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/ordered_groups/".
18 include "ordered_sets.ma".
20 record pre_ordered_abelian_group : Type ≝
21 { og_abelian_group:> abelian_group;
22 og_tordered_set_: tordered_set;
23 og_with: exc_carr og_tordered_set_ = og_abelian_group
26 lemma og_tordered_set: pre_ordered_abelian_group → tordered_set.
27 intro G; apply mk_tordered_set;
28 [1: apply mk_pordered_set;
29 [1: apply (mk_excedence G);
30 [1: cases G; clear G; simplify; rewrite < H; clear H;
31 cases og_tordered_set_; clear og_tordered_set_; simplify;
32 cases tos_poset; simplify; cases pos_carr; simplify; assumption;
33 |2: cases G; simplify; cases H; simplify; clear H;
34 cases og_tordered_set_; simplify; clear og_tordered_set_;
35 cases tos_poset; simplify; cases pos_carr; simplify;
37 |3: cases G; simplify; cases H; simplify; cases og_tordered_set_; simplify;
38 cases tos_poset; simplify; cases pos_carr; simplify;
39 intros; apply c; assumption]
40 |2: cases G; simplify;
41 cases H; simplify; clear H; cases og_tordered_set_; simplify;
42 cases tos_poset; simplify; assumption;]
43 |2: simplify; (* SLOW, senza la simplify il widget muore *)
45 generalize in match (tos_totality og_tordered_set_);
46 unfold total_order_property;
47 cases H; simplify; cases og_tordered_set_; simplify;
48 cases tos_poset; simplify; cases pos_carr; simplify;
49 intros; apply f; assumption;]
52 coercion cic:/matita/ordered_groups/og_tordered_set.con.
54 definition is_ordered_abelian_group ≝
55 λG:pre_ordered_abelian_group. ∀f,g,h:G. f≤g → f+h≤g+h.
57 record ordered_abelian_group : Type ≝
58 { og_pre_ordered_abelian_group:> pre_ordered_abelian_group;
59 og_ordered_abelian_group_properties:
60 is_ordered_abelian_group og_pre_ordered_abelian_group
63 lemma le_zero_x_to_le_opp_x_zero:
64 ∀G:ordered_abelian_group.∀x:G.0 ≤ x → -x ≤ 0.
66 generalize in match (og_ordered_abelian_group_properties ? ? ? (-x) Px); intro;
67 (* ma cazzo, qui bisogna rifare anche i gruppi con ≈ ? *)
68 rewrite > zero_neutral in H;
69 rewrite > plus_comm in H;
70 rewrite > opp_inverse in H;
74 lemma le_x_zero_to_le_zero_opp_x: ∀G:ordered_abelian_group.∀x:G. x ≤ 0 → 0 ≤ -x.
76 generalize in match (og_ordered_abelian_group_properties ? ? ? (-x) H); intro;
77 rewrite > zero_neutral in H1;
78 rewrite > plus_comm in H1;
79 rewrite > opp_inverse in H1;