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\tilde 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.
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.
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".
∀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.