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[helm.git] / helm / software / matita / dama / divisible_group.ma

diff --git a/helm/software/matita/dama/divisible_group.ma b/helm/software/matita/dama/divisible_group.ma
index 44db35a15ca0dfbef614568c71daeac17607f3d0..3a79b11bbd969a884307f4a6a86d08e63cbcfff8 100644 (file)
--- a/helm/software/matita/dama/divisible_group.ma
+++ b/helm/software/matita/dama/divisible_group.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/divisible_group/".
+
 
 include "nat/orders.ma".
 include "group.ma".
@@ -50,8 +50,8 @@ 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.
@@ -83,3 +83,17 @@ 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.