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
- }.
+(**************** Riesz Spaces ********************)
-record lattice (C:Type) : Type \def
- { join: C → C → C;
- meet: C → C → C;
- l_lattice_properties: is_lattice ? join meet
+record pre_riesz_space (K:ordered_field_ch0) : Type \def
+ { rs_vector_space:> vector_space K;
+ rs_lattice_: lattice;
+ rs_ordered_abelian_group_: ordered_abelian_group;
+ rs_with1:
+ og_abelian_group rs_ordered_abelian_group_ = vs_abelian_group ? rs_vector_space;
+ rs_with2:
+ og_ordered_set rs_ordered_abelian_group_ = ordered_set_of_lattice rs_lattice_
}.
-definition le \def λC:Type.λL:lattice C.λf,g. meet ? L f g = f.
+lemma rs_lattice: ∀K.pre_riesz_space K → lattice.
+ intros (K V);
+ cut (os_carrier (rs_lattice_ ? V) = V);
+ [ apply mk_lattice;
+ [ apply (carrier V)
+ | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? Hcut);
+ apply l_join
+ | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? Hcut);
+ apply l_meet
+ | apply
+ (eq_rect' ? ?
+ (λa:Type.λH:os_carrier (rs_lattice_ ? V)=a.
+ is_lattice a
+ (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C)
+ (l_join (rs_lattice_ K V)) a H)
+ (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C)
+ (l_meet (rs_lattice_ K V)) a H))
+ ? ? Hcut);
+ simplify;
+ apply l_lattice_properties
+ ]
+ | transitivity (os_carrier (rs_ordered_abelian_group_ ? V));
+ [ apply (eq_f ? ? os_carrier);
+ symmetry;
+ apply rs_with2
+ | apply (eq_f ? ? carrier);
+ apply rs_with1
+ ]
+ ].
+qed.
-interpretation "Lattice le" 'leq a b =
- (cic:/matita/integration_algebras/le.con _ _ a b).
+coercion cic:/matita/integration_algebras/rs_lattice.con.
+
+lemma rs_ordered_abelian_group: ∀K.pre_riesz_space K → ordered_abelian_group.
+ intros (K V);
+ apply mk_ordered_abelian_group;
+ [ apply mk_pre_ordered_abelian_group;
+ [ apply (vs_abelian_group ? (rs_vector_space ? V))
+ | apply (ordered_set_of_lattice (rs_lattice ? V))
+ | reflexivity
+ ]
+ | simplify;
+ generalize in match
+ (og_ordered_abelian_group_properties (rs_ordered_abelian_group_ ? V));
+ intro P;
+ unfold in P;
+ elim daemon(*
+ apply
+ (eq_rect ? ?
+ (λO:ordered_set.
+ ∀f,g,h.
+ os_le O f g →
+ os_le O
+ (plus (abelian_group_OF_pre_riesz_space K V) f h)
+ (plus (abelian_group_OF_pre_riesz_space K V) g h))
+ ? ? (rs_with2 ? V));
+ apply
+ (eq_rect ? ?
+ (λG:abelian_group.
+ ∀f,g,h.
+ os_le (ordered_set_OF_pre_riesz_space K V) f g →
+ os_le (ordered_set_OF_pre_riesz_space K V)
+ (plus (abelian_group_OF_pre_riesz_space K V) f h)
+ (plus (abelian_group_OF_pre_riesz_space K V) g h))
+ ? ? (rs_with1 ? V));
+ simplify;
+ apply og_ordered_abelian_group_properties*)
+ ]
+qed.
-definition lt \def λC:Type.λL:lattice C.λf,g. le ? L f g ∧ f ≠ g.
+coercion cic:/matita/integration_algebras/rs_ordered_abelian_group.con.
-interpretation "Lattice lt" 'lt a b =
- (cic:/matita/integration_algebras/lt.con _ _ a b).
+record is_riesz_space (K:ordered_field_ch0) (V:pre_riesz_space K) : Prop ≝
+ { rs_compat_le_times: ∀a:K.∀f:V. 0≤a → 0≤f → 0≤a*f
+ }.
-definition carrier_of_lattice ≝
- λC:Type.λL:lattice C.C.
+record riesz_space (K:ordered_field_ch0) : Type \def
+ { rs_pre_riesz_space:> pre_riesz_space K;
+ rs_riesz_space_properties: is_riesz_space ? rs_pre_riesz_space
+ }.
-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 is_positive_linear (K) (V:riesz_space K) (T:V→K) : Prop ≝
+ { positive: ∀u:V. 0≤u → 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 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 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).
+definition absolute_value ≝ λK.λS:riesz_space K.λf:S.f ∨ -f.
+
+(**************** Normed Riesz spaces ****************************)
-(*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).
+ λR:real.λV:riesz_space R.λnorm:norm R V.
+ ∀f,g:V. absolute_value ? V f ≤ absolute_value ? V g →
+ 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
- (absolute_value ? S a)
- ((inv ? (sum_field K n) (not_eq_sum_field_zero ? n p))* u) →
+ { ars_archimedean: ∃u:S.∀n.∀a.∀p:n > O.
+ absolute_value ? S a ≤
+ (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
3. Fremlin proves u > 0 implies x /\ u > 0 > 0 for Archimedean spaces
only. We pick this definition for now.
*) λR:real.λV:archimedean_riesz_space R.λe:V.
- ∀v:V. lt ? V 0 v → lt ? V 0 (meet ? V v e).
+ ∀v:V. 0<v → 0 < (v ∧ e).
(* Here we are avoiding a construction (the quotient space to define
f=g iff I(|f-g|)=0 *)
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 ?
- (λn.integral (meet ? irs_archimedean_riesz_space f
- ((sum_field R n)*irs_unit)))
+ (λn.integral (f ∧ ((sum_field R n)*irs_unit)))
(integral f);
irs_limit2:
∀f:irs_archimedean_riesz_space.
tends_to ?
(λn.
- integral (meet ? irs_archimedean_riesz_space f
+ integral (f ∧
((inv ? (sum_field R (S n))
(not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))
) * irs_unit))) 0;
irs_quotient_space1:
∀f,g:irs_archimedean_riesz_space.
- integral (absolute_value irs_archimedean_riesz_space (f - g)) = 0 → f=g
+ integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0 → f=g
}.
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 ********************************)
a_associative_right: ∀a,f,g. a * (mult f g) = mult f (a * g)
}.
-record algebra (K: field) (V:vector_space K) (a_one:V) : Type \def
- { a_mult: V → V → V;
+record algebra (K: field) : Type \def
+ { a_vector_space:> vector_space K;
+ a_one: a_vector_space;
+ a_mult: a_vector_space → a_vector_space → a_vector_space;
a_algebra_properties: is_algebra ? ? a_mult a_one
}.
interpretation "Algebra product" 'times a b =
- (cic:/matita/integration_algebras/a_mult.con _ _ _ a b).
+ (cic:/matita/integration_algebras/a_mult.con _ a b).
definition ring_of_algebra ≝
- λK.λV:vector_space K.λone:V.λA:algebra ? V one.
- mk_ring V (a_mult ? ? ? A) one
- (a_ring ? ? ? ? (a_algebra_properties ? ? ? A)).
+ λK.λA:algebra K.
+ mk_ring A (a_mult ? A) (a_one ? A)
+ (a_ring ? ? ? ? (a_algebra_properties ? A)).
coercion cic:/matita/integration_algebras/ring_of_algebra.con.
-record is_f_algebra (K) (S:archimedean_riesz_space K) (one: S)
- (A:algebra ? S one) : Prop
-\def
-{ compat_mult_le:
- ∀f,g:S.
- le ? S 0 f → le ? S 0 g → le ? S 0 (a_mult ? ? ? A f g);
+record pre_f_algebra (K:ordered_field_ch0) : Type ≝
+ { fa_archimedean_riesz_space:> archimedean_riesz_space K;
+ fa_algebra_: algebra K;
+ fa_with: a_vector_space ? fa_algebra_ = rs_vector_space ? fa_archimedean_riesz_space
+ }.
+
+lemma fa_algebra: ∀K:ordered_field_ch0.pre_f_algebra K → algebra K.
+ intros (K A);
+ apply mk_algebra;
+ [ apply (rs_vector_space ? A)
+ | elim daemon
+ | elim daemon
+ | elim daemon
+ ]
+ qed.
+
+coercion cic:/matita/integration_algebras/fa_algebra.con.
+
+record is_f_algebra (K) (A:pre_f_algebra K) : Prop ≝
+{ compat_mult_le: ∀f,g:A.0 ≤ f → 0 ≤ g → 0 ≤ f*g;
compat_mult_meet:
- ∀f,g,h:S.
- meet ? S f g = 0 → meet ? S (a_mult ? ? ? A h f) g = 0
+ ∀f,g,h:A.(f ∧ g) = 0 → ((h*f) ∧ g) = 0
}.
-record f_algebra (K:ordered_field_ch0) (R:archimedean_riesz_space K) (one:R) :
-Type \def
-{ fa_algebra:> algebra ? R one;
- fa_f_algebra_properties: is_f_algebra ? ? ? fa_algebra
+record f_algebra (K:ordered_field_ch0) : Type ≝
+{ fa_pre_f_algebra:> pre_f_algebra K;
+ fa_f_algebra_properties: is_f_algebra ? fa_pre_f_algebra
}.
(* to be proved; see footnote 2 in the paper by Spitters *)
axiom symmetric_a_mult:
- ∀K,R,one.∀A:f_algebra K R one. symmetric ? (a_mult ? ? ? A).
+ ∀K.∀A:f_algebra K. symmetric ? (a_mult ? A).
record integration_f_algebra (R:real) : Type \def
{ ifa_integration_riesz_space:> integration_riesz_space R;
- ifa_f_algebra:>
- f_algebra ? ifa_integration_riesz_space
- (irs_unit ? ifa_integration_riesz_space)
+ ifa_f_algebra_: f_algebra R;
+ ifa_with:
+ fa_archimedean_riesz_space ? ifa_f_algebra_ =
+ irs_archimedean_riesz_space ? ifa_integration_riesz_space
}.
+
+axiom ifa_f_algebra: ∀R:real.integration_f_algebra R → f_algebra R.
+
+coercion cic:/matita/integration_algebras/ifa_f_algebra.con.
projects / helm.git / blobdiff