--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/algebra/groups/".
+
+include "algebra/monoids.ma".
+include "nat/le_arith.ma".
+include "datatypes/bool.ma".
+include "nat/compare.ma".
+
+record PreGroup : Type ≝
+ { premonoid:> PreMonoid;
+ inv: premonoid -> premonoid
+ }.
+
+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)
+ }.
+
+record Group : Type ≝
+ { pregroup:> PreGroup;
+ 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).
+
+definition left_cancellable ≝
+ λT:Type. λop: T -> T -> T.
+ ∀x. injective ? ? (op x).
+
+definition right_cancellable ≝
+ λT:Type. λop: T -> T -> T.
+ ∀x. injective ? ? (λz.op z x).
+
+theorem eq_op_x_y_op_x_z_to_eq:
+ ∀G:Group. left_cancellable G (op G).
+intros;
+unfold left_cancellable;
+unfold injective;
+intros (x y z);
+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);
+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 > H;
+reflexivity.
+qed.
+
+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);
+rewrite > (inv_is_left_inverse ? G);
+reflexivity.
+qed.
+
+theorem eq_opxy_e_to_eq_x_invy:
+ ∀G:Group. ∀x,y:G. x·y=1 → x=y \sup -1.
+intros;
+apply (eq_op_x_y_op_z_y_to_eq ? y);
+rewrite > (inv_is_left_inverse ? G);
+assumption.
+qed.
+
+theorem eq_opxy_e_to_eq_invx_y:
+ ∀G:Group. ∀x,y:G. x·y=1 → x \sup -1=y.
+intros;
+apply (eq_op_x_y_op_x_z_to_eq ? x);
+rewrite > (inv_is_right_inverse ? G);
+symmetry;
+assumption.
+qed.
+
+theorem eq_opxy_z_to_eq_x_opzinvy:
+ ∀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 > (inv_is_left_inverse ? G);
+rewrite > (e_is_right_unit ? G);
+assumption.
+qed.
+
+theorem eq_opxy_z_to_eq_y_opinvxz:
+ ∀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 > (inv_is_right_inverse ? 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 ≝
+ { image: 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.
+intros;
+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.
+
+theorem eq_image_inv_inv_image:
+ ∀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 ? (f˜x));
+rewrite > (inv_is_left_inverse ? G');
+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';
+ injective: injective ? ? (image ? ? morphism)
+ }.
+
+(* Subgroups *)
+
+record subgroup (G:Group) : Type ≝
+ { 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_OF_subgroup.con _ 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
+for @{ 'member_of $x $H }.
+
+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 *)
+
+record left_coset (G:Group) : Type ≝
+ { element: G;
+ subgrp: subgroup G
+ }.
+
+(* 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).
+
+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).
+
+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').
+
+definition left_coset_disjoint ≝
+ λG.λC,C':left_coset G.
+ ∀x.¬(((element ? C)·x \sub (subgrp ? C)) ∈ C').
+
+notation "hvbox(a break ∥ 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').
+
+(* 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;
+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 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.
+
+(* Normal Subgroups *)
+
+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
+ }.
+
+(*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