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
}.
-interpretation "Group inverse" 'invert x = (inv _ x).
+definition Monoid_of_Group: Group → Monoid ≝
+ λG. mk_Monoid ? (is_group G).
+
+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.
∀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);
for @{ 'subgroupimage $H $x }.
interpretation "Subgroup image" 'subgroupimage H x =
- (image _ _ (morphism_OF_subgroup _ 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 =
- (member_of_subgroup _ H x).
+ (member_of_subgroup ? H x).
interpretation "Not member of subgroup" 'not_member_of x H =
- (Not (member_of_subgroup _ 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 =
- (mk_left_coset _ 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 =
- (member_of_left_coset _ C x).
+ (member_of_left_coset ? C x).
definition left_coset_eq ≝
λG.λC,C':left_coset G.
for @{ 'disjoint $a $b }.
interpretation "Left cosets disjoint" 'disjoint C C' =
- (left_coset_disjoint _ 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);