}.
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)
}.
group_properties:> isGroup pregroup
}.
-(*notation < "G"
-for @{ 'monoid $G }.
-
-interpretation "Monoid coercion" 'monoid G =
- (cic:/matita/algebra/groups/monoid.con G).*)
-
-notation < "G"
-for @{ 'type_of_group $G }.
-
-interpretation "Type_of_group coercion" 'type_of_group G =
- (cic:/matita/algebra/groups/Type_of_Group.con G).
-
-notation < "G"
-for @{ 'magma_of_group $G }.
-
-interpretation "magma_of_group coercion" 'magma_of_group G =
- (cic:/matita/algebra/groups/Magma_of_Group.con G).
-
notation "hvbox(x \sup (-1))" with precedence 89
for @{ 'ginv $x }.
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 > (op_associative ? G);
+rewrite > (op_associative ? 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 < (op_associative ? G);
+rewrite < (op_associative ? G);
rewrite > H;
reflexivity.
qed.
-theorem inv_inv: ∀G:Group. ∀x:G. x \sup -1 \sup -1 = x.
+theorem eq_inv_inv_x_x: ∀G:Group. ∀x:G. x \sup -1 \sup -1 = x.
intros;
apply (eq_op_x_y_op_z_y_to_eq ? (x \sup -1));
rewrite > (inv_is_right_inverse ? G);
∀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 > (op_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 < (op_associative ? G);
rewrite > (inv_is_right_inverse ? G);
-rewrite > (e_is_left_unit ? (is_monoid ? G));
+rewrite > (e_is_left_unit ? 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 < (op_associative ? G);
+rewrite > (op_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 ≝
theorem morphism_to_eq_f_1_1:
∀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 < (f_morph ? ? f);
-rewrite > (e_is_left_unit ? (is_monoid ? G));
+apply (eq_op_x_y_op_z_y_to_eq ? (f˜1));
+rewrite > (e_is_left_unit ? G');
+rewrite < f_morph;
+rewrite > (e_is_left_unit ? G);
reflexivity.
qed.
∀G,G'.∀f:morphism G G'.
∀x.f˜(x \sup -1) = (f˜x) \sup -1.
intros;
-apply (eq_op_x_y_op_z_y_to_eq G' (f˜x));
+apply (eq_op_x_y_op_z_y_to_eq ? (f˜x));
rewrite > (inv_is_left_inverse ? G');
-rewrite < (f_morph ? ? f);
+rewrite < f_morph;
rewrite > (inv_is_left_inverse ? G);
apply (morphism_to_eq_f_1_1 ? ? f).
qed.
record monomorphism (G,G':Group) : Type ≝
- { morphism: morphism G G';
+ { morphism:> morphism G G';
injective: injective ? ? (image ? ? morphism)
}.
(* Subgroups *)
record subgroup (G:Group) : Type ≝
- { group: Group;
- embed: monomorphism group G
+ { group:> Group;
+ embed:> monomorphism group G
}.
-
+
notation "hvbox(x \sub H)" with precedence 79
for @{ 'subgroupimage $H $x }.
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).
+notation "hvbox(x break ∉ 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).
+
+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)).
(* 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 ⊢ (? % → ?);
-intros;
-unfold belongs_to_left_coset;
-simplify in ⊢ (? ? (λy:?.? ? ? (? ? ? (? ? ? (? ? ? (? ? %)) ?))));
-simplify in ⊢ (? ? (λy:? %.?));
-simplify in ⊢ (? ? (λy:?.? ? ? (? ? % ?)));
-unfold belongs_to_subgroup in H1;
+simplify;
+intro;
+unfold member_of_subgroup in H1;
elim H1;
clear H1;
exists;
-[
-|
+[ apply (a\sup-1 · x1)
+| rewrite > f_morph;
+ rewrite > eq_image_inv_inv_image;
+ 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 > (inv_is_right_inverse ? G);
+ rewrite > (e_is_left_unit ? G);
+ reflexivity
].
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 ? ? 1);
+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