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: exc_carr og_tordered_set_ = og_abelian_group
  24  }.
  25
  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;
  36             intros; apply H;
  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 *)
  44     cases G; simplify;
  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;]
  50 qed.
  51
  52 coercion cic:/matita/ordered_groups/og_tordered_set.con.
  53
  54 definition is_ordered_abelian_group ≝
  55  λG:pre_ordered_abelian_group. ∀f,g,h:G. f≤g → f+h≤g+h.
  56
  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
  61  }.
  62
  63 lemma le_zero_x_to_le_opp_x_zero:
  64   ∀G:ordered_abelian_group.∀x:G.0 ≤ x → -x ≤ 0.
  65 intros (G x Px);
  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 H1;
  69  rewrite > plus_comm in H1;
  70  rewrite > opp_inverse in H1;
  71  assumption.
  72 qed.
  73
  74 lemma le_x_zero_to_le_zero_opp_x: ∀G:ordered_abelian_group.∀x:G. x ≤ 0 → 0 ≤ -x.
  75  intros;
  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;
  80  assumption.
  81 qed.