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 (**************************************************************************)
17 include "attic/reals.ma".
19 record is_vector_space (K: field) (G:abelian_group) (emult:K→G→G) : Prop
21 { vs_nilpotent: ∀v. emult 0 v = 0;
22 vs_neutral: ∀v. emult 1 v = v;
23 vs_distributive: ∀a,b,v. emult (a + b) v = (emult a v) + (emult b v);
24 vs_associative: ∀a,b,v. emult (a * b) v = emult a (emult b v)
27 record vector_space (K:field): Type \def
28 { vs_abelian_group :> abelian_group;
29 emult: K → vs_abelian_group → vs_abelian_group;
30 vs_vector_space_properties :> is_vector_space ? vs_abelian_group emult
33 interpretation "Vector space external product" 'times a b = (emult ? ? a b).
35 record is_semi_norm (R:real) (V: vector_space R) (semi_norm:V→R) : Prop \def
36 { sn_positive: ∀x:V. zero R ≤ semi_norm x;
37 sn_omogeneous: ∀a:R.∀x:V. semi_norm (a*x) = (abs ? a) * semi_norm x;
38 sn_triangle_inequality: ∀x,y:V. semi_norm (x + y) ≤ semi_norm x + semi_norm y
41 theorem eq_semi_norm_zero_zero:
42 ∀R:real.∀V:vector_space R.∀semi_norm:V→R.
43 is_semi_norm ? ? semi_norm →
50 record is_norm (R:real) (V:vector_space R) (norm:V → R) : Prop ≝
51 { n_semi_norm:> is_semi_norm ? ? norm;
52 n_properness: ∀x:V. norm x = 0 → x = 0
55 record norm (R:real) (V:vector_space R) : Type ≝
57 n_norm_properties: is_norm ? ? n_function
60 record is_semi_distance (R:real) (C:Type) (semi_d: C→C→R) : Prop ≝
61 { sd_positive: ∀x,y:C. zero R ≤ semi_d x y;
62 sd_properness: ∀x:C. semi_d x x = 0;
63 sd_triangle_inequality: ∀x,y,z:C. semi_d x z ≤ semi_d x y + semi_d z y
66 record is_distance (R:real) (C:Type) (d:C→C→R) : Prop ≝
67 { d_semi_distance:> is_semi_distance ? ? d;
68 d_properness: ∀x,y:C. d x y = 0 → x=y
71 record distance (R:real) (V:vector_space R) : Type ≝
72 { d_function:2> V→V→R;
73 d_distance_properties: is_distance ? ? d_function
76 definition induced_distance_fun ≝
77 λR:real.λV:vector_space R.λnorm:norm ? V.
80 theorem induced_distance_is_distance:
81 ∀R:real.∀V:vector_space R.∀norm:norm ? V.
82 is_distance ? ? (induced_distance_fun ? ? norm).
86 [ apply mk_is_semi_distance;
87 [ unfold induced_distance_fun;
91 apply (n_norm_properties ? ? norm)
92 | unfold induced_distance_fun;
96 rewrite > opp_inverse;
97 apply eq_semi_norm_zero_zero;
99 apply (n_norm_properties ? ? norm)
100 | unfold induced_distance_fun;
105 | unfold induced_distance_fun;
107 generalize in match (n_properness ? ? norm ? ? H);
111 | apply (n_norm_properties ? ? norm)
116 definition induced_distance ≝
117 λR:real.λV:vector_space R.λnorm:norm ? V.
118 mk_distance ? ? (induced_distance_fun ? ? norm)
119 (induced_distance_is_distance ? ? norm).
121 definition tends_to :
122 ∀R:real.∀V:vector_space R.∀d:distance ? V.∀f:nat→V.∀l:V.Prop.
124 (λR:real.λV:vector_space R.λd:distance ? V.λf:nat→V.λl:V.
125 ∀n:nat.∃m:nat.∀j:nat. m ≤ j →
126 d (f j) l ≤ inv R (sum_field ? (S n)) ?);
127 apply not_eq_sum_field_zero;
132 definition is_cauchy_seq : ∀R:real.\forall V:vector_space R.
133 \forall d:distance ? V.∀f:nat→V.Prop.
135 (λR:real.λV: vector_space R. \lambda d:distance ? V.
139 -(inv R (sum_field ? (S m)) ?) ≤ d (f N) (f n) ∧
140 d (f N) (f n)≤ inv R (sum_field R (S m)) ?);
141 apply not_eq_sum_field_zero;
146 definition is_complete ≝
147 λR:real.λV:vector_space R.
149 ∀f:nat→V. is_cauchy_seq ? ? d f→
150 ex V (λl:V. tends_to ? ? d f l).