Library fingroup

 This file defines the main interface for finite groups :                   
          finGroupType == the structure for finite types with a group law.  
           {group gT}  == type of groups with elements of type gT.          
      baseFinGroupType == the structure for finite types with a monoid law  
                          and an involutive antimorphism; finGroupType is   
                          derived from baseFinGroupType (via a telescope).  
    FinGroupType mulVg == the finGroupType structure for an existing        
                          baseFinGroupType structure, built from a proof of 
                          the left inverse group axiom for that structure's 
                          operations.                                       
  BaseFinGroupType bgm == the baseFingroupType structure built by packaging 
                          bgm : FinGroup.mixin_of T for a type T with an    
                          existing finType structure.                       
 FinGroup.BaseMixin mulA mul1x invK invM ==                                 
                          the mixin for a baseFinGroupType structure, built 
                          from proofs of the baseFinGroupType axioms.       
 FinGroup.Mixin mulA mul1x mulVg ==                                         
                          the mixin for a baseFinGroupType structure, built 
                          from proofs of the group axioms.                  
 [baseFinGroupType of T] == a clone of an existing baseFinGroupType         
                          structure on T, for T (the existing structure     
                          might be for som delta-expansion of T).           
   [finGroupType of T] == a clone of an existing finGroupType structure on  
                          T, for the canonical baseFinGroupType structure   
                          of T (the existing structure might be for the     
                          baseFinGroupType of some delta-expansion of T).   
          [group of G] == a clone for an existing {group gT} structure on   
                          G : {set gT} (the existing structure might be for 
                          some delta-expansion of G).                       
 If gT implements finGroupType, then we can form {set gT}, the type of      
 finite sets with elements of type gT (as finGroupType extends finType).    
 The group law extends pointwise to {set gT}, which thus implements a sub-  
 interface baseFinGroupType of finGroupType. To be consistent with the      
 predType interface, this is done by coercion to FinGroup.arg_sort, an      
 alias for FinGroup.sort. Accordingly, all pointwise group operations below 
 have arguments of type (FinGroup.arg_sort) gT and return results of type   
 FinGroup.sort gT.                                                          
   The notations below are declared in two scopes:                          
      group_scope (delimiter %g) for point operations and set constructs.   
   subgroup_scope (delimiter %G) for explicit {group gT} structures.        
 These scopes should not be opened globally, athough group_scope is often   
 opened locally in group-theory files (via Import GroupScope).              
   As {group gT} is both a subtype and an interface structure for {set gT}, 
 the fact that a given G : {set gT} is a group can (and usually should) be  
 inferred by type inference with Canonical Structures. This means that all  
 `group' constructions (e.g., the normaliser 'N_G(H)) actually define sets  
 with a canonical {group gT} structure; the %G delimiter can be used to     
 specify the actual {group gT} structure (e.g., 'N_G(H)%G).                 
  Operations on elements of a group:                                        
                x * y == the group product of x and y.                      
               x ^+ n == the nth power of x, i.e., x * ... * x (n times).   
                 x^-1 == the group inverse of x.                            
               x ^- n == the inverse of x ^+ n (notation for (x ^+ n)^-1).  
                    1 == the unit element.                                  
                x ^ y == the conjugate of x by y.                           
    \prod_(i ...) x i == the product of the x i (order-sensitive).          
         commute x y  <-> x and y commute.                                  
      centralises x A <-> x centralises A.                                  
                'C[x] == the set of elements that commute with x.           
              'C_G[x] == the set of elements of G that commute with x.      
                <[x]> == the cyclic subgroup generated by the element x.    
                 #[x] == the order of the element x, i.e., #|<[x]>|.        
     [~ x1, ..., xn]  == the commutator of x1, ..., xn.                     
  Operations on subsets/subgroups of a finite group:                        
                H * G == {xy | x \in H, y \in G}.                           
   1 or [1] or [1 gT] == the unit group.                                    
          [set: gT]%G == the group of all x : gT (in subgroup_scope).       
             [subg G] == the subtype, set, or group of all x \in G: this    
                         notation is defined simultaneously in %type, %g    
                         and %G scopes, and G must denote a {group gT}      
                         structure (G is in the %G scope).                  
          subg, sgval == the projection into and injection from [subg G].   
                  H^# == the set H minus the unit element                   
               repr H == some element of H if 1 \notin H != set0, else 1.   
                         (repr is defined over sets of a baseFinGroupType,  
                         so it can be used, e.g., to pick right cosets.)    
               x *: H == left coset of H by x.                              
          lcosets H G == the set of the left cosets of H by elements of G.  
               H :* x == right coset of H by x.                             
          rcosets H G == the set of the right cosets of H by elements of G. 
             #|G : H| == the index of H in G, i.e., #|rcosets G : H|.       
               H :^ x == the conjugate of H by x.                           
               x ^: H == the conjugate class of x in H.                     
            classes G == the set of all conjugate classes of G.             
              G :^: H == {G :^ x | x \in H}.                                
    class_support G H == {x ^ y | x \in G, y \in H}.                        
     [~: H1, ..., Hn] == commutator subgroup of H1, ..., Hn.                
{in G, centralised H} <-> G centralises H.                                  
 {in G, normalised H} <-> G normalises H.                                   
                      <-> forall x, x \in G -> H :^ x = H.                  
                'N(H) == the normaliser of H.                               
              'N_G(H) == the normaliser of H in G.                          
               H <| G <=> H is normal in G.                                 
                'C(H) == the centraliser of H.                              
              'C_G(H) == the centraliser of H in G.                         
                <<H>> == the subgroup generated by the set H.               
              H <*> G == the subgroup generated by H and G (their join).    
 (\prod_(i ...) H i)%G == the group generated by the H i.                   
            gcore H G == the largest subgroup of H normalised by G.         
                         If H is a subgroup of G, this is the largest       
                         normal subgroup of G contained in H).              
            abelian H <=> H is abelian.                                     
          subgroups G == the set of subgroups of G, i.e., the set of all    
                         H : {group gT} such that H \subset G.              
 In the notation below G is a variable that is bound in P.                  
          [max G | P] <=> G is the largest group such that P holds.         
     [max H of G | P] <=> H is the largest group G such that P holds.       
          [min G | P] <=> G is the smallest group such that P holds.        
     [min H of G | P] <=> H is the smallest group G such that P holds.      
 In addition to the generic suffixes described in ssrbool.v and finset.v,   
 we associate the following suffixes to group operations:                   
   M - multiplication, as is invMg : (x * y)^-1 = x^-1 * y^-1.              
   V - inverse, as in mulgV : x * x^-1 = 1.                                 
   X - exponentiation, as in conjXg : (x + n) ^ y = (x ^ y) ^+ n.           
   J - conjugation, as in orderJ : #[x ^ y] = #[x].                         
   R - commutator, as in conjRg : [~ x, y] ^ z = [~ x ^ z, y ^ z].          
   Y - join, as in centY : 'C(G <*> H) = 'C(G) :&: 'C(H).                   
 We sometimes prefix these with an `s' to indicate a set-lifted operation,  
 e.g., conjsMg : (A * B) :^ x = A :^ x * B :^ x.                            


Delimit Scope group_scope with g.
Delimit Scope subgroup_scope with G.

 This module can be imported to open the scope for group element 
 operations locally to a file, without exporing the Open to      
 clients of that file (as Open would do).                        
Module GroupScope.
Open Scope group_scope.
End GroupScope.
Import GroupScope.

 These are the operation notations introduced by this file. 
Reserved Notation "[ ~ x1 , x2 , .. , xn ]" (at level 0,
  format "'[ ' [ ~ x1 , '/' x2 , '/' .. , '/' xn ] ']'").
Reserved Notation "[ 1 gT ]" (at level 0, format "[ 1 gT ]").
Reserved Notation "[ 1 ]" (at level 0, format "[ 1 ]").
Reserved Notation "[ 'subg' G ]" (at level 0, format "[ 'subg' G ]").
Reserved Notation "A ^#" (at level 2, format "A ^#").
Reserved Notation "A :^ x" (at level 35, right associativity).
Reserved Notation "x ^: B" (at level 35, right associativity).
Reserved Notation "A :^: B" (at level 35, right associativity).
Reserved Notation "#| B : A |" (at level 0, B, A at level 99,
  format "#| B : A |").
Reserved Notation "''N' ( A )" (at level 8, format "''N' ( A )").
Reserved Notation "''N_' G ( A )" (at level 8, G at level 2,
  format "''N_' G ( A )").
Reserved Notation "A <| B" (at level 70, no associativity).
Reserved Notation "''C' [ x ]" (at level 8, format "''C' [ x ]").
Reserved Notation "''C_' G [ x ]" (at level 8, G at level 2,
  format "''C_' G [ x ]").
Reserved Notation "''C_' ( G ) [ x ]" (at level 8, only parsing).
Reserved Notation "<< A >>" (at level 0, format "<< A >>").
Reserved Notation "<[ x ] >" (at level 0, format "<[ x ] >").
Reserved Notation "#[ x ]" (at level 0, format "#[ x ]").
Reserved Notation "A <*> B" (at level 40, left associativity).
Reserved Notation "[ ~: A1 , A2 , .. , An ]" (at level 0,
  format "[ ~: '[' A1 , '/' A2 , '/' .. , '/' An ']' ]").
Reserved Notation "[ 'max' A 'of' G | gP ]" (at level 0,
  format "[ '[hv' 'max' A 'of' G '/ ' | gP ']' ]").
Reserved Notation "[ 'max' G | gP ]" (at level 0,
  format "[ '[hv' 'max' G '/ ' | gP ']' ]").
Reserved Notation "[ 'min' A 'of' G | gP ]" (at level 0,
  format "[ '[hv' 'min' A 'of' G '/ ' | gP ']' ]").
Reserved Notation "[ 'min' G | gP ]" (at level 0,
  format "[ '[hv' 'min' G '/ ' | gP ']' ]").

Module FinGroup.

 We split the group axiomatisation in two. We define a  
 class of "base groups", which are basically monoids    
 with an involutive antimorphism, from which we derive  
 the class of groups proper. This allows use to reuse   
 much of the group notation and algebraic axioms for    
 group subsets, by defining a base group class on them. 
   We use class/mixins here rather than telescopes to   
 be able to interoperate with the type coercions.       
 Another potential benefit (not exploited here) would   
 be to define a class for infinite groups, which could  
 share all of the algebraic laws.                       
Record mixin_of (T : Type) : Type := BaseMixin {
  mul : T -> T -> T;
  one : T;
  inv : T -> T;
  _ : associative mul;
  _ : left_id one mul;
  _ : involutive inv;
  _ : {morph inv : x y / mul x y >-> mul y x}
}.

Structure base_type : Type := PackBase {
  sort : Type;
   _ : mixin_of sort;
   _ : Finite.class_of sort
}.

 We want to use sort as a coercion class, both to infer         
 argument scopes properly, and to allow groups and cosets to    
 coerce to the base group of group subsets.                     
   However, the return type of group operations should NOT be a 
 coercion class, since this would trump the real (head-normal)  
 coercion class for concrete group types, thus spoiling the     
 coercion of A * B to pred_sort in x \in A * B, or rho * tau to 
 ffun and Funclass in (rho * tau) x, when rho tau : perm T.     
   Therefore we define an alias of sort for argument types, and 
 make it the default coercion FinGroup.base_class >-> Sortclass 
 so that arguments of a functions whose parameters are of type, 
 say, gT : finGroupType, can be coerced to the coercion class   
 of arg_sort. Care should be taken, however, to declare the     
 return type of functions and operators as FinGroup.sort gT     
 rather than gT, e.g., mulg : gT -> gT -> FinGroup.sort gT.     
 Note that since we do this here and in normal.v for all the    
 basic functions, the inferred return type should generally be  
 correct.                                                       
Definition arg_sort := sort.

Definition mixin T :=
  let: PackBase _ m _ := T return mixin_of (sort T) in m.

Definition finClass T :=
  let: PackBase _ _ m := T return Finite.class_of (sort T) in m.

Structure type : Type := Pack {
  base : base_type;
  _ : left_inverse (one (mixin base)) (inv (mixin base)) (mul (mixin base))
}.

 We only need three axioms to make a true group. 

Section Mixin.

Variables (T : Type) (one : T) (mul : T -> T -> T) (inv : T -> T).

Hypothesis mulA : associative mul.
Hypothesis mul1 : left_id one mul.
Hypothesis mulV : left_inverse one inv mul.
Notation "1" := one.
Infix "*" := mul.
Notation "x ^-1" := (inv x).

Lemma mk_invgK : involutive inv.

Lemma mk_invMg : {morph inv : x y / x * y >-> y * x}.

Definition Mixin := BaseMixin mulA mul1 mk_invgK mk_invMg.

End Mixin.

Definition pack_base T m :=
  fun c cT & phant_id (Finite.class cT) c => @PackBase T m c.

Definition clone_base T :=
  fun bT & sort bT -> T =>
  fun m c (bT' := @PackBase T m c) & phant_id bT' bT => bT'.

Definition clone T :=
  fun bT gT & sort bT * sort (base gT) -> T * T =>
  fun m (gT' := @Pack bT m) & phant_id gT' gT => gT'.

Section InheritedClasses.

Variable bT : base_type.
Local Notation T := (arg_sort bT).
Local Notation rT := (sort bT).
Local Notation class := (finClass bT).

Canonical Structure eqType := Equality.Pack class rT.
Canonical Structure choiceType := Choice.Pack class rT.
Canonical Structure countType := Countable.Pack class rT.
Canonical Structure finType := Finite.Pack class rT.
Definition arg_eqType := Eval hnf in [eqType of T].
Definition arg_choiceType := Eval hnf in [choiceType of T].
Definition arg_countType := Eval hnf in [countType of T].
Definition arg_finType := Eval hnf in [finType of T].

End InheritedClasses.

Module Import Exports.
 Declaring sort as a Coercion is clearly redundant; it only     
 serves the purpose of eliding FinGroup.sort in the display of  
 return types. The warning could be eliminated by using the     
 functor trick to replace Sortclass by a dummy target.          
Coercion arg_sort : base_type >-> Sortclass.
Coercion sort : base_type >-> Sortclass.
Coercion mixin : base_type >-> mixin_of.
Coercion base : type >-> base_type.
Canonical Structure eqType.
Canonical Structure choiceType.
Canonical Structure countType.
Canonical Structure finType.
Coercion arg_eqType : base_type >-> Equality.type.
Canonical Structure arg_eqType.
Coercion arg_choiceType : base_type >-> Choice.type.
Canonical Structure arg_choiceType.
Coercion arg_countType : base_type >-> Countable.type.
Canonical Structure arg_countType.
Coercion arg_finType : base_type >-> Finite.type.
Canonical Structure arg_finType.
Notation baseFinGroupType := base_type.
Notation finGroupType := type.
Notation BaseFinGroupType T m := (@pack_base T m _ _ id).
Notation FinGroupType := Pack.
Notation "[ 'baseFinGroupType' 'of' T ]" := (@clone_base T _ id _ _ id)
  (at level 0, format "[ 'baseFinGroupType' 'of' T ]") : form_scope.
Notation "[ 'finGroupType' 'of' T ]" := (@clone T _ _ id _ id)
  (at level 0, format "[ 'finGroupType' 'of' T ]") : form_scope.
End Exports.

End FinGroup.
Export FinGroup.Exports.

Section ElementOps.

Variable T : baseFinGroupType.
Notation rT := (FinGroup.sort T).

Definition oneg : rT := FinGroup.one T.
Definition mulg : T -> T -> rT := FinGroup.mul T.
Definition invg : T -> rT := FinGroup.inv T.
Definition expgn_rec (x : T) n : rT := iterop n mulg x oneg.

End ElementOps.

Definition expgn := nosimpl expgn_rec.

Notation "1" := (oneg _) : group_scope.
Notation "x1 * x2" := (mulg x1 x2) : group_scope.
Notation "x ^-1" := (invg x) : group_scope.
Notation "x ^+ n" := (expgn x n) : group_scope.
Notation "x ^- n" := (x ^+ n)^-1 : group_scope.

 Arguments of conjg are restricted to true groups to avoid an 
 improper interpretation of A ^ B with A and B sets, namely:  
       {x^-1 * (y * z) | y \in A, x, z \in B}                 
Definition conjg (T : finGroupType) (x y : T) := y^-1 * (x * y).
Notation "x1 ^ x2" := (conjg x1 x2) : group_scope.

Definition commg (T : finGroupType) (x y : T) := x^-1 * x ^ y.
Notation "[ ~ x1 , x2 , .. , xn ]" := (commg .. (commg x1 x2) .. xn)
  : group_scope.


Notation "\prod_ ( <- r | P ) F" :=
  (\big[mulg/1]_(<- r | P%B) F%g) : group_scope.
Notation "\prod_ ( i <- r | P ) F" :=
  (\big[mulg/1]_(i <- r | P%B) F%g) : group_scope.
Notation "\prod_ ( i <- r ) F" :=
  (\big[mulg/1]_(i <- r) F%g) : group_scope.
Notation "\prod_ ( m <= i < n | P ) F" :=
  (\big[mulg/1]_(m <= i < n | P%B) F%g) : group_scope.
Notation "\prod_ ( m <= i < n ) F" :=
  (\big[mulg/1]_(m <= i < n) F%g) : group_scope.
Notation "\prod_ ( i | P ) F" :=
  (\big[mulg/1]_(i | P%B) F%g) : group_scope.
Notation "\prod_ i F" :=
  (\big[mulg/1]_i F%g) : group_scope.
Notation "\prod_ ( i : t | P ) F" :=
  (\big[mulg/1]_(i : t | P%B) F%g) (only parsing) : group_scope.
Notation "\prod_ ( i : t ) F" :=
  (\big[mulg/1]_(i : t) F%g) (only parsing) : group_scope.
Notation "\prod_ ( i < n | P ) F" :=
  (\big[mulg/1]_(i < n | P%B) F%g) : group_scope.
Notation "\prod_ ( i < n ) F" :=
  (\big[mulg/1]_(i < n) F%g) : group_scope.
Notation "\prod_ ( i \in A | P ) F" :=
  (\big[mulg/1]_(i \in A | P%B) F%g) : group_scope.
Notation "\prod_ ( i \in A ) F" :=
  (\big[mulg/1]_(i \in A) F%g) : group_scope.

Section PreGroupIdentities.

Variable T : baseFinGroupType.
Implicit Types x y z : T.
Local Notation mulgT := (@mulg T).

Lemma mulgA : associative mulgT.

Lemma mul1g : left_id 1 mulgT.

Lemma invgK : @involutive T invg.

Lemma invMg : forall x y, (x * y)^-1 = y^-1 * x^-1.

Lemma invg_inj : @injective T T invg.

Lemma eq_invg_sym : forall x y, (x^-1 == y :> T) = (x == y^-1).

Lemma invg1 : 1^-1 = 1 :> T.

Lemma eq_invg1 : forall x, (x^-1 == 1 :> T) = (x == 1).

Lemma mulg1 : right_id 1 mulgT.

Canonical Structure finGroup_law := Monoid.Law mulgA mul1g mulg1.

Lemma expgnE : forall x n, x ^+ n = expgn_rec x n.

Lemma expg0 : forall x, x ^+ 0 = 1.

Lemma expg1 : forall x, x ^+ 1 = x.

Lemma expgS : forall x n, x ^+ n.+1 = x * x ^+ n.

Lemma exp1gn : forall n, 1 ^+ n = 1 :> T.

Lemma expgn_add : forall x n m, x ^+ (n + m) = x ^+ n * x ^+ m.

Lemma expgSr : forall x n, x ^+ n.+1 = x ^+ n * x.

Lemma expgn_mul : forall x n m, x ^+ (n * m) = x ^+ n ^+ m.

Lemma expgnC : forall x m n, x ^+ m ^+ n = x ^+ n ^+ m.

Definition commute x y := x * y = y * x.

Lemma commute_refl : forall x, commute x x.

Lemma commute_sym : forall x y, commute x y -> commute y x.

Lemma commute1 : forall x, commute x 1.

Lemma commuteM : forall x y z,
  commute x y -> commute x z -> commute x (y * z).

Lemma commuteX : forall x y n, commute x y -> commute x (y ^+ n).

Lemma commuteX2 : forall x y m n, commute x y -> commute (x ^+ m) (y ^+ n).

Lemma expVgn : forall x n, x^-1 ^+ n = x ^- n.

Lemma expMgn : forall x y n, commute x y -> (x * y) ^+ n = x ^+ n * y ^+ n.

End PreGroupIdentities.

Hint Resolve commute1.
Implicit Arguments invg_inj [T].

Section GroupIdentities.

Variable T : finGroupType.
Implicit Types x y z : T.
Local Notation mulgT := (@mulg T).

Lemma mulVg : left_inverse 1 invg mulgT.

Lemma mulgV : right_inverse 1 invg mulgT.

Lemma mulKg : left_loop invg mulgT.

Lemma mulKVg : rev_left_loop invg mulgT.

Lemma mulgI : right_injective mulgT.

Lemma mulgK : right_loop invg mulgT.

Lemma mulgKV : rev_right_loop invg mulgT.

Lemma mulIg : left_injective mulgT.

Lemma eq_invg_mul : forall x y, (x^-1 == y :> T) = (x * y == 1 :> T).

Lemma eq_mulgV1 : forall x y, (x == y) = (x * y^-1 == 1 :> T).

Lemma eq_mulVg1 : forall x y, (x == y) = (x^-1 * y == 1 :> T).

Lemma commuteV : forall x y, commute x y -> commute x y^-1.

Lemma conjgE : forall x y, x ^ y = y^-1 * (x * y).

Lemma conjgC : forall x y, x * y = y * x ^ y.

Lemma conjgCV : forall x y, x * y = y ^ x^-1 * x.

Lemma conjg1 : forall x, x ^ 1 = x.

Lemma conj1g : forall x, 1 ^ x = 1.

Lemma conjMg : forall x y z, (x * y) ^ z = x ^ z * y ^ z.

Lemma conjgM : forall x y z, x ^ (y * z) = (x ^ y) ^ z.

Lemma conjVg : forall x y, x^-1 ^ y = (x ^ y)^-1.

Lemma conjJg : forall x y z, (x ^ y) ^ z = (x ^ z) ^ y ^ z.

Lemma conjXg : forall x y n, (x ^+ n) ^ y = (x ^ y) ^+ n.

Lemma conjgK : @right_loop T T invg conjg.

Lemma conjgKV : @rev_right_loop T T invg conjg.

Lemma conjg_inj : @left_injective T T T conjg.

Lemma commgEl : forall x y, [~ x, y] = x^-1 * x ^ y.

Lemma commgEr : forall x y, [~ x, y] = y^-1 ^ x * y.

Lemma commgC : forall x y, x * y = y * x * [~ x, y].

Lemma commgCV : forall x y, x * y = [~ x^-1, y^-1] * (y * x).

Lemma conjRg : forall x y z, [~ x, y] ^ z = [~ x ^ z, y ^ z].

Lemma invg_comm : forall x y, [~ x, y]^-1 = [~ y, x].

Lemma commgP : forall x y, reflect (commute x y) ([~ x, y] == 1 :> T).

Lemma conjg_fixP : forall x y, reflect (x ^ y = x) ([~ x, y] == 1 :> T).

Lemma commg1_sym : forall x y, ([~ x, y] == 1 :> T) = ([~ y, x] == 1 :> T).

Lemma commg1 : forall x, [~ x, 1] = 1.

Lemma comm1g : forall x, [~ 1, x] = 1.

Lemma commgg : forall x, [~ x, x] = 1.

Lemma commgXg : forall x n, [~ x, x ^+ n] = 1.

Lemma commgVg : forall x, [~ x, x^-1] = 1.

Lemma commgXVg : forall x n, [~ x, x ^- n] = 1.

 Other commg identities should slot in here. 

End GroupIdentities.

Hint Rewrite mulg1 mul1g invg1 mulVg mulgV (@invgK) mulgK mulgKV
             invMg mulgA : gsimpl.

Ltac gsimpl := autorewrite with gsimpl; try done.

Definition gsimp := (mulg1 , mul1g, (invg1, @invgK), (mulgV, mulVg)).
Definition gnorm := (gsimp, (mulgK, mulgKV, (mulgA, invMg))).

Implicit Arguments mulgI [T].
Implicit Arguments mulIg [T].
Implicit Arguments conjg_inj [T].
Implicit Arguments commgP [T x y].
Implicit Arguments conjg_fixP [T x y].

Section Repr.
 Plucking a set representative. 

Variable gT : baseFinGroupType.
Implicit Type A : {set gT}.

Definition repr A :=
  if 1 \in A then 1 else if [pick x \in A] is Some x then x else 1.

Lemma mem_repr : forall A x, x \in A -> repr A \in A.

Lemma card_mem_repr : forall A, #|A| > 0 -> repr A \in A.

Lemma repr_set1 : forall x, repr [set x] = x.

Lemma repr_set0 : repr set0 = 1.

End Repr.

Implicit Arguments mem_repr [gT A].

Section BaseSetMulDef.
 We only assume a baseFinGroupType to allow this construct to be iterated. 
Variable gT : baseFinGroupType.
Implicit Types A B : {set gT}.

 Set-lifted group operations. 

Definition set_mulg A B := mulg @2: (A, B).
Definition set_invg A := invg @^-1: A.

 The pre-group structure of group subsets. 

Lemma set_mul1g : left_id [set 1] set_mulg.

Lemma set_mulgA : associative set_mulg.

Lemma set_invgK : involutive set_invg.

Lemma set_invgM : {morph set_invg : A B / set_mulg A B >-> set_mulg B A}.

Definition group_set_baseGroupMixin : FinGroup.mixin_of (set_type gT) :=
  FinGroup.BaseMixin set_mulgA set_mul1g set_invgK set_invgM.

Canonical Structure group_set_baseGroupType :=
  Eval hnf in BaseFinGroupType (set_type gT) group_set_baseGroupMixin.

Canonical Structure group_set_of_baseGroupType :=
  Eval hnf in [baseFinGroupType of {set gT}].

End BaseSetMulDef.

 Time to open the bag of dirty tricks. When we define groups down below 
 as a subtype of {set gT}, we need them to be able to coerce to sets in 
 both set-style contexts (x \in G) and monoid-style contexts (G * H),   
 and we need the coercion function to be EXACTLY the structure          
 projection in BOTH cases -- otherwise the canonical unification breaks.
   Alas, Coq doesn't let us use the same coercion function twice, even  
 when the targets are convertible. Our workaround (ab)uses the module   
 system to declare two different identity coercions on an alias class.  

Module GroupSet.
Definition sort (gT : baseFinGroupType) := {set gT}.
End GroupSet.
Identity Coercion GroupSet_of_sort : GroupSet.sort >-> set_of.

Module Type GroupSetBaseGroupSig.
Definition sort gT := group_set_of_baseGroupType gT : Type.
End GroupSetBaseGroupSig.

Module MakeGroupSetBaseGroup (Gset_base : GroupSetBaseGroupSig).
Identity Coercion of_sort : Gset_base.sort >-> FinGroup.arg_sort.
End MakeGroupSetBaseGroup.

Module Export GroupSetBaseGroup := MakeGroupSetBaseGroup GroupSet.

Canonical Structure group_set_eqType gT :=
  Eval hnf in [eqType of GroupSet.sort gT].
Canonical Structure group_set_choiceType gT :=
  Eval hnf in [choiceType of GroupSet.sort gT].
Canonical Structure group_set_countType gT :=
  Eval hnf in [countType of GroupSet.sort gT].
Canonical Structure group_set_finType gT :=
  Eval hnf in [finType of GroupSet.sort gT].

Section GroupSetMulDef.
 Some of these constructs could be defined on a baseFinGroupType. 
 We restrict them to proper finGroupType because we only develop  
 the theory for that case.                                        
Variable gT : finGroupType.
Implicit Types A B : {set gT}.
Implicit Type x y : gT.

Definition lcoset A x := mulg x @: A.
Definition rcoset A x := mulg^~ x @: A.
Definition lcosets A B := lcoset A @: B.
Definition rcosets A B := rcoset A @: B.
Definition indexg B A := #|rcosets A B|.

Definition conjugate A x := conjg^~ x @: A.
Definition conjugates A B := conjugate A @: B.
Definition class x B := conjg x @: B.
Definition classes A := class^~ A @: A.
Definition class_support A B := conjg @2: (A, B).

Definition commg_set A B := commg @2: (A, B).

 These will only be used later, but are defined here so that we can 
 keep all the Notation together.                                    
Definition normaliser A := [set x | conjugate A x \subset A].
Definition centraliser A := \bigcap_(x \in A) normaliser [set x].
Definition abelian A := A \subset centraliser A.
Definition normal A B := (A \subset B) && (B \subset normaliser A).

 "normalised" and "centralise[s|d]" are intended to be used with   
 the {in ...} form, as in abelian below.                           
Definition normalised A := forall x, conjugate A x = A.
Definition centralises x A := forall y, y \in A -> commute x y.
Definition centralised A := forall x, centralises x A.

End GroupSetMulDef.


Notation "[ 1 gT ]" := (1 : {set gT}) : group_scope.
Notation "[ 1 ]" := [1 FinGroup.sort _] : group_scope.

Notation "A ^#" := (A :\ 1) : group_scope.

Notation "x *: A" := ([set x%g] * A) : group_scope.
Notation "A :* x" := (A * [set x%g]) : group_scope.
Notation "A :^ x" := (conjugate A x) : group_scope.
Notation "x ^: B" := (class x B) : group_scope.
Notation "A :^: B" := (conjugates A B) : group_scope.

Notation "#| B : A |" := (indexg B A) : group_scope.

 No notation for lcoset and rcoset, which are to be used mostly  
 in curried form; x *: B and A :* 1 denote singleton products,   
 so thus we can use mulgA, mulg1, etc, on, say, A :* 1 * B :* x. 
 No notation for the set commutator generator set set_commg.     

Notation "''N' ( A )" := (normaliser A) : group_scope.
Notation "''N_' G ( A )" := (G%g :&: 'N(A)) : group_scope.
Notation "A <| B" := (normal A B) : group_scope.
Notation "''C' ( A )" := (centraliser A) : group_scope.
Notation "''C_' G ( A )" := (G%g :&: 'C(A)) : group_scope.
Notation "''C_' ( G ) ( A )" := 'C_G(A) (only parsing) : group_scope.
Notation "''C' [ x ]" := 'N([set x%g]) : group_scope.
Notation "''C_' G [ x ]" := 'N_G([set x%g]) : group_scope.
Notation "''C_' ( G ) [ x ]" := 'C_G[x] (only parsing) : group_scope.


Section BaseSetMulProp.
 Properties of the purely multiplicative structure. 
Variable gT : baseFinGroupType.
Implicit Types A B C D : {set gT}.
Implicit Type x y z : gT.

 Set product. We already have all the pregroup identities, so we 
 only need to add the monotonicity rules.                        

Lemma mulsgP : forall A B x,
  reflect (imset2_spec mulg (mem A) (fun _ => mem B) x) (x \in A * B).

Lemma mem_mulg : forall A B x y, x \in A -> y \in B -> x * y \in A * B.

Lemma prodsgP : forall (I : finType) (P : pred I) (A : I -> {set gT}) x,
  reflect (exists2 c, forall i, P i -> c i \in A i & x = \prod_(i | P i) c i)
          (x \in \prod_(i | P i) A i).

Lemma mem_prodg : forall (I : finType) (P : pred I) (A : I -> {set gT}) c,
    (forall i, P i -> c i \in A i) ->
  \prod_(i | P i) c i \in \prod_(i | P i) A i.

Lemma mulSg : forall A B C, A \subset B -> A * C \subset B * C.

Lemma mulgS : forall A B C, B \subset C -> A * B \subset A * C.

Lemma mulgSS : forall A B C D,
  A \subset B -> C \subset D -> A * C \subset B * D.

Lemma mulg_subl : forall A B, 1 \in B -> A \subset A * B.

Lemma mulg_subr : forall A B, 1 \in A -> B \subset A * B.

Lemma mulUg : forall A B C, (A :|: B) * C = (A * C) :|: (B * C).

Lemma mulgU : forall A B C, A * (B :|: C) = (A * B) :|: (A * C).

 Set (pointwise) inverse. 

Lemma invUg : forall A B, (A :|: B)^-1 = A^-1 :|: B^-1.

Lemma invIg : forall A B, (A :&: B)^-1 = A^-1 :&: B^-1.

Lemma invDg : forall A B, (A :\: B)^-1 = A^-1 :\: B^-1.

Lemma invCg : forall A, (~: A)^-1 = ~: A^-1.

Lemma invSg : forall A B, (A^-1 \subset B^-1) = (A \subset B).

Lemma mem_invg : forall x A, (x \in A^-1) = (x^-1 \in A).

Lemma memV_invg : forall x A, (x^-1 \in A^-1) = (x \in A).

Lemma card_invg : forall A, #|A^-1| = #|A|.

 Product with singletons. 

Lemma set1gE : 1 = [set 1] :> {set gT}.

Lemma set1gP : forall x, reflect (x = 1) (x \in [1]).

Lemma mulg_set1 : forall x y, [set x] :* y = [set x * y].

Lemma invg_set1 : forall x, [set x]^-1 = [set x^-1].

End BaseSetMulProp.

Implicit Arguments set1gP [gT x].
Implicit Arguments mulsgP [gT A B x].
Implicit Arguments prodsgP [gT I P A x].

Section GroupSetMulProp.
 Constructs that need a finGroupType 
Variable gT : finGroupType.
Implicit Types A B C D : {set gT}.
Implicit Type x y z : gT.

 Left cosets. 

Lemma lcosetE : forall A x, lcoset A x = x *: A.

Lemma card_lcoset : forall A x, #|x *: A| = #|A|.

Lemma mem_lcoset : forall A x y, (y \in x *: A) = (x^-1 * y \in A).

Lemma lcosetP : forall A x y,
  reflect (exists2 a, a \in A & y = x * a) (y \in x *: A).

Lemma lcosetsP : forall A B C,
  reflect (exists2 x, x \in B & C = x *: A) (C \in lcosets A B).

Lemma lcosetM : forall A x y, (x * y) *: A = x *: (y *: A).

Lemma lcoset1 : forall A, 1 *: A = A.

Lemma lcosetK : left_loop invg (fun x A => x *: A).

Lemma lcosetKV : rev_left_loop invg (fun x A => x *: A).

Lemma lcoset_inj : right_injective (fun x A => x *: A).

Lemma lcosetS : forall x A B, (x *: A \subset x *: B) = (A \subset B).

Lemma sub_lcoset : forall x A B, (A \subset x *: B) = (x^-1 *: A \subset B).

Lemma sub_lcosetV : forall x A B, (A \subset x^-1 *: B) = (x *: A \subset B).

 Right cosets. 

Lemma rcosetE : forall A x, rcoset A x = A :* x.

Lemma card_rcoset : forall A x, #|A :* x| = #|A|.

Lemma mem_rcoset : forall A x y, (y \in A :* x) = (y * x^-1 \in A).

Lemma rcosetP : forall A x y,
  reflect (exists2 a, a \in A & y = a * x) (y \in A :* x).

Lemma rcosetsP : forall A B C,
  reflect (exists2 x, x \in B & C = A :* x) (C \in rcosets A B).

Lemma rcosetM : forall A x y, A :* (x * y) = A :* x :* y.

Lemma rcoset1 : forall A, A :* 1 = A.

Lemma rcosetK : right_loop invg (fun A x => A :* x).

Lemma rcosetKV : rev_right_loop invg (fun A x => A :* x).

Lemma rcoset_inj : left_injective (fun A x => A :* x).

Lemma rcosetS : forall x A B, (A :* x \subset B :* x) = (A \subset B).

Lemma sub_rcoset : forall x A B, (A \subset B :* x) = (A :* x ^-1 \subset B).

Lemma sub_rcosetV : forall x A B, (A \subset B :* x^-1) = (A :* x \subset B).

 Inverse map lcosets to rcosets 

Lemma lcosets_invg : forall A B, lcosets A^-1 B^-1 = invg @^-1: rcosets A B.

 Conjugates. 

Lemma conjg_preim : forall A x, A :^ x = (conjg^~ x^-1) @^-1: A.

Lemma mem_conjg : forall A x y, (y \in A :^ x) = (y ^ x^-1 \in A).

Lemma mem_conjgV : forall A x y, (y \in A :^ x^-1) = (y ^ x \in A).

Lemma memJ_conjg : forall A x y, (y ^ x \in A :^ x) = (y \in A).

Lemma conjsgE : forall A x, A :^ x = x^-1 *: (A :* x).

Lemma conjsg1 : forall A, A :^ 1 = A.

Lemma conjsgM : forall A x y, A :^ (x * y) = (A :^ x) :^ y.

Lemma conjsgK : @right_loop _ gT invg conjugate.

Lemma conjsgKV : @rev_right_loop _ gT invg conjugate.

Lemma conjsg_inj : @left_injective _ gT _ conjugate.

Lemma cardJg : forall A x, #|A :^ x| = #|A|.

Lemma conjSg : forall A B x, (A :^ x \subset B :^ x) = (A \subset B).

Lemma properJ : forall A B x, (A :^ x \proper B :^ x) = (A \proper B).

Lemma sub_conjg : forall A B x, (A :^ x \subset B) = (A \subset B :^ x^-1).

Lemma sub_conjgV : forall A B x, (A :^ x^-1 \subset B) = (A \subset B :^ x).

Lemma conjg_set1 : forall x y, [set x] :^ y = [set x ^ y].

Lemma conjs1g : forall x, 1 :^ x = 1.

Lemma conjsMg : forall A B x, (A * B) :^ x = A :^ x * B :^ x.

Lemma conjIg : forall A B x, (A :&: B) :^ x = A :^ x :&: B :^ x.

Lemma conjUg : forall A B x, (A :|: B) :^ x = A :^ x :|: B :^ x.

Lemma conjCg : forall A x, (~: A) :^ x = ~: A :^ x.

Lemma conjDg : forall A B x, (A :\: B) :^ x = A :^ x :\: B :^ x.

Lemma conjD1g : forall A x, A^# :^ x = (A :^ x)^#.

 Classes; not much for now. 

Lemma memJ_class : forall x y A, y \in A -> x ^ y \in x ^: A.

Lemma classS : forall x A B, A \subset B -> x ^: A \subset x ^: B.

Lemma class_set1 : forall x y, x ^: [set y] = [set x ^ y].

Lemma class1g : forall x A, x \in A -> 1 ^: A = 1.

Lemma mem_classes : forall x A, x \in A -> x ^: A \in classes A.

Lemma class_supportM : forall A B C,
  class_support A (B * C) = class_support (class_support A B) C.

Lemma class_support_set1l : forall A x, class_support [set x] A = x ^: A.

Lemma class_support_set1r : forall A x, class_support A [set x] = A :^ x.

Lemma classM : forall x A B, x ^: (A * B) = class_support (x ^: A) B.

Lemma class_lcoset : forall x y A, x ^: (y *: A) = (x ^ y) ^: A.

Lemma class_rcoset : forall x A y, x ^: (A :* y) = (x ^: A) :^ y.

 Conjugate set. 

Lemma conjugatesS : forall A B C, B \subset C -> A :^: B \subset A :^: C.

Lemma conjugates_set1 : forall A x, A :^: [set x] = [set A :^ x].

Lemma conjugates_conj : forall A x B, (A :^ x) :^: B = A :^: (x *: B).

 Class support. 

Lemma class_supportEl : forall A B,
  class_support A B = \bigcup_(x \in A) x ^: B.

Lemma class_supportEr : forall A B,
  class_support A B = \bigcup_(x \in B) A :^ x.

 Groups (at last!) 

Definition group_set A := (1 \in A) && (A * A \subset A).

Lemma group_setP : forall A,
  reflect (1 \in A /\ {in A & A, forall x y, x * y \in A}) (group_set A).

Structure group_type : Type := Group {
  gval :> GroupSet.sort gT;
  _ : group_set gval
}.

Definition group_of of phant gT : predArgType := group_type.
Notation Local groupT := (group_of (Phant gT)).
Identity Coercion type_of_group : group_of >-> group_type.

Canonical Structure group_subType :=
  Eval hnf in [subType for gval by group_type_rect].
Definition group_eqMixin := Eval hnf in [eqMixin of group_type by <:].
Canonical Structure group_eqType := Eval hnf in EqType group_type group_eqMixin.
Definition group_choiceMixin := [choiceMixin of group_type by <:].
Canonical Structure group_choiceType :=
  Eval hnf in ChoiceType group_type group_choiceMixin.
Definition group_countMixin := [countMixin of group_type by <:].
Canonical Structure group_countType :=
  Eval hnf in CountType group_type group_countMixin.
Canonical Structure group_subCountType :=
  Eval hnf in [subCountType of group_type].
Definition group_finMixin := [finMixin of group_type by <:].
Canonical Structure group_finType :=
  Eval hnf in FinType group_type group_finMixin.
Canonical Structure group_subFinType := Eval hnf in [subFinType of group_type].

 No predType or baseFinGroupType structures, as these would hide the 
 group-to-set coercion and thus spoil unification.                  

Canonical Structure group_of_subType := Eval hnf in [subType of groupT].
Canonical Structure group_of_eqType := Eval hnf in [eqType of groupT].
Canonical Structure group_of_choiceType := Eval hnf in [choiceType of groupT].
Canonical Structure group_of_countType := Eval hnf in [countType of groupT].
Canonical Structure group_of_subCountType :=
  Eval hnf in [subCountType of groupT].
Canonical Structure group_of_finType := Eval hnf in [finType of groupT].
Canonical Structure group_of_subFinType := Eval hnf in [subFinType of groupT].

Definition group (A : {set gT}) gA : groupT := @Group A gA.

Definition clone_group G :=
  let: Group _ gP := G return {type of Group for G} -> groupT in fun k => k gP.

Lemma group_inj : injective gval.

Lemma groupP : forall G : groupT, group_set G.

Lemma congr_group : forall H K : groupT, H = K -> H :=: K.

Lemma isgroupP : forall A, reflect (exists G : groupT, A = G) (group_set A).

Lemma group_set_one : group_set 1.

Canonical Structure one_group := group group_set_one.
Canonical Structure set1_group := @group [set 1] group_set_one.

Lemma group_setT : forall phT : phant gT, group_set (setTfor phT).

Canonical Structure setT_group phT := group (group_setT phT).

 These definitions come early so we can establish the Notation. 
Definition generated A := \bigcap_(G : groupT | A \subset G) G.
Definition gcore A B := \bigcap_(x \in B) A :^ x.
Definition joing A B := generated (A :|: B).
Definition commutator A B := generated (commg_set A B).
Definition cycle x := generated [set x].
Definition order x := #|cycle x|.

End GroupSetMulProp.

Implicit Arguments lcosetP [gT A x y].
Implicit Arguments lcosetsP [gT A B C].
Implicit Arguments rcosetP [gT A x y].
Implicit Arguments rcosetsP [gT A B C].
Implicit Arguments group_setP [gT A].


Notation "{ 'group' gT }" := (group_of (Phant gT))
  (at level 0, format "{ 'group' gT }") : type_scope.

Notation "[ 'group' 'of' G ]" := (clone_group (@group _ G))
  (at level 0, format "[ 'group' 'of' G ]") : form_scope.

Notation "1" := (one_group _) : subgroup_scope.
Notation "[ 1 gT ]" := (1%G : {group gT}) : subgroup_scope.
Notation "[ 'set' : gT ]" := (setT_group (Phant gT)) : subgroup_scope.

 Helper notation for defining new groups that need a bespoke finGroupType. 
 The actual group for such a type (say, my_gT) will be the full group,     
 i.e., [set: my_gT] or [set: my_gT]%G, but Coq will not recognize          
 specific notation for these because of the coercions inserted during type 
 inference, unless they are defined as [set: gsort my_gT] using the        
 Notation below.                                                           
Notation gsort gT := (FinGroup.arg_sort (FinGroup.base gT%type)) (only parsing).
Notation "<< A >>" := (generated A) : group_scope.
Notation "<[ x ] >" := (cycle x) : group_scope.
Notation "#[ x ]" := (order x) : group_scope.
Notation "A <*> B" := (joing A B) : group_scope.
Notation "[ ~: A1 , A2 , .. , An ]" :=
  (commutator .. (commutator A1 A2) .. An) : group_scope.


Section GroupProp.

Variable gT : finGroupType.
Notation sT := {set gT}.
Implicit Types A B C D : sT.
Implicit Types x y z : gT.
Implicit Types G H K : {group gT}.

Section OneGroup.

Variable G : {group gT}.

Lemma valG : val G = G.

 Non-triviality. 

Lemma group1 : 1 \in G.

Hint Resolve group1.

 Loads of silly variants to placate the incompleteness of trivial. 
 An alternative would be to upgrade done, pending better support   
 in the ssreflect ML code.                                         
Notation gTr := (FinGroup.sort gT).
Notation Gcl := (pred_of_set G : pred gTr).
Lemma group1_class1 : (1 : gTr) \in G.

Lemma group1_class2 : 1 \in Gcl.

Lemma group1_class12 : (1 : gTr) \in Gcl.

Lemma group1_eqType : (1 : gT : eqType) \in G.

Lemma group1_finType : (1 : gT : finType) \in G.

Lemma group1_contra : forall x, x \notin G -> x != 1.

Lemma sub1G : [1 gT] \subset G.

Lemma subG1 : (G \subset [1]) = (G :==: 1).

Lemma setI1g : 1 :&: G = 1.

Lemma setIg1 : G :&: 1 = 1.

Lemma subG1_contra : forall H, G \subset H -> G :!=: 1 -> H :!=: 1.

Lemma repr_group : repr G = 1.

Lemma cardG_gt0 : 0 < #|G|.
 Workaround for the fact that the simple matching used by Trivial does not  
 always allow conversion. In particular cardG_gt0 always fails to apply to  
 subgoals that have been simplified (by /=) because type inference in the   
 notation #|G| introduces redexes of the form                               
    Finite.sort (arg_finGroupType (FinGroup.base gT))                       
 which get collapsed to Fingroup.arg_sort (FinGroup.base gT).               
Definition cardG_gt0_reduced : 0 < card (@mem gT (predPredType gT) G)
  := cardG_gt0.

Lemma indexg_gt0 : forall A, 0 < #|G : A|.

Lemma trivgP : reflect (G :=: 1) (G \subset [1]).

Lemma trivGP : reflect (G = 1%G) (G \subset [1]).

Lemma proper1G : ([1] \proper G) = (G :!=: 1).

Lemma trivgPn : reflect (exists2 x, x \in G & x != 1) (G :!=: 1).

Lemma trivg_card_le1 : (G :==: 1) = (#|G| <= 1).

Lemma trivg_card1 : (G :==: 1) = (#|G| == 1%N).

Lemma cardG_gt1 : (#|G| > 1) = (G :!=: 1).

Lemma card_le1_trivg : #|G| <= 1 -> G :=: 1.

Lemma card1_trivg : #|G| = 1%N -> G :=: 1.

 Inclusion and product. 

Lemma mulG_subl : forall A, A \subset A * G.

Lemma mulG_subr : forall A, A \subset G * A.

Lemma mulGid : G * G = G.

Lemma mulGS : forall A B, (G * A \subset G * B) = (A \subset G * B).

Lemma mulSG : forall A B, (A * G \subset B * G) = (A \subset B * G).

Lemma mul_subG : forall A B, A \subset G -> B \subset G -> A * B \subset G.

 Membership lemmas 

Lemma groupM : forall x y, x \in G -> y \in G -> x * y \in G.

Lemma groupX : forall x n, x \in G -> x ^+ n \in G.

Lemma groupVr : forall x, x \in G -> x^-1 \in G.

Lemma groupVl : forall x, x^-1 \in G -> x \in G.

Lemma groupV : forall x, (x^-1 \in G) = (x \in G).

Lemma groupMl : forall x y, x \in G -> (x * y \in G) = (y \in G).

Lemma groupMr : forall x y, x \in G -> (y * x \in G) = (y \in G).

Definition in_group := (group1, groupV, (groupMl, groupX)).

Lemma groupJ : forall x y, x \in G -> y \in G -> x ^ y \in G.

Lemma groupJr : forall x y, y \in G -> (x ^ y \in G) = (x \in G).

Lemma groupR : forall x y, x \in G -> y \in G -> [~ x, y] \in G.

Lemma group_prod : forall I r (P : pred I) F,
  (forall i, P i -> F i \in G) -> \prod_(i <- r | P i) F i \in G.

 Inverse is an anti-morphism. 

Lemma invGid : G^-1 = G.

Lemma inv_subG : forall A, (A^-1 \subset G) = (A \subset G).

Lemma invg_lcoset : forall x, (x *: G)^-1 = G :* x^-1.

Lemma invg_rcoset : forall x, (G :* x)^-1 = x^-1 *: G.

Lemma memV_lcosetV : forall x y, (y^-1 \in x^-1 *: G) = (y \in G :* x).

Lemma memV_rcosetV : forall x y, (y^-1 \in G :* x^-1) = (y \in x *: G).

 Product idempotence 

Lemma mulSgGid : forall A x, x \in A -> A \subset G -> A * G = G.

Lemma mulGSgid : forall A x, x \in A -> A \subset G -> G * A = G.

 Left cosets 

Lemma lcoset_refl : forall x, x \in x *: G.

Lemma lcoset_sym : forall x y, (x \in y *: G) = (y \in x *: G).

Lemma lcoset_transl : forall x y, x \in y *: G -> x *: G = y *: G.

Lemma lcoset_transr : forall x y z,
  x \in y *: G -> (x \in z *: G) = (y \in z *: G).

Lemma lcoset_trans : forall x y z,
  x \in y *: G -> y \in z *: G -> x \in z *: G.

Lemma lcoset_id : forall x, x \in G -> x *: G = G.

 Right cosets, with an elimination form for repr. 

Lemma rcoset_refl : forall x, x \in G :* x.

Lemma rcoset_sym : forall x y, (x \in G :* y) = (y \in G :* x).

Lemma rcoset_transl : forall x y, x \in G :* y -> G :* x = G :* y.

Lemma rcoset_transr : forall x y z,
  x \in G :* y -> (x \in G :* z) = (y \in G :* z).

Lemma rcoset_trans : forall x y z,
  y \in G :* x -> z \in G :* y -> z \in G :* x.

Lemma rcoset_id : forall x, x \in G -> G :* x = G.

 Elimination form. 

CoInductive rcoset_repr_spec x : gT -> Type :=
  RcosetReprSpec g : g \in G -> rcoset_repr_spec x (g * x).

Lemma mem_repr_rcoset : forall x, repr (G :* x) \in G :* x.

 This form sometimes fails because ssreflect 1.1 delegates matching to the 
 (weaker) primitive Coq algorithm for general (co)inductive type families. 
Lemma repr_rcosetP : forall x, rcoset_repr_spec x (repr (G :* x)).

Lemma rcoset_repr : forall x, G :* (repr (G :* x)) = G :* x.

 Coset spaces. 

Lemma mem_lcosets : forall A x, (x *: G \in lcosets G A) = (x \in A * G).

Lemma mem_rcosets : forall A x, (G :* x \in rcosets G A) = (x \in G * A).

 Conjugates. 

Lemma group_set_conjG : forall x, group_set (G :^ x).

Canonical Structure conjG_group x := group (group_set_conjG x).

Lemma conjGid : {in G, normalised G}.

Lemma conj_subG : forall x A, x \in G -> A \subset G -> A :^ x \subset G.

 Classes 

Lemma class1G : 1 ^: G = 1.

Lemma classes1 : [1] \in classes G.

Lemma classGidl : forall x y, y \in G -> (x ^ y) ^: G = x ^: G.

Lemma classGidr : forall x, {in G, normalised (x ^: G)}.

Lemma class_refl : forall x, x \in x ^: G.
Hint Resolve class_refl.

Lemma class_transr : forall x y, x \in y ^: G -> x ^: G = y ^: G.

Lemma class_sym : forall x y, (x \in y ^: G) = (y \in x ^: G).

Lemma class_transl : forall x y z,
   x \in y ^: G -> (x \in z ^: G) = (y \in z ^: G).

Lemma class_trans : forall x y z,
   x \in y ^: G -> y \in z ^: G -> x \in z ^: G.

Lemma repr_class : forall x, {y | y \in G & repr (x ^: G) = x ^ y}.

Lemma classG_eq1 : forall x, (x ^: G == 1) = (x == 1).

Lemma class_subG : forall x A, x \in G -> A \subset G -> x ^: A \subset G.

Lemma class_supportGidl : forall A x,
  x \in G -> class_support (A :^ x) G = class_support A G.

Lemma class_supportGidr : forall A, {in G, normalised (class_support A G)}.

Lemma class_support_subG : forall A, A \subset G -> class_support A G \subset G.

Lemma sub_class_support : forall A, A \subset class_support A G.

Lemma class_support_id : class_support G G = G.

Lemma class_supportD1 : forall A, (class_support A G)^# = cover (A^# :^: G).

 Subgroup Type construction. 
 We only expect to use this for abstract groups, so we don't project 
 the argument to a set.                                              

Inductive subg_of : predArgType := Subg x & x \in G.
Definition sgval u := let: Subg x _ := u in x.
Canonical Structure subg_subType :=
  Eval hnf in [subType for sgval by subg_of_rect].
Definition subg_eqMixin := Eval hnf in [eqMixin of subg_of by <:].
Canonical Structure subg_eqType := Eval hnf in EqType subg_of subg_eqMixin.
Definition subg_choiceMixin := [choiceMixin of subg_of by <:].
Canonical Structure subg_choiceType :=
  Eval hnf in ChoiceType subg_of subg_choiceMixin.
Definition subg_countMixin := [countMixin of subg_of by <:].
Canonical Structure subg_countType :=
  Eval hnf in CountType subg_of subg_countMixin.
Canonical Structure subg_subCountType := Eval hnf in [subCountType of subg_of].
Definition subg_finMixin := [finMixin of subg_of by <:].
Canonical Structure subg_finType := Eval hnf in FinType subg_of subg_finMixin.
Canonical Structure subg_subFinType := Eval hnf in [subFinType of subg_of].

Lemma subgP : forall u, sgval u \in G.
Lemma subg_inj : injective sgval.
Lemma congr_subg : forall u v, u = v -> sgval u = sgval v.

Definition subg_one := Subg group1.
Definition subg_inv u := Subg (groupVr (subgP u)).
Definition subg_mul u v := Subg (groupM (subgP u) (subgP v)).
Lemma subg_oneP : left_id subg_one subg_mul.

Lemma subg_invP : left_inverse subg_one subg_inv subg_mul.
Lemma subg_mulP : associative subg_mul.

Definition subFinGroupMixin := FinGroup.Mixin subg_mulP subg_oneP subg_invP.
Canonical Structure subBaseFinGroupType :=
  Eval hnf in BaseFinGroupType subg_of subFinGroupMixin.
Canonical Structure subFinGroupType := FinGroupType subg_invP.

Lemma sgvalM : {in setT &, {morph sgval : x y / x * y}}.

Lemma valgM : {in setT &, {morph val : x y / (x : subg_of) * y >-> x * y}}.

Definition subg : gT -> subg_of := insubd (1 : subg_of).
Lemma subgK : forall x, x \in G -> val (subg x) = x.
Lemma sgvalK : cancel sgval subg.
Lemma subg_default : forall x, (x \in G) = false -> val (subg x) = 1.
Lemma subgM : {in G &, {morph subg : x y / x * y}}.

End OneGroup.

Hint Resolve group1.

Lemma groupD1_inj : forall G H, G^# = H^# -> G :=: H.

Lemma invMG : forall G H, (G * H)^-1 = H * G.

Lemma mulSGid : forall G H, H \subset G -> H * G = G.

Lemma mulGSid : forall G H, H \subset G -> G * H = G.

Lemma mulGidPl : forall G H, reflect (G * H = G) (H \subset G).

Lemma mulGidPr : forall G H, reflect (G * H = H) (G \subset H).

Lemma comm_group_setP : forall G H, reflect (commute G H) (group_set (G * H)).

Lemma card_lcosets : forall G H, #|lcosets H G| = #|G : H|.

 Group Modularity equations 

Lemma group_modl : forall A B G, A \subset G -> A * (B :&: G) = A * B :&: G.

Lemma group_modr : forall A B G, B \subset G -> (G :&: A) * B = G :&: A * B.

End GroupProp.

Hint Resolve group1 group1_class1 group1_class12 group1_class12.
Hint Resolve group1_eqType group1_finType.
Hint Resolve cardG_gt0 cardG_gt0_reduced indexg_gt0.

Notation "G :^ x" := (conjG_group G x) : subgroup_scope.

Notation "[ 'subg' G ]" := (subg_of G) : type_scope.
Notation "[ 'subg' G ]" := [set: subg_of G] : group_scope.
Notation "[ 'subg' G ]" := [set: subg_of G]%G : subgroup_scope.


Implicit Arguments trivgP [gT G].
Implicit Arguments trivGP [gT G].
Implicit Arguments mulGidPl [gT G H].
Implicit Arguments mulGidPr [gT G H].
Implicit Arguments comm_group_setP [gT G H].

Section GroupInter.

Variable gT : finGroupType.
Implicit Types A B : {set gT}.
Implicit Types G H : {group gT}.

Lemma group_setI : forall G H, group_set (G :&: H).

Canonical Structure setI_group G H := group (group_setI G H).

Section Nary.

Variables (I : finType) (P : pred I) (F : I -> {group gT}).

Lemma group_set_bigcap : group_set (\bigcap_(i | P i) F i).

Canonical Structure bigcap_group := group group_set_bigcap.

End Nary.

Canonical Structure generated_group A : {group _} :=
  Eval hnf in [group of <<A>>].
Canonical Structure gcore_group G A : {group _} :=
  Eval hnf in [group of gcore G A].
Canonical Structure commutator_group A B : {group _} :=
  Eval hnf in [group of [~: A, B]].
Canonical Structure joing_group A B : {group _} :=
  Eval hnf in [group of A <*> B].
Canonical Structure cycle_group x : {group _} :=
  Eval hnf in [group of <[x]>].

Lemma order_gt0 : forall x : gT, 0 < #[x].

End GroupInter.

Hint Resolve order_gt0.

Definition joinG (gT : finGroupType) (G H : {group gT}) :=
  nosimpl (joing_group G H).

Definition subgroups (gT : finGroupType) (G : {set gT}) :=
  [set H : {group gT} | H \subset G].


Notation "G :&: H" := (setI_group G H) : subgroup_scope.
Notation "<< A >>" := (generated_group A) : subgroup_scope.
Notation "<[ x ] >" := (cycle_group x) : subgroup_scope.
Notation "[ ~: A1 , A2 , .. , An ]" :=
  (commutator_group .. (commutator_group A1 A2) .. An) : subgroup_scope.
Notation "G <*> H" := (joinG G H) : subgroup_scope.

Notation "\prod_ ( <- r | P ) F" :=
  (\big[joinG/1%G]_(<- r | P%B) F%G) : subgroup_scope.
Notation "\prod_ ( i <- r | P ) F" :=
  (\big[joinG/1%G]_(i <- r | P%B) F%G) : subgroup_scope.
Notation "\prod_ ( i <- r ) F" :=
  (\big[joinG/1%G]_(i <- r) F%G) : subgroup_scope.
Notation "\prod_ ( m <= i < n | P ) F" :=
  (\big[joinG/1%G]_(m <= i < n | P%B) F%G) : subgroup_scope.
Notation "\prod_ ( m <= i < n ) F" :=
  (\big[joinG/1%G]_(m <= i < n) F%G) : subgroup_scope.
Notation "\prod_ ( i | P ) F" :=
  (\big[joinG/1%G]_(i | P%B) F%G) : subgroup_scope.
Notation "\prod_ i F" :=
  (\big[joinG/1%G]_i F%G) : subgroup_scope.
Notation "\prod_ ( i : t | P ) F" :=
  (\big[joinG/1%G]_(i : t | P%B) F%G) (only parsing) : subgroup_scope.
Notation "\prod_ ( i : t ) F" :=
  (\big[joinG/1%G]_(i : t) F%G) (only parsing) : subgroup_scope.
Notation "\prod_ ( i < n | P ) F" :=
  (\big[joinG/1%G]_(i < n | P%B) F%G) : subgroup_scope.
Notation "\prod_ ( i < n ) F" :=
  (\big[joinG/1%G]_(i < n) F%G) : subgroup_scope.
Notation "\prod_ ( i \in A | P ) F" :=
  (\big[joinG/1%G]_(i \in A | P%B) F%G) : subgroup_scope.
Notation "\prod_ ( i \in A ) F" :=
  (\big[joinG/1%G]_(i \in A) F%G) : subgroup_scope.

Section LaGrange.

Variable gT : finGroupType.
Implicit Types G H K : {group gT}.

Lemma LaGrangeI : forall G H, (#|G :&: H| * #|G : H|)%N = #|G|.

Lemma divgI : forall G H, #|G| %/ #|G :&: H| = #|G : H|.

Lemma divg_index : forall G H, #|G| %/ #|G : H| = #|G :&: H|.

Lemma dvdn_indexg : forall G H, #|G : H| %| #|G|.

Theorem LaGrange : forall G H, H \subset G -> (#|H| * #|G : H|)%N = #|G|.

Lemma cardSg : forall G H, H \subset G -> #|H| %| #|G|.

Lemma lognSg : forall p G H, G \subset H -> logn p #|G| <= logn p #|H|.

Lemma piSg : forall G H, G \subset H -> {subset \pi(gval G) <= \pi(gval H)}.

Lemma divgS : forall G H, H \subset G -> #|G| %/ #|H| = #|G : H|.

Lemma coprimeSg : forall G H p, H \subset G -> coprime #|G| p -> coprime #|H| p.

Lemma coprimegS : forall G H p, H \subset G -> coprime p #|G| -> coprime p #|H|.

Lemma indexJg : forall G H x, #|G :^ x : H :^ x| = #|G : H|.

Lemma indexgg : forall G, #|G : G| = 1%N.

Lemma LaGrange_index : forall G H K,
  H \subset G -> K \subset H -> (#|G : H| * #|H : K|)%N = #|G : K|.

Lemma indexgI : forall G H, #|G : G :&: H| = #|G : H|.

Lemma indexgS : forall G H K, H \subset K -> #|G : K| %| #|G : H|.

Lemma indexSg : forall G H K,
  H \subset K -> K \subset G -> #|K : H| %| #|G : H|.

Lemma indexg_eq1 : forall G H, (#|G : H| == 1%N) = (G \subset H).

Lemma indexg_gt1 : forall G H, (#|G : H| > 1) = ~~ (G \subset H).

Lemma index1g : forall G H, H \subset G -> #|G : H| = 1%N -> H :=: G.

Lemma indexg1 : forall G, #|G : 1| = #|G|.

Lemma indexMg : forall G A, #|G * A : G| = #|A : G|.

Lemma LaGrangeMl : forall G H, (#|G| * #|H : G|)%N = #|G * H|.

Lemma LaGrangeMr : forall G H, (#|G : H| * #|H|)%N = #|G * H|.

Lemma mul_cardG : forall G H, (#|G| * #|H| = #|G * H|%g * #|G :&: H|)%N.

Lemma dvdn_cardMg : forall G H, #|G * H| %| #|G| * #|H|.

Lemma cardMg_divn : forall G H, #|G * H| = (#|G| * #|H|) %/ #|G :&: H|.

Lemma cardIg_divn : forall G H, #|G :&: H| = (#|G| * #|H|) %/ #|G * H|.

Lemma TI_cardMg : forall G H, G :&: H = 1 -> #|G * H| = (#|G| * #|H|)%N.

Lemma cardMg_TI : forall G H, #|G| * #|H| <= #|G * H| -> G :&: H = 1.

Lemma coprime_TIg : forall G H, coprime #|G| #|H| -> G :&: H = 1.

Lemma prime_TIg : forall G H, prime #|G| -> ~~ (G \subset H) -> G :&: H = 1.

Lemma prime_meetG : forall G H, prime #|G| -> G :&: H != 1 -> G \subset H.

Lemma coprime_cardMg : forall G H,
  coprime #|G| #|H| -> #|G * H| = (#|G| * #|H|)%N.

Lemma coprime_index_mulG : forall G H K,
  H \subset G -> K \subset G -> coprime #|G : H| #|G : K| -> H * K = G.

End LaGrange.

Section GeneratedGroup.

Variable gT : finGroupType.
Implicit Types x y z : gT.
Implicit Types A B C D : {set gT}.
Implicit Types G H K : {group gT}.

Lemma subset_gen : forall A, A \subset <<A>>.

Lemma sub_gen : forall A B, A \subset B -> A \subset <<B>>.

Lemma mem_gen : forall x A, x \in A -> x \in <<A>>.

Lemma generatedP : forall x A,
  reflect (forall G, A \subset G -> x \in G) (x \in <<A>>).

Lemma gen_subG : forall A G, (<<A>> \subset G) = (A \subset G).

Lemma genGid : forall G, <<G>> = G.

Lemma genGidG : forall G, <<G>>%G = G.

Lemma gen_set_id : forall A, group_set A -> <<A>> = A.

Lemma genS : forall A B, A \subset B -> <<A>> \subset <<B>>.

Lemma gen0 : <<set0>> = 1 :> {set gT}.

Lemma gen_expgs : forall A, {n | <<A>> = (1 |: A) ^+ n}.

Lemma gen_prodgP : forall A x,
  reflect (exists n, exists2 c, forall i : 'I_n, c i \in A & x = \prod_i c i)
          (x \in <<A>>).

Lemma genD : forall A B, A \subset <<A :\: B>> -> <<A :\: B>> = <<A>>.

Lemma genV : forall A, <<A^-1>> = <<A>>.

Lemma genJ : forall A z, <<A :^z>> = <<A>> :^ z.

Lemma conjYg : forall A B z, (A <*> B) :^z = A :^ z <*> B :^ z.

Lemma genD1 : forall A x, x \in <<A :\ x>> -> <<A :\ x>> = <<A>>.

Lemma genD1id : forall A, <<A^#>> = <<A>>.

Notation joingT := (@joing gT) (only parsing).
Notation joinGT := (@joinG gT) (only parsing).

Lemma joingE : forall A B, A <*> B = <<A :|: B>>.

Lemma joinGE : forall G H, (G <*> H)%G :=: G <*> H.

Lemma joingC : commutative joingT.

Lemma joing_idr : forall A B, A <*> <<B>> = A <*> B.

Lemma joing_idl : forall A B, <<A>> <*> B = A <*> B.

Lemma joing_subl : forall A B, A \subset A <*> B.

Lemma joing_subr : forall A B, B \subset A <*> B.

Lemma join_subG : forall A B G,
  (A <*> B \subset G) = (A \subset G) && (B \subset G).

Lemma joing_idPl : forall G A, reflect (G <*> A = G) (A \subset G).

Lemma joing_idPr : forall A G, reflect (A <*> G = G) (A \subset G).

Lemma joing_subP : forall A B G,
  reflect (A \subset G /\ B \subset G) (A <*> B \subset G).

Lemma joing_sub : forall A B C, A <*> B = C -> A \subset C /\ B \subset C.

Lemma genDU : forall A B C,
  A \subset C -> <<C :\: A>> = <<B>> -> <<A :|: B>> = <<C>>.

Lemma joingA : associative joingT.

Lemma joing1G : forall G, 1 <*> G = G.

Lemma joingG1 : forall G, G <*> 1 = G.

Lemma genM_join : forall G H, <<G * H>> = G <*> H.

Lemma mulG_subG : forall G H K,
  (G * H \subset K) = (G \subset K) && (H \subset K).

Lemma mulGsubP : forall K H G,
  reflect (K \subset G /\ H \subset G) (K * H \subset G).

Lemma mulG_sub : forall K H A, K * H = A -> K \subset A /\ H \subset A.

Lemma trivMg : forall G H, (G * H == 1) = (G :==: 1) && (H :==: 1).

Lemma comm_joingE : forall G H, commute G H -> G <*> H = G * H.

Lemma joinGC : commutative joinGT.

Lemma joinGA : associative joinGT.

Lemma join1G : left_id 1%G joinGT.

Lemma joinG1 : right_id 1%G joinGT.

Canonical Structure joinG_law := Monoid.Law joinGA join1G joinG1.
Canonical Structure joinG_abelaw := Monoid.ComLaw joinGC.

Lemma bigprodGEgen : forall I r (P : pred I) (F : I -> {set gT}),
  (\prod_(i <- r | P i) <<F i>>)%G :=: << \bigcup_(i <- r | P i) F i >>.

Lemma bigprodGE : forall I r (P : pred I) (F : I -> {group gT}),
  (\prod_(i <- r | P i) F i)%G :=: << \bigcup_(i <- r | P i) F i >>.

Lemma mem_commg : forall A B x y, x \in A -> y \in B -> [~ x, y] \in [~: A, B].

Lemma commSg : forall A B C, A \subset B -> [~: A, C] \subset [~: B, C].

Lemma commgS : forall A B C, B \subset C -> [~: A, B] \subset [~: A, C].

Lemma commgSS : forall A B C D,
  A \subset B -> C \subset D -> [~: A, C] \subset [~: B, D].

Lemma der1_subG : forall G, [~: G, G] \subset G.

Lemma comm_subG : forall A B G,
  A \subset G -> B \subset G -> [~: A, B] \subset G.

Lemma commGC : forall A B, [~: A, B] = [~: B, A].

Lemma conjsRg : forall A B x, [~: A, B] :^ x = [~: A :^ x, B :^ x].

End GeneratedGroup.

Implicit Arguments gen_prodgP [gT A x].
Implicit Arguments joing_idPl [gT G A].
Implicit Arguments joing_idPr [gT A G].
Implicit Arguments mulGsubP [gT K H G].
Implicit Arguments joing_subP [gT A B G].

Section Cycles.

 Elementary properties of cycles and order, needed in perm.v.  
 More advanced results on the structure of cyclic groups will  
 be given in cyclic.v.                                         

Variable gT : finGroupType.
Implicit Types x y : gT.
Implicit Types G : {group gT}.

Import Monoid.Theory.

Lemma cycle1 : <[1]> = [1 gT].

Lemma order1 : #[1 : gT] = 1%N.

Lemma cycle_id : forall x, x \in <[x]>.

Lemma mem_cycle : forall x i, x ^+ i \in <[x]>.

Lemma cycle_subG : forall x G, (<[x]> \subset G) = (x \in G).

Lemma cycle_eq1 : forall x, (<[x]> == 1) = (x == 1).

Lemma orderE : forall x, #[x] = #|<[x]>|.

Lemma order_eq1 : forall x, (#[x] == 1%N) = (x == 1).

Lemma order_gt1 : forall x, (#[x] > 1) = (x != 1).

Lemma cycle_traject : forall x, <[x]> =i traject (mulg x) 1 #[x].

Lemma cycle2g : forall x, #[x] = 2 -> <[x]> = [set 1; x].

Lemma cyclePmin : forall x y, y \in <[x]> -> {i | i < #[x] & y = x ^+ i}.

Lemma cycleP : forall x y, reflect (exists i, y = x ^+ i) (y \in <[x]>).

Lemma expg_order : forall x, x ^+ #[x] = 1.

Lemma expg_mod : forall p k x, x ^+ p = 1 -> x ^+ (k %% p) = x ^+ k.

Lemma expg_mod_order : forall x i, x ^+ (i %% #[x]) = x ^+ i.

Lemma invg_expg : forall x, x^-1 = x ^+ #[x].-1.

Lemma invg2id : forall x, #[x] = 2 -> x^-1 = x.

Lemma cycleX : forall x i, <[x ^+ i]> \subset <[x]>.

Lemma cycleV : forall x, <[x^-1]> = <[x]>.

Lemma orderV : forall x, #[x^-1] = #[x].

Lemma cycleJ : forall x y, <[x ^ y]> = <[x]> :^ y.

Lemma orderJ : forall x y, #[x ^ y] = #[x].

End Cycles.

Section Normaliser.

Variable gT : finGroupType.
Implicit Types x y z : gT.
Implicit Types A B C D : {set gT}.
Implicit Type G H K : {group gT}.

Lemma normP : forall x A, reflect (A :^ x = A) (x \in 'N(A)).
Implicit Arguments normP [x A].

Lemma group_set_normaliser : forall A, group_set 'N(A).

Canonical Structure normaliser_group A := group (group_set_normaliser A).

Lemma normsP : forall A B, reflect {in A, normalised B} (A \subset 'N(B)).
Implicit Arguments normsP [A B].

Lemma memJ_norm : forall x y A, x \in 'N(A) -> (y ^ x \in A) = (y \in A).

Lemma norms_cycle : forall x y,
  (<[y]> \subset 'N(<[x]>)) = (x ^ y \in <[x]>).

Lemma norm1 : 'N(1) = setT :> {set gT}.

Lemma norms1 : forall A, A \subset 'N(1).

Lemma normCs : forall A, 'N(~: A) = 'N(A).

Lemma normG : forall G, G \subset 'N(G).

Lemma normsG : forall A G, A \subset G -> A \subset 'N(G).

Lemma normC : forall A B, A \subset 'N(B) -> commute A B.

Lemma norm_joinEl : forall G H, G \subset 'N(H) -> G <*> H = G * H.

Lemma norm_joinEr : forall G H, H \subset 'N(G) -> G <*> H = G * H.

Lemma norm_rlcoset : forall G x, x \in 'N(G) -> G :* x = x *: G.

Lemma rcoset_mul : forall G x y,
  x \in 'N(G) -> (G :* x) * (G :* y) = G :* (x * y).

Lemma normJ : forall A x, 'N(A :^ x) = 'N(A) :^ x.

Lemma norm_conj_norm : forall x A B,
  x \in 'N(A) -> (A \subset 'N(B :^ x)) = (A \subset 'N(B)).

Lemma norm_gen : forall A, 'N(A) \subset 'N(<<A>>).

Lemma class_normG : forall x G, G \subset 'N(x ^: G).

Lemma class_sub_norm : forall G A x,
  G \subset 'N(A) -> (x ^: G \subset A) = (x \in A).

Lemma class_support_normG : forall A G, G \subset 'N(class_support A G).

Lemma class_support_sub_norm : forall A B G,
  A \subset G -> B \subset 'N(G) -> class_support A B \subset G.

Section norm_trans.

Variables (A B C D : {set gT}).
Hypotheses (nBA : A \subset 'N(B)) (nCA : A \subset 'N(C)).

Lemma norms_gen : A \subset 'N(<<B>>).

Lemma norms_norm : A \subset 'N('N(B)).

Lemma normsI : A \subset 'N(B :&: C).

Lemma normsU : A \subset 'N(B :|: C).

Lemma normsIs : B \subset 'N(D) -> A :&: B \subset 'N(C :&: D).

Lemma normsD : A \subset 'N(B :\: C).

Lemma normsM : A \subset 'N(B * C).

Lemma normsY : A \subset 'N(B <*> C).

Lemma normsR : A \subset 'N([~: B, C]).

Lemma norms_class_support : A \subset 'N(class_support B C).

End norm_trans.

Lemma normsIG : forall A B G, A \subset 'N(B) -> A :&: G \subset 'N(B :&: G).

Lemma normsGI : forall A B G, A \subset 'N(B) -> G :&: A \subset 'N(G :&: B).

Lemma norms_bigcap : forall (I : finType) (P : pred I) A (B_ : I -> {set gT}),
  A \subset \bigcap_(i | P i) 'N(B_ i) -> A \subset 'N(\bigcap_(i | P i) B_ i).

Lemma norms_bigcup : forall (I : finType) (P : pred I) A (B_ : I -> {set gT}),
  A \subset \bigcap_(i | P i) 'N(B_ i) -> A \subset 'N(\bigcup_(i | P i) B_ i).

Lemma normD1 : forall A, 'N(A^#) = 'N(A).

Lemma normalP : forall A B,
  reflect (A \subset B /\ {in B, normalised A}) (A <| B).

Lemma normal_norm : forall A B, A <| B -> B \subset 'N(A).

Lemma normal_sub : forall A B, A <| B -> A \subset B.

Lemma normalS : forall G H K,
  K \subset H -> H \subset G -> K <| G -> K <| H.

Lemma normal1 : forall G, 1 <| G.

Lemma normal_refl : forall G, G <| G.

Lemma normalG : forall G, G <| 'N(G).

Lemma normalSG : forall G H, H \subset G -> H <| 'N_G(H).

Lemma normalJ : forall A B x, (A :^ x <| B :^ x) = (A <| B).

Lemma normalM : forall G H K, H <| G -> K <| G -> H * K <| G.

Lemma normalI : forall G H K, H <| G -> K <| G -> H :&: K <| G.

Lemma norm_normalI : forall G H, G \subset 'N(H) -> G :&: H <| G.

Lemma normalGI : forall G H K, H \subset G -> K <| G -> H :&: K <| H.

Lemma normal_subnorm : forall G H, (H <| 'N_G(H)) = (H \subset G).

Lemma normalYG : forall G H, (H <| H <*> G) = (G \subset 'N(H)).

Lemma normalGY : forall G H, (H <| G <*> H) = (G \subset 'N(H)).

Lemma gcore_sub : forall A G, gcore A G \subset A.

Lemma gcore_norm : forall A G, G \subset 'N(gcore A G).

Lemma gcore_normal : forall A G, A \subset G -> gcore A G <| G.

Lemma gcore_max : forall A B G,
  B \subset A -> G \subset 'N(B) -> B \subset gcore A G.

 An elementary proof that subgroups of index 2 are normal; it is almost as  
 short as the "advanced" proof using group actions; besides, the fact that  
 the coset is equal to the complement is used in extremal.v.                
Lemma rcoset_index2 : forall G H x,
  H \subset G -> #|G : H| = 2 -> x \in G :\: H -> H :* x = G :\: H.

Lemma index2_normal : forall G H, H \subset G -> #|G : H| = 2 -> H <| G.

Lemma cent1P : forall x y, reflect (commute x y) (x \in 'C[y]).

Lemma cent1id : forall x, x \in 'C[x].

Lemma cent1E : forall x y, (x \in 'C[y]) = (x * y == y * x).

Lemma cent1C : forall x y, (x \in 'C[y]) = (y \in 'C[x]).

Canonical Structure centraliser_group A : {group _} :=
  Eval hnf in [group of 'C(A)].

Lemma cent_set1 : forall x, 'C([set x]) = 'C[x].

Lemma centP : forall A x, reflect (centralises x A) (x \in 'C(A)).

Lemma centsP : forall A B, reflect {in A, centralised B} (A \subset 'C(B)).

Lemma centsC : forall A B, (A \subset 'C(B)) = (B \subset 'C(A)).

Lemma cents1 : forall A, A \subset 'C(1).

Lemma cent1T : 'C(1) = setT :> {set gT}.

Lemma cent11T : 'C[1] = setT :> {set gT}.

Lemma cent_sub : forall A, 'C(A) \subset 'N(A).

Lemma cents_norm : forall A B, A \subset 'C(B) -> A \subset 'N(B).

Lemma centC : forall A B, A \subset 'C(B) -> commute A B.

Lemma cent_joinEl : forall G H, G \subset 'C(H) -> G <*> H = G * H.

Lemma cent_joinEr : forall G H, H \subset 'C(G) -> G <*> H = G * H.

Lemma centJ : forall A x, 'C(A :^ x) = 'C(A) :^ x.

Lemma cent_norm : forall A, 'N(A) \subset 'N('C(A)).

Lemma norms_cent : forall A B, A \subset 'N(B) -> A \subset 'N('C(B)).

Lemma cent_normal : forall A, 'C(A) <| 'N(A).

Lemma centS : forall A B, B \subset A -> 'C(A) \subset 'C(B).

Lemma centsS : forall A B C, A \subset B -> C \subset 'C(B) -> C \subset 'C(A).

Lemma centSS : forall A B C D,
  A \subset C -> B \subset D -> C \subset 'C(D) -> A \subset 'C(B).

Lemma centI : forall A B, 'C(A) <*> 'C(B) \subset 'C(A :&: B).

Lemma centU : forall A B, 'C(A :|: B) = 'C(A) :&: 'C(B).

Lemma cent_gen : forall A, 'C(<<A>>) = 'C(A).

Lemma cent_cycle : forall x, 'C(<[x]>) = 'C[x].

Lemma sub_cent1 : forall A x, (A \subset 'C[x]) = (x \in 'C(A)).

Lemma cents_cycle : forall x y, commute x y -> <[x]> \subset 'C(<[y]>).

Lemma cycle_abelian : forall x, abelian <[x]>.

Lemma centY : forall A B, 'C(A <*> B) = 'C(A) :&: 'C(B).

Lemma centM : forall G H, 'C(G * H) = 'C(G) :&: 'C(H).

Lemma cent_classP : forall x G, reflect (x ^: G = [set x]) (x \in 'C(G)).

Lemma commG1P : forall A B, reflect ([~: A, B] = 1) (A \subset 'C(B)).

Lemma abelianE : forall A, abelian A = (A \subset 'C(A)).

Lemma abelian1 : abelian [1 gT].

Lemma abelianS : forall A B, A \subset B -> abelian B -> abelian A.

Lemma abelianJ : forall A x, abelian (A :^ x) = abelian A.

Lemma abelian_gen : forall A, abelian <<A>> = abelian A.

Lemma abelianM : forall G H,
  abelian (G * H) = [&& abelian G, abelian H & G \subset 'C(H)].

Section SubAbelian.

Variable A B C : {set gT}.
Hypothesis cAA : abelian A.

Lemma sub_abelian_cent : C \subset A -> A \subset 'C(C).

Lemma sub_abelian_cent2 : B \subset A -> C \subset A -> B \subset 'C(C).

Lemma sub_abelian_norm : C \subset A -> A \subset 'N(C).

Lemma sub_abelian_normal : (C \subset A) = (C <| A).

End SubAbelian.

End Normaliser.

Implicit Arguments normP [gT x A].
Implicit Arguments centP [gT x A].
Implicit Arguments normsP [gT A B].
Implicit Arguments cent1P [gT x y].
Implicit Arguments normalP [gT A B].
Implicit Arguments centsP [gT A B].
Implicit Arguments commG1P [gT A B].



Notation "''N' ( A )" := (normaliser_group A) : subgroup_scope.
Notation "''C' ( A )" := (centraliser_group A) : subgroup_scope.
Notation "''C' [ x ]" := (normaliser_group [set x%g]) : subgroup_scope.
Notation "''N_' G ( A )" := (setI_group G 'N(A)) : subgroup_scope.
Notation "''C_' G ( A )" := (setI_group G 'C(A)) : subgroup_scope.
Notation "''C_' ( G ) ( A )" := (setI_group G 'C(A))
  (only parsing) : subgroup_scope.
Notation "''C_' G [ x ]" := (setI_group G 'C[x]) : subgroup_scope.
Notation "''C_' ( G ) [ x ]" := (setI_group G 'C[x])
  (only parsing) : subgroup_scope.

Hint Resolve normG normal_refl.

Section MinMaxGroup.

Variable gT : finGroupType.
Variable gP : pred {group gT}.

Definition maxgroup := maxset (fun A => group_set A && gP <<A>>).
Definition mingroup := minset (fun A => group_set A && gP <<A>>).

Lemma ex_maxgroup : (exists G, gP G) -> {G : {group gT} | maxgroup G}.

Lemma ex_mingroup : (exists G, gP G) -> {G : {group gT} | mingroup G}.

Variable G : {group gT}.

Lemma mingroupP :
  reflect (gP G /\ forall H, gP H -> H \subset G -> H :=: G) (mingroup G).

Lemma maxgroupP :
  reflect (gP G /\ forall H, gP H -> G \subset H -> H :=: G) (maxgroup G).

Lemma maxgroupp : maxgroup G -> gP G.

Lemma mingroupp : mingroup G -> gP G.

Hypothesis gPG : gP G.

Lemma maxgroup_exists : {H : {group gT} | maxgroup H & G \subset H}.

Lemma mingroup_exists : {H : {group gT} | mingroup H & H \subset G}.

End MinMaxGroup.

Notation "[ 'max' A 'of' G | gP ]" :=
  (maxgroup (fun G : {group _} => gP) A) : group_scope.
Notation "[ 'max' G | gP ]" := [max gval G of G | gP] : group_scope.
Notation "[ 'min' A 'of' G | gP ]" :=
  (mingroup (fun G : {group _} => gP) A) : group_scope.
Notation "[ 'min' G | gP ]" := [min gval G of G | gP] : group_scope.

Implicit Arguments mingroupP [gT gP G].
Implicit Arguments maxgroupP [gT gP G].