(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/algebra/groups/".
-
include "algebra/monoids.ma".
include "nat/le_arith.ma".
include "datatypes/bool.ma".
group_properties:> isGroup pregroup
}.
-notation "hvbox(x \sup (-1))" with precedence 89
-for @{ 'ginv $x }.
-
-interpretation "Group inverse" 'ginv x =
- (cic:/matita/algebra/groups/inv.con _ x).
+interpretation "Group inverse" 'invert x = (inv _ x).
definition left_cancellable ≝
λT:Type. λop: T -> T -> T.
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);
(* 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);
interpretation "Subgroup image" 'subgroupimage H x =
(cic:/matita/algebra/groups/image.con _ _
- (cic:/matita/algebra/groups/morphism_of_subgroup.con _ H) x).
+ (cic:/matita/algebra/groups/morphism_OF_subgroup.con _ 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 =
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 }.
(*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.
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.
\ No newline at end of file
+qed.
+*)