X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Falgebra%2Fgroups.ma;h=fd08a95dacb504e2cee3bfdf1a06948175c993f7;hb=647b419e96770d90a82d7a9e5e8843566a9f93ee;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..fd08a95da 100644
--- a/helm/software/matita/library/algebra/groups.ma
+++ b/helm/software/matita/library/algebra/groups.ma
@@ -12,34 +12,33 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/algebra/groups/".
-
 include "algebra/monoids.ma".
 include "nat/le_arith.ma".
 include "datatypes/bool.ma".
 include "nat/compare.ma".
 
 record PreGroup : Type ≝
- { premonoid:> PreMonoid;
-   inv: premonoid -> premonoid
+ { pre_monoid:> PreMonoid;
+   inv: pre_monoid -> pre_monoid
  }.
 
+interpretation "Group inverse" 'invert x = (inv ? x).
+
 record isGroup (G:PreGroup) : Prop ≝
- { is_monoid:> isMonoid G;
-   inv_is_left_inverse: is_left_inverse (mk_Monoid ? is_monoid) (inv G);
-   inv_is_right_inverse: is_right_inverse (mk_Monoid ? is_monoid) (inv G)
+ { is_monoid           :> IsMonoid G;
+   inv_is_left_inverse :  is_left_inverse G (inv G);
+   inv_is_right_inverse:  is_right_inverse G (inv G)
  }.
- 
+
 record Group : Type ≝
- { pregroup:> PreGroup;
-   group_properties:> isGroup pregroup
+ { pre_group:> PreGroup;
+   is_group:> isGroup pre_group
  }.
 
-notation "hvbox(x \sup (-1))" with precedence 89
-for @{ 'ginv $x }.
+definition Monoid_of_Group: Group → Monoid ≝
+ λG. mk_Monoid ? (is_group G).
 
-interpretation "Group inverse" 'ginv x =
- (cic:/matita/algebra/groups/inv.con _ x).
+coercion Monoid_of_Group nocomposites.
 
 definition left_cancellable ≝
  λT:Type. λop: T -> T -> T.
@@ -58,25 +57,23 @@ intros (x y z);
 rewrite < (e_is_left_unit ? G);
 rewrite < (e_is_left_unit ? G z);
 rewrite < (inv_is_left_inverse ? G x);
-rewrite > (op_associative ? G);
-rewrite > (op_associative ? G);
+rewrite > (op_is_associative ? G);
+rewrite > (op_is_associative ? G);
 apply eq_f;
 assumption.
 qed.
 
-
 theorem eq_op_x_y_op_z_y_to_eq:
  ∀G:Group. right_cancellable G (op G).
 intros;
 unfold right_cancellable;
 unfold injective;
-simplify;fold simplify (op G); 
 intros (x y z);
 rewrite < (e_is_right_unit ? G);
 rewrite < (e_is_right_unit ? G z);
 rewrite < (inv_is_right_inverse ? G x);
-rewrite < (op_associative ? G);
-rewrite < (op_associative ? G);
+rewrite < (op_is_associative ? G);
+rewrite < (op_is_associative ? G);
 rewrite > H;
 reflexivity.
 qed.
@@ -90,7 +87,7 @@ 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 +95,7 @@ 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);
@@ -110,7 +107,7 @@ theorem eq_opxy_z_to_eq_x_opzinvy:
  ∀G:Group. ∀x,y,z:G. x·y=z → x = z·y \sup -1.
 intros;
 apply (eq_op_x_y_op_z_y_to_eq ? y);
-rewrite > (op_associative ? G);
+rewrite > (op_is_associative ? G);
 rewrite > (inv_is_left_inverse ? G);
 rewrite > (e_is_right_unit ? G);
 assumption.
@@ -120,7 +117,7 @@ theorem eq_opxy_z_to_eq_y_opinvxz:
  ∀G:Group. ∀x,y,z:G. x·y=z → y = x \sup -1·z.
 intros;
 apply (eq_op_x_y_op_x_z_to_eq ? x);
-rewrite < (op_associative ? G);
+rewrite < (op_is_associative ? G);
 rewrite > (inv_is_right_inverse ? G);
 rewrite > (e_is_left_unit ? G);
 assumption.
@@ -131,8 +128,8 @@ theorem eq_inv_op_x_y_op_inv_y_inv_x:
 intros;
 apply (eq_op_x_y_op_z_y_to_eq ? (x·y));
 rewrite > (inv_is_left_inverse ? G);
-rewrite < (op_associative ? G);
-rewrite > (op_associative ? G (y \sup -1));
+rewrite < (op_is_associative ? G);
+rewrite > (op_is_associative ? G (y \sup -1));
 rewrite > (inv_is_left_inverse ? G);
 rewrite > (e_is_right_unit ? G);
 rewrite > (inv_is_left_inverse ? G);
@@ -142,20 +139,14 @@ 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 +155,9 @@ 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);
@@ -189,24 +180,22 @@ notation "hvbox(x \sub H)" with precedence 79
 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).
+ (image ?? (morphism_OF_subgroup ? 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 =
- (cic:/matita/algebra/groups/member_of_subgroup.con _ H x).
+ (member_of_subgroup ? H x).
  
 interpretation "Not member of subgroup" 'not_member_of x H =
- (cic:/matita/logic/connectives/Not.con
-  (cic:/matita/algebra/groups/member_of_subgroup.con _ H x)).
+ (Not (member_of_subgroup ? H x)).
 
 (* Left cosets *)
 
@@ -217,32 +206,31 @@ record left_coset (G:Group) : Type ≝
 
 (* Here I would prefer 'magma_op, but this breaks something in the next definition *)
 interpretation "Left_coset" 'times x C =
- (cic:/matita/algebra/groups/left_coset.ind#xpointer(1/1/1) _ x C).
+ (mk_left_coset ? x C).
 
 definition member_of_left_coset ≝
  λG:Group.λC:left_coset G.λx:G.
   ∃y.x=(element ? C)·y \sub (subgrp ? C).
 
 interpretation "Member of left_coset" 'member_of x C =
- (cic:/matita/algebra/groups/member_of_left_coset.con _ C x).
+ (member_of_left_coset ? C x).
 
 definition left_coset_eq ≝
  λG.λC,C':left_coset G.
   ∀x.((element ? C)·x \sub (subgrp ? C)) ∈ C'.
   
-interpretation "Left cosets equality" 'eq C C' =
- (cic:/matita/algebra/groups/left_coset_eq.con _ C C').
+interpretation "Left cosets equality" 'eq t C C' = (left_coset_eq t C C').
 
 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 }.
 
 interpretation "Left cosets disjoint" 'disjoint C C' =
- (cic:/matita/algebra/groups/left_coset_disjoint.con _ C C').
+ (left_coset_disjoint ? C C').
 
 (* The following should be a one-shot alias! *)
 alias symbol "member_of" (instance 0) = "Member of subgroup".
@@ -261,8 +249,8 @@ exists;
   rewrite < H2;
   rewrite > eq_inv_op_x_y_op_inv_y_inv_x;
   rewrite > eq_inv_inv_x_x;
-  rewrite < (op_associative ? G);
-  rewrite < (op_associative ? G);
+  rewrite < (op_is_associative ? G);
+  rewrite < (op_is_associative ? G);
   rewrite > (inv_is_right_inverse ? G);
   rewrite > (e_is_left_unit ? G);
   reflexivity
@@ -279,14 +267,14 @@ apply H1;
 unfold member_of_subgroup;
 elim H2;
 apply (ex_intro ? ? (x'·a \sup -1));
-rewrite > f_morph;
+rewrite > f_morph; 
 apply (eq_op_x_y_op_z_y_to_eq ? (a \sub H));
-rewrite > (op_associative ? G);
+rewrite > (op_is_associative ? G);
 rewrite < H3;
-rewrite > (op_associative ? G);
+rewrite > (op_is_associative ? G);
 rewrite < f_morph;
 rewrite > (inv_is_left_inverse ? H);
-rewrite < (op_associative ? G);
+rewrite < (op_is_associative ? G);
 rewrite > (inv_is_left_inverse ? G);
 rewrite > (e_is_left_unit ? G);
 rewrite < (f_morph ? ? H);
@@ -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.
+*)