set "baseuri" "cic:/matita/integration_algebras/".
-include "higher_order_defs/functions.ma".
-include "nat/nat.ma".
-include "nat/orders.ma".
-
-definition left_neutral \def λC,op.λe:C. ∀x:C. op e x = x.
-
-definition right_neutral \def λC,op. λe:C. ∀x:C. op x e=x.
-
-definition left_inverse \def λC,op.λe:C.λinv:C→C. ∀x:C. op (inv x) x = e.
-
-definition right_inverse \def λC,op.λe:C.λ inv: C→ C. ∀x:C. op x (inv x)=e.
-
-definition distributive_left ≝
- λA:Type.λf:A→A→A.λg:A→A→A.
- ∀x,y,z. f x (g y z) = g (f x y) (f x z).
-
-definition distributive_right ≝
- λA:Type.λf:A→A→A.λg:A→A→A.
- ∀x,y,z. f (g x y) z = g (f x z) (f y z).
-
-record is_abelian_group (C:Type) (plus:C→C→C) (zero:C) (opp:C→C) : Prop \def
- { (* abelian additive semigroup properties *)
- plus_assoc_: associative ? plus;
- plus_comm_: symmetric ? plus;
- (* additive monoid properties *)
- zero_neutral_: left_neutral ? plus zero;
- (* additive group properties *)
- opp_inverse_: left_inverse ? plus zero opp
- }.
-
-record abelian_group : Type \def
- { carrier:> Type;
- plus: carrier → carrier → carrier;
- zero: carrier;
- opp: carrier → carrier;
- ag_abelian_group_properties: is_abelian_group ? plus zero opp
- }.
-
-notation "0" with precedence 89
-for @{ 'zero }.
-
-interpretation "Ring zero" 'zero =
- (cic:/matita/integration_algebras/zero.con _).
-
-interpretation "Ring plus" 'plus a b =
- (cic:/matita/integration_algebras/plus.con _ a b).
-
-interpretation "Ring opp" 'uminus a =
- (cic:/matita/integration_algebras/opp.con _ a).
-
-theorem plus_assoc: ∀G:abelian_group. associative ? (plus G).
- intro;
- apply (plus_assoc_ ? ? ? ? (ag_abelian_group_properties G)).
-qed.
-
-theorem plus_comm: ∀G:abelian_group. symmetric ? (plus G).
- intro;
- apply (plus_comm_ ? ? ? ? (ag_abelian_group_properties G)).
-qed.
-
-theorem zero_neutral: ∀G:abelian_group. left_neutral ? (plus G) 0.
- intro;
- apply (zero_neutral_ ? ? ? ? (ag_abelian_group_properties G)).
-qed.
-
-theorem opp_inverse: ∀G:abelian_group. left_inverse ? (plus G) 0 (opp G).
- intro;
- apply (opp_inverse_ ? ? ? ? (ag_abelian_group_properties G)).
-qed.
-
-lemma cancellationlaw: ∀G:abelian_group.∀x,y,z:G. x+y=x+z → y=z.
-intros;
-generalize in match (eq_f ? ? (λa.-x +a) ? ? H);
-intros; clear H;
-rewrite < plus_assoc in H1;
-rewrite < plus_assoc in H1;
-rewrite > opp_inverse in H1;
-rewrite > zero_neutral in H1;
-rewrite > zero_neutral in H1;
-assumption.
-qed.
-
-(****************************** rings *********************************)
-
-record is_ring (G:abelian_group) (mult:G→G→G) (one:G) : Prop
-≝
- { (* multiplicative monoid properties *)
- mult_assoc_: associative ? mult;
- one_neutral_left_: left_neutral ? mult one;
- one_neutral_right_: right_neutral ? mult one;
- (* ring properties *)
- mult_plus_distr_left_: distributive_left ? mult (plus G);
- mult_plus_distr_right_: distributive_right ? mult (plus G);
- not_eq_zero_one_: (0 ≠ one)
- }.
-
-record ring : Type \def
- { r_abelian_group:> abelian_group;
- mult: r_abelian_group → r_abelian_group → r_abelian_group;
- one: r_abelian_group;
- r_ring_properties: is_ring r_abelian_group mult one
- }.
-
-theorem mult_assoc: ∀R:ring.associative ? (mult R).
- intros;
- apply (mult_assoc_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem one_neutral_left: ∀R:ring.left_neutral ? (mult R) (one R).
- intros;
- apply (one_neutral_left_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem one_neutral_right: ∀R:ring.right_neutral ? (mult R) (one R).
- intros;
- apply (one_neutral_right_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem mult_plus_distr_left: ∀R:ring.distributive_left ? (mult R) (plus R).
- intros;
- apply (mult_plus_distr_left_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem mult_plus_distr_right: ∀R:ring.distributive_right ? (mult R) (plus R).
- intros;
- apply (mult_plus_distr_right_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem not_eq_zero_one: ∀R:ring.0 ≠ one R.
- intros;
- apply (not_eq_zero_one_ ? ? ? (r_ring_properties R)).
-qed.
-
-interpretation "Ring mult" 'times a b =
- (cic:/matita/integration_algebras/mult.con _ a b).
-
-notation "1" with precedence 89
-for @{ 'one }.
-
-interpretation "Field one" 'one =
- (cic:/matita/integration_algebras/one.con _).
-
-lemma eq_mult_zero_x_zero: ∀R:ring.∀x:R.0*x=0.
- intros;
- generalize in match (zero_neutral R 0); intro;
- generalize in match (eq_f ? ? (λy.y*x) ? ? H); intro; clear H;
- rewrite > mult_plus_distr_right in H1;
- generalize in match (eq_f ? ? (λy.-(0*x)+y) ? ? H1); intro; clear H1;
- rewrite < plus_assoc in H;
- rewrite > opp_inverse in H;
- rewrite > zero_neutral in H;
- assumption.
-qed.
-
-lemma eq_mult_x_zero_zero: ∀R:ring.∀x:R.x*0=0.
-intros;
-generalize in match (zero_neutral R 0);
-intro;
-generalize in match (eq_f ? ? (\lambda y.x*y) ? ? H); intro; clear H;
-rewrite > mult_plus_distr_left in H1;
-generalize in match (eq_f ? ? (\lambda y. (-(x*0)) +y) ? ? H1);intro;
-clear H1;
-rewrite < plus_assoc in H;
-rewrite > opp_inverse in H;
-rewrite > zero_neutral in H;
-assumption.
-qed.
-
-record is_field (R:ring) (inv:∀x:R.x ≠ 0 → R) : Prop
-≝
- { (* multiplicative abelian properties *)
- mult_comm_: symmetric ? (mult R);
- (* multiplicative group properties *)
- inv_inverse_: ∀x.∀p: x ≠ 0. mult ? (inv x p) x = 1
- }.
-
-lemma opp_opp: \forall R:ring. \forall x:R. (-(-x))=x.
-intros;
-apply (cancellationlaw ? (-x) ? ?);
-rewrite > (opp_inverse R x);
-rewrite > plus_comm;
-rewrite > opp_inverse;
-reflexivity.
-qed.
-
-
-let rec sum (C:Type) (plus:C→C→C) (zero,one:C) (n:nat) on n ≝
- match n with
- [ O ⇒ zero
- | (S m) ⇒ plus one (sum C plus zero one m)
- ].
-
-record field : Type \def
- { f_ring:> ring;
- inv: ∀x:f_ring. x ≠ 0 → f_ring;
- field_properties: is_field f_ring inv
- }.
-
-theorem mult_comm: ∀F:field.symmetric ? (mult F).
- intro;
- apply (mult_comm_ ? ? (field_properties F)).
-qed.
-
-theorem inv_inverse: ∀F:field.∀x.∀p: x ≠ 0. mult ? (inv F x p) x = 1.
- intro;
- apply (inv_inverse_ ? ? (field_properties F)).
-qed.
-
-definition sum_field ≝
- λF:field. sum ? (plus F) (zero F) (one F).
-
-record is_ordered_field_ch0 (F:field) (le:F→F→Prop) : Prop \def
- { of_mult_compat: ∀a,b. le 0 a → le 0 b → le 0 (a*b);
- of_plus_compat: ∀a,b,c. le a b → le (a+c) (b+c);
- of_weak_tricotomy : ∀a,b. a≠b → le a b ∨ le b a;
- (* 0 characteristics *)
- of_char0: ∀n. n > O → sum ? (plus F) 0 1 n ≠ 0
- }.
-
-record ordered_field_ch0 : Type \def
- { of_field:> field;
- of_le: of_field → of_field → Prop;
- of_ordered_field_properties:> is_ordered_field_ch0 of_field of_le
- }.
-
-interpretation "Ordered field le" 'leq a b =
- (cic:/matita/integration_algebras/of_le.con _ a b).
-
-definition lt \def λF:ordered_field_ch0.λa,b:F.a ≤ b ∧ a ≠ b.
-
-interpretation "Ordered field lt" 'lt a b =
- (cic:/matita/integration_algebras/lt.con _ a b).
-
-lemma le_zero_x_to_le_opp_x_zero: ∀F:ordered_field_ch0.∀x:F. 0 ≤ x → -x ≤ 0.
-intros;
- generalize in match (of_plus_compat ? ? F ? ? (-x) H); intro;
- rewrite > zero_neutral in H1;
- rewrite > plus_comm in H1;
- rewrite > opp_inverse in H1;
- assumption.
-qed.
-
-lemma le_x_zero_to_le_zero_opp_x: ∀F:ordered_field_ch0.∀x:F. x ≤ 0 → 0 ≤ -x.
- intros;
- generalize in match (of_plus_compat ? ? F ? ? (-x) H); intro;
- rewrite > zero_neutral in H1;
- rewrite > plus_comm in H1;
- rewrite > opp_inverse in H1;
- assumption.
-qed.
-
-(*
-lemma eq_opp_x_times_opp_one_x: ∀F:ordered_field_ch0.∀x:F.-x = -1*x.
- intros;
-
-lemma not_eq_x_zero_to_lt_zero_mult_x_x:
- ∀F:ordered_field_ch0.∀x:F. x ≠ 0 → 0 < x * x.
- intros;
- elim (of_weak_tricotomy ? ? ? ? ? ? ? ? F ? ? H);
- [ generalize in match (le_x_zero_to_le_zero_opp_x F ? H1); intro;
- generalize in match (of_mult_compat ? ? ? ? ? ? ? ? F ? ? H2 H2); intro;
-*)
-
-axiom not_eq_sum_field_zero: ∀F,n. n > O → sum_field F n ≠ 0.
-
-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).
+include "vector_spaces.ma".
record is_lattice (C:Type) (join,meet:C→C→C) : Prop \def
{ (* abelian semigroup properties *)
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.
rs_compat_le_times: ∀a:K.∀f. of_le ? 0 a → le ? L 0 f → le ? L 0 (a*f)
}.
-record riesz_space : Type \def
- { rs_ordered_field_ch0: ordered_field_ch0;
- rs_vector_space:> vector_space rs_ordered_field_ch0;
+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
}.
-definition absolute_value \def λS:riesz_space.λf.join ? S f (-f).
+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 (S:riesz_space) : Prop
+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 (rs_ordered_field_ch0 S) n) (not_eq_sum_field_zero ? n p))
- u) →
+ (absolute_value ? S a)
+ ((inv ? (sum_field K n) (not_eq_sum_field_zero ? n p))* u) →
a = 0
}.
-record archimedean_riesz_space : Type \def
- { ars_riesz_space:> riesz_space;
- ars_archimedean_property: is_archimedean_riesz_space ars_riesz_space
+record archimedean_riesz_space (K:ordered_field_ch0) : Type \def
+ { ars_riesz_space:> riesz_space K;
+ 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
+ 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_integral_properties: is_integral ? ? 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 ? ? (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 *)
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 "Field 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 (S:archimedean_riesz_space)
- (A:algebra (rs_ordered_field_ch0 (ars_riesz_space S)) 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 : Type \def
-{ fa_archimedean_riesz_space:> archimedean_riesz_space;
- 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: ∀A:f_algebra. symmetric ? (a_mult ? ? A).
+axiom symmetric_a_mult:
+ ∀K,R,one.∀A:f_algebra K R one. 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)
+ }.
projects / helm.git / blobdiff