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;
+ plus_assoc_: associative ? plus;
+ plus_comm_: symmetric ? plus;
(* additive monoid properties *)
- zero_neutral: left_neutral ? plus zero;
+ zero_neutral_: left_neutral ? plus zero;
(* additive group properties *)
- opp_inverse: left_inverse ? plus zero opp
+ 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
}.
-record is_ring (C:Type) (plus:C→C→C) (mult:C→C→C) (zero:C) (opp:C→C) : Prop
+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
≝
- { (* abelian group properties *)
- abelian_group:> is_abelian_group ? plus zero opp;
- (* multiplicative semigroup properties *)
- mult_assoc: associative ? mult;
+ { (* 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 ? mult plus;
- mult_plus_distr_right: distributive_right C mult plus
+ 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_carrier:> Type;
- r_plus: r_carrier → r_carrier → r_carrier;
- r_mult: r_carrier → r_carrier → r_carrier;
- r_zero: r_carrier;
- r_opp: r_carrier → r_carrier;
- r_ring_properties:> is_ring ? r_plus r_mult r_zero r_opp
+ { 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
}.
-notation "0" with precedence 89
-for @{ 'zero }.
+theorem mult_assoc: ∀R:ring.associative ? (mult R).
+ intros;
+ apply (mult_assoc_ ? ? ? (r_ring_properties R)).
+qed.
-interpretation "Ring zero" 'zero =
- (cic:/matita/integration_algebras/r_zero.con _).
+theorem one_neutral_left: ∀R:ring.left_neutral ? (mult R) (one R).
+ intros;
+ apply (one_neutral_left_ ? ? ? (r_ring_properties R)).
+qed.
-interpretation "Ring plus" 'plus a b =
- (cic:/matita/integration_algebras/r_plus.con _ a b).
+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/r_mult.con _ a b).
+ (cic:/matita/integration_algebras/mult.con _ a b).
-interpretation "Ring opp" 'uminus a =
- (cic:/matita/integration_algebras/r_opp.con _ a).
+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 (zero_neutral R 0); intro;
generalize in match (eq_f ? ? (λy.y*x) ? ? H); intro; clear H;
- rewrite > (mult_plus_distr_right ? ? ? ? ? R) in H1;
+ rewrite > mult_plus_distr_right in H1;
generalize in match (eq_f ? ? (λy.-(0*x)+y) ? ? H1); intro; clear H1;
- rewrite < (plus_assoc ? ? ? ? R) in H;
- rewrite > (opp_inverse ? ? ? ? R) in H;
- rewrite > (zero_neutral ? ? ? ? R) in H;
+ rewrite < plus_assoc in H;
+ rewrite > opp_inverse in H;
+ rewrite > zero_neutral in H;
assumption.
qed.
-record is_field (C:Type) (plus:C→C→C) (mult:C→C→C) (zero,one:C) (opp:C→C)
- (inv:∀x:C.x ≠ zero →C) : Prop
+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
≝
- { (* ring properties *)
- ring_properties: is_ring ? plus mult zero opp;
- (* multiplicative abelian properties *)
- mult_comm: symmetric ? mult;
- (* multiplicative monoid properties *)
- one_neutral: left_neutral ? mult one;
+ { (* multiplicative abelian properties *)
+ mult_comm_: symmetric ? (mult R);
(* multiplicative group properties *)
- inv_inverse: ∀x.∀p: x ≠ zero. mult (inv x p) x = one;
- (* integral domain *)
- not_eq_zero_one: zero ≠ one
+ 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
record field : Type \def
{ f_ring:> ring;
- one: f_ring;
inv: ∀x:f_ring. x ≠ 0 → f_ring;
- field_properties:
- is_field ? (r_plus f_ring) (r_mult f_ring) (r_zero f_ring) one
- (r_opp f_ring) inv
+ field_properties: is_field f_ring inv
}.
-
-definition sum_field ≝
- λF:field. sum ? (r_plus F) (r_zero F) (one F).
-notation "1" with precedence 89
-for @{ 'one }.
+theorem mult_comm: ∀F:field.symmetric ? (mult F).
+ intro;
+ apply (mult_comm_ ? ? (field_properties F)).
+qed.
-interpretation "Field one" 'one =
- (cic:/matita/integration_algebras/one.con _).
+theorem inv_inverse: ∀F:field.∀x.∀p: x ≠ 0. mult ? (inv F x p) x = 1.
+ intro;
+ apply (inv_inverse_ ? ? (field_properties F)).
+qed.
-record is_ordered_field_ch0 (C:Type) (plus,mult:C→C→C) (zero,one:C) (opp:C→C)
- (inv:∀x:C.x ≠ zero → C) (le:C→C→Prop) : Prop \def
- { (* field properties *)
- of_is_field:> is_field C plus mult zero one opp inv;
- of_mult_compat: ∀a,b. le zero a → le zero b → le zero (mult a b);
- of_plus_compat: ∀a,b,c. le a b → le (plus a c) (plus b c);
+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 zero one n ≠ zero
+ 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 ? (r_plus of_field) (r_mult of_field) (r_zero of_field)
- (one of_field) (r_opp of_field) (inv of_field) of_le
+ of_ordered_field_properties:> is_ordered_field_ch0 of_field of_le
}.
interpretation "Ordered field le" 'leq a b =
interpretation "Ordered field lt" 'lt a b =
(cic:/matita/integration_algebras/lt.con _ a b).
-axiom le_zero_x_to_le_opp_x_zero: ∀F:ordered_field_ch0.∀x:F. 0 ≤ x → -x ≤ 0.
-(* intros;
+(*incompleto
+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 ? ? ? ? F) in H1;
- rewrite > (plus_comm ? ? ? ? F) in H1;
+ rewrite > (plus_comm ? ? ? ? F) in H1;
rewrite > (opp_inverse ? ? ? ? F) in H1;
+
assumption.
qed.*)
axiom not_eq_sum_field_zero: ∀F,n. n > O → sum_field F n ≠ 0.
-record is_vector_space (K: field) (C:Type) (plus:C→C→C) (zero:C) (opp:C→C)
- (emult:K→C→C) : Prop
+record is_vector_space (K: field) (G:abelian_group) (emult:K→G→G) : Prop
≝
- { (* abelian group properties *)
- vs_abelian_group: is_abelian_group ? plus zero opp;
- (* other properties *)
- vs_nilpotent: ∀v. emult 0 v = zero;
+ { vs_nilpotent: ∀v. emult 0 v = 0;
vs_neutral: ∀v. emult 1 v = v;
- vs_distributive: ∀a,b,v. emult (a + b) v = plus (emult a v) (emult b 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_adsorb_m_j: ∀f,g. meet f (join f g) = f
}.
-definition le \def λC.λmeet:C→C→C.λf,g. meet f g = f.
+record lattice (C:Type) : Type \def
+ { join: C → C → C;
+ meet: C → C → C;
+ l_lattice_properties: is_lattice ? join meet
+ }.
-record is_riesz_space (K:ordered_field_ch0) (C:Type) (plus:C→C→C) (zero:C)
- (opp:C→C) (emult:K→C→C) (join,meet:C→C→C) : Prop \def
- { (* vector space properties *)
- rs_vector_space: is_vector_space K C plus zero opp emult;
- (* lattice properties *)
- rs_lattice: is_lattice C join meet;
- (* other properties *)
- rs_compat_le_plus: ∀f,g,h. le ? meet f g →le ? meet (plus f h) (plus g h);
- rs_compat_le_times: ∀a,f. 0≤a → le ? meet zero f → le ? meet zero (emult a f)
+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.
+
+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)
}.
-definition absolute_value \def λC:Type.λopp.λjoin:C→C→C.λf.join f (opp f).
+record riesz_space : Type \def
+ { rs_ordered_field_ch0: ordered_field_ch0;
+ rs_vector_space:> vector_space rs_ordered_field_ch0;
+ rs_lattice:> lattice rs_vector_space;
+ rs_riesz_space_properties: is_riesz_space ? rs_vector_space rs_lattice
+ }.
-record is_archimedean_riesz_space (K:ordered_field_ch0) (C:Type) (plus:C→C→C)
- (zero:C) (opp:C→C) (mult:Type_OF_ordered_field_ch0 K→C→C) (join,meet:C→C→C)
- :Prop \def
- { ars_riesz_space: is_riesz_space ? ? plus zero opp mult join meet;
- ars_archimedean: ∃u.∀n,a.∀p:n > O.
- le C meet (absolute_value ? opp join a)
- (mult (inv K (sum_field K n) (not_eq_sum_field_zero K n p)) u) →
- a = zero
+definition absolute_value \def λS:riesz_space.λf.join ? S f (-f).
+
+record is_archimedean_riesz_space (S:riesz_space) : 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) →
+ a = 0
}.
-record is_algebra (K: field) (C:Type) (plus:C→C→C) (zero:C) (opp:C→C)
- (emult:K→C→C) (mult:C→C→C) : Prop
+record archimedean_riesz_space : Type \def
+ { ars_riesz_space:> riesz_space;
+ ars_archimedean_property: is_archimedean_riesz_space ars_riesz_space
+ }.
+
+record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
≝
- { (* vector space properties *)
- a_vector_space_properties: is_vector_space ? ? plus zero opp emult;
- (* ring properties *)
- a_ring: is_ring ? plus mult zero opp;
+ { (* ring properties *)
+ a_ring: is_ring V mult one;
(* algebra properties *)
- a_associative_left: ∀a,f,g. emult a (mult f g) = mult (emult a f) g;
- a_associative_right: ∀a,f,g. emult a (mult f g) = mult f (emult a g)
- }.
\ No newline at end of file
+ a_associative_left: ∀a,f,g. a * (mult f g) = mult (a * f) g;
+ a_associative_right: ∀a,f,g. a * (mult f g) = mult f (a * g)
+ }.
+
+record algebra (K: field) (V:vector_space K) : Type \def
+ { a_mult: V → V → V;
+ a_one: V;
+ a_algebra_properties: is_algebra K V 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)).
+
+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
+\def
+{ compat_mult_le:
+ ∀f,g:S.
+ 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
+}.
+
+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
+}.
+
+(* to be proved; see footnote 2 in the paper by Spitters *)
+axiom symmetric_a_mult: ∀A:f_algebra. symmetric ? (a_mult ? ? A).
projects / helm.git / blobdiff