matita/dama/ordered_groups.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  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 "ordered_sets.ma".
  18 include "groups.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_rewl: ∀E:excedence.∀x,z,y:E. x ≈ y → x ≤ z → y ≤ z.
  45 intros (E x z y); apply (le_transitive ???? ? H1);
  46 clear H1 z; unfold in H; unfold; intro H1; apply H; clear H;
  47 lapply ap_cotransitive;
  48 intros (G x z y); intro Eyz;
  49
  50
  51 lemma plus_cancr_le:
  52   ∀G:ordered_abelian_group.∀x,y,z:G.x+z ≤ y + z → x ≤ y.
  53 intros 5 (G x y z L);
  54
  55  apply L; clear L; elim (exc_cotransitive ???z Exy);
  56
  57 lemma le_zero_x_to_le_opp_x_zero:
  58   ∀G:ordered_abelian_group.∀x:G.0 ≤ x → -x ≤ 0.
  59 intros (G x Px);
  60 generalize in match (og_ordered_abelian_group_properties ? ? ? (-x) Px); intro;
  61 (* ma cazzo, qui bisogna rifare anche i gruppi con ≈ ? *)
  62  rewrite > zero_neutral in H;
  63  rewrite > plus_comm in H;
  64  rewrite > opp_inverse in H;
  65  assumption.
  66 qed.
  67
  68 lemma le_x_zero_to_le_zero_opp_x: ∀G:ordered_abelian_group.∀x:G. x ≤ 0 → 0 ≤ -x.
  69  intros;
  70  generalize in match (og_ordered_abelian_group_properties ? ? ? (-x) H); intro;
  71  rewrite > zero_neutral in H1;
  72  rewrite > plus_comm in H1;
  73  rewrite > opp_inverse in H1;
  74  assumption.
  75 qed.