(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/divisible_group/".
+
include "nat/orders.ma".
include "group.ma".
-let rec pow (G : abelian_group) (x : G) (n : nat) on n : G ≝
- match n with [ O ⇒ 0 | S m ⇒ x + pow ? x m].
+let rec gpow (G : abelian_group) (x : G) (n : nat) on n : G ≝
+ match n with [ O ⇒ 0 | S m ⇒ x + gpow ? x m].
-interpretation "abelian group pow" 'exp x n =
- (cic:/matita/divisible_group/pow.con _ x n).
+interpretation "additive abelian group pow" 'times n x =
+ (cic:/matita/divisible_group/gpow.con _ x n).
record dgroup : Type ≝ {
dg_carr:> abelian_group;
- dg_prop: ∀x:dg_carr.∀n:nat.∃y.pow ? y (S n) ≈ x
+ dg_prop: ∀x:dg_carr.∀n:nat.∃y.S n * y ≈ x
}.
lemma divide: ∀G:dgroup.G → nat → G.
(cic:/matita/divisible_group/divide.con _ x n).
lemma divide_divides:
- ∀G:dgroup.∀x:G.∀n. pow ? (x / n) (S n) ≈ x.
+ ∀G:dgroup.∀x:G.∀n. S n * (x / n) ≈ x.
intro G; cases G; unfold divide; intros (x n); simplify;
cases (f x n); simplify; exact H;
qed.
-lemma feq_pow: ∀G:dgroup.∀x,y:G.∀n.x≈y → pow ? x n ≈ pow ? y n.
+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 + (pow ? y n1)) H1);
-apply (Eq≈ (y+pow ? y n1) H (eq_reflexive ??));
+simplify; apply (Eq≈ (x + (n1 * y)) H1);
+apply (Eq≈ (y+n1*y) H (eq_reflexive ??));
qed.
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 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 f; assumption;]
+apply (plus_cancr_ap ??? 0); assumption;
+qed.
+
+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 (Eq≈ (x+(n1*x+(y+(n1*y))))); [
+ apply eq_sym; apply plus_assoc;]
+ apply (Eq≈ (x+((n1*x+y+(n1*y))))); [
+ apply feq_plusl; apply plus_assoc;]
+ apply (Eq≈ (x+(y+n1*x+n1*y))); [
+ apply feq_plusl; apply feq_plusr; apply plus_comm;]
+ apply (Eq≈ (x+(y+(n1*x+n1*y)))); [
+ apply feq_plusl; apply eq_sym; apply plus_assoc;]
+ apply plus_assoc;]
+apply feq_plusl; apply eq_sym; assumption;
+qed.
+
+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.