(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/divisible_group/".
+
include "nat/orders.ma".
include "group.ma".
cases (f x n); simplify; exact H;
qed.
-lemma feq_pow: ∀G:dgroup.∀x,y:G.∀n.x≈y → n * x ≈ n * y.
+lemma feq_mul: ∀G:dgroup.∀x,y:G.∀n.x≈y → n * x ≈ n * y.
intros (G x y n H); elim n; [apply eq_reflexive]
simplify; apply (Eq≈ (x + (n1 * y)) H1);
apply (Eq≈ (y+n1*y) H (eq_reflexive ??));
lemma div1: ∀G:dgroup.∀x:G.x/O ≈ x.
intro G; cases G; unfold divide; intros; simplify;
cases (f x O); simplify; simplify in H; intro; apply H;
-apply (ap_rewl ???? (plus_comm ???));
-apply (ap_rewl ???w (zero_neutral ??)); assumption;
+apply (Ap≪ ? (plus_comm ???));
+apply (Ap≪ w (zero_neutral ??)); assumption;
qed.
-lemma appow_ap: ∀G:dgroup.∀x,y:G.∀n.S n * x # S n * y → x # y.
+lemma apmul_ap: ∀G:dgroup.∀x,y:G.∀n.S n * x # S n * y → x # y.
intros 4 (G x y n); elim n; [2:
simplify in a;
cases (applus ????? a); [assumption]
apply (plus_cancr_ap ??? 0); assumption;
qed.
-lemma pluspow: ∀G:dgroup.∀x,y:G.∀n.n * (x+y) ≈ n * x + n * y.
+lemma plusmul: ∀G:dgroup.∀x,y:G.∀n.n * (x+y) ≈ n * x + n * y.
intros (G x y n); elim n; [
simplify; apply (Eq≈ 0 ? (zero_neutral ? 0)); apply eq_reflexive]
simplify; apply eq_sym; apply (Eq≈ (x+y+(n1*x+n1*y))); [
apply feq_plusl; apply eq_sym; assumption;
qed.
-lemma powzero: ∀G:dgroup.∀n.n*0 ≈ 0. [2: apply dg_carr; apply G]
+lemma mulzero: ∀G:dgroup.∀n.n*0 ≈ 0. [2: apply dg_carr; apply G]
intros; elim n; [simplify; apply eq_reflexive]
simplify; apply (Eq≈ ? (zero_neutral ??)); assumption;
qed.
+let rec gpowS (G : abelian_group) (x : G) (n : nat) on n : G ≝
+ match n with [ O ⇒ x | S m ⇒ gpowS ? x m + x].
+
+lemma gpowS_gpow: ∀G:dgroup.∀e:G.∀n. S n * e ≈ gpowS ? e n.
+intros (G e n); elim n; simplify; [
+ apply (Eq≈ ? (plus_comm ???));apply zero_neutral]
+apply (Eq≈ ?? (plus_comm ???));
+apply (Eq≈ (e+S n1*e) ? H); clear H; simplify; apply eq_reflexive;
+qed.
+
+lemma divpow: ∀G:dgroup.∀e:G.∀n. e ≈ gpowS ? (e/n) n.
+intros (G e n); apply (Eq≈ ?? (gpowS_gpow ?(e/n) n));
+apply eq_sym; apply divide_divides;
+qed.