(* *)
(**************************************************************************)
-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);
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 \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 =
+ (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".
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;
intro;
unfold member_of_subgroup in H1;
].
qed.
-(*theorem foo:
- \forall G:Group. \forall x1,x2:G. \forall H:subgroup G.
- x1*x2^-1 \nin H \to x1*H does_not_overlap x2*H
-
-theorem foo:
- \forall x:G. \forall H:subgroup G. x \in x*H
-
-definition disjoinct
- (T: Type) (n:nat) (S: \forall x:nat. x < n -> {S:Type * (S -> T)})
-:=
- \forall i,j:nat. i < n \to j < n \to ...
+theorem Not_member_of_subgroup_to_left_coset_disjoint:
+ ∀G.∀x,y.∀H:subgroup G.(x \sup -1 ·y) ∉ H → x*H ∥ y*H.
+intros;
+simplify;
+unfold Not;
+intros (x');
+apply H1;
+unfold member_of_subgroup;
+elim H2;
+apply (ex_intro ? ? (x'·a \sup -1));
+rewrite > f_morph;
+apply (eq_op_x_y_op_z_y_to_eq ? (a \sub H));
+rewrite > (op_associative ? G);
+rewrite < H3;
+rewrite > (op_associative ? G);
+rewrite < f_morph;
+rewrite > (inv_is_left_inverse ? H);
+rewrite < (op_associative ? G);
+rewrite > (inv_is_left_inverse ? G);
+rewrite > (e_is_left_unit ? G);
+rewrite < (f_morph ? ? H);
+rewrite > (e_is_right_unit ? 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).
+intros;
+simplify;
+apply (ex_intro ? ? ⅇ);
+rewrite > morphism_to_eq_f_1_1;
+rewrite > (e_is_right_unit ? G);
+reflexivity.
+qed.
-check
- (λG.λH,H':left_coset G.λx:Type_of_Group (group ? (subgrp ? H)). (embed ? (subgrp ? H) x)).
+(* Normal Subgroups *)
-definition left_coset_eq ≝
- λG.λH,H':left_coset G.
- ∀x:group ? (subgrp ? H).
- ex (group ? (subgroup ? H')) (λy.
- (element ? H)·(embed ? (subgrp ? H) x) =
- (element ? H')·(embed ? (subgrp ? H') y)).
-
-(*record left_coset (G:Group) : Type ≝
- { subgroup: Group;
- subgroup_is_subgroup: subgroup ≤ G;
- element: G
+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
}.
-definition left_coset_eq ≝
- λG.λH,H':left_coset G.
- ∀x:subgroup ? H.
- ex (subgroup ? H') (λy.
- (element ? H)·(embed ? ? (subgroup_is_subgroup ? H) ˜ x) =
- (element ? H')·(embed ? ? (subgroup_is_subgroup ? H') ˜ y)).
+(*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.
*)
projects / helm.git / blobdiff