Library mathcomp.fingroup.presentation

(* (c) Copyright 2006-2015 Microsoft Corporation and Inria.                  
 Distributed under the terms of CeCILL-B.                                  *)
Require Import mathcomp.ssreflect.ssreflect.

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.

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.

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 *)

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 ⇒
    [∃ 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 :=
  ∀ 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.

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.

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.

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).

Lemma homGrp_trans rT gT (H : {set rT}) (G : {group gT}) p :
  H \homg G → G \homg Grp p → H \homg Grp p.

Lemma eq_homGrp gT rT (G : {group gT}) (H : {group rT}) p :
  G \isog H → (G \homg Grp p) = (H \homg Grp p).

Lemma isoGrp_trans gT rT (G : {group gT}) (H : {group rT}) p :
  G \isog H → H \isog Grp p → G \isog Grp p.

Lemma intro_isoGrp gT (G : {group gT}) p :
    G \homg Grp p → (∀ rT (H : {group rT}), H \homg Grp p → H \homg G) →
  G \isog Grp p.

End PresentationTheory.