X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Falgebra%2Fgroups.ma;h=f257951bc4a031abb0a1640ac0c673da3e6b340c;hb=0080faad4e730c227b6bbb2549407b23703b477a;hp=360e75c9f06d52c85e931e84f42eed4fa68bee76;hpb=5ce2b9b14c6b76021c780080e31b283a5e061d28;p=helm.git

diff --git a/helm/software/matita/library/algebra/groups.ma b/helm/software/matita/library/algebra/groups.ma
index 360e75c9f..f257951bc 100644
--- a/helm/software/matita/library/algebra/groups.ma
+++ b/helm/software/matita/library/algebra/groups.ma
@@ -12,8 +12,6 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/algebra/groups/".
-
 include "algebra/monoids.ma".
 include "nat/le_arith.ma".
 include "datatypes/bool.ma".
@@ -35,11 +33,7 @@ record Group : Type ≝
    group_properties:> isGroup pregroup
  }.
 
-notation "hvbox(x \sup (-1))" with precedence 89
-for @{ 'ginv $x }.
-
-interpretation "Group inverse" 'ginv x =
- (cic:/matita/algebra/groups/inv.con _ x).
+interpretation "Group inverse" 'invert x = (inv _ x).
 
 definition left_cancellable ≝
  λT:Type. λop: T -> T -> T.
@@ -90,7 +84,7 @@ reflexivity.
 qed.
 
 theorem eq_opxy_e_to_eq_x_invy:
- ∀G:Group. ∀x,y:G. x·y=1 → x=y \sup -1.
+ ∀G:Group. ∀x,y:G. x·y=ⅇ → x=y \sup -1.
 intros;
 apply (eq_op_x_y_op_z_y_to_eq ? y);
 rewrite > (inv_is_left_inverse ? G);
@@ -98,7 +92,7 @@ assumption.
 qed.
 
 theorem eq_opxy_e_to_eq_invx_y:
- ∀G:Group. ∀x,y:G. x·y=1 → x \sup -1=y.
+ ∀G:Group. ∀x,y:G. x·y=ⅇ → x \sup -1=y.
 intros;
 apply (eq_op_x_y_op_x_z_to_eq ? x);
 rewrite > (inv_is_right_inverse ? G);
@@ -142,20 +136,14 @@ qed.
 (* Morphisms *)
 
 record morphism (G,G':Group) : Type ≝
- { image: G → G';
+ { image:1> G → G';
    f_morph: ∀x,y:G.image(x·y) = image x · image y
  }.
  
-notation "hvbox(f˜ x)" with precedence 79
-for @{ 'morimage $f $x }.
-
-interpretation "Morphism image" 'morimage f x =
- (cic:/matita/algebra/groups/image.con _ _ f x).
- 
 theorem morphism_to_eq_f_1_1:
- ∀G,G'.∀f:morphism G G'.f˜1 = 1.
+ ∀G,G'.∀f:morphism G G'.f ⅇ  = ⅇ.
 intros;
-apply (eq_op_x_y_op_z_y_to_eq ? (f˜1));
+apply (eq_op_x_y_op_z_y_to_eq ? (f ⅇ));
 rewrite > (e_is_left_unit ? G');
 rewrite < f_morph;
 rewrite > (e_is_left_unit ? G);
@@ -164,9 +152,9 @@ qed.
  
 theorem eq_image_inv_inv_image:
  ∀G,G'.∀f:morphism G G'.
-  ∀x.f˜(x \sup -1) = (f˜x) \sup -1.
+  ∀x.f (x \sup -1) = (f x) \sup -1.
 intros;
-apply (eq_op_x_y_op_z_y_to_eq ? (f˜x));
+apply (eq_op_x_y_op_z_y_to_eq ? (f x));
 rewrite > (inv_is_left_inverse ? G');
 rewrite < f_morph;
 rewrite > (inv_is_left_inverse ? G);
@@ -190,15 +178,15 @@ for @{ 'subgroupimage $H $x }.
 
 interpretation "Subgroup image" 'subgroupimage H x =
  (cic:/matita/algebra/groups/image.con _ _
-   (cic:/matita/algebra/groups/morphism_of_subgroup.con _ H) x).
+   (cic:/matita/algebra/groups/morphism_OF_subgroup.con _ H) x).
 
 definition member_of_subgroup ≝
  λG.λH:subgroup G.λx:G.∃y.x=y \sub H.
 
-notation "hvbox(x break ∈ H)" with precedence 79
+notation "hvbox(x break \in H)" with precedence 79
 for @{ 'member_of $x $H }.
 
-notation "hvbox(x break ∉ H)" with precedence 79
+notation "hvbox(x break \notin H)" with precedence 79
 for @{ 'not_member_of $x $H }.
 
 interpretation "Member of subgroup" 'member_of x H =
@@ -237,7 +225,7 @@ definition left_coset_disjoint ≝
  λG.λC,C':left_coset G.
   ∀x.¬(((element ? C)·x \sub (subgrp ? C)) ∈ C'). 
 
-notation "hvbox(a break ∥ b)"
+notation "hvbox(a break \par b)"
  non associative with precedence 45
 for @{ 'disjoint $a $b }.
 
@@ -294,12 +282,33 @@ rewrite > (e_is_right_unit ? H);
 reflexivity.
 qed.
 
+(*CSC: here the coercion Type_of_Group cannot be omitted. Why? *)
 theorem in_x_mk_left_coset_x_H:
- ∀G.∀x:Type_of_Group G.∀H:subgroup G.x ∈ (x*H).
+ ∀G.∀x:Type_OF_Group G.∀H:subgroup G.x ∈ (x*H).
 intros;
 simplify;
-apply (ex_intro ? ? 1);
+apply (ex_intro ? ? ⅇ);
 rewrite > morphism_to_eq_f_1_1;
 rewrite > (e_is_right_unit ? G);
 reflexivity.
-qed.
\ No newline at end of file
+qed.
+
+(* Normal Subgroups *)
+
+record normal_subgroup (G:Group) : Type ≝
+ { ns_subgroup:> subgroup G;
+   normal:> ∀x:G.∀y:ns_subgroup.(x·y \sub ns_subgroup·x \sup -1) ∈ ns_subgroup
+ }.
+
+(*CSC: I have not defined yet right cosets 
+theorem foo:
+ ∀G.∀H:normal_subgroup G.∀x.x*H=H*x.
+*)
+(*
+theorem member_of_left_coset_mk_left_coset_x_H_a_to_member_of_left_coset_mk_left_coset_y_H_b_to_member_of_left_coset_mk_left_coset_op_x_y_H_op_a_b:
+ ∀G.∀H:normal_subgroup G.∀x,y,a,b.
+  a ∈ (x*H) → b ∈ (y*H) → (a·b) ∈ ((x·y)*H).
+intros;
+simplify;
+qed.
+*)