1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/algebra/groups/".
17 include "algebra/monoids.ma".
18 include "nat/le_arith.ma".
19 include "datatypes/bool.ma".
20 include "nat/compare.ma".
22 record PreGroup : Type ≝
23 { premonoid:> PreMonoid;
24 inv: premonoid -> premonoid
27 record isGroup (G:PreGroup) : Prop ≝
28 { is_monoid: isMonoid G;
29 inv_is_left_inverse: is_left_inverse (mk_Monoid ? is_monoid) (inv G);
30 inv_is_right_inverse: is_right_inverse (mk_Monoid ? is_monoid) (inv G)
34 { pregroup:> PreGroup;
35 group_properties:> isGroup pregroup
41 interpretation "Monoid coercion" 'monoid G =
42 (cic:/matita/algebra/groups/monoid.con G).*)
45 for @{ 'type_of_group $G }.
47 interpretation "Type_of_group coercion" 'type_of_group G =
48 (cic:/matita/algebra/groups/Type_of_Group.con G).
51 for @{ 'magma_of_group $G }.
53 interpretation "magma_of_group coercion" 'magma_of_group G =
54 (cic:/matita/algebra/groups/Magma_of_Group.con G).
56 notation "hvbox(x \sup (-1))" with precedence 89
59 interpretation "Group inverse" 'ginv x =
60 (cic:/matita/algebra/groups/inv.con _ x).
62 definition left_cancellable ≝
63 λT:Type. λop: T -> T -> T.
64 ∀x. injective ? ? (op x).
66 definition right_cancellable ≝
67 λT:Type. λop: T -> T -> T.
68 ∀x. injective ? ? (λz.op z x).
70 theorem eq_op_x_y_op_x_z_to_eq:
71 ∀G:Group. left_cancellable G (op G).
73 unfold left_cancellable;
76 rewrite < (e_is_left_unit ? (is_monoid ? G));
77 rewrite < (e_is_left_unit ? (is_monoid ? G) z);
78 rewrite < (inv_is_left_inverse ? G x);
79 rewrite > (associative ? (is_semi_group ? (is_monoid ? G)));
80 rewrite > (associative ? (is_semi_group ? (is_monoid ? G)));
86 theorem eq_op_x_y_op_z_y_to_eq:
87 ∀G:Group. right_cancellable G (op G).
89 unfold right_cancellable;
91 simplify;fold simplify (op G);
93 rewrite < (e_is_right_unit ? (is_monoid ? G));
94 rewrite < (e_is_right_unit ? (is_monoid ? G) z);
95 rewrite < (inv_is_right_inverse ? G x);
96 rewrite < (associative ? (is_semi_group ? (is_monoid ? G)));
97 rewrite < (associative ? (is_semi_group ? (is_monoid ? G)));
102 theorem inv_inv: ∀G:Group. ∀x:G. x \sup -1 \sup -1 = x.
104 apply (eq_op_x_y_op_z_y_to_eq ? (x \sup -1));
105 rewrite > (inv_is_right_inverse ? G);
106 rewrite > (inv_is_left_inverse ? G);
110 theorem eq_opxy_e_to_eq_x_invy:
111 ∀G:Group. ∀x,y:G. x·y=1 → x=y \sup -1.
113 apply (eq_op_x_y_op_z_y_to_eq ? y);
114 rewrite > (inv_is_left_inverse ? G);
118 theorem eq_opxy_e_to_eq_invx_y:
119 ∀G:Group. ∀x,y:G. x·y=1 → x \sup -1=y.
121 apply (eq_op_x_y_op_x_z_to_eq ? x);
122 rewrite > (inv_is_right_inverse ? G);
127 theorem eq_opxy_z_to_eq_x_opzinvy:
128 ∀G:Group. ∀x,y,z:G. x·y=z → x = z·y \sup -1.
130 apply (eq_op_x_y_op_z_y_to_eq ? y);
131 rewrite > (associative ? (is_semi_group ? (is_monoid ? G)));
132 rewrite > (inv_is_left_inverse ? G);
133 rewrite > (e_is_right_unit ? (is_monoid ? G));
137 theorem eq_opxy_z_to_eq_y_opinvxz:
138 ∀G:Group. ∀x,y,z:G. x·y=z → y = x \sup -1·z.
140 apply (eq_op_x_y_op_x_z_to_eq ? x);
141 rewrite < (associative ? (is_semi_group ? (is_monoid ? G)));
142 rewrite > (inv_is_right_inverse ? G);
143 rewrite > (e_is_left_unit ? (is_monoid ? G));
149 record morphism (G,G':Group) : Type ≝
151 f_morph: ∀x,y:G.image(x·y) = image x · image y
154 notation "hvbox(f˜ x)" with precedence 79
155 for @{ 'morimage $f $x }.
157 interpretation "Morphism image" 'morimage f x =
158 (cic:/matita/algebra/groups/image.con _ _ f x).
160 theorem morphism_to_eq_f_1_1:
161 ∀G,G'.∀f:morphism G G'.f˜1 = 1.
163 apply (eq_op_x_y_op_z_y_to_eq G' (f˜1));
164 rewrite > (e_is_left_unit ? (is_monoid ? G') ?);
165 rewrite < (f_morph ? ? f);
166 rewrite > (e_is_left_unit ? (is_monoid ? G));
170 theorem eq_image_inv_inv_image:
171 ∀G,G'.∀f:morphism G G'.
172 ∀x.f˜(x \sup -1) = (f˜x) \sup -1.
174 apply (eq_op_x_y_op_z_y_to_eq G' (f˜x));
175 rewrite > (inv_is_left_inverse ? G');
176 rewrite < (f_morph ? ? f);
177 rewrite > (inv_is_left_inverse ? G);
178 apply (morphism_to_eq_f_1_1 ? ? f).
181 record monomorphism (G,G':Group) : Type ≝
182 { morphism: morphism G G';
183 injective: injective ? ? (image ? ? morphism)
188 record subgroup (G:Group) : Type ≝
190 embed: monomorphism group G
193 notation "hvbox(x \sub H)" with precedence 79
194 for @{ 'subgroupimage $H $x }.
196 interpretation "Subgroup image" 'subgroupimage H x =
197 (cic:/matita/algebra/groups/image.con _ _
198 (cic:/matita/algebra/groups/morphism.con _ _
199 (cic:/matita/algebra/groups/embed.con _ H))
202 definition belongs_to_subgroup ≝
203 λG.λH:subgroup G.λx:G.∃y.x=y \sub H.
205 notation "hvbox(x ∈ H)" with precedence 79
206 for @{ 'belongs_to $x $H }.
208 interpretation "Belongs to subgroup" 'belongs_to x H =
209 (cic:/matita/algebra/groups/belongs_to_subgroup.con _ H x).
213 record left_coset (G:Group) : Type ≝
218 (* Here I would prefer 'magma_op, but this breaks something in the next definition *)
219 interpretation "Left_coset" 'times x C =
220 (cic:/matita/algebra/groups/left_coset.ind#xpointer(1/1/1) _ x C).
222 definition belongs_to_left_coset ≝
223 λG:Group.λC:left_coset G.λx:G.
224 ∃y.x=(element ? C)·y \sub (subgrp ? C).
226 interpretation "Belongs to left_coset" 'belongs_to x C =
227 (cic:/matita/algebra/groups/belongs_to_left_coset.con _ C x).
229 definition left_coset_eq ≝
230 λG.λC,C':left_coset G.
231 ∀x.((element ? C)·x \sub (subgrp ? C)) ∈ C'.
233 interpretation "Left cosets equality" 'eq C C' =
234 (cic:/matita/algebra/groups/left_coset_eq.con _ C C').
236 definition left_coset_disjoint ≝
237 λG.λC,C':left_coset G.
238 ∀x.¬(((element ? C)·x \sub (subgrp ? C)) ∈ C').
240 notation "hvbox(a break ∥ b)"
241 non associative with precedence 45
242 for @{ 'disjoint $a $b }.
244 interpretation "Left cosets disjoint" 'disjoint C C' =
245 (cic:/matita/algebra/groups/left_coset_disjoint.con _ C C').
248 (* The following should be a one-shot alias! *)
249 alias symbol "belongs_to" (instance 0) = "Belongs to subgroup".
251 ∀G.∀x,y:(Type_of_Group G).∀H:subgroup G.
252 (x \sup -1 ·y) ∈ H → (mk_left_coset ? x H) = (mk_left_coset ? y H).
254 unfold left_coset_eq;
255 simplify in ⊢ (? → ? ? ? (? ? ? (? ? ? (? ? ? (? ? %)) ?)));
256 simplify in ⊢ (? → ? ? ? (? ? % ?));
257 simplify in ⊢ (? % → ?);
259 unfold belongs_to_left_coset;
260 simplify in ⊢ (? ? (λy:?.? ? ? (? ? ? (? ? ? (? ? ? (? ? %)) ?))));
261 simplify in ⊢ (? ? (λy:? %.?));
262 simplify in ⊢ (? ? (λy:?.? ? ? (? ? % ?)));
263 unfold belongs_to_subgroup in H1;
274 \forall G:Group. \forall x1,x2:G. \forall H:subgroup G.
275 x1*x2^-1 \nin H \to x1*H does_not_overlap x2*H
278 \forall x:G. \forall H:subgroup G. x \in x*H
281 (T: Type) (n:nat) (S: \forall x:nat. x < n -> {S:Type * (S -> T)})
283 \forall i,j:nat. i < n \to j < n \to ...
287 (λG.λH,H':left_coset G.λx:Type_of_Group (group ? (subgrp ? H)). (embed ? (subgrp ? H) x)).
289 definition left_coset_eq ≝
290 λG.λH,H':left_coset G.
291 ∀x:group ? (subgrp ? H).
292 ex (group ? (subgroup ? H')) (λy.
293 (element ? H)·(embed ? (subgrp ? H) x) =
294 (element ? H')·(embed ? (subgrp ? H') y)).
296 (*record left_coset (G:Group) : Type ≝
298 subgroup_is_subgroup: subgroup ≤ G;
302 definition left_coset_eq ≝
303 λG.λH,H':left_coset G.
305 ex (subgroup ? H') (λy.
306 (element ? H)·(embed ? ? (subgroup_is_subgroup ? H) ˜ x) =
307 (element ? H')·(embed ? ? (subgroup_is_subgroup ? H') ˜ y)).