helm/software/matita/dama/ordered_groups.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 set "baseuri" "cic:/matita/ordered_groups/".
  16
  17 include "groups.ma".
  18 include "ordered_sets2.ma".
  19
  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
  24  }.
  25
  26 lemma og_ordered_set: pre_ordered_abelian_group → ordered_set.
  27  intro G;
  28  apply mk_ordered_set;
  29   [ apply (carrier (og_abelian_group G))
  30   | apply (eq_rect ? ? (λC:Type.C→C→Prop) ? ? (og_with G));
  31     apply os_le
  32   | apply
  33      (eq_rect' ? ?
  34        (λa:Type.λH:os_carrier (og_ordered_set_ G) = a.
  35         is_order_relation a
  36          (eq_rect Type (og_ordered_set_ G) (λC:Type.C→C→Prop)
  37           (os_le (og_ordered_set_ G)) a H))
  38        ? ? (og_with G));
  39     simplify;
  40     apply (os_order_relation_properties (og_ordered_set_ G))
  41   ]
  42 qed.
  43
  44 coercion cic:/matita/ordered_groups/og_ordered_set.con.
  45
  46 definition is_ordered_abelian_group ≝
  47  λG:pre_ordered_abelian_group. ∀f,g,h:G. f≤g → f+h≤g+h.
  48
  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
  53  }.
  54
  55 lemma le_zero_x_to_le_opp_x_zero: ∀G:ordered_abelian_group.∀x:G.0 ≤ x → -x ≤ 0.
  56  intros;
  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;
  61  assumption.
  62 qed.
  63
  64 lemma le_x_zero_to_le_zero_opp_x: ∀G:ordered_abelian_group.∀x:G. x ≤ 0 → 0 ≤ -x.
  65  intros;
  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;
  70  assumption.
  71 qed.