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=1301d34f923725ea583d1ea4fe47403e869fa33c;hpb=93422dfbcbf88ad0a7d3cac7ec79def2ff2a6b30;p=helm.git

diff --git a/helm/software/matita/library/algebra/groups.ma b/helm/software/matita/library/algebra/groups.ma
index 1301d34f9..fd08a95da 100644
--- a/helm/software/matita/library/algebra/groups.ma
+++ b/helm/software/matita/library/algebra/groups.ma
@@ -12,52 +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 < "G"
-for @{ 'monoid $G }.
-
-interpretation "Monoid coercion" 'monoid G =
- (cic:/matita/algebra/groups/monoid.con G).*)
-
-notation < "G"
-for @{ 'type_of_group $G }.
-
-interpretation "Type_of_group coercion" 'type_of_group G =
- (cic:/matita/algebra/groups/Type_of_Group.con G).
-
-notation < "G"
-for @{ 'magma_of_group $G }.
-
-interpretation "magma_of_group coercion" 'magma_of_group G =
- (cic:/matita/algebra/groups/Magma_of_Group.con G).
-
-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.
@@ -76,30 +57,28 @@ 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 > (associative ? (is_semi_group ? ( G)));
-rewrite > (associative ? (is_semi_group ? ( 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 < (e_is_right_unit ? G);
+rewrite < (e_is_right_unit ? G z);
 rewrite < (inv_is_right_inverse ? G x);
-rewrite < (associative ? (is_semi_group ? ( G)));
-rewrite < (associative ? (is_semi_group ? ( G)));
+rewrite < (op_is_associative ? G);
+rewrite < (op_is_associative ? G);
 rewrite > H;
 reflexivity.
 qed.
 
-theorem inv_inv: ∀G:Group. ∀x:G. x \sup -1 \sup -1 = x.
+theorem eq_inv_inv_x_x: ∀G:Group. ∀x:G. x \sup -1 \sup -1 = x.
 intros;
 apply (eq_op_x_y_op_z_y_to_eq ? (x \sup -1));
 rewrite > (inv_is_right_inverse ? G);
@@ -108,7 +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);
@@ -116,7 +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);
@@ -128,7 +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 > (associative ? G);
+rewrite > (op_is_associative ? G);
 rewrite > (inv_is_left_inverse ? G);
 rewrite > (e_is_right_unit ? G);
 assumption.
@@ -138,9 +117,9 @@ 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 ? G);
+rewrite < (op_is_associative ? G);
 rewrite > (inv_is_right_inverse ? G);
-rewrite > (e_is_left_unit ? (is_monoid ? G));
+rewrite > (e_is_left_unit ? G);
 assumption.
 qed.
 
@@ -149,8 +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 < (associative ? G);
-rewrite > (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);
@@ -160,33 +139,27 @@ 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 G' (f˜1));
-rewrite > (e_is_left_unit ? G' ?);
-rewrite < (f_morph ? ? f);
+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);
 reflexivity.
 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 G' (f˜x));
+apply (eq_op_x_y_op_z_y_to_eq ? (f x));
 rewrite > (inv_is_left_inverse ? G');
-rewrite < (f_morph ? ? f);
+rewrite < f_morph;
 rewrite > (inv_is_left_inverse ? G);
 apply (morphism_to_eq_f_1_1 ? ? f).
 qed.
@@ -199,25 +172,30 @@ record monomorphism (G,G':Group) : Type ≝
 (* Subgroups *)
 
 record subgroup (G:Group) : Type ≝
- { group: Group;
+ { group:> Group;
    embed:> monomorphism group G
  }.
- 
+
 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 \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 =
+ (Not (member_of_subgroup ? H x)).
 
 (* Left cosets *)
 
@@ -228,93 +206,109 @@ 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".
 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 ⊢ (? % → ?);
-intros;
-unfold member_of_left_coset;
-simplify in ⊢ (? ? (λy:?.? ? ? (? ? ? (? ? ? (? ? %) ?))));
-simplify in ⊢ (? ? (λy:? %.?));
-simplify in ⊢ (? ? (λy:?.? ? ? (? ? % ?)));
+simplify;
+intro;
 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 > f_morph;
+  rewrite > eq_image_inv_inv_image; 
   rewrite < H2;
-  rewrite > (eq_inv_op_x_y_op_inv_y_inv_x ? ? ? ? H2);
+  rewrite > eq_inv_op_x_y_op_inv_y_inv_x;
+  rewrite > eq_inv_inv_x_x;
+  rewrite < (op_is_associative ? G);
+  rewrite < (op_is_associative ? G);
+  rewrite > (inv_is_right_inverse ? G);
+  rewrite > (e_is_left_unit ? G);
+  reflexivity
 ].
 qed.
 
-(*theorem foo:
- \forall G:Group. \forall x1,x2:G. \forall H:subgroup G.
-  x1*x2^-1 \nin H \to x1*H does_not_overlap x2*H
-
-theorem foo:
- \forall x:G. \forall H:subgroup G. x \in x*H
-
-definition disjoinct
- (T: Type) (n:nat) (S: \forall x:nat. x < n -> {S:Type * (S -> T)})
-:=
- \forall i,j:nat. i < n \to j < n \to ...
+theorem Not_member_of_subgroup_to_left_coset_disjoint:
+ ∀G.∀x,y.∀H:subgroup G.(x \sup -1 ·y) ∉ H → x*H ∥ y*H.
+intros;
+simplify;
+unfold Not;
+intros (x');
+apply H1;
+unfold member_of_subgroup;
+elim H2;
+apply (ex_intro ? ? (x'·a \sup -1));
+rewrite > f_morph; 
+apply (eq_op_x_y_op_z_y_to_eq ? (a \sub H));
+rewrite > (op_is_associative ? G);
+rewrite < H3;
+rewrite > (op_is_associative ? G);
+rewrite < f_morph;
+rewrite > (inv_is_left_inverse ? H);
+rewrite < (op_is_associative ? G);
+rewrite > (inv_is_left_inverse ? G);
+rewrite > (e_is_left_unit ? G);
+rewrite < (f_morph ? ? H);
+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).
+intros;
+simplify;
+apply (ex_intro ? ? ⅇ);
+rewrite > morphism_to_eq_f_1_1;
+rewrite > (e_is_right_unit ? G);
+reflexivity.
+qed.
 
-check
- (λG.λH,H':left_coset G.λx:Type_of_Group (group ? (subgrp ? H)). (embed ? (subgrp ? H) x)).
+(* Normal Subgroups *)
 
-definition left_coset_eq ≝
- λG.λH,H':left_coset G.
-  ∀x:group ? (subgrp ? H).
-   ex (group ? (subgroup ? H')) (λy.
-    (element ? H)·(embed ? (subgrp ? H) x) =
-    (element ? H')·(embed ? (subgrp ? H') y)).
- 
-(*record left_coset (G:Group) : Type ≝
- { subgroup: Group;
-   subgroup_is_subgroup: subgroup ≤ G;
-   element: G
+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
  }.
 
-definition left_coset_eq ≝
- λG.λH,H':left_coset G.
-  ∀x:subgroup ? H.
-   ex (subgroup ? H') (λy.
-    (element ? H)·(embed ? ? (subgroup_is_subgroup ? H) ˜ x) =
-    (element ? H')·(embed ? ? (subgroup_is_subgroup ? H') ˜ y)).
+(*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.
 *)