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=f72711aaab9d33880f00caf8b312d6a780e79121;hpb=4dc47c9675ffd5fa50296ffaa9b5997501518c98;p=helm.git

diff --git a/helm/software/matita/library/algebra/groups.ma b/helm/software/matita/library/algebra/groups.ma
index f72711aaa..fd08a95da 100644
--- a/helm/software/matita/library/algebra/groups.ma
+++ b/helm/software/matita/library/algebra/groups.ma
@@ -18,22 +18,27 @@ 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
  }.
 
-interpretation "Group inverse" 'invert x = (inv _ x).
+definition Monoid_of_Group: Group → Monoid ≝
+ λG. mk_Monoid ? (is_group G).
+
+coercion Monoid_of_Group nocomposites.
 
 definition left_cancellable ≝
  λT:Type. λop: T -> T -> T.
@@ -52,25 +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.
@@ -104,7 +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.
@@ -114,7 +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.
@@ -125,8 +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);
@@ -177,7 +180,7 @@ notation "hvbox(x \sub H)" with precedence 79
 for @{ 'subgroupimage $H $x }.
 
 interpretation "Subgroup image" 'subgroupimage H x =
- (image _ _ (morphism_OF_subgroup _ H) x).
+ (image ?? (morphism_OF_subgroup ? H) x).
 
 definition member_of_subgroup ≝
  λG.λH:subgroup G.λx:G.∃y.x=y \sub H.
@@ -189,10 +192,10 @@ notation "hvbox(x break \notin H)" with precedence 79
 for @{ 'not_member_of $x $H }.
 
 interpretation "Member of subgroup" 'member_of x H =
- (member_of_subgroup _ H x).
+ (member_of_subgroup ? H x).
  
 interpretation "Not member of subgroup" 'not_member_of x H =
- (Not (member_of_subgroup _ H x)).
+ (Not (member_of_subgroup ? H x)).
 
 (* Left cosets *)
 
@@ -203,14 +206,14 @@ 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 =
- (mk_left_coset _ 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 =
- (member_of_left_coset _ C x).
+ (member_of_left_coset ? C x).
 
 definition left_coset_eq ≝
  λG.λC,C':left_coset G.
@@ -227,7 +230,7 @@ notation "hvbox(a break \par b)"
 for @{ 'disjoint $a $b }.
 
 interpretation "Left cosets disjoint" 'disjoint C C' =
- (left_coset_disjoint _ C C').
+ (left_coset_disjoint ? C C').
 
 (* The following should be a one-shot alias! *)
 alias symbol "member_of" (instance 0) = "Member of subgroup".
@@ -246,8 +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
@@ -264,14 +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);