matita/dama/ordered_groups.ma

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  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 set "baseuri" "cic:/matita/ordered_groups/".
  16
  17 include "groups.ma".
  18 include "ordered_sets.ma".
  19
  20 record pre_ordered_abelian_group : Type ≝
  21  { og_abelian_group_: abelian_group;
  22    og_tordered_set:> tordered_set;
  23    og_with: carr og_abelian_group_ = og_tordered_set
  24  }.
  25
  26 lemma og_abelian_group: pre_ordered_abelian_group → abelian_group.
  27 intro G; apply (mk_abelian_group G); [1,2,3: rewrite < (og_with G)]
  28 [apply (plus (og_abelian_group_ G));|apply zero;|apply opp]
  29 unfold apartness_OF_pre_ordered_abelian_group; cases (og_with G); simplify;
  30 [apply plus_assoc|apply plus_comm|apply zero_neutral|apply opp_inverse|apply plus_strong_ext]
  31 qed.
  32
  33 coercion cic:/matita/ordered_groups/og_abelian_group.con.
  34
  35 definition is_ordered_abelian_group ≝
  36  λG:pre_ordered_abelian_group. ∀f,g,h:G. f≤g → f+h≤g+h.
  37
  38 record ordered_abelian_group : Type ≝
  39  { og_pre_ordered_abelian_group:> pre_ordered_abelian_group;
  40    og_ordered_abelian_group_properties:
  41     is_ordered_abelian_group og_pre_ordered_abelian_group
  42  }.
  43
  44 lemma le_zero_x_to_le_opp_x_zero:
  45   ∀G:ordered_abelian_group.∀x:G.0 ≤ x → -x ≤ 0.
  46 intros (G x Px);
  47 generalize in match (og_ordered_abelian_group_properties ? ? ? (-x) Px); intro;
  48 (* ma cazzo, qui bisogna rifare anche i gruppi con ≈ ? *)
  49  rewrite > zero_neutral in H;
  50  rewrite > plus_comm in H;
  51  rewrite > opp_inverse in H;
  52  assumption.
  53 qed.
  54
  55 lemma le_x_zero_to_le_zero_opp_x: ∀G:ordered_abelian_group.∀x:G. x ≤ 0 → 0 ≤ -x.
  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.