X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fdama%2Fintegration_algebras.ma;h=46bebc0e9da6745255d64691b99f5fe6a00eb7e2;hb=31bcb2d7cb6fc4c01af9d62b6ede25a9b82308aa;hp=1ebfad8344ec62f45b73933bad58ce4fd1810a04;hpb=4a073df57b56bf6d89ec0f806d5471388940deda;p=helm.git diff --git a/matita/dama/integration_algebras.ma b/matita/dama/integration_algebras.ma index 1ebfad834..46bebc0e9 100644 --- a/matita/dama/integration_algebras.ma +++ b/matita/dama/integration_algebras.ma @@ -14,38 +14,7 @@ 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 K 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:Type_OF_vector_space ? 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_norm (R:real) (V:vector_space R) (norm:Type_OF_vector_space ? V → R) - : Prop \def - { n_semi_norm:> is_semi_norm ? ? norm; - n_properness: ∀x:V. norm x = 0 → x = 0 - }. +include "vector_spaces.ma". record is_lattice (C:Type) (join,meet:C→C→C) : Prop \def { (* abelian semigroup properties *) @@ -69,6 +38,11 @@ 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. @@ -88,14 +62,39 @@ record riesz_space (K:ordered_field_ch0) : Type \def 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. le ? S (absolute_value ? S a) - (emult ? S - (inv ? (sum_field K n) (not_eq_sum_field_zero ? n p)) - u) → + ((inv ? (sum_field K n) (not_eq_sum_field_zero ? n p))* u) → a = 0 }. @@ -104,11 +103,11 @@ record archimedean_riesz_space (K:ordered_field_ch0) : Type \def ars_archimedean_property: is_archimedean_riesz_space ? ars_riesz_space }. -record is_integral (K) (R:archimedean_riesz_space K) (I:Type_OF_archimedean_riesz_space ? R→K) : Prop +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 (emult ? R k f) = k*(I f) + i_linear2: ∀f:R.∀k:K. I (k*f) = k*(I f) }. definition is_weak_unit ≝ @@ -118,14 +117,19 @@ definition is_weak_unit ≝ 2. Fremlin proves |x|/\u=0 \to u=0. How do we remove the absolute value? λR:real.λV:archimedean_riesz_space R.λunit: V. ∀x:V. meet x unit = 0 → u = 0. -*) λR:real.λV:archimedean_riesz_space R.λe:V.True. + 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). +(* Here we are avoiding a construction (the quotient space to define + f=g iff I(|f-g|)=0 *) record integration_riesz_space (R:real) : Type \def { irs_archimedean_riesz_space:> archimedean_riesz_space R; - irs_unit: Type_OF_archimedean_riesz_space ? irs_archimedean_riesz_space; + irs_unit: irs_archimedean_riesz_space; irs_weak_unit: is_weak_unit ? ? irs_unit; - integral: Type_OF_archimedean_riesz_space ? irs_archimedean_riesz_space → R; - irs_integral_properties: is_integral R irs_archimedean_riesz_space integral; + integral: irs_archimedean_riesz_space → R; + irs_integral_properties: is_integral ? ? integral; irs_limit1: ∀f:irs_archimedean_riesz_space. tends_to ? @@ -142,9 +146,101 @@ record integration_riesz_space (R:real) : Type \def ) * irs_unit))) 0; irs_quotient_space1: ∀f,g:irs_archimedean_riesz_space. - f=g → integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0 + 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). + +lemma induced_norm_is_norm: + ∀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_fun; + intros; + apply i_positive; + [ apply (irs_integral_properties ? V) + | (* difficile *) + elim daemon + ] + | intros; + unfold induced_norm_fun; + (* facile *) + elim daemon + | intros; + unfold induced_norm_fun; + (* difficile *) + elim daemon + ] + | intros; + 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 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). + +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 *) @@ -156,14 +252,14 @@ record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop record algebra (K: field) (V:vector_space K) (a_one:V) : Type \def { a_mult: V → V → V; - a_algebra_properties: is_algebra K V a_mult a_one + a_algebra_properties: is_algebra ? ? a_mult a_one }. interpretation "Algebra product" 'times a b = (cic:/matita/integration_algebras/a_mult.con _ _ _ a b). definition ring_of_algebra ≝ - λK.λV:vector_space K.λone:Type_OF_vector_space ? V.λA:algebra ? V one. + λK.λV:vector_space K.λone:V.λA:algebra ? V one. mk_ring V (a_mult ? ? ? A) one (a_ring ? ? ? ? (a_algebra_properties ? ? ? A)). @@ -180,8 +276,7 @@ record is_f_algebra (K) (S:archimedean_riesz_space K) (one: S) meet ? S f g = 0 → meet ? S (a_mult ? ? ? A h f) g = 0 }. -record f_algebra (K:ordered_field_ch0) (R:archimedean_riesz_space K) - (one:Type_OF_archimedean_riesz_space ? R) : +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 @@ -191,11 +286,9 @@ Type \def axiom symmetric_a_mult: ∀K,R,one.∀A:f_algebra K R one. symmetric ? (a_mult ? ? ? A). -(* Here we are avoiding a construction (the quotient space to define - f=g iff I(|f-g|)=0 *) 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) - }. \ No newline at end of file + }.