X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Falgebra%2Fgroups.ma;h=1301d34f923725ea583d1ea4fe47403e869fa33c;hb=3cc1aec78e4c51aa766127487f19a3d38a4b56ae;hp=970a5e38892ae9a3b2c46c9ce92b81f7520fb2b2;hpb=43d3607afd27248d8df1bca20908307330a3871a;p=helm.git

diff --git a/helm/software/matita/library/algebra/groups.ma b/helm/software/matita/library/algebra/groups.ma
index 970a5e388..1301d34f9 100644
--- a/helm/software/matita/library/algebra/groups.ma
+++ b/helm/software/matita/library/algebra/groups.ma
@@ -25,7 +25,7 @@ record PreGroup : Type ≝
  }.
 
 record isGroup (G:PreGroup) : Prop ≝
- { is_monoid: isMonoid G;
+ { 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)
  }.
@@ -73,11 +73,11 @@ intros;
 unfold left_cancellable;
 unfold injective;
 intros (x y z);
-rewrite < (e_is_left_unit ? (is_monoid ? G));
-rewrite < (e_is_left_unit ? (is_monoid ? G) z);
+rewrite < (e_is_left_unit ? G);
+rewrite < (e_is_left_unit ? G z);
 rewrite < (inv_is_left_inverse ? G x);
-rewrite > (associative ? (is_semi_group ? (is_monoid ? G)));
-rewrite > (associative ? (is_semi_group ? (is_monoid ? G)));
+rewrite > (associative ? (is_semi_group ? ( G)));
+rewrite > (associative ? (is_semi_group ? ( G)));
 apply eq_f;
 assumption.
 qed.
@@ -90,11 +90,11 @@ unfold right_cancellable;
 unfold injective;
 simplify;fold simplify (op G); 
 intros (x y z);
-rewrite < (e_is_right_unit ? (is_monoid ? G));
-rewrite < (e_is_right_unit ? (is_monoid ? G) z);
+rewrite < (e_is_right_unit ? ( G));
+rewrite < (e_is_right_unit ? ( G) z);
 rewrite < (inv_is_right_inverse ? G x);
-rewrite < (associative ? (is_semi_group ? (is_monoid ? G)));
-rewrite < (associative ? (is_semi_group ? (is_monoid ? G)));
+rewrite < (associative ? (is_semi_group ? ( G)));
+rewrite < (associative ? (is_semi_group ? ( G)));
 rewrite > H;
 reflexivity.
 qed.
@@ -128,9 +128,9 @@ 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 > (associative ? (is_semi_group ? (is_monoid ? G)));
+rewrite > (associative ? G);
 rewrite > (inv_is_left_inverse ? G);
-rewrite > (e_is_right_unit ? (is_monoid ? G));
+rewrite > (e_is_right_unit ? G);
 assumption.
 qed.
 
@@ -138,12 +138,25 @@ 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 < (associative ? (is_semi_group ? (is_monoid ? G)));
+rewrite < (associative ? G);
 rewrite > (inv_is_right_inverse ? G);
 rewrite > (e_is_left_unit ? (is_monoid ? G));
 assumption.
 qed.
 
+theorem eq_inv_op_x_y_op_inv_y_inv_x:
+ ∀G:Group. ∀x,y:G. (x·y) \sup -1 = y \sup -1 · x \sup -1.
+intros;
+apply (eq_op_x_y_op_z_y_to_eq ? (x·y));
+rewrite > (inv_is_left_inverse ? G);
+rewrite < (associative ? G);
+rewrite > (associative ? G (y \sup -1));
+rewrite > (inv_is_left_inverse ? G);
+rewrite > (e_is_right_unit ? G);
+rewrite > (inv_is_left_inverse ? G);
+reflexivity.
+qed.
+
 (* Morphisms *)
 
 record morphism (G,G':Group) : Type ≝
@@ -161,9 +174,9 @@ theorem morphism_to_eq_f_1_1:
  ∀G,G'.∀f:morphism G G'.f˜1 = 1.
 intros;
 apply (eq_op_x_y_op_z_y_to_eq G' (f˜1));
-rewrite > (e_is_left_unit ? (is_monoid ? G') ?);
+rewrite > (e_is_left_unit ? G' ?);
 rewrite < (f_morph ? ? f);
-rewrite > (e_is_left_unit ? (is_monoid ? G));
+rewrite > (e_is_left_unit ? G);
 reflexivity.
 qed.
  
@@ -179,7 +192,7 @@ apply (morphism_to_eq_f_1_1 ? ? f).
 qed.
 
 record monomorphism (G,G':Group) : Type ≝
- { morphism: morphism G G';
+ { morphism:> morphism G G';
    injective: injective ? ? (image ? ? morphism)
  }.
 
@@ -187,7 +200,7 @@ record monomorphism (G,G':Group) : Type ≝
 
 record subgroup (G:Group) : Type ≝
  { group: Group;
-   embed: monomorphism group G
+   embed:> monomorphism group G
  }.
  
 notation "hvbox(x \sub H)" with precedence 79
@@ -195,18 +208,16 @@ for @{ 'subgroupimage $H $x }.
 
 interpretation "Subgroup image" 'subgroupimage H x =
  (cic:/matita/algebra/groups/image.con _ _
-   (cic:/matita/algebra/groups/morphism.con _ _
-     (cic:/matita/algebra/groups/embed.con _ H))
-   x).
+   (cic:/matita/algebra/groups/morphism_of_subgroup.con _ H) x).
 
-definition belongs_to_subgroup ≝
+definition member_of_subgroup ≝
  λG.λH:subgroup G.λx:G.∃y.x=y \sub H.
 
-notation "hvbox(x ∈ H)" with precedence 79
-for @{ 'belongs_to $x $H }.
+notation "hvbox(x break ∈ H)" with precedence 79
+for @{ 'member_of $x $H }.
 
-interpretation "Belongs to subgroup" 'belongs_to x H =
- (cic:/matita/algebra/groups/belongs_to_subgroup.con _ H x).
+interpretation "Member of subgroup" 'member_of x H =
+ (cic:/matita/algebra/groups/member_of_subgroup.con _ H x).
 
 (* Left cosets *)
 
@@ -219,12 +230,12 @@ record left_coset (G:Group) : Type ≝
 interpretation "Left_coset" 'times x C =
  (cic:/matita/algebra/groups/left_coset.ind#xpointer(1/1/1) _ x C).
 
-definition belongs_to_left_coset ≝
+definition member_of_left_coset ≝
  λG:Group.λC:left_coset G.λx:G.
   ∃y.x=(element ? C)·y \sub (subgrp ? C).
 
-interpretation "Belongs to left_coset" 'belongs_to x C =
- (cic:/matita/algebra/groups/belongs_to_left_coset.con _ C x).
+interpretation "Member of left_coset" 'member_of x C =
+ (cic:/matita/algebra/groups/member_of_left_coset.con _ C x).
 
 definition left_coset_eq ≝
  λG.λC,C':left_coset G.
@@ -244,31 +255,31 @@ for @{ 'disjoint $a $b }.
 interpretation "Left cosets disjoint" 'disjoint C C' =
  (cic:/matita/algebra/groups/left_coset_disjoint.con _ C C').
 
-(*
 (* The following should be a one-shot alias! *)
-alias symbol "belongs_to" (instance 0) = "Belongs to subgroup".
-theorem foo:
- ∀G.∀x,y:(Type_of_Group G).∀H:subgroup G.
-  (x \sup -1 ·y) ∈ H → (mk_left_coset ? x H) = (mk_left_coset ? y H).
+alias symbol "member_of" (instance 0) = "Member of subgroup".
+theorem member_of_subgroup_op_inv_x_y_to_left_coset_eq:
+ ∀G.∀x,y.∀H:subgroup G. (x \sup -1 ·y) ∈ H → x*H = y*H.
 intros;
 unfold left_coset_eq;
-simplify in ⊢ (? → ? ? ? (? ? ? (? ? ? (? ? ? (? ? %)) ?)));
 simplify in ⊢ (? → ? ? ? (? ? % ?));
+simplify in ⊢ (? → ? ? ? (? ? ? (? ? ? (? ? %) ?)));
 simplify in ⊢ (? % → ?);
 intros;
-unfold belongs_to_left_coset;
-simplify in ⊢ (? ? (λy:?.? ? ? (? ? ? (? ? ? (? ? ? (? ? %)) ?))));
+unfold member_of_left_coset;
+simplify in ⊢ (? ? (λy:?.? ? ? (? ? ? (? ? ? (? ? %) ?))));
 simplify in ⊢ (? ? (λy:? %.?));
 simplify in ⊢ (? ? (λy:?.? ? ? (? ? % ?)));
-unfold belongs_to_subgroup in H1;
+unfold member_of_subgroup in H1;
 elim H1;
 clear H1;
 exists;
-[
-| 
+[ apply (a\sup-1 · x1)
+| rewrite > (f_morph ? ? (morphism ? ? H));
+  rewrite > (eq_image_inv_inv_image ? ? 
+  rewrite < H2;
+  rewrite > (eq_inv_op_x_y_op_inv_y_inv_x ? ? ? ? H2);
 ].
 qed.
-*)
 
 (*theorem foo:
  \forall G:Group. \forall x1,x2:G. \forall H:subgroup G.