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_sets2.ma".
20 record pre_ordered_abelian_group : Type ≝
21 { og_abelian_group:> abelian_group;
22 og_ordered_set_: ordered_set;
23 og_with: os_carrier og_ordered_set_ = og_abelian_group
26 lemma og_ordered_set: pre_ordered_abelian_group → ordered_set.
29 [ apply (carrier (og_abelian_group G))
30 | apply (eq_rect ? ? (λC:Type.C→C→Prop) ? ? (og_with G));
34 (λa:Type.λH:os_carrier (og_ordered_set_ G) = a.
36 (eq_rect Type (og_ordered_set_ G) (λC:Type.C→C→Prop)
37 (os_le (og_ordered_set_ G)) a H))
40 apply (os_order_relation_properties (og_ordered_set_ G))
44 coercion cic:/matita/ordered_groups/og_ordered_set.con.
46 definition is_ordered_abelian_group ≝
47 λG:pre_ordered_abelian_group. ∀f,g,h:G. f≤g → f+h≤g+h.
49 record ordered_abelian_group : Type ≝
50 { og_pre_ordered_abelian_group:> pre_ordered_abelian_group;
51 og_ordered_abelian_group_properties:
52 is_ordered_abelian_group og_pre_ordered_abelian_group
55 lemma le_zero_x_to_le_opp_x_zero: ∀G:ordered_abelian_group.∀x:G.0 ≤ x → -x ≤ 0.
57 generalize in match (og_ordered_abelian_group_properties ? ? ? (-x) H); intro;
58 rewrite > zero_neutral in H1;
59 rewrite > plus_comm in H1;
60 rewrite > opp_inverse in H1;
64 lemma le_x_zero_to_le_zero_opp_x: ∀G:ordered_abelian_group.∀x:G. x ≤ 0 → 0 ≤ -x.
66 generalize in match (og_ordered_abelian_group_properties ? ? ? (-x) H); intro;
67 rewrite > zero_neutral in H1;
68 rewrite > plus_comm in H1;
69 rewrite > opp_inverse in H1;