(* *)
(**************************************************************************)
-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.
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.
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);
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);
∀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.
∀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.
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);
(* 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);
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);
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 *)
(* 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".
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
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);
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.
+*)