(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
From mathcomp
From mathcomp
(******************************************************************************)
(* 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.
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.
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.

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