1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
|
(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
From mathcomp Require Import fintype finset fingroup morphism.
(******************************************************************************)
(* Support for generator-and-relation presentations of groups. We provide the *)
(* syntax: *)
(* G \homg Grp (x_1 : ... x_n : (s_1 = t_1, ..., s_m = t_m)) *)
(* <=> G is generated by elements x_1, ..., x_m satisfying the relations *)
(* s_1 = t_1, ..., s_m = t_m, i.e., G is a homomorphic image of the *)
(* group generated by the x_i, subject to the relations s_j = t_j. *)
(* G \isog Grp (x_1 : ... x_n : (s_1 = t_1, ..., s_m = t_m)) *)
(* <=> G is isomorphic to the group generated by the x_i, subject to the *)
(* relations s_j = t_j. This is an intensional predicate (in Prop), as *)
(* even the non-triviality of a generated group is undecidable. *)
(* Syntax details: *)
(* - Grp is a litteral constant. *)
(* - There must be at least one generator and one relation. *)
(* - A relation s_j = 1 can be abbreviated as simply s_j (a.k.a. a relator). *)
(* - Two consecutive relations s_j = t, s_j+1 = t can be abbreviated *)
(* s_j = s_j+1 = t. *)
(* - The s_j and t_j are terms built from the x_i and the standard group *)
(* operators *, 1, ^-1, ^+, ^-, ^, [~ u_1, ..., u_k]; no other operator or *)
(* abbreviation may be used, as the notation is implemented using static *)
(* overloading. *)
(* - This is the closest we could get to the notation used in Aschbacher, *)
(* Grp (x_1, ... x_n : t_1,1 = ... = t_1,k1, ..., t_m,1 = ... = t_m,km) *)
(* under the current limitations of the Coq Notation facility. *)
(* Semantics details: *)
(* - G \isog Grp (...) : Prop expands to the statement *)
(* forall rT (H : {group rT}), (H \homg G) = (H \homg Grp (...)) *)
(* (with rT : finGroupType). *)
(* - G \homg Grp (x_1 : ... x_n : (s_1 = t_1, ..., s_m = t_m)) : bool, with *)
(* G : {set gT}, is convertible to the boolean expression *)
(* [exists t : gT * ... gT, let: (x_1, ..., x_n) := t in *)
(* (<[x_1]> <*> ... <*> <[x_n]>, (s_1, ... (s_m-1, s_m) ...)) *)
(* == (G, (t_1, ... (t_m-1, t_m) ...))] *)
(* where the tuple comparison above is convertible to the conjunction *)
(* [&& <[x_1]> <*> ... <*> <[x_n]> == G, s_1 == t_1, ... & s_m == t_m] *)
(* Thus G \homg Grp (...) can be easily exploited by destructing the tuple *)
(* created case/existsP, then destructing the tuple equality with case/eqP. *)
(* Conversely it can be proved by using apply/existsP, providing the tuple *)
(* with a single exists (u_1, ..., u_n), then using rewrite !xpair_eqE /= *)
(* to expose the conjunction, and optionally using an apply/and{m+1}P view *)
(* to split it into subgoals (in that case, the rewrite is in principle *)
(* redundant, but necessary in practice because of the poor performance of *)
(* conversion in the Coq unifier). *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Declare Scope group_presentation.
Declare Scope nt_group_presentation.
Import GroupScope.
Module Presentation.
Section Presentation.
Implicit Types gT rT : finGroupType.
Implicit Type vT : finType. (* tuple value type *)
Inductive term :=
| Cst of nat
| Idx
| Inv of term
| Exp of term & nat
| Mul of term & term
| Conj of term & term
| Comm of term & term.
Fixpoint eval {gT} e t : gT :=
match t with
| Cst i => nth 1 e i
| Idx => 1
| Inv t1 => (eval e t1)^-1
| Exp t1 n => eval e t1 ^+ n
| Mul t1 t2 => eval e t1 * eval e t2
| Conj t1 t2 => eval e t1 ^ eval e t2
| Comm t1 t2 => [~ eval e t1, eval e t2]
end.
Inductive formula := Eq2 of term & term | And of formula & formula.
Definition Eq1 s := Eq2 s Idx.
Definition Eq3 s1 s2 t := And (Eq2 s1 t) (Eq2 s2 t).
Inductive rel_type := NoRel | Rel vT of vT & vT.
Definition bool_of_rel r := if r is Rel vT v1 v2 then v1 == v2 else true.
Local Coercion bool_of_rel : rel_type >-> bool.
Definition and_rel vT (v1 v2 : vT) r :=
if r is Rel wT w1 w2 then Rel (v1, w1) (v2, w2) else Rel v1 v2.
Fixpoint rel {gT} (e : seq gT) f r :=
match f with
| Eq2 s t => and_rel (eval e s) (eval e t) r
| And f1 f2 => rel e f1 (rel e f2 r)
end.
Inductive type := Generator of term -> type | Formula of formula.
Definition Cast p : type := p. (* syntactic scope cast *)
Local Coercion Formula : formula >-> type.
Inductive env gT := Env of {set gT} & seq gT.
Definition env1 {gT} (x : gT : finType) := Env <[x]> [:: x].
Fixpoint sat gT vT B n (s : vT -> env gT) p :=
match p with
| Formula f =>
[exists v, let: Env A e := s v in and_rel A B (rel (rev e) f NoRel)]
| Generator p' =>
let s' v := let: Env A e := s v.1 in Env (A <*> <[v.2]>) (v.2 :: e) in
sat B n.+1 s' (p' (Cst n))
end.
Definition hom gT (B : {set gT}) p := sat B 1 env1 (p (Cst 0)).
Definition iso gT (B : {set gT}) p :=
forall rT (H : {group rT}), (H \homg B) = hom H p.
End Presentation.
End Presentation.
Import Presentation.
Coercion bool_of_rel : rel_type >-> bool.
Coercion Eq1 : term >-> formula.
Coercion Formula : formula >-> type.
(* Declare (implicitly) the argument scope tags. *)
Notation "1" := Idx : group_presentation.
Arguments Inv _%group_presentation.
Arguments Exp _%group_presentation _%N.
Arguments Mul _%group_presentation _%group_presentation.
Arguments Conj _%group_presentation _%group_presentation.
Arguments Comm _%group_presentation _%group_presentation.
Arguments Eq1 _%group_presentation.
Arguments Eq2 _%group_presentation _%group_presentation.
Arguments Eq3 _%group_presentation _%group_presentation _%group_presentation.
Arguments And _%group_presentation _%group_presentation.
Arguments Formula _%group_presentation.
Arguments Cast _%group_presentation.
Infix "*" := Mul : group_presentation.
Infix "^+" := Exp : group_presentation.
Infix "^" := Conj : group_presentation.
Notation "x ^-1" := (Inv x) : group_presentation.
Notation "x ^- n" := (Inv (x ^+ n)) : group_presentation.
Notation "[ ~ x1 , x2 , .. , xn ]" :=
(Comm .. (Comm x1 x2) .. xn) : group_presentation.
Notation "x = y" := (Eq2 x y) : group_presentation.
Notation "x = y = z" := (Eq3 x y z) : group_presentation.
Notation "( r1 , r2 , .. , rn )" :=
(And .. (And r1 r2) .. rn) : group_presentation.
(* Declare (implicitly) the argument scope tags. *)
Notation "x : p" := (fun x => Cast p) : nt_group_presentation.
Arguments Generator _%nt_group_presentation.
Arguments hom _ _%group_scope _%nt_group_presentation.
Arguments iso _ _%group_scope _%nt_group_presentation.
Notation "x : p" := (Generator (x : p)) : group_presentation.
Notation "H \homg 'Grp' p" := (hom H p)
(at level 70, p at level 0, format "H \homg 'Grp' p") : group_scope.
Notation "H \isog 'Grp' p" := (iso H p)
(at level 70, p at level 0, format "H \isog 'Grp' p") : group_scope.
Notation "H \homg 'Grp' ( x : p )" := (hom H (x : p))
(at level 70, x at level 0,
format "'[hv' H '/ ' \homg 'Grp' ( x : p ) ']'") : group_scope.
Notation "H \isog 'Grp' ( x : p )" := (iso H (x : p))
(at level 70, x at level 0,
format "'[hv' H '/ ' \isog 'Grp' ( x : p ) ']'") : group_scope.
Section PresentationTheory.
Implicit Types gT rT : finGroupType.
Import Presentation.
Lemma isoGrp_hom gT (G : {group gT}) p : G \isog Grp p -> G \homg Grp p.
Proof. by move <-; apply: homg_refl. Qed.
Lemma isoGrpP gT (G : {group gT}) p rT (H : {group rT}) :
G \isog Grp p -> reflect (#|H| = #|G| /\ H \homg Grp p) (H \isog G).
Proof.
move=> isoGp; apply: (iffP idP) => [isoGH | [oH homHp]].
by rewrite (card_isog isoGH) -isoGp isog_hom.
by rewrite isogEcard isoGp homHp /= oH.
Qed.
Lemma homGrp_trans rT gT (H : {set rT}) (G : {group gT}) p :
H \homg G -> G \homg Grp p -> H \homg Grp p.
Proof.
case/homgP=> h <-{H}; rewrite /hom; move: {p}(p _) => p.
have evalG e t: all (mem G) e -> eval (map h e) t = h (eval e t).
move=> Ge; apply: (@proj2 (eval e t \in G)); elim: t => /=.
- move=> i; case: (leqP (size e) i) => [le_e_i | lt_i_e].
by rewrite !nth_default ?size_map ?morph1.
by rewrite (nth_map 1) // [_ \in G](allP Ge) ?mem_nth.
- by rewrite morph1.
- by move=> t [Gt ->]; rewrite groupV morphV.
- by move=> t [Gt ->] n; rewrite groupX ?morphX.
- by move=> t1 [Gt1 ->] t2 [Gt2 ->]; rewrite groupM ?morphM.
- by move=> t1 [Gt1 ->] t2 [Gt2 ->]; rewrite groupJ ?morphJ.
by move=> t1 [Gt1 ->] t2 [Gt2 ->]; rewrite groupR ?morphR.
have and_relE xT x1 x2 r: @and_rel xT x1 x2 r = (x1 == x2) && r :> bool.
by case: r => //=; rewrite andbT.
have rsatG e f: all (mem G) e -> rel e f NoRel -> rel (map h e) f NoRel.
move=> Ge; have: NoRel -> NoRel by []; move: NoRel {2 4}NoRel.
elim: f => [x1 x2 | f1 IH1 f2 IH2] r hr IHr; last by apply: IH1; apply: IH2.
by rewrite !and_relE !evalG //; case/andP; move/eqP->; rewrite eqxx.
set s := env1; set vT := gT : finType in s *.
set s' := env1; set vT' := rT : finType in s' *.
have (v): let: Env A e := s v in
A \subset G -> all (mem G) e /\ exists v', s' v' = Env (h @* A) (map h e).
- rewrite /= cycle_subG andbT => Gv; rewrite morphim_cycle //.
by split; last exists (h v).
elim: p 1%N vT vT' s s' => /= [p IHp | f] n vT vT' s s' Gs.
apply: IHp => [[v x]] /=; case: (s v) {Gs}(Gs v) => A e /= Gs.
rewrite join_subG cycle_subG; case/andP=> sAG Gx; rewrite Gx.
have [//|-> [v' def_v']] := Gs; split=> //; exists (v', h x); rewrite def_v'.
by congr (Env _ _); rewrite morphimY ?cycle_subG // morphim_cycle.
case/existsP=> v; case: (s v) {Gs}(Gs v) => /= A e Gs.
rewrite and_relE => /andP[/eqP defA rel_f].
have{Gs} [|Ge [v' def_v']] := Gs; first by rewrite defA.
apply/existsP; exists v'; rewrite def_v' and_relE defA eqxx /=.
by rewrite -map_rev rsatG ?(eq_all_r (mem_rev e)).
Qed.
Lemma eq_homGrp gT rT (G : {group gT}) (H : {group rT}) p :
G \isog H -> (G \homg Grp p) = (H \homg Grp p).
Proof.
by rewrite isogEhom => /andP[homGH homHG]; apply/idP/idP; apply: homGrp_trans.
Qed.
Lemma isoGrp_trans gT rT (G : {group gT}) (H : {group rT}) p :
G \isog H -> H \isog Grp p -> G \isog Grp p.
Proof. by move=> isoGH isoHp kT K; rewrite -isoHp; apply: eq_homgr. Qed.
Lemma intro_isoGrp gT (G : {group gT}) p :
G \homg Grp p -> (forall rT (H : {group rT}), H \homg Grp p -> H \homg G) ->
G \isog Grp p.
Proof.
move=> homGp freeG rT H.
by apply/idP/idP=> [homHp|]; [apply: homGrp_trans homGp | apply: freeG].
Qed.
End PresentationTheory.
|
