1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/integration_algebras/".
17 include "higher_order_defs/functions.ma".
19 include "nat/orders.ma".
21 definition left_neutral \def λC,op.λe:C. ∀x:C. op e x = x.
23 definition left_inverse \def λC,op.λe:C.λinv:C→C. ∀x:C. op (inv x) x = e.
25 record is_abelian_group (C:Type) (plus:C→C→C) (zero:C) (opp:C→C) : Prop \def
26 { (* abelian additive semigroup properties *)
27 plus_assoc: associative ? plus;
28 plus_comm: symmetric ? plus;
29 (* additive monoid properties *)
30 zero_neutral: left_neutral ? plus zero;
31 (* additive group properties *)
32 opp_inverse: left_inverse ? plus zero opp
35 record is_field (C:Type) (plus:C→C→C) (mult:C→C→C) (zero,one:C) (opp:C→C)
36 (inv:∀x:C.x ≠ zero →C) : Prop
38 { (* abelian group properties *)
39 abelian_group:> is_abelian_group ? plus zero opp;
40 (* abelian multiplicative semigroup properties *)
41 mult_assoc: associative ? mult;
42 mult_comm: symmetric ? mult;
43 (* multiplicative monoid properties *)
44 one_neutral: left_neutral ? mult one;
45 (* multiplicative group properties *)
46 inv_inverse: ∀x.∀p: x ≠ zero. mult (inv x p) x = one;
48 mult_plus_distr: distributive ? mult plus;
50 not_eq_zero_one: zero ≠ one
53 let rec sum (C:Type) (plus:C→C→C) (zero,one:C) (n:nat) on n ≝
56 | (S m) ⇒ plus one (sum C plus zero one m)
59 record field : Type \def
61 plus: carrier → carrier → carrier;
62 mult: carrier → carrier → carrier;
65 opp: carrier → carrier;
66 inv: ∀x:carrier. x ≠ zero → carrier;
67 field_properties: is_field ? plus mult zero one opp inv
70 definition sum_field ≝
71 λF:field. sum ? (plus F) (zero F) (one F).
73 notation "0" with precedence 89
76 interpretation "Field zero" 'zero =
77 (cic:/matita/integration_algebras/zero.con _).
79 notation "1" with precedence 89
82 interpretation "Field one" 'one =
83 (cic:/matita/integration_algebras/one.con _).
85 interpretation "Field plus" 'plus a b =
86 (cic:/matita/integration_algebras/plus.con _ a b).
88 interpretation "Field mult" 'times a b =
89 (cic:/matita/integration_algebras/mult.con _ a b).
91 interpretation "Field opp" 'uminus a =
92 (cic:/matita/integration_algebras/opp.con _ a).
94 record is_ordered_field_ch0 (C:Type) (plus,mult:C→C→C) (zero,one:C) (opp:C→C)
95 (inv:∀x:C.x ≠ zero → C) (le:C→C→Prop) : Prop \def
96 { (* field properties *)
97 of_is_field:> is_field C plus mult zero one opp inv;
98 of_mult_compat: ∀a,b. le zero a → le zero b → le zero (mult a b);
99 of_plus_compat: ∀a,b,c. le a b → le (plus a c) (plus b c);
100 of_weak_tricotomy : ∀a,b. a≠b → le a b ∨ le b a;
101 (* 0 characteristics *)
102 of_char0: ∀n. n > O → sum ? plus zero one n ≠ zero
105 record ordered_field_ch0 : Type \def
107 of_le: of_field → of_field → Prop;
108 of_ordered_field_properties:>
109 is_ordered_field_ch0 ? (plus of_field) (mult of_field) (zero of_field)
110 (one of_field) (opp of_field) (inv of_field) of_le
113 interpretation "Ordered field le" 'leq a b =
114 (cic:/matita/integration_algebras/of_le.con _ a b).
116 definition lt \def λF:ordered_field_ch0.λa,b:F.a ≤ b ∧ a ≠ b.
118 interpretation "Ordered field lt" 'lt a b =
119 (cic:/matita/integration_algebras/lt.con _ a b).
121 lemma le_zero_x_to_le_opp_x_zero: ∀F:ordered_field_ch0.∀x:F. 0 ≤ x → -x ≤ 0.
123 generalize in match (of_plus_compat ? ? ? ? ? ? ? ? F ? ? (-x) H); intro;
124 rewrite > (zero_neutral ? ? ? ? F) in H1;
125 rewrite > (plus_comm ? ? ? ? F) in H1;
126 rewrite > (opp_inverse ? ? ? ? F) in H1;
130 lemma le_x_zero_to_le_zero_opp_x: ∀F:ordered_field_ch0.∀x:F. x ≤ 0 → 0 ≤ -x.
132 generalize in match (of_plus_compat ? ? ? ? ? ? ? ? F ? ? (-x) H); intro;
133 rewrite > (zero_neutral ? ? ? ? F) in H1;
134 rewrite > (plus_comm ? ? ? ? F) in H1;
135 rewrite > (opp_inverse ? ? ? ? F) in H1;
139 (* To be proved for rings only *)
140 lemma eq_mult_zero_x_zero: ∀F:ordered_field_ch0.∀x:F.0*x=0.
142 generalize in match (zero_neutral ? ? ? ? F 0); intro;
143 generalize in match (eq_f ? ? (λy.x*y) ? ? H); intro; clear H;
144 rewrite > (mult_plus_distr ? ? ? ? ? ? ? F) in H1;
145 generalize in match (eq_f ? ? (λy.-(x*0)+y) ? ? H1); intro; clear H1;
146 rewrite < (plus_assoc ? ? ? ? F) in H;
147 rewrite > (opp_inverse ? ? ? ? F) in H;
148 rewrite > (zero_neutral ? ? ? ? F) in H;
149 rewrite > (mult_comm ? ? ? ? ? ? ? F) in H;
154 lemma eq_opp_x_times_opp_one_x: ∀F:ordered_field_ch0.∀x:F.-x = -1*x.
157 lemma not_eq_x_zero_to_lt_zero_mult_x_x:
158 ∀F:ordered_field_ch0.∀x:F. x ≠ 0 → 0 < x * x.
160 elim (of_weak_tricotomy ? ? ? ? ? ? ? ? F ? ? H);
161 [ generalize in match (le_x_zero_to_le_zero_opp_x F ? H1); intro;
162 generalize in match (of_mult_compat ? ? ? ? ? ? ? ? F ? ? H2 H2); intro;
165 axiom not_eq_sum_field_zero: ∀F,n. n > O → sum_field F n ≠ 0.
167 record is_vector_space (K: field) (C:Type) (plus:C→C→C) (zero:C) (opp:C→C)
170 { (* abelian group properties *)
171 vs_abelian_group: is_abelian_group ? plus zero opp;
172 (* other properties *)
173 vs_nilpotent: ∀v. mult 0 v = zero;
174 vs_neutral: ∀v. mult 1 v = v;
175 vs_distributive: ∀a,b,v. mult (a + b) v = plus (mult a v) (mult b v);
176 vs_associative: ∀a,b,v. mult (a * b) v = mult a (mult b v)
179 record is_lattice (C:Type) (join,meet:C→C→C) : Prop \def
180 { (* abelian semigroup properties *)
181 l_comm_j: symmetric ? join;
182 l_associative_j: associative ? join;
183 l_comm_m: symmetric ? meet;
184 l_associative_m: associative ? meet;
185 (* other properties *)
186 l_adsorb_j_m: ∀f,g. join f (meet f g) = f;
187 l_adsorb_m_j: ∀f,g. meet f (join f g) = f
190 definition le \def λC.λmeet:C→C→C.λf,g. meet f g = f.
192 record is_riesz_space (K:ordered_field_ch0) (C:Type) (plus:C→C→C) (zero:C)
193 (opp:C→C) (mult:K→C→C) (join,meet:C→C→C) : Prop \def
194 { (* vector space properties *)
195 rs_vector_space: is_vector_space K C plus zero opp mult;
196 (* lattice properties *)
197 rs_lattice: is_lattice C join meet;
198 (* other properties *)
199 rs_compat_le_plus: ∀f,g,h. le ? meet f g →le ? meet (plus f h) (plus g h);
200 rs_compat_le_times: ∀a,f. 0≤a → le ? meet zero f → le ? meet zero (mult a f)
203 definition absolute_value \def λC:Type.λopp.λjoin:C→C→C.λf.join f (opp f).
205 record is_archimedean_riesz_space (K:ordered_field_ch0) (C:Type) (plus:C→C→C)
206 (zero:C) (opp:C→C) (mult:Type_OF_ordered_field_ch0 K→C→C) (join,meet:C→C→C) : Prop \def
207 { ars_riesz_space: is_riesz_space ? ? plus zero opp mult join meet;
208 ars_archimedean: ∃u.∀n,a.∀p:n > O.
209 le C meet (absolute_value ? opp join a)
210 (mult (inv K (sum_field K n) (not_eq_sum_field_zero K n p)) u) →