X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2Fdama%2Fdama%2Fordered_divisible_group.ma;fp=matita%2Fcontribs%2Fdama%2Fdama%2Fordered_divisible_group.ma;h=15dd52cdb853ba822b7d35e4a1ab959b0bf6e311;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+
+
+include "nat/orders.ma".
+include "nat/times.ma".
+include "ordered_group.ma".
+include "divisible_group.ma".
+
+record todgroup : Type ≝ {
+  todg_order:> togroup;
+  todg_division_: dgroup;
+  todg_with_: dg_carr todg_division_ = og_abelian_group todg_order
+}.
+
+lemma todg_division: todgroup → dgroup.
+intro G; apply (mk_dgroup G); unfold abelian_group_OF_todgroup; 
+cases (todg_with_ G); exact (dg_prop (todg_division_ G));
+qed.
+
+coercion cic:/matita/ordered_divisible_group/todg_division.con.
+
+lemma mul_ge: ∀G:todgroup.∀x:G.∀n.0 ≤ x → 0 ≤ n * x.
+intros (G x n); elim n; simplify; [apply le_reflexive]
+apply (le_transitive ???? H1); 
+apply (Le≪ (0+(n1*x)) (zero_neutral ??));
+apply fle_plusr; assumption;
+qed. 
+
+lemma lt_ltmul: ∀G:todgroup.∀x,y:G.∀n. x < y → S n * x < S n * y.
+intros; elim n; [simplify; apply flt_plusr; assumption]
+simplify; apply (ltplus); [assumption] assumption;
+qed.
+
+lemma ltmul_lt: ∀G:todgroup.∀x,y:G.∀n. S n * x < S n * y → x < y.
+intros 4; elim n; [apply (plus_cancr_lt ??? 0); assumption]
+simplify in l; cases (ltplus_orlt ????? l); [assumption]
+apply f; assumption;
+qed.
+
+lemma divide_preserves_lt: ∀G:todgroup.∀e:G.∀n.0<e → 0 < e/n.
+intros; elim n; [apply (Lt≫ ? (div1 ??));assumption]
+unfold divide; elim (dg_prop G e (S n1)); simplify; simplify in f;
+apply (ltmul_lt ??? (S n1)); simplify; apply (Lt≫ ? f);
+apply (Lt≪ ? (zero_neutral ??)); (* bug se faccio repeat *)
+apply (Lt≪ ? (zero_neutral ??));  
+apply (Lt≪ ? (mulzero ?n1));
+assumption;
+qed.
+
+lemma muleqplus_lt: ∀G:todgroup.∀x,y:G.∀n,m.
+   0<x → 0<y → S n * x ≈ S (n + S m) * y → y < x.
+intros (G x y n m H1 H2 H3); apply (ltmul_lt ??? n); apply (Lt≫ ? H3);
+clear H3; elim m; [
+  rewrite > sym_plus; simplify; apply (Lt≪ (0+(y+n*y))); [
+    apply eq_sym; apply zero_neutral]
+  apply flt_plusr; assumption;]
+apply (lt_transitive ???? l); rewrite > sym_plus; simplify;  
+rewrite > (sym_plus n); simplify; repeat apply flt_plusl;
+apply (Lt≪ (0+(n1+n)*y)); [apply eq_sym; apply zero_neutral]
+apply flt_plusr; assumption;
+qed.  
+