}.
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)
}.
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.
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.
∀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.
∀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 ≝
∀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.
qed.
record monomorphism (G,G':Group) : Type ≝
- { morphism: morphism G G';
+ { morphism:> morphism G G';
injective: injective ? ? (image ? ? morphism)
}.
record subgroup (G:Group) : Type ≝
{ group: Group;
- embed: monomorphism group G
+ embed:> monomorphism group G
}.
notation "hvbox(x \sub H)" with precedence 79
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 *)
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.
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.