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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+
+
+include "attic/reals.ma".
+
+record is_vector_space (K: field) (G:abelian_group) (emult:K→G→G) : Prop
+≝
+ { vs_nilpotent: ∀v. emult 0 v = 0;
+   vs_neutral: ∀v. emult 1 v = 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 ? vs_abelian_group emult
+}.
+
+interpretation "Vector space external product" 'times a b =
+ (cic:/matita/attic/vector_spaces/emult.con _ _ a b).
+
+record is_semi_norm (R:real) (V: vector_space R) (semi_norm:V→R) : Prop \def
+ { sn_positive: ∀x:V. zero R ≤ 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 ≝
+ { n_semi_norm:> is_semi_norm ? ? norm;
+   n_properness: ∀x:V. norm x = 0 → x = 0
+ }.
+
+record norm (R:real) (V:vector_space R) : Type ≝
+ { n_function:1> V→R;
+   n_norm_properties: is_norm ? ? n_function
+ }.
+
+record is_semi_distance (R:real) (C:Type) (semi_d: C→C→R) : Prop ≝
+ { sd_positive: ∀x,y:C. zero R ≤ semi_d x y;
+   sd_properness: ∀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 ≝
+ { d_semi_distance:> is_semi_distance ? ? d;
+   d_properness: ∀x,y:C. d x y = 0 → x=y
+ }.
+
+record distance (R:real) (V:vector_space R) : Type ≝
+ { d_function:2> V→V→R;
+   d_distance_properties: is_distance ? ? d_function
+ }.
+
+definition induced_distance_fun ≝
+ λR:real.λV:vector_space R.λnorm:norm ? V.
+  λf,g:V.norm (f - g).
+
+theorem induced_distance_is_distance:
+ ∀R:real.∀V:vector_space R.∀norm:norm ? V.
+  is_distance ? ? (induced_distance_fun ? ? norm).
+elim daemon.(*
+ intros;
+ apply mk_is_distance;
+  [ apply mk_is_semi_distance;
+    [ unfold induced_distance_fun;
+      intros;
+      apply sn_positive;
+      apply n_semi_norm;
+      apply (n_norm_properties ? ? norm)
+    | unfold induced_distance_fun;
+      intros;
+      unfold minus;
+      rewrite < plus_comm;
+      rewrite > opp_inverse;
+      apply eq_semi_norm_zero_zero;
+      apply n_semi_norm;
+      apply (n_norm_properties ? ? norm)
+    | unfold induced_distance_fun;
+      intros;
+      (* ??? *)
+      elim daemon
+    ]
+  | unfold induced_distance_fun;
+    intros;
+    generalize in match (n_properness ? ? norm ? ? H);
+     [ intro;
+       (* facile *)
+       elim daemon
+     | apply (n_norm_properties ? ? norm)
+     ]
+  ].*)
+qed.
+
+definition induced_distance ≝
+ λR:real.λV:vector_space R.λnorm:norm ? V.
+  mk_distance ? ? (induced_distance_fun ? ? norm)
+   (induced_distance_is_distance ? ? norm).
+
+definition tends_to :
+ ∀R:real.∀V:vector_space R.∀d:distance ? V.∀f:nat→V.∀l:V.Prop.
+apply
+  (λR:real.λV:vector_space R.λd:distance ? V.λf:nat→V.λl:V.
+    ∀n:nat.∃m:nat.∀j:nat. m ≤ j →
+     d (f j) l ≤ inv R (sum_field ? (S n)) ?);
+ apply not_eq_sum_field_zero;
+ unfold;
+ autobatch.
+qed.
+
+definition is_cauchy_seq : ∀R:real.\forall V:vector_space R.
+\forall d:distance ? V.∀f:nat→V.Prop.
+ apply
+  (λR:real.λV: vector_space R. \lambda d:distance ? V.
+   \lambda f:nat→V.
+    ∀m:nat.
+     ∃n:nat.∀N. n ≤ N →
+      -(inv R (sum_field ? (S m)) ?) ≤ d (f N)  (f n)  ∧
+      d (f N)  (f n)≤ inv R (sum_field R (S m)) ?);
+ apply not_eq_sum_field_zero;
+ unfold;
+ autobatch.
+qed.
+
+definition is_complete ≝
+ λR:real.λV:vector_space R. 
+ λd:distance ? V.
+  ∀f:nat→V. is_cauchy_seq ? ? d f→
+   ex V (λl:V. tends_to ? ? d f l).