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
+ 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.
+
record is_lattice (C:Type) (join,meet:C→C→C) : Prop \def
{ (* abelian semigroup properties *)
l_comm_j: symmetric ? join;
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.
{ 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
}.
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 ≝
+ 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).
+ intros;
+ apply mk_is_norm;
+ [ apply mk_is_semi_norm;
+ [ unfold induced_norm;
+ intros;
+ apply i_positive;
+ [ apply (irs_integral_properties ? V)
+ | (* difficile *)
+ elim daemon
+ ]
+ | intros;
+ unfold induced_norm;
+ (* facile *)
+ elim daemon
+ | intros;
+ unfold induced_norm;
+ (* difficile *)
+ elim daemon
+ ]
+ | intros;
+ unfold induced_norm in H;
+ apply irs_quotient_space1;
+ unfold minus;
+ rewrite < plus_comm;
+ rewrite < eq_zero_opp_zero;
+ rewrite > zero_neutral;
+ assumption
+ ].
+qed.
+
+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).
+ intros;
+ unfold distance_induced_by_integral;
+ apply induced_distance_is_distance;
+ apply induced_norm_is_norm.
+qed.
+
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 "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)
+ }.
\ No newline at end of file
projects / helm.git / blobdiff