--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+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 left_inverse \def λC,op.λe:C.λinv:C→C. ∀x:C. op (inv x) x = e.
+
+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 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
+≝
+ { (* abelian group properties *)
+ abelian_group:> is_abelian_group ? plus zero opp;
+ (* abelian multiplicative semigroup properties *)
+ mult_assoc: associative ? mult;
+ mult_comm: symmetric ? mult;
+ (* multiplicative monoid properties *)
+ one_neutral: left_neutral ? mult one;
+ (* multiplicative group properties *)
+ inv_inverse: ∀x.∀p: x ≠ zero. mult (inv x p) x = one;
+ (* ring properties *)
+ mult_plus_distr: distributive ? mult plus;
+ (* integral domain *)
+ not_eq_zero_one: zero ≠ one
+ }.
+
+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
+ { carrier:> Type;
+ plus: carrier → carrier → carrier;
+ mult: carrier → carrier → carrier;
+ zero: carrier;
+ one: carrier;
+ opp: carrier → carrier;
+ inv: ∀x:carrier. x ≠ zero → carrier;
+ field_properties: is_field ? plus mult zero one opp inv
+ }.
+
+definition sum_field ≝
+ λF:field. sum ? (plus F) (zero F) (one F).
+
+notation "0" with precedence 89
+for @{ 'zero }.
+
+interpretation "Field zero" 'zero =
+ (cic:/matita/integration_algebras/zero.con _).
+
+notation "1" with precedence 89
+for @{ 'one }.
+
+interpretation "Field one" 'one =
+ (cic:/matita/integration_algebras/one.con _).
+
+interpretation "Field plus" 'plus a b =
+ (cic:/matita/integration_algebras/plus.con _ a b).
+
+interpretation "Field mult" 'times a b =
+ (cic:/matita/integration_algebras/mult.con _ a b).
+
+interpretation "Field opp" 'uminus a =
+ (cic:/matita/integration_algebras/opp.con _ a).
+
+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);
+ 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
+ }.
+
+record ordered_field_ch0 : Type \def
+ { of_field:> field;
+ of_le: of_field → of_field → Prop;
+ of_ordered_field_properties:>
+ is_ordered_field_ch0 ? (plus of_field) (mult of_field) (zero of_field)
+ (one of_field) (opp of_field) (inv 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 ? ? ? ? F) in H1;
+ rewrite > (plus_comm ? ? ? ? F) in H1;
+ rewrite > (opp_inverse ? ? ? ? F) 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 ? ? ? ? F) in H1;
+ rewrite > (plus_comm ? ? ? ? F) in H1;
+ rewrite > (opp_inverse ? ? ? ? F) in H1;
+ assumption.
+qed.
+
+(* To be proved for rings only *)
+lemma eq_mult_zero_x_zero: ∀F:ordered_field_ch0.∀x:F.0*x=0.
+ intros;
+ generalize in match (zero_neutral ? ? ? ? F 0); intro;
+ generalize in match (eq_f ? ? (λy.x*y) ? ? H); intro; clear H;
+ rewrite > (mult_plus_distr ? ? ? ? ? ? ? F) in H1;
+ generalize in match (eq_f ? ? (λy.-(x*0)+y) ? ? H1); intro; clear H1;
+ rewrite < (plus_assoc ? ? ? ? F) in H;
+ rewrite > (opp_inverse ? ? ? ? F) in H;
+ rewrite > (zero_neutral ? ? ? ? F) in H;
+ rewrite > (mult_comm ? ? ? ? ? ? ? F) in H;
+ 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) (C:Type) (plus:C→C→C) (zero:C) (opp:C→C)
+ (mult:K→C→C) : Prop
+≝
+ { (* abelian group properties *)
+ vs_abelian_group: is_abelian_group ? plus zero opp;
+ (* other properties *)
+ vs_nilpotent: ∀v. mult 0 v = zero;
+ vs_neutral: ∀v. mult 1 v = v;
+ vs_distributive: ∀a,b,v. mult (a + b) v = plus (mult a v) (mult b v);
+ vs_associative: ∀a,b,v. mult (a * b) v = mult a (mult b v)
+ }.
+
+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
+ }.
+
+definition le \def λC.λmeet:C→C→C.λf,g. meet f g = f.
+
+record is_riesz_space (K:ordered_field_ch0) (C:Type) (plus:C→C→C) (zero:C)
+ (opp:C→C) (mult:K→C→C) (join,meet:C→C→C) : Prop \def
+ { (* vector space properties *)
+ rs_vector_space: is_vector_space K C plus zero opp mult;
+ (* 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 (mult a f)
+ }.
+
+definition absolute_value \def λC:Type.λopp.λjoin:C→C→C.λf.join f (opp f).
+
+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
+ }.
\ No newline at end of file
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