorm:V→R.
- λf,g:V.norm (f - g).
-
-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.
+include "vector_spaces.ma".
record is_lattice (C:Type) (join,meet:C→C→C) : Prop \def
{ (* abelian semigroup properties *)
definition absolute_value \def λK.λS:riesz_space K.λf.join ? S f (-f).
+(*CSC: qui la notazione non fa capire!!! *)
+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).
+
+record riesz_norm (R:real) (V:riesz_space R) : Type ≝
+ { rn_norm:> norm ? 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).
+
+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.
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).
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 ? V (induced_norm_fun ? V).
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)
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;
].
qed.
+definition induced_norm ≝
+ λR:real.λV:integration_riesz_space R.
+ mk_norm ? ? (induced_norm_fun ? V) (induced_norm_is_norm ? V).
+
+lemma is_riesz_norm_induced_norm:
+ ∀R:real.∀V:integration_riesz_space R.
+ is_riesz_norm ? ? (induced_norm ? V).
+ intros;
+ 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).
-theorem distance_induced_by_integral_is_distance:
- ∀R:real.∀V:integration_riesz_space R.
- is_distance ? ? (distance_induced_by_integral ? 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;
- unfold distance_induced_by_integral;
- apply induced_distance_is_distance;
- apply induced_norm_is_norm.
+ 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
+ }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/vector_spaces/".
+
+include "reals.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
+}.
+
+interpretation "Vector space external product" 'times a b =
+ (cic:/matita/vector_spaces/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
+ }.
+
+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 ≝
+ { n_semi_norm:> is_semi_norm ? ? norm;
+ n_properness: ∀x:V. norm x = 0 → x = 0
+ }.
+
+record norm (R:real) (V:vector_space R) : Type ≝
+ { n_function:1> V→R;
+ n_norm_properties: is_norm ? ? n_function
+ }.
+
+record is_semi_distance (R:real) (C:Type) (semi_d: C→C→R) : Prop ≝
+ { sd_positive: ∀x,y:C. 0 ≤ semi_d x y;
+ sd_properness: ∀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_distance (R:real) (C:Type) (d:C→C→R) : Prop ≝
+ { d_semi_distance:> is_semi_distance ? ? d;
+ d_properness: ∀x,y:C. d x y = 0 → x=y
+ }.
+
+record distance (R:real) (V:vector_space R) : Type ≝
+ { d_function:2> V→V→R;
+ d_distance_properties: is_distance ? ? d_function
+ }.
+
+definition induced_distance_fun ≝
+ λR:real.λV:vector_space R.λnorm:norm ? V.
+ λf,g:V.norm (f - g).
+
+theorem induced_distance_is_distance:
+ ∀R:real.∀V:vector_space R.∀norm:norm ? V.
+ is_distance ? ? (induced_distance_fun ? ? norm).
+ intros;
+ apply mk_is_distance;
+ [ apply mk_is_semi_distance;
+ [ unfold induced_distance_fun;
+ intros;
+ apply sn_positive;
+ apply n_semi_norm;
+ apply (n_norm_properties ? ? norm)
+ | unfold induced_distance_fun;
+ intros;
+ unfold minus;
+ rewrite < plus_comm;
+ rewrite > opp_inverse;
+ apply eq_semi_norm_zero_zero;
+ apply n_semi_norm;
+ apply (n_norm_properties ? ? norm)
+ | unfold induced_distance_fun;
+ intros;
+ (* ??? *)
+ elim daemon
+ ]
+ | unfold induced_distance_fun;
+ intros;
+ generalize in match (n_properness ? ? norm ? ? H);
+ [ intro;
+ (* facile *)
+ elim daemon
+ | apply (n_norm_properties ? ? norm)
+ ]
+ ].
+qed.
+
+definition induced_distance ≝
+ λR:real.λV:vector_space R.λnorm:norm ? V.
+ mk_distance ? ? (induced_distance_fun ? ? norm)
+ (induced_distance_is_distance ? ? norm).
+
+definition tends_to :
+ ∀R:real.∀V:vector_space R.∀d:distance ? V.∀f:nat→V.∀l:V.Prop.
+ alias symbol "leq" = "Ordered field le".
+ alias id "le" = "cic:/matita/nat/orders/le.ind#xpointer(1/1)".
+apply
+ (λR:real.λV:vector_space R.λd:distance ? V.λf:nat→V.λl:V.
+ ∀n:nat.∃m:nat.∀j:nat. le m j →
+ d (f j) l ≤ inv R (sum_field ? (S n)) ?);
+ apply not_eq_sum_field_zero;
+ unfold;
+ auto new.
+qed.
+
+definition is_cauchy_seq : ∀R:real.\forall V:vector_space R.
+\forall d:distance ? V.∀f:nat→V.Prop.
+ apply
+ (λR:real.λV: vector_space R. \lambda d:distance ? V.
+ \lambda f:nat→V.
+ ∀m:nat.
+ ∃n:nat.∀N. le n N →
+ -(inv R (sum_field ? (S m)) ?) ≤ d (f N) (f n) ∧
+ d (f N) (f n)≤ inv R (sum_field R (S m)) ?);
+ apply not_eq_sum_field_zero;
+ unfold;
+ auto.
+qed.
+
+definition is_complete ≝
+ λR:real.λV:vector_space R.
+ λd:distance ? V.
+ ∀f:nat→V. is_cauchy_seq ? ? d f→
+ ex V (λl:V. tends_to ? ? d f l).
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