X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fdama%2Fintegration_algebras.ma;h=ca43093807d3eceabebda8b75c0dfc8a068a5215;hb=6968ba1ad67ba19e9d794260dc21ba2246d31dac;hp=2a2aae51dbcde98f5c4b773f941865fbb6554e6e;hpb=348acb421355c345a9af0754c1c16508a43eeea5;p=helm.git diff --git a/matita/dama/integration_algebras.ma b/matita/dama/integration_algebras.ma index 2a2aae51d..ca4309380 100644 --- a/matita/dama/integration_algebras.ma +++ b/matita/dama/integration_algebras.ma @@ -14,56 +14,17 @@ 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 K vs_abelian_group emult -}. - -interpretation "Vector space external product" 'times a b = - (cic:/matita/integration_algebras/emult.con _ _ a b). - -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 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 @@ -72,16 +33,54 @@ 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). +(**************** Normed Riesz spaces ****************************) + +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 riesz_norm (R:real) (V:riesz_space R) : Type ≝ + { 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 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) - (emult ? S - (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 }. @@ -90,6 +89,138 @@ record archimedean_riesz_space (K:ordered_field_ch0) : Type \def ars_archimedean_property: is_archimedean_riesz_space ? ars_riesz_space }. +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 + defining it on Riesz spaces) + 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. + 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: irs_archimedean_riesz_space; + irs_weak_unit: is_weak_unit ? ? irs_unit; + integral: irs_archimedean_riesz_space → R; + 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))) + (integral f); + irs_limit2: + ∀f:irs_archimedean_riesz_space. + tends_to ? + (λn. + integral (meet ? irs_archimedean_riesz_space 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 + }. + +definition induced_norm_fun ≝ + λR:real.λV:integration_riesz_space R.λf:V. + integral ? V (absolute_value ? ? f). + +lemma induced_norm_is_norm: + ∀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 positive; + [ apply (irs_positive_linear ? 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 *) @@ -99,66 +230,45 @@ record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop a_associative_right: ∀a,f,g. a * (mult f g) = mult f (a * g) }. -record algebra (K: field) (V:vector_space K) : Type \def +record algebra (K: field) (V:vector_space K) (a_one:V) : Type \def { a_mult: V → V → V; - a_one: 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). -interpretation "Algebra one" 'one = - (cic:/matita/integration_algebras/a_one.con _ _ _). - definition ring_of_algebra ≝ - λK.λV:vector_space K.λA:algebra ? V. - mk_ring V (a_mult ? ? A) (a_one ? ? A) - (a_ring ? ? ? ? (a_algebra_properties ? ? A)). + λK.λV:vector_space K.λone:V.λA:algebra ? V one. + mk_ring V (a_mult ? ? ? A) one + (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) (A:algebra ? S) : Prop +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); + le ? S 0 f → le ? S 0 g → le ? S 0 (a_mult ? ? ? A f g); compat_mult_meet: ∀f,g,h:S. - meet ? S f g = 0 → meet ? S (a_mult ? ? A h f) g = 0 + meet ? S f g = 0 → meet ? S (a_mult ? ? ? A h f) g = 0 }. -record f_algebra (K:ordered_field_ch0) : Type \def -{ fa_archimedean_riesz_space:> archimedean_riesz_space K; - fa_algebra:> algebra ? fa_archimedean_riesz_space; - fa_f_algebra_properties: is_f_algebra ? fa_archimedean_riesz_space fa_algebra +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 }. (* to be proved; see footnote 2 in the paper by Spitters *) -axiom symmetric_a_mult: ∀K.∀A:f_algebra K. symmetric ? (a_mult ? ? A). +axiom symmetric_a_mult: + ∀K,R,one.∀A:f_algebra K R one. symmetric ? (a_mult ? ? ? A). -record is_integral (K) (A:f_algebra K) (I:Type_OF_f_algebra ? A→K) : Prop -\def - { i_positive: ∀f:Type_OF_f_algebra ? A. le ? (lattice_OF_f_algebra ? A) 0 f → of_le K 0 (I f); - i_linear1: ∀f,g:Type_OF_f_algebra ? A. I (f + g) = I f + I g; - i_linear2: ∀f:A.∀k:K. I (emult ? A k f) = k*(I f) - }. - -(* Here we are avoiding a construction (the quotient space to define - f=g iff I(|f-g|)=0 *) -record is_integration_f_algebra (K) (A:f_algebra K) (I:Type_OF_f_algebra ? A→K) : Prop -\def - { ifa_integral: is_integral ? ? I; - ifa_limit1: - ∀f:A. tends_to ? (λn.I(meet ? A f ((sum_field K n)*(a_one ? ? A)))) (I f); - ifa_limit2: - ∀f:A. - tends_to ? - (λn. - I (meet ? A f - ((inv ? (sum_field K (S n)) - (not_eq_sum_field_zero K (S n) (le_S_S O n (le_O_n n))) - ) * (a_one ? ? A)))) 0; - ifa_quotient_space1: - ∀f,g:A. f=g → I(absolute_value ? A (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) }.