matita/contribs/dama/dama/ordered_divisible_group.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                  *)
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  14
  15
  16
  17 include "nat/orders.ma".
  18 include "nat/times.ma".
  19 include "ordered_group.ma".
  20 include "divisible_group.ma".
  21
  22 record todgroup : Type ≝ {
  23   todg_order:> togroup;
  24   todg_division_: dgroup;
  25   todg_with_: dg_carr todg_division_ = og_abelian_group todg_order
  26 }.
  27
  28 lemma todg_division: todgroup → dgroup.
  29 intro G; apply (mk_dgroup G); unfold abelian_group_OF_todgroup;
  30 cases (todg_with_ G); exact (dg_prop (todg_division_ G));
  31 qed.
  32
  33 coercion cic:/matita/ordered_divisible_group/todg_division.con.
  34
  35 lemma mul_ge: ∀G:todgroup.∀x:G.∀n.0 ≤ x → 0 ≤ n * x.
  36 intros (G x n); elim n; simplify; [apply le_reflexive]
  37 apply (le_transitive ???? H1);
  38 apply (Le≪ (0+(n1*x)) (zero_neutral ??));
  39 apply fle_plusr; assumption;
  40 qed.
  41
  42 lemma lt_ltmul: ∀G:todgroup.∀x,y:G.∀n. x < y → S n * x < S n * y.
  43 intros; elim n; [simplify; apply flt_plusr; assumption]
  44 simplify; apply (ltplus); [assumption] assumption;
  45 qed.
  46
  47 lemma ltmul_lt: ∀G:todgroup.∀x,y:G.∀n. S n * x < S n * y → x < y.
  48 intros 4; elim n; [apply (plus_cancr_lt ??? 0); assumption]
  49 simplify in l; cases (ltplus_orlt ????? l); [assumption]
  50 apply f; assumption;
  51 qed.
  52
  53 lemma divide_preserves_lt: ∀G:todgroup.∀e:G.∀n.0<e → 0 < e/n.
  54 intros; elim n; [apply (Lt≫ ? (div1 ??));assumption]
  55 unfold divide; elim (dg_prop G e (S n1)); simplify; simplify in f;
  56 apply (ltmul_lt ??? (S n1)); simplify; apply (Lt≫ ? f);
  57 apply (Lt≪ ? (zero_neutral ??)); (* bug se faccio repeat *)
  58 apply (Lt≪ ? (zero_neutral ??));
  59 apply (Lt≪ ? (mulzero ?n1));
  60 assumption;
  61 qed.
  62
  63 lemma muleqplus_lt: ∀G:todgroup.∀x,y:G.∀n,m.
  64    0<x → 0<y → S n * x ≈ S (n + S m) * y → y < x.
  65 intros (G x y n m H1 H2 H3); apply (ltmul_lt ??? n); apply (Lt≫ ? H3);
  66 clear H3; elim m; [
  67   rewrite > sym_plus; simplify; apply (Lt≪ (0+(y+n*y))); [
  68     apply eq_sym; apply zero_neutral]
  69   apply flt_plusr; assumption;]
  70 apply (lt_transitive ???? l); rewrite > sym_plus; simplify;
  71 rewrite > (sym_plus n); simplify; repeat apply flt_plusl;
  72 apply (Lt≪ (0+(n1+n)*y)); [apply eq_sym; apply zero_neutral]
  73 apply flt_plusr; assumption;
  74 qed.
  75