Require Import Basics Types HProp HFiber HSet.
[Loading ML file number_string_notation_plugin.cmxs (using legacy method) ... done]
Require Import PathAny.
Require Import (notations) Classes.interfaces.canonical_names.
Require Export (hints) Classes.interfaces.abstract_algebra.
Require Export (hints) Classes.interfaces.canonical_names.
(** We only export the parts of these that will be most useful to users of this file. *)
Require Export Classes.interfaces.canonical_names (SgOp, sg_op, One, one,
    MonUnit, mon_unit, LeftIdentity, left_identity, RightIdentity, right_identity,
    Negate, negate, Associative, simple_associativity, associativity,
    LeftInverse, left_inverse, RightInverse, right_inverse, Commutative, commutativity).
Export canonical_names.BinOpNotations.
Require Export Classes.interfaces.abstract_algebra (IsGroup(..), group_monoid, negate_l, negate_r,
    IsSemiGroup(..), sg_set, sg_ass,
    IsMonoid(..), monoid_left_id, monoid_right_id, monoid_semigroup,
    IsMonoidPreserving(..), monmor_unitmor, monmor_sgmor,
    IsSemiGroupPreserving, preserves_sg_op, IsUnitPreserving, preserves_mon_unit).
Require Export Classes.theory.groups.
Require Import Pointed.Core.
Require Import WildCat.
Require Import Spaces.Nat.Core.
Require Import Truncations.Core.

Local Set Polymorphic Inductive Cumulativity.

Generalizable Variables G H A B C f g.

Declare Scope group_scope.

(** * Groups *)

(** A group is an abstraction of several common situations in mathematics. For example, consider the symmetries of an object.  Two symmetries can be combined; there is a symmetry that does nothing; and any symmetry can be reversed. Such situations arise in geometry, algebra and, importantly for us, homotopy theory. *)

Local Open Scope pointed_scope.
Local Open Scope mc_mult_scope.
Local Open Scope wc_iso_scope.

(** ** Definition of a Group *)

(** A group consists of a type, an operation on that type, a unit and an inverse that satisfy the group axioms in [IsGroup]. *)
Record Group := {
  group_type :> Type;
  group_sgop :: SgOp group_type;
  group_unit :: MonUnit group_type;
  group_inverse :: Negate group_type;
  group_isgroup :: IsGroup group_type;
}.

Arguments group_sgop {_}.
Arguments group_unit {_}.
Arguments group_inverse {_}.
Arguments group_isgroup {_}.
(** We should never need to unfold the proof that something is a group. *)
Global Opaque group_isgroup.

Definition issig_group : _ <~> Group
  := ltac:(issig).

(** ** Proof automation *)
(** Many times in group theoretic proofs we want some form of automation for obvious identities. Here we implement such a behavior. *)

(** We create a database of hints for the group theory library *)
Create HintDb group_db.

(** Our group laws can be proven easily with tactics such as [rapply associativity]. However this requires a typeclass search on more general algebraic structures. Therefore we explicitly list many groups laws here so that coq can use them. We also create hints for each law in our groups database. *)
Section GroupLaws.
  Context {G : Group} (x y z : G).

  Definition grp_assoc := associativity x y z.
  Definition grp_unit_l := left_identity x.
  Definition grp_unit_r := right_identity x.
  Definition grp_inv_l := left_inverse x.
  Definition grp_inv_r := right_inverse x.

End GroupLaws.

#[export] Hint Immediate grp_assoc  : group_db.
#[export] Hint Immediate grp_unit_l : group_db.
#[export] Hint Immediate grp_unit_r : group_db.
#[export] Hint Immediate grp_inv_l  : group_db.
#[export] Hint Immediate grp_inv_r  : group_db.

(** Given path types in a product we may want to decompose. *)
#[export] Hint Extern 5 (@paths (_ * _) _ _) => (apply path_prod) : group_db.
(** Given path types in a sigma type of a hprop family (i.e. a subset) we may want to decompose. *)
#[export] Hint Extern 6 (@paths (sig _) _ _) => (rapply path_sigma_hprop) : group_db.

(** We also declare a tactic (notation) for automatically solving group laws *)
(** TODO: improve this tactic so that it also rewrites and is able to solve basic group lemmas. *)
Tactic Notation "grp_auto" := hnf; intros; eauto with group_db.

(** * Some basic properties of groups *)

(** Groups are pointed sets with point the identity. *)
Global Instance ispointed_group (G : Group)
  : IsPointed G := @mon_unit G _.

Definition ptype_group : Group -> pType
  := fun G => [G, _].

Coercion ptype_group : Group >-> pType.
(** An element acting like the identity is unique. *)
Definition identity_unique {A : Type} {Aop : SgOp A}
  (x y : A) {p : LeftIdentity Aop x} {q : RightIdentity Aop y}
  : x = y := (q x)^ @ p y.

Definition identity_unique' {A : Type} {Aop : SgOp A}
  (x y : A) {p : LeftIdentity Aop x} {q : RightIdentity Aop y}
  : y = x := (identity_unique x y)^.

(** An element acting like an inverse is unique. *)
Definition inverse_unique `{IsMonoid A}
  (a x y : A) {p : x * a = mon_unit} {q : a * y = mon_unit}
  : x = y.
A: Type
Aop: SgOp A
Aunit: MonUnit A
H: IsMonoid A
a, x, y: A
p: x * a = mon_unit
q: a * y = mon_unit
x = y
Proof.
A: Type
Aop: SgOp A
Aunit: MonUnit A
H: IsMonoid A
a, x, y: A
p: x * a = mon_unit
q: a * y = mon_unit
x = y
  refine ((right_identity x)^ @ ap _ q^ @ _).
A: Type
Aop: SgOp A
Aunit: MonUnit A
H: IsMonoid A
a, x, y: A
p: x * a = mon_unit
q: a * y = mon_unit
x * (a * y) = y
  refine (associativity _ _ _ @ _).
A: Type
Aop: SgOp A
Aunit: MonUnit A
H: IsMonoid A
a, x, y: A
p: x * a = mon_unit
q: a * y = mon_unit
x * a * y = y
  refine (ap (fun x => x * y) p @ _).
A: Type
Aop: SgOp A
Aunit: MonUnit A
H: IsMonoid A
a, x, y: A
p: x * a = mon_unit
q: a * y = mon_unit
mon_unit * y = y
  apply left_identity.
Defined.

(** ** Group homomorphisms *)

(** Group homomorphisms are maps between groups that preserve the group operation. They allow us to compare groups and map their structure to one another. This is useful for determining if two groups are really the same, in which case we say they are "isomorphic". *)

(** A group homomorphism consists of a map between groups and a proof that the map preserves the group operation. *)
Record GroupHomomorphism (G H : Group) := Build_GroupHomomorphism' {
  grp_homo_map :> group_type G -> group_type H;
  grp_homo_ishomo :: IsMonoidPreserving grp_homo_map;
}.

(** Group homomorphisms are pointed maps. *)
Definition pmap_GroupHomomorphism {G H : Group} (f : GroupHomomorphism G H) : G ->* H
  := Build_pMap G H f (@monmor_unitmor _ _ _ _ _ _ _ (@grp_homo_ishomo G H f)).
Coercion pmap_GroupHomomorphism : GroupHomomorphism >-> pForall.

Definition issig_GroupHomomorphism (G H : Group) : _ <~> GroupHomomorphism G H
  := ltac:(issig).

(** Function extensionality for group homomorphisms. *)
Definition equiv_path_grouphomomorphism {F : Funext} {G H : Group}
  {g h : GroupHomomorphism G H} : g == h <~> g = h.
F: Funext
G, H: Group
g, h: GroupHomomorphism G H
g == h <~> g = h
Proof.
F: Funext
G, H: Group
g, h: GroupHomomorphism G H
g == h <~> g = h
  refine ((equiv_ap (issig_GroupHomomorphism G H)^-1 _ _)^-1 oE _).
F: Funext
G, H: Group
g, h: GroupHomomorphism G H
g == h <~>
(issig_GroupHomomorphism G H)^-1 g =
(issig_GroupHomomorphism G H)^-1 h
  refine (equiv_path_sigma_hprop _ _ oE _).
F: Funext
G, H: Group
g, h: GroupHomomorphism G H
g == h <~>
((issig_GroupHomomorphism G H)^-1 g).1 =
((issig_GroupHomomorphism G H)^-1 h).1
  apply equiv_path_forall.
Defined.

(** Group homomorphisms are sets, in the presence of funext. *)
Global Instance ishset_grouphomomorphism {F : Funext} {G H : Group}
  : IsHSet (GroupHomomorphism G H).
F: Funext
G, H: Group
IsHSet (GroupHomomorphism G H)
Proof.
F: Funext
G, H: Group
IsHSet (GroupHomomorphism G H)
  apply istrunc_S.
F: Funext
G, H: Group
is_mere_relation (GroupHomomorphism G H) paths
  intros f g; apply (istrunc_equiv_istrunc _ equiv_path_grouphomomorphism).
Defined.

(** ** Basic properties of group homomorphisms *)

(** Group homomorphisms preserve identities. *)
Definition grp_homo_unit {G H} (f : GroupHomomorphism G H)
  : f (mon_unit) = mon_unit.
G, H: Group
f: GroupHomomorphism G H
f mon_unit = mon_unit
Proof.
G, H: Group
f: GroupHomomorphism G H
f mon_unit = mon_unit
  apply monmor_unitmor.
Defined.
#[export] Hint Immediate grp_homo_unit : group_db.

(** Group homomorphisms preserve group operations. *)
Definition grp_homo_op {G H} (f : GroupHomomorphism G H)
  : forall x y : G, f (x * y) = f x * f y.
G, H: Group
f: GroupHomomorphism G H
forall x y : G, f (x * y) = f x * f y
Proof.
G, H: Group
f: GroupHomomorphism G H
forall x y : G, f (x * y) = f x * f y
  apply monmor_sgmor.
Defined.
#[export] Hint Immediate grp_homo_op : group_db.

(** Group homomorphisms preserve inverses. *)
Definition grp_homo_inv {G H} (f : GroupHomomorphism G H)
  : forall x, f (- x) = -(f x).
G, H: Group
f: GroupHomomorphism G H
forall x : G, f (- x) = - f x
Proof.
G, H: Group
f: GroupHomomorphism G H
forall x : G, f (- x) = - f x
  intro x.
G, H: Group
f: GroupHomomorphism G H
x: G
f (- x) = - f x
  apply (inverse_unique (f x)).
G, H: Group
f: GroupHomomorphism G H
x: G
f (- x) * f x = mon_unit
G, H: Group
f: GroupHomomorphism G H
x: G
f x * - f x = mon_unit
  +
G, H: Group
f: GroupHomomorphism G H
x: G
f (- x) * f x = mon_unit refine (_ @ grp_homo_unit f).
G, H: Group
f: GroupHomomorphism G H
x: G
f (- x) * f x = f mon_unit
    refine ((grp_homo_op f (-x) x)^ @ _).
G, H: Group
f: GroupHomomorphism G H
x: G
f (- x * x) = f mon_unit
    apply ap.
G, H: Group
f: GroupHomomorphism G H
x: G
- x * x = mon_unit
    apply grp_inv_l.
  +
G, H: Group
f: GroupHomomorphism G H
x: G
f x * - f x = mon_unit apply grp_inv_r.
Defined.
#[export] Hint Immediate grp_homo_inv : group_db.

(** When building a group homomorphism we only need that it preserves the group operation, since we can prove that the identity is preserved. *)
Definition Build_GroupHomomorphism {G H : Group}
  (f : G -> H) {h : IsSemiGroupPreserving f}
  : GroupHomomorphism G H.
G, H: Group
f: G -> H
h: IsSemiGroupPreserving f
GroupHomomorphism G H
Proof.
G, H: Group
f: G -> H
h: IsSemiGroupPreserving f
GroupHomomorphism G H
  srapply (Build_GroupHomomorphism' _ _ f).
G, H: Group
f: G -> H
h: IsSemiGroupPreserving f
IsMonoidPreserving f
  split.
G, H: Group
f: G -> H
h: IsSemiGroupPreserving f
IsSemiGroupPreserving f
G, H: Group
f: G -> H
h: IsSemiGroupPreserving f
IsUnitPreserving f
  1: exact h.
G, H: Group
f: G -> H
h: IsSemiGroupPreserving f
IsUnitPreserving f
  unfold IsUnitPreserving.
G, H: Group
f: G -> H
h: IsSemiGroupPreserving f
f mon_unit = mon_unit
  apply (group_cancelL (f mon_unit)).
G, H: Group
f: G -> H
h: IsSemiGroupPreserving f
f mon_unit * f mon_unit = f mon_unit * mon_unit
  refine (_ @ (grp_unit_r _)^).
G, H: Group
f: G -> H
h: IsSemiGroupPreserving f
f mon_unit * f mon_unit = f mon_unit
  refine (_ @ ap _ (monoid_left_id _ mon_unit)).
G, H: Group
f: G -> H
h: IsSemiGroupPreserving f
f mon_unit * f mon_unit = f (mon_unit * mon_unit)
  symmetry.
G, H: Group
f: G -> H
h: IsSemiGroupPreserving f
f (mon_unit * mon_unit) = f mon_unit * f mon_unit
  apply h.
Defined.

(** The identity map is a group homomorphism. *)
Definition grp_homo_id {G : Group} : GroupHomomorphism G G
  := Build_GroupHomomorphism idmap.

(** The composition of the underlying functions of two group homomorphisms is also a group homomorphism. *)
Definition grp_homo_compose {G H K : Group}
  : GroupHomomorphism H K -> GroupHomomorphism G H -> GroupHomomorphism G K.
G, H, K: Group
GroupHomomorphism H K ->
GroupHomomorphism G H -> GroupHomomorphism G K
Proof.
G, H, K: Group
GroupHomomorphism H K ->
GroupHomomorphism G H -> GroupHomomorphism G K
  intros f g.
G, H, K: Group
f: GroupHomomorphism H K
g: GroupHomomorphism G H
GroupHomomorphism G K
  srapply (Build_GroupHomomorphism (f o g)).
Defined.

(** ** Group Isomorphisms *)

(** Group isomorphsims are group homomorphisms whose underlying map happens to be an equivalence. They allow us to consider two groups to be the "same". They can be inverted and composed just like equivalences. *)

(** An isomorphism of groups is defined as group homomorphism that is an equivalence. *)
Record GroupIsomorphism (G H : Group) := Build_GroupIsomorphism {
  grp_iso_homo :> GroupHomomorphism G H;
  isequiv_group_iso :: IsEquiv grp_iso_homo;
}.

(** We can build an isomorphism from an operation-preserving equivalence. *)
Definition Build_GroupIsomorphism' {G H : Group}
  (f : G <~> H) (h : IsSemiGroupPreserving f)
  : GroupIsomorphism G H.
G, H: Group
f: G <~> H
h: IsSemiGroupPreserving f
GroupIsomorphism G H
Proof.
G, H: Group
f: G <~> H
h: IsSemiGroupPreserving f
GroupIsomorphism G H
  srapply Build_GroupIsomorphism.
G, H: Group
f: G <~> H
h: IsSemiGroupPreserving f
GroupHomomorphism G H
G, H: Group
f: G <~> H
h: IsSemiGroupPreserving f
IsEquiv ?grp_iso_homo
  1: srapply Build_GroupHomomorphism.
G, H: Group
f: G <~> H
h: IsSemiGroupPreserving f
IsEquiv (Build_GroupHomomorphism f)
  exact _.
Defined.

Definition issig_GroupIsomorphism (G H : Group)
  : _ <~> GroupIsomorphism G H := ltac:(issig).

(** The underlying equivalence of a group isomorphism. *)
Definition equiv_groupisomorphism {G H : Group}
  : GroupIsomorphism G H -> G <~> H
  := fun f => Build_Equiv G H f _.
Coercion equiv_groupisomorphism : GroupIsomorphism >-> Equiv.

(** The underlying pointed equivalence of a group isomorphism. *)
Definition pequiv_groupisomorphism {A B : Group}
  : GroupIsomorphism A B -> (A <~>* B)
  := fun f => Build_pEquiv _ _ f _.
Coercion pequiv_groupisomorphism : GroupIsomorphism >-> pEquiv.

(** Funext for group isomorphisms. *)
Definition equiv_path_groupisomorphism `{F : Funext} {G H : Group}
  (f g : GroupIsomorphism G H)
  : f == g <~> f = g.
F: Funext
G, H: Group
f, g: GroupIsomorphism G H
f == g <~> f = g
Proof.
F: Funext
G, H: Group
f, g: GroupIsomorphism G H
f == g <~> f = g
  refine ((equiv_ap (issig_GroupIsomorphism G H)^-1 _ _)^-1 oE _).
F: Funext
G, H: Group
f, g: GroupIsomorphism G H
f == g <~>
(issig_GroupIsomorphism G H)^-1 f =
(issig_GroupIsomorphism G H)^-1 g
  refine (equiv_path_sigma_hprop _ _ oE _).
F: Funext
G, H: Group
f, g: GroupIsomorphism G H
f == g <~>
((issig_GroupIsomorphism G H)^-1 f).1 =
((issig_GroupIsomorphism G H)^-1 g).1
  apply equiv_path_grouphomomorphism.
Defined.

(** Group isomorphisms form a set. *)
Definition ishset_groupisomorphism `{F : Funext} {G H : Group}
  : IsHSet (GroupIsomorphism G H).
F: Funext
G, H: Group
IsHSet (GroupIsomorphism G H)
Proof.
F: Funext
G, H: Group
IsHSet (GroupIsomorphism G H)
  apply istrunc_S.
F: Funext
G, H: Group
is_mere_relation (GroupIsomorphism G H) paths
  intros f g; apply (istrunc_equiv_istrunc _ (equiv_path_groupisomorphism _ _)).
Defined.

(** The identity map is an equivalence and therefore a group isomorphism. *)
Definition grp_iso_id {G : Group} : GroupIsomorphism G G
  := Build_GroupIsomorphism _ _ grp_homo_id _.

(** Group isomorphisms can be composed by composing the underlying group homomorphism. *)
Definition grp_iso_compose {G H K : Group}
  (g : GroupIsomorphism H K) (f : GroupIsomorphism G H)
  : GroupIsomorphism G K
  := Build_GroupIsomorphism _ _ (grp_homo_compose g f) _.

(** Group isomorphisms can be inverted. The inverse map of the underlying equivalence also preserves the group operation and unit. *)
Definition grp_iso_inverse {G H : Group}
  : GroupIsomorphism G H -> GroupIsomorphism H G.
G, H: Group
GroupIsomorphism G H -> GroupIsomorphism H G
Proof.
G, H: Group
GroupIsomorphism G H -> GroupIsomorphism H G
  intros [f e].
G, H: Group
f: GroupHomomorphism G H
e: IsEquiv f
GroupIsomorphism H G
  srapply Build_GroupIsomorphism.
G, H: Group
f: GroupHomomorphism G H
e: IsEquiv f
GroupHomomorphism H G
G, H: Group
f: GroupHomomorphism G H
e: IsEquiv f
IsEquiv ?grp_iso_homo
  -
G, H: Group
f: GroupHomomorphism G H
e: IsEquiv f
GroupHomomorphism H G srapply (Build_GroupHomomorphism f^-1).
  -
G, H: Group
f: GroupHomomorphism G H
e: IsEquiv f
IsEquiv (Build_GroupHomomorphism f^-1) exact _.
Defined.

(** Group isomorphism is a reflexive relation. *)
Global Instance reflexive_groupisomorphism
  : Reflexive GroupIsomorphism
  := fun G => grp_iso_id.

(** Group isomorphism is a symmetric relation. *)
Global Instance symmetric_groupisomorphism
  : Symmetric GroupIsomorphism
  := fun G H => grp_iso_inverse.

(** Group isomorphism is a transitive relation. *)
Global Instance transitive_groupisomorphism
  : Transitive GroupIsomorphism
  := fun G H K f g => grp_iso_compose g f.

(** Under univalence, equality of groups is equivalent to isomorphism of groups. This is the structure identity principle for groups. *)
Definition equiv_path_group' {U : Univalence} {G H : Group}
  : GroupIsomorphism G H <~> G = H.
U: Univalence
G, H: Group
GroupIsomorphism G H <~> G = H
Proof.
U: Univalence
G, H: Group
GroupIsomorphism G H <~> G = H
  refine (equiv_compose'
    (B := sig (fun f : G <~> H => IsMonoidPreserving f)) _ _).
U: Univalence
G, H: Group
{f : G <~> H & IsMonoidPreserving f} <~> G = H
U: Univalence
G, H: Group
GroupIsomorphism G H <~>
{f : G <~> H & IsMonoidPreserving f}
  {
U: Univalence
G, H: Group
{f : G <~> H & IsMonoidPreserving f} <~> G = H revert G H; apply (equiv_path_issig_contr issig_group).
U: Univalence
forall
b : {H : Type &
    {H0 : SgOp H &
    {H1 : MonUnit H & {H2 : Negate H & IsGroup H}}}},
{f : issig_group b <~> issig_group b &
IsMonoidPreserving f}
U: Univalence
forall
b1 : {H : Type &
     {H0 : SgOp H &
     {H1 : MonUnit H & {H2 : Negate H & IsGroup H}}}},
Contr
  {b2
  : {H : Type &
    {H0 : SgOp H &
    {H1 : MonUnit H & {H2 : Negate H & IsGroup H}}}} &
  {f : issig_group b1 <~> issig_group b2 &
  IsMonoidPreserving f}}
    +
U: Univalence
forall
b : {H : Type &
    {H0 : SgOp H &
    {H1 : MonUnit H & {H2 : Negate H & IsGroup H}}}},
{f : issig_group b <~> issig_group b &
IsMonoidPreserving f} intros [G [? [? [? ?]]]].
U: Univalence
G: Type
proj1: SgOp G
proj0: MonUnit G
proj2: Negate G
proj3: IsGroup G
{f
: issig_group (G; proj1; proj0; proj2; proj3) <~>
  issig_group (G; proj1; proj0; proj2; proj3) &
IsMonoidPreserving f}
      exists 1%equiv.
U: Univalence
G: Type
proj1: SgOp G
proj0: MonUnit G
proj2: Negate G
proj3: IsGroup G
IsMonoidPreserving 1%equiv
      exact _.
    +
U: Univalence
forall
b1 : {H : Type &
     {H0 : SgOp H &
     {H1 : MonUnit H & {H2 : Negate H & IsGroup H}}}},
Contr
  {b2
  : {H : Type &
    {H0 : SgOp H &
    {H1 : MonUnit H & {H2 : Negate H & IsGroup H}}}} &
  {f : issig_group b1 <~> issig_group b2 &
  IsMonoidPreserving f}} intros [G [op [unit [neg ax]]]]; cbn.
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
Contr
  {b2
  : {H : Type &
    {H0 : SgOp H &
    {H1 : MonUnit H & {H2 : Negate H & IsGroup H}}}} &
  {f : G <~> b2.1 & IsMonoidPreserving f}}
      contr_sigsig G (equiv_idmap G).
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
Contr
  {y
  : {H0 : SgOp G &
    {H1 : MonUnit G & {H2 : Negate G & IsGroup G}}} &
  IsMonoidPreserving 1%equiv}
      srefine (Build_Contr _ ((_;(_;(_;_)));_) _); cbn.
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
IsGroup G
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
IsMonoidPreserving idmap
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
forall
y : {y
    : {H0 : SgOp G &
      {H1 : MonUnit G & {H2 : Negate G & IsGroup G}}}
    & IsMonoidPreserving idmap},
((op; unit; neg; ?Goal0); ?Goal1) = y
      1: assumption.
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
IsMonoidPreserving idmap
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
forall
y : {y
    : {H0 : SgOp G &
      {H1 : MonUnit G & {H2 : Negate G & IsGroup G}}}
    & IsMonoidPreserving idmap},
((op; unit; neg; ax); ?Goal0) = y
      1: exact _.
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
forall
y : {y
    : {H0 : SgOp G &
      {H1 : MonUnit G & {H2 : Negate G & IsGroup G}}}
    & IsMonoidPreserving idmap},
((op; unit; neg; ax); id_monoid_morphism) = y
      intros [[op' [unit' [neg' ax']]] eq].
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
op': SgOp G
unit': MonUnit G
neg': Negate G
ax': IsGroup G
eq: IsMonoidPreserving idmap
((op; unit; neg; ax); id_monoid_morphism) =
((op'; unit'; neg'; ax'); eq)
      apply path_sigma_hprop; cbn.
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
op': SgOp G
unit': MonUnit G
neg': Negate G
ax': IsGroup G
eq: IsMonoidPreserving idmap
(op; unit; neg; ax) = (op'; unit'; neg'; ax')
      refine (@ap _ _ (fun x : { oun :
        { oo : SgOp G & { u : MonUnit G & Negate G}}
        & @IsGroup G oun.1 oun.2.1 oun.2.2}
        => (x.1.1 ; x.1.2.1 ; x.1.2.2 ; x.2))
        ((op;unit;neg);ax) ((op';unit';neg');ax') _).
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
op': SgOp G
unit': MonUnit G
neg': Negate G
ax': IsGroup G
eq: IsMonoidPreserving idmap
((op; unit; neg); ax) = ((op'; unit'; neg'); ax')
      apply path_sigma_hprop; cbn.
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
op': SgOp G
unit': MonUnit G
neg': Negate G
ax': IsGroup G
eq: IsMonoidPreserving idmap
(op; unit; neg) = (op'; unit'; neg')
      srefine (path_sigma' _ _ _).
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
op': SgOp G
unit': MonUnit G
neg': Negate G
ax': IsGroup G
eq: IsMonoidPreserving idmap
op = op'
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
op': SgOp G
unit': MonUnit G
neg': Negate G
ax': IsGroup G
eq: IsMonoidPreserving idmap
transport
  (fun _ : SgOp G => {_ : MonUnit G & Negate G}) 
  ?p (unit; neg) = (unit'; neg')
      1: funext x y; apply eq.
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
op': SgOp G
unit': MonUnit G
neg': Negate G
ax': IsGroup G
eq: IsMonoidPreserving idmap
transport
  (fun _ : SgOp G => {_ : MonUnit G & Negate G})
  (path_forall op op'
     ((fun x : G =>
       path_forall (op x) (op' x)
         ((fun y : G => let X := monmor_sgmor in X x y)
          :
          op x == op' x))
      :
      op == op')) (unit; neg) = (unit'; neg')
      rewrite transport_const.
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
op': SgOp G
unit': MonUnit G
neg': Negate G
ax': IsGroup G
eq: IsMonoidPreserving idmap
(unit; neg) = (unit'; neg')
      srefine (path_sigma' _ _ _).
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
op': SgOp G
unit': MonUnit G
neg': Negate G
ax': IsGroup G
eq: IsMonoidPreserving idmap
unit = unit'
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
op': SgOp G
unit': MonUnit G
neg': Negate G
ax': IsGroup G
eq: IsMonoidPreserving idmap
transport (fun _ : MonUnit G => Negate G) ?p neg =
neg'
      1: apply eq.
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
op': SgOp G
unit': MonUnit G
neg': Negate G
ax': IsGroup G
eq: IsMonoidPreserving idmap
transport (fun _ : MonUnit G => Negate G)
  (let X := monmor_unitmor in X) neg = neg'
      rewrite transport_const.
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
op': SgOp G
unit': MonUnit G
neg': Negate G
ax': IsGroup G
eq: IsMonoidPreserving idmap
neg = neg'
      funext x.
U: Univalence
G: Type
op: SgOp G
unit: MonUnit G
neg: Negate G
ax: IsGroup G
op': SgOp G
unit': MonUnit G
neg': Negate G
ax': IsGroup G
eq: IsMonoidPreserving idmap
x: G
neg x = neg' x
      exact (preserves_negate (f:=idmap) _). }
U: Univalence
G, H: Group
GroupIsomorphism G H <~>
{f : G <~> H & IsMonoidPreserving f}
  make_equiv.
Defined.

(** A version with nicer universe variables. *)
Definition equiv_path_group@{u v | u < v} {U : Univalence} {G H : Group@{u}}
  : GroupIsomorphism G H <~> (paths@{v} G H)
  := equiv_path_group'.

(** ** Simple group equivalences *)

(** Left multiplication is an equivalence. *)
Global Instance isequiv_group_left_op {G : Group}
  : forall (x : G), IsEquiv (x *.).
G: Group
forall x : G, IsEquiv (sg_op x)
Proof.
G: Group
forall x : G, IsEquiv (sg_op x)
  intro x.
G: Group
x: G
IsEquiv (sg_op x)
  srapply isequiv_adjointify.
G: Group
x: G
G -> G
G: Group
x: G
sg_op x o ?g == idmap
G: Group
x: G
?g o sg_op x == idmap
  1: exact (-x *.).
G: Group
x: G
sg_op x o sg_op (- x) == idmap
G: Group
x: G
sg_op (- x) o sg_op x == idmap
  all: intro y.
G: Group
x, y: G
x * (- x * y) = y
G: Group
x, y: G
- x * (x * y) = y
  all: refine (grp_assoc _ _ _ @ _ @ grp_unit_l y).
G: Group
x, y: G
x * - x * y = mon_unit * y
G: Group
x, y: G
- x * x * y = mon_unit * y
  all: refine (ap (fun x => x * y) _).
G: Group
x, y: G
x * - x = mon_unit
G: Group
x, y: G
- x * x = mon_unit
  1: apply grp_inv_r.
G: Group
x, y: G
- x * x = mon_unit
  apply grp_inv_l.
Defined.

(** Right multiplication is an equivalence. *)
Global Instance isequiv_group_right_op (G : Group)
  : forall (x : G), IsEquiv (fun y => y * x).
G: Group
forall x : G, IsEquiv (fun y : G => y * x)
Proof.
G: Group
forall x : G, IsEquiv (fun y : G => y * x)
  intro x.
G: Group
x: G
IsEquiv (fun y : G => y * x)
  srapply isequiv_adjointify.
G: Group
x: G
G -> G
G: Group
x: G
(fun y : G => y * x) o ?g == idmap
G: Group
x: G
?g o (fun y : G => y * x) == idmap
  1: exact (fun y => y * - x).
G: Group
x: G
(fun y : G => y * x) o (fun y : G => y * - x) == idmap
G: Group
x: G
(fun y : G => y * - x) o (fun y : G => y * x) == idmap
  all: intro y.
G: Group
x, y: G
y * - x * x = y
G: Group
x, y: G
y * x * - x = y
  all: refine ((grp_assoc _ _ _)^ @ _ @ grp_unit_r y).
G: Group
x, y: G
y * (- x * x) = y * mon_unit
G: Group
x, y: G
y * (x * - x) = y * mon_unit
  all: refine (ap (y *.) _).
G: Group
x, y: G
- x * x = mon_unit
G: Group
x, y: G
x * - x = mon_unit
  1: apply grp_inv_l.
G: Group
x, y: G
x * - x = mon_unit
  apply grp_inv_r.
Defined.

(** The operation inverting group elements is an equivalence. Note that, since the order of the operation will change after inversion, this isn't a group homomorphism. *)
Global Instance isequiv_group_inverse {G : Group}
  : IsEquiv ((-) : G -> G).
G: Group
IsEquiv (negate : G -> G)
Proof.
G: Group
IsEquiv (negate : G -> G)
  srapply isequiv_adjointify.
G: Group
G -> G
G: Group
negate o ?g == idmap
G: Group
?g o negate == idmap
  1: apply (-).
G: Group
negate o negate == idmap
G: Group
negate o negate == idmap
  all: intro; apply negate_involutive.
Defined.

(** ** Reasoning with equations in groups. *)

Section GroupEquations.

  Context {G : Group} (x y z : G).

  (** Inverses are involutive. *)
  Definition grp_inv_inv : --x = x := negate_involutive x.

  (** Inverses distribute over the group operation. *)
  Definition grp_inv_op : - (x * y) = -y * -x := negate_sg_op x y.

End GroupEquations.

(** ** Cancelation lemmas *)

(** Group elements can be cancelled both on the left and the right. *)
Definition grp_cancelL {G : Group} {x y : G} z : x = y <~> z * x = z * y
  := equiv_ap (fun x => z * x) _ _.
Definition grp_cancelR {G : Group} {x y : G} z : x = y <~> x * z = y * z
  := equiv_ap (fun x => x * z) _ _.

(** ** Group movement lemmas *)

Section GroupMovement.

  (** Since left/right multiplication is an equivalence, we can use lemmas about moving equivalences around to prove group movement lemmas. *)

  Context {G : Group} {x y z : G}.

  (** *** Moving group elements *)

  Definition grp_moveL_gM : x * -z = y <~> x = y * z
    := equiv_moveL_equiv_M (f := fun t => t * z) _ _.

  Definition grp_moveL_Mg : -y * x = z <~> x = y * z
    := equiv_moveL_equiv_M (f := fun t => y * t) _ _.

  Definition grp_moveR_gM : x = z * -y <~> x * y = z
    := equiv_moveR_equiv_M (f := fun t => t * y) _ _.

  Definition grp_moveR_Mg : y = -x * z <~> x * y = z
    := equiv_moveR_equiv_M (f := fun t => x * t) _ _.

  (** *** Moving inverses.*)
  (** These are the inverses of the previous but are included here for completeness*)
  Definition grp_moveR_gV : x = y * z <~> x * -z = y
    := equiv_moveR_equiv_V (f := fun t => t * z) _ _.

  Definition grp_moveR_Vg : x = y * z <~> -y * x = z 
    := equiv_moveR_equiv_V (f := fun t => y * t) _ _.

  Definition grp_moveL_gV :  x * y = z <~> x = z * -y
    := equiv_moveL_equiv_V (f := fun t => t * y) _ _.

  Definition grp_moveL_Vg :  x * y = z <~> y = -x * z
    := equiv_moveL_equiv_V (f := fun t => x * t) _ _.

(** We close the section here so the previous lemmas generalise their assumptions. *)
End GroupMovement.

Section GroupMovement.

  Context {G : Group} {x y z : G}.

  (** *** Moving elements equal to unit. *)

  Definition grp_moveL_1M : x * -y = mon_unit <~> x = y
    := equiv_concat_r (grp_unit_l _) _ oE grp_moveL_gM.

  Definition grp_moveL_1V : x * y = mon_unit <~> x = -y
    := equiv_concat_r (grp_unit_l _) _ oE grp_moveL_gV.

  Definition grp_moveL_M1 : -y * x = mon_unit <~> x = y
    := equiv_concat_r (grp_unit_r _) _ oE grp_moveL_Mg.

  Definition grp_moveR_1M : mon_unit = y * (-x) <~> x = y
    := (equiv_concat_l (grp_unit_l _) _)^-1%equiv oE grp_moveR_gM.

  Definition grp_moveR_M1 : mon_unit = -x * y <~> x = y
    := (equiv_concat_l (grp_unit_r _) _)^-1%equiv oE grp_moveR_Mg.

  (** *** Cancelling elements equal to unit. *)

  Definition grp_cancelL1 : x = mon_unit <~> z * x = z
    := (equiv_concat_r (grp_unit_r _) _ oE grp_cancelL z).

  Definition grp_cancelR1 : x = mon_unit <~> x * z = z
    := (equiv_concat_r (grp_unit_l _) _) oE grp_cancelR z.

End GroupMovement.

(** ** Power operation *)

(** For a given [n : nat] we can define the [n]th power of a group element. *)
Definition grp_pow {G : Group} (g : G) (n : nat) : G := nat_iter n (g *.) mon_unit.

(** Any homomorphism respects [grp_pow]. *)
Lemma grp_pow_homo {G H : Group} (f : GroupHomomorphism G H)
  (n : nat) (g : G) : f (grp_pow g n) = grp_pow (f g) n.
G, H: Group
f: GroupHomomorphism G H
n: nat
g: G
f (grp_pow g n) = grp_pow (f g) n
Proof.
G, H: Group
f: GroupHomomorphism G H
n: nat
g: G
f (grp_pow g n) = grp_pow (f g) n
  induction n.
G, H: Group
f: GroupHomomorphism G H
g: G
f (grp_pow g 0) = grp_pow (f g) 0
G, H: Group
f: GroupHomomorphism G H
n: nat
g: G
IHn: f (grp_pow g n) = grp_pow (f g) n
f (grp_pow g n.+1) = grp_pow (f g) n.+1
  +
G, H: Group
f: GroupHomomorphism G H
g: G
f (grp_pow g 0) = grp_pow (f g) 0 cbn.
G, H: Group
f: GroupHomomorphism G H
g: G
f mon_unit = mon_unit apply grp_homo_unit.
  +
G, H: Group
f: GroupHomomorphism G H
n: nat
g: G
IHn: f (grp_pow g n) = grp_pow (f g) n
f (grp_pow g n.+1) = grp_pow (f g) n.+1 cbn.
G, H: Group
f: GroupHomomorphism G H
n: nat
g: G
IHn: f (grp_pow g n) = grp_pow (f g) n
f (g * nat_iter n (sg_op g) mon_unit) =
f g * nat_iter n (sg_op (f g)) mon_unit refine ((grp_homo_op f g (grp_pow g n)) @ _).
G, H: Group
f: GroupHomomorphism G H
n: nat
g: G
IHn: f (grp_pow g n) = grp_pow (f g) n
f g * f (grp_pow g n) =
f g * nat_iter n (sg_op (f g)) mon_unit
    exact (ap (fun m => f g + m) IHn).
Defined.

(** ** The category of Groups *)

(** ** Groups together with homomorphisms form a 1-category whose equivalences are the group isomorphisms. *)

Global Instance isgraph_group : IsGraph Group
  := Build_IsGraph Group GroupHomomorphism.

Global Instance is01cat_group : Is01Cat Group :=
  Build_Is01Cat Group _ (@grp_homo_id) (@grp_homo_compose).

(** Helper notation so that the wildcat instances can easily be inferred. *)
Local Notation grp_homo_map' A B := (@grp_homo_map A B : _ -> (group_type A $-> _)).

Global Instance is2graph_group : Is2Graph Group
  := fun A B => isgraph_induced (grp_homo_map' A B).

Global Instance isgraph_grouphomomorphism {A B : Group} : IsGraph (A $-> B)
  := isgraph_induced (grp_homo_map' A B).

Global Instance is01cat_grouphomomorphism {A B : Group} : Is01Cat (A $-> B)
  := is01cat_induced (grp_homo_map' A B).

Global Instance is0gpd_grouphomomorphism {A B : Group}: Is0Gpd (A $-> B)
  := is0gpd_induced (grp_homo_map' A B).

Global Instance is0functor_postcomp_grouphomomorphism {A B C : Group} (h : B $-> C)
  : Is0Functor (@cat_postcomp Group _ _ A B C h).
A, B, C: Group
h: B $-> C
Is0Functor (cat_postcomp A h)
Proof.
A, B, C: Group
h: B $-> C
Is0Functor (cat_postcomp A h)
  apply Build_Is0Functor.
A, B, C: Group
h: B $-> C
forall a b : A $-> B,
(a $-> b) -> cat_postcomp A h a $-> cat_postcomp A h b
  intros [f ?] [g ?] p a ; exact (ap h (p a)).
Defined.

Global Instance is0functor_precomp_grouphomomorphism
       {A B C : Group} (h : A $-> B)
  : Is0Functor (@cat_precomp Group _ _ A B C h).
A, B, C: Group
h: A $-> B
Is0Functor (cat_precomp C h)
Proof.
A, B, C: Group
h: A $-> B
Is0Functor (cat_precomp C h)
  apply Build_Is0Functor.
A, B, C: Group
h: A $-> B
forall a b : B $-> C,
(a $-> b) -> cat_precomp C h a $-> cat_precomp C h b
  intros [f ?] [g ?] p a ; exact (p (h a)).
Defined.

(** Group forms a 1Cat *)
Global Instance is1cat_group : Is1Cat Group.
Is1Cat Group
Proof.
Is1Cat Group
  by rapply Build_Is1Cat.
Defined.

(** Under [Funext], the category of groups has morphism extensionality. *)
Global Instance hasmorext_group `{Funext} : HasMorExt Group.
H: Funext
HasMorExt Group
Proof.
H: Funext
HasMorExt Group
  srapply Build_HasMorExt.
H: Funext
forall (a b : Group) (f g : a $-> b),
IsEquiv GpdHom_path
  intros A B f g; cbn in *.
H: Funext
A, B: Group
f, g: GroupHomomorphism A B
IsEquiv GpdHom_path
  snrapply @isequiv_homotopic.
H: Funext
A, B: Group
f, g: GroupHomomorphism A B
f = g -> f == g
H: Funext
A, B: Group
f, g: GroupHomomorphism A B
IsEquiv ?f
H: Funext
A, B: Group
f, g: GroupHomomorphism A B
?f == GpdHom_path
  1: exact (equiv_path_grouphomomorphism^-1%equiv).
H: Funext
A, B: Group
f, g: GroupHomomorphism A B
IsEquiv (equiv_path_grouphomomorphism^-1)%equiv
H: Funext
A, B: Group
f, g: GroupHomomorphism A B
(equiv_path_grouphomomorphism^-1)%equiv == GpdHom_path
  1: exact _.
H: Funext
A, B: Group
f, g: GroupHomomorphism A B
(equiv_path_grouphomomorphism^-1)%equiv == GpdHom_path
  intros []; reflexivity. 
Defined.

(** Group isomorphisms become equivalences in the category of groups. *)
Global Instance hasequivs_group
  : HasEquivs Group.
HasEquivs Group
Proof.
HasEquivs Group
  unshelve econstructor.
Group -> Group -> Type
forall a b : Group, (a $-> b) -> Type
forall a b : Group, ?CatEquiv' a b -> a $-> b
forall (a b : Group) (f : a $-> b),
?CatIsEquiv' a b f -> ?CatEquiv' a b
forall a b : Group, ?CatEquiv' a b -> b $-> a
forall (a b : Group) (f : ?CatEquiv' a b),
?CatIsEquiv' a b (?cate_fun' a b f)
forall (a b : Group) (f : a $-> b)
(fe : ?CatIsEquiv' a b f),
?cate_fun' a b (?cate_buildequiv' a b f fe) $== f
forall (a b : Group) (f : ?CatEquiv' a b),
?cate_inv' a b f $o ?cate_fun' a b f $== Id a
forall (a b : Group) (f : ?CatEquiv' a b),
?cate_fun' a b f $o ?cate_inv' a b f $== Id b
forall (a b : Group) (f : a $-> b) (g : b $-> a),
f $o g $== Id b ->
g $o f $== Id a -> ?CatIsEquiv' a b f
  +
Group -> Group -> Type exact GroupIsomorphism.
  +
forall a b : Group, (a $-> b) -> Type exact (fun G H f => IsEquiv f).
  +
forall a b : Group, GroupIsomorphism a b -> a $-> b intros G H f; exact f.
  +
forall (a b : Group) (f : a $-> b),
(fun (G H : Group) (f0 : G $-> H) => IsEquiv f0) a b f ->
GroupIsomorphism a b exact Build_GroupIsomorphism.
  +
forall a b : Group, GroupIsomorphism a b -> b $-> a intros G H; exact grp_iso_inverse.
  +
forall (a b : Group) (f : GroupIsomorphism a b),
(fun (G H : Group) (f0 : G $-> H) => IsEquiv f0) a b
  ((fun (G H : Group) (f0 : GroupIsomorphism G H) =>
    grp_iso_homo G H f0) a b f) cbn; exact _.
  +
forall (a b : Group) (f : a $-> b)
(fe : (fun (G H : Group) (f0 : G $-> H) => IsEquiv f0)
        a b f),
(fun (G H : Group) (f0 : GroupIsomorphism G H) =>
 grp_iso_homo G H f0) a b
  {| grp_iso_homo := f; isequiv_group_iso := fe |} $==
f reflexivity.
  +
forall (a b : Group) (f : GroupIsomorphism a b),
(fun (G H : Group) (x : GroupIsomorphism G H) =>
 grp_iso_homo H G (grp_iso_inverse x)) a b f $o
(fun (G H : Group) (f0 : GroupIsomorphism G H) =>
 grp_iso_homo G H f0) a b f $== Id a intros ????; apply eissect.
  +
forall (a b : Group) (f : GroupIsomorphism a b),
(fun (G H : Group) (f0 : GroupIsomorphism G H) =>
 grp_iso_homo G H f0) a b f $o
(fun (G H : Group) (x : GroupIsomorphism G H) =>
 grp_iso_homo H G (grp_iso_inverse x)) a b f $== Id b intros ????; apply eisretr.
  +
forall (a b : Group) (f : a $-> b) (g : b $-> a),
f $o g $== Id b ->
g $o f $== Id a ->
(fun (G H : Group) (f0 : G $-> H) => IsEquiv f0) a b f intros G H f g p q.
G, H: Group
f: G $-> H
g: H $-> G
p: f $o g $== Id H
q: g $o f $== Id G
(fun (G H : Group) (f : G $-> H) => IsEquiv f) G H f
    exact (isequiv_adjointify f g p q).
Defined.

Global Instance is1cat_strong `{Funext} : Is1Cat_Strong Group.
H: Funext
Is1Cat_Strong Group
Proof.
H: Funext
Is1Cat_Strong Group
  rapply Build_Is1Cat_Strong.
H: Funext
forall (a b c d : Group) (f : a $-> b) (g : b $-> c)
(h : c $-> d), h $o g $o f = h $o (g $o f)
H: Funext
forall (a b c d : Group) (f : a $-> b) (g : b $-> c)
(h : c $-> d), h $o (g $o f) = h $o g $o f
H: Funext
forall (a b : Group) (f : a $-> b), Id b $o f = f
H: Funext
forall (a b : Group) (f : a $-> b), f $o Id a = f
  all: intros; apply equiv_path_grouphomomorphism; intro; reflexivity.
Defined.

(** The [group_type] map is a 1-functor. *)

Global Instance is0functor_type_group : Is0Functor group_type.
Is0Functor group_type
Proof.
Is0Functor group_type
  apply Build_Is0Functor.
forall a b : Group, (a $-> b) -> a $-> b
  rapply @grp_homo_map.
Defined.

Global Instance is1functor_type_group : Is1Functor group_type.
Is1Functor group_type
Proof.
Is1Functor group_type
  by apply Build_Is1Functor.
Defined.

(** The [ptype_group] map is a 1-functor. *)

Global Instance is0functor_ptype_group : Is0Functor ptype_group.
Is0Functor ptype_group
Proof.
Is0Functor ptype_group
  apply Build_Is0Functor.
forall a b : Group, (a $-> b) -> a $-> b
  rapply @pmap_GroupHomomorphism.
Defined.

Global Instance is1functor_ptype_group : Is1Functor ptype_group.
Is1Functor ptype_group
Proof.
Is1Functor ptype_group
  apply Build_Is1Functor; intros; by apply phomotopy_homotopy_hset.
Defined.

(** Given a group element [a0 : A] over [b : B], multiplication by [a] establishes an equivalence between the kernel and the fiber over [b]. *)
Lemma equiv_grp_hfiber {A B : Group} (f : GroupHomomorphism A B) (b : B)
  : forall (a0 : hfiber f b), hfiber f b <~> hfiber f mon_unit.
A, B: Group
f: GroupHomomorphism A B
b: B
hfiber f b -> hfiber f b <~> hfiber f mon_unit
Proof.
A, B: Group
f: GroupHomomorphism A B
b: B
hfiber f b -> hfiber f b <~> hfiber f mon_unit
  intros [a0 p].
A, B: Group
f: GroupHomomorphism A B
b: B
a0: A
p: f a0 = b
hfiber f b <~> hfiber f mon_unit
  refine (equiv_transport (hfiber f) (right_inverse b) oE _).
A, B: Group
f: GroupHomomorphism A B
b: B
a0: A
p: f a0 = b
hfiber f b <~> hfiber f (b * - b)
  snrapply Build_Equiv.
A, B: Group
f: GroupHomomorphism A B
b: B
a0: A
p: f a0 = b
hfiber f b -> hfiber f (b * - b)
A, B: Group
f: GroupHomomorphism A B
b: B
a0: A
p: f a0 = b
IsEquiv ?equiv_fun
  {
A, B: Group
f: GroupHomomorphism A B
b: B
a0: A
p: f a0 = b
hfiber f b -> hfiber f (b * - b) srapply (functor_hfiber (h := fun t => t * -a0) (k := fun t => t * -b)).
A, B: Group
f: GroupHomomorphism A B
b: B
a0: A
p: f a0 = b
(fun t : B => t * - b) o f ==
f o (fun t : A => t * - a0)
    intro a; cbn; symmetry.
A, B: Group
f: GroupHomomorphism A B
b: B
a0: A
p: f a0 = b
a: A
f (a * - a0) = f a * - b
    refine (_ @ ap (fun x => f a * (- x)) p).
A, B: Group
f: GroupHomomorphism A B
b: B
a0: A
p: f a0 = b
a: A
f (a * - a0) = f a * - f a0
    exact (grp_homo_op f _ _ @ ap (fun x => f a * x) (grp_homo_inv f a0)). }
A, B: Group
f: GroupHomomorphism A B
b: B
a0: A
p: f a0 = b
IsEquiv
  (functor_hfiber
     ((fun a : A =>
       ((grp_homo_op f a (- a0) @
         ap (fun x : B => f a * x) (grp_homo_inv f a0)) @
        ap (fun x : B => f a * - x) p)^
       :
       f a * - b = f (a * - a0))
      :
      (fun t : B => t * - b) o f ==
      f o (fun t : A => t * - a0)) b)
  srapply isequiv_functor_hfiber.
Defined.

(** ** The trivial group *)

Definition grp_trivial : Group.
Group
Proof.
Group
  refine (Build_Group Unit (fun _ _ => tt) tt (fun _ => tt) _).
IsGroup Unit
  repeat split; try exact _; by intros [].
Defined.

(** Map out of trivial group. *)
Definition grp_trivial_rec (G : Group) : GroupHomomorphism grp_trivial G.
G: Group
GroupHomomorphism grp_trivial G
Proof.
G: Group
GroupHomomorphism grp_trivial G
  snrapply Build_GroupHomomorphism.
G: Group
grp_trivial -> G
G: Group
IsSemiGroupPreserving ?f
  1: exact (fun _ => group_unit).
G: Group
IsSemiGroupPreserving
  (fun _ : grp_trivial => group_unit)
  intros ??; symmetry; apply grp_unit_l.
Defined.

(** Map into trivial group. *)
Definition grp_trivial_corec (G : Group) : GroupHomomorphism G grp_trivial.
G: Group
GroupHomomorphism G grp_trivial
Proof.
G: Group
GroupHomomorphism G grp_trivial
  snrapply Build_GroupHomomorphism.
G: Group
G -> grp_trivial
G: Group
IsSemiGroupPreserving ?f
  1: exact (fun _ => tt).
G: Group
IsSemiGroupPreserving (fun _ : G => tt)
  intros ??; symmetry; exact (grp_unit_l _).
Defined.

(** Group is a pointed category. *)
Global Instance ispointedcat_group : IsPointedCat Group.
IsPointedCat Group
Proof.
IsPointedCat Group
  snrapply Build_IsPointedCat.
Group
IsInitial ?zero_object
IsTerminal ?zero_object
  -
Group exact grp_trivial.
  -
IsInitial grp_trivial intro G.
G: Group
{f : grp_trivial $-> G &
forall g : grp_trivial $-> G, f $== g}
    exists (grp_trivial_rec G).
G: Group
forall g : grp_trivial $-> G, grp_trivial_rec G $== g
    intros g []; cbn.
G: Group
g: grp_trivial $-> G
group_unit = g tt
    exact (grp_homo_unit g)^.
  -
IsTerminal grp_trivial intro G.
G: Group
{f : G $-> grp_trivial &
forall g : G $-> grp_trivial, f $== g}
    exists (grp_trivial_corec G).
G: Group
forall g : G $-> grp_trivial,
grp_trivial_corec G $== g
    intros g x; cbn.
G: Group
g: G $-> grp_trivial
x: G
tt = g x
    apply path_unit.
Defined.

Definition grp_homo_const {G H : Group} : GroupHomomorphism G H
  := zero_morphism.

(** ** The direct product of groups *)

(** The cartesian product of the underlying sets of two groups has a natural group structure. We call this the direct product of groups. *)
Definition grp_prod : Group -> Group -> Group.
Group -> Group -> Group
Proof.
Group -> Group -> Group
  intros G H.
G, H: Group
Group
  srapply (Build_Group (G * H)).
G, H: Group
SgOp (G * H)
G, H: Group
MonUnit (G * H)
G, H: Group
Negate (G * H)
G, H: Group
IsGroup (G * H)
  (** Operation *)
  {
G, H: Group
SgOp (G * H) intros [g1 h1] [g2 h2].
G, H: Group
g1: G
h1: H
g2: G
h2: H
(G * H)%type
    exact (g1 * g2, h1 * h2). }
G, H: Group
MonUnit (G * H)
G, H: Group
Negate (G * H)
G, H: Group
IsGroup (G * H)
  (** Unit *)
  1: exact (mon_unit, mon_unit).
G, H: Group
Negate (G * H)
G, H: Group
IsGroup (G * H)
  (** Inverse *)
  {
G, H: Group
Negate (G * H) intros [g h].
G, H: Group
g: G
h: H
(G * H)%type
    exact (-g, -h). }
G, H: Group
IsGroup (G * H)
  repeat split.
G, H: Group
IsHSet (G * H)
G, H: Group
Associative sg_op
G, H: Group
LeftIdentity sg_op mon_unit
G, H: Group
RightIdentity sg_op mon_unit
G, H: Group
LeftInverse sg_op negate mon_unit
G, H: Group
RightInverse sg_op negate mon_unit
  1: exact _.
G, H: Group
Associative sg_op
G, H: Group
LeftIdentity sg_op mon_unit
G, H: Group
RightIdentity sg_op mon_unit
G, H: Group
LeftInverse sg_op negate mon_unit
G, H: Group
RightInverse sg_op negate mon_unit
  all: grp_auto.
Defined.

(** Maps into the direct product can be built by mapping separately into each factor. *)
Proposition grp_prod_corec {G H K : Group}
            (f : GroupHomomorphism K G)
            (g : GroupHomomorphism K H)
  : GroupHomomorphism K (grp_prod G H).
G, H, K: Group
f: GroupHomomorphism K G
g: GroupHomomorphism K H
GroupHomomorphism K (grp_prod G H)
Proof.
G, H, K: Group
f: GroupHomomorphism K G
g: GroupHomomorphism K H
GroupHomomorphism K (grp_prod G H)
  snrapply Build_GroupHomomorphism.
G, H, K: Group
f: GroupHomomorphism K G
g: GroupHomomorphism K H
K -> grp_prod G H
G, H, K: Group
f: GroupHomomorphism K G
g: GroupHomomorphism K H
IsSemiGroupPreserving ?f
  -
G, H, K: Group
f: GroupHomomorphism K G
g: GroupHomomorphism K H
K -> grp_prod G H exact (fun x:K => (f x, g x)).
  -
G, H, K: Group
f: GroupHomomorphism K G
g: GroupHomomorphism K H
IsSemiGroupPreserving (fun x : K => (f x, g x)) intros x y.
G, H, K: Group
f: GroupHomomorphism K G
g: GroupHomomorphism K H
x, y: K
(f (x * y), g (x * y)) = (f x, g x) * (f y, g y)
    refine (path_prod' _ _ ); try apply grp_homo_op.
Defined.

(** The left factor injects into the direct product. *)
Definition grp_prod_inl {H K : Group}
  : GroupHomomorphism H (grp_prod H K)
  := grp_prod_corec grp_homo_id grp_homo_const.

(** The left injection is an embedding. *)
Global Instance isembedding_grp_prod_inl {H K : Group}
  : IsEmbedding (@grp_prod_inl H K).
H, K: Group
IsEmbedding grp_prod_inl
Proof.
H, K: Group
IsEmbedding grp_prod_inl
  apply isembedding_isinj_hset.
H, K: Group
isinj grp_prod_inl
  intros h0 h1 p; cbn in p.
H, K: Group
h0, h1: H
p: (h0, group_unit) = (h1, group_unit)
h0 = h1
  exact (fst ((equiv_path_prod _ _)^-1 p)).
Defined.

(** The right factor injects into the direct product. *)
Definition grp_prod_inr {H K : Group}
  : GroupHomomorphism K (grp_prod H K)
  := grp_prod_corec grp_homo_const grp_homo_id.

(** The right injection is an embedding. *)
Global Instance isembedding_grp_prod_inr {H K : Group}
  : IsEmbedding (@grp_prod_inr H K).
H, K: Group
IsEmbedding grp_prod_inr
Proof.
H, K: Group
IsEmbedding grp_prod_inr
  apply isembedding_isinj_hset.
H, K: Group
isinj grp_prod_inr
  intros k0 k1 q; cbn in q.
H, K: Group
k0, k1: K
q: (group_unit, k0) = (group_unit, k1)
k0 = k1
  exact (snd ((equiv_path_prod _ _)^-1 q)).
Defined.

(** Given two pairs of isomorphic groups, their pairwise direct products are isomorphic. *)
Definition grp_iso_prod {A B C D : Group}
  : A ≅ B -> C ≅ D -> (grp_prod A C) ≅ (grp_prod B D).
A, B, C, D: Group
A ≅ B -> C ≅ D -> grp_prod A C ≅ grp_prod B D
Proof.
A, B, C, D: Group
A ≅ B -> C ≅ D -> grp_prod A C ≅ grp_prod B D
  intros f g.
A, B, C, D: Group
f: A ≅ B
g: C ≅ D
grp_prod A C ≅ grp_prod B D
  srapply Build_GroupIsomorphism'.
A, B, C, D: Group
f: A ≅ B
g: C ≅ D
grp_prod A C <~> grp_prod B D
A, B, C, D: Group
f: A ≅ B
g: C ≅ D
IsSemiGroupPreserving ?f
  1: srapply (equiv_functor_prod (f:=f) (g:=g)).
A, B, C, D: Group
f: A ≅ B
g: C ≅ D
IsSemiGroupPreserving equiv_functor_prod
  simpl.
A, B, C, D: Group
f: A ≅ B
g: C ≅ D
IsSemiGroupPreserving (functor_prod f g)
  unfold functor_prod.
A, B, C, D: Group
f: A ≅ B
g: C ≅ D
IsSemiGroupPreserving
  (fun z : A * C => (f (fst z), g (snd z)))
  intros x y.
A, B, C, D: Group
f: A ≅ B
g: C ≅ D
x, y: (A * C)%type
(f (fst (x * y)), g (snd (x * y))) =
(f (fst x), g (snd x)) * (f (fst y), g (snd y))
  apply path_prod.
A, B, C, D: Group
f: A ≅ B
g: C ≅ D
x, y: (A * C)%type
fst (f (fst (x * y)), g (snd (x * y))) =
fst ((f (fst x), g (snd x)) * (f (fst y), g (snd y)))
A, B, C, D: Group
f: A ≅ B
g: C ≅ D
x, y: (A * C)%type
snd (f (fst (x * y)), g (snd (x * y))) =
snd ((f (fst x), g (snd x)) * (f (fst y), g (snd y)))
  1,2: apply grp_homo_op.
Defined.

(** The first projection of the direct product. *)
Definition grp_prod_pr1 {G H : Group}
  : GroupHomomorphism (grp_prod G H) G.
G, H: Group
GroupHomomorphism (grp_prod G H) G
Proof.
G, H: Group
GroupHomomorphism (grp_prod G H) G
  snrapply Build_GroupHomomorphism.
G, H: Group
grp_prod G H -> G
G, H: Group
IsSemiGroupPreserving ?f
  1: exact fst.
G, H: Group
IsSemiGroupPreserving fst
  intros ? ?; reflexivity.
Defined.

(** The first projection is a surjection. *)
Global Instance issurj_grp_prod_pr1 {G H : Group}
  : IsSurjection (@grp_prod_pr1 G H)
  := issurj_retr grp_prod_inl (fun _ => idpath).

(** The second projection of the direct product. *)
Definition grp_prod_pr2 {G H : Group}
  : GroupHomomorphism (grp_prod G H) H.
G, H: Group
GroupHomomorphism (grp_prod G H) H
Proof.
G, H: Group
GroupHomomorphism (grp_prod G H) H
  snrapply Build_GroupHomomorphism.
G, H: Group
grp_prod G H -> H
G, H: Group
IsSemiGroupPreserving ?f
  1: exact snd.
G, H: Group
IsSemiGroupPreserving snd
  intros ? ?; reflexivity.
Defined.

(** The second projection is a surjection. *)
Global Instance issurj_grp_prod_pr2 {G H : Group}
  : IsSurjection (@grp_prod_pr2 G H)
  := issurj_retr grp_prod_inr (fun _ => idpath).

(** [Group] is a category with binary products given by the direct product. *)
Global Instance hasbinaryproducts_group : HasBinaryProducts Group.
HasBinaryProducts Group
Proof.
HasBinaryProducts Group
  intros G H.
G, H: Group
BinaryProduct G H
  snrapply Build_BinaryProduct.
G, H: Group
Group
G, H: Group
?cat_binprod $-> G
G, H: Group
?cat_binprod $-> H
G, H: Group
forall z : Group,
(z $-> G) -> (z $-> H) -> z $-> ?cat_binprod
G, H: Group
forall (z : Group) (f : z $-> G) (g : z $-> H),
?cat_pr1 $o ?cat_binprod_corec z f g $== f
G, H: Group
forall (z : Group) (f : z $-> G) (g : z $-> H),
?cat_pr2 $o ?cat_binprod_corec z f g $== g
G, H: Group
forall (z : Group) (f g : z $-> ?cat_binprod),
?cat_pr1 $o f $== ?cat_pr1 $o g ->
?cat_pr2 $o f $== ?cat_pr2 $o g -> f $== g
  -
G, H: Group
Group exact (grp_prod G H).
  -
G, H: Group
grp_prod G H $-> G exact grp_prod_pr1.
  -
G, H: Group
grp_prod G H $-> H exact grp_prod_pr2.
  -
G, H: Group
forall z : Group,
(z $-> G) -> (z $-> H) -> z $-> grp_prod G H intros K.
G, H, K: Group
(K $-> G) -> (K $-> H) -> K $-> grp_prod G H
    exact grp_prod_corec.
  -
G, H: Group
forall (z : Group) (f : z $-> G) (g : z $-> H),
grp_prod_pr1 $o
(fun K : Group => grp_prod_corec) z f g $== f intros K f g.
G, H, K: Group
f: K $-> G
g: K $-> H
grp_prod_pr1 $o
(fun K : Group => grp_prod_corec) K f g $== f
    exact (Id _).
  -
G, H: Group
forall (z : Group) (f : z $-> G) (g : z $-> H),
grp_prod_pr2 $o
(fun K : Group => grp_prod_corec) z f g $== g intros K f g.
G, H, K: Group
f: K $-> G
g: K $-> H
grp_prod_pr2 $o
(fun K : Group => grp_prod_corec) K f g $== g
    exact (Id _).
  -
G, H: Group
forall (z : Group) (f g : z $-> grp_prod G H),
grp_prod_pr1 $o f $== grp_prod_pr1 $o g ->
grp_prod_pr2 $o f $== grp_prod_pr2 $o g -> f $== g intros K f g p q a.
G, H, K: Group
f, g: K $-> grp_prod G H
p: grp_prod_pr1 $o f $== grp_prod_pr1 $o g
q: grp_prod_pr2 $o f $== grp_prod_pr2 $o g
a: K
f a = g a
    exact (path_prod' (p a) (q a)).
Defined.

(** *** Properties of maps to and from the trivial group *)

Global Instance isinitial_grp_trivial : IsInitial grp_trivial.
IsInitial grp_trivial
Proof.
IsInitial grp_trivial
  intro G.
G: Group
{f : grp_trivial $-> G &
forall g : grp_trivial $-> G, f $== g}
  exists (grp_trivial_rec _).
G: Group
forall g : grp_trivial $-> G, grp_trivial_rec G $== g
  intros g [].
G: Group
g: grp_trivial $-> G
grp_trivial_rec G tt = g tt
  apply (grp_homo_unit g)^.
Defined.

Global Instance contr_grp_homo_trivial_source `{Funext} G
  : Contr (GroupHomomorphism grp_trivial G).
H: Funext
G: Group
Contr (GroupHomomorphism grp_trivial G)
Proof.
H: Funext
G: Group
Contr (GroupHomomorphism grp_trivial G)
  snrapply Build_Contr.
H: Funext
G: Group
GroupHomomorphism grp_trivial G
H: Funext
G: Group
forall y : GroupHomomorphism grp_trivial G,
?center = y
  1: exact (grp_trivial_rec _).
H: Funext
G: Group
forall y : GroupHomomorphism grp_trivial G,
grp_trivial_rec G = y
  intros g.
H: Funext
G: Group
g: GroupHomomorphism grp_trivial G
grp_trivial_rec G = g
  rapply equiv_path_grouphomomorphism.
H: Funext
G: Group
g: GroupHomomorphism grp_trivial G
grp_trivial_rec G == g
  intros [].
H: Funext
G: Group
g: GroupHomomorphism grp_trivial G
grp_trivial_rec G tt = g tt
  symmetry.
H: Funext
G: Group
g: GroupHomomorphism grp_trivial G
g tt = grp_trivial_rec G tt
  rapply grp_homo_unit.
Defined.

Global Instance isterminal_grp_trivial : IsTerminal grp_trivial.
IsTerminal grp_trivial
Proof.
IsTerminal grp_trivial
  intro G.
G: Group
{f : G $-> grp_trivial &
forall g : G $-> grp_trivial, f $== g}
  exists (grp_trivial_corec _).
G: Group
forall g : G $-> grp_trivial,
grp_trivial_corec G $== g
  intros g x.
G: Group
g: G $-> grp_trivial
x: G
grp_trivial_corec G x = g x
  apply path_contr.
Defined.

Global Instance contr_grp_homo_trivial_target `{Funext} G
  : Contr (GroupHomomorphism G grp_trivial).
H: Funext
G: Group
Contr (GroupHomomorphism G grp_trivial)
Proof.
H: Funext
G: Group
Contr (GroupHomomorphism G grp_trivial)
  snrapply Build_Contr.
H: Funext
G: Group
GroupHomomorphism G grp_trivial
H: Funext
G: Group
forall y : GroupHomomorphism G grp_trivial,
?center = y
  1: exact (pr1 (isterminal_grp_trivial _)).
H: Funext
G: Group
forall y : GroupHomomorphism G grp_trivial,
(isterminal_grp_trivial G).1 = y
  intros g.
H: Funext
G: Group
g: GroupHomomorphism G grp_trivial
(isterminal_grp_trivial G).1 = g
  rapply equiv_path_grouphomomorphism.
H: Funext
G: Group
g: GroupHomomorphism G grp_trivial
(isterminal_grp_trivial G).1 == g
  intros x.
H: Funext
G: Group
g: GroupHomomorphism G grp_trivial
x: G
(isterminal_grp_trivial G).1 x = g x
  apply path_contr.
Defined.

Global Instance ishprop_grp_iso_trivial `{Funext} (G : Group)
  : IsHProp (G ≅ grp_trivial).
H: Funext
G: Group
IsHProp (G ≅ grp_trivial)
Proof.
H: Funext
G: Group
IsHProp (G ≅ grp_trivial)
  apply equiv_hprop_allpath.
H: Funext
G: Group
forall x y : G ≅ grp_trivial, x = y
  intros f g.
H: Funext
G: Group
f, g: G ≅ grp_trivial
f = g
  apply equiv_path_groupisomorphism; intro; apply path_ishprop.
Defined.

(** ** Free groups *)

Definition FactorsThroughFreeGroup (S : Type) (F_S : Group)
  (i : S -> F_S) (A : Group) (g : S -> A) : Type
  := {f : F_S $-> A & f o i == g}.

(** Universal property of a free group on a set (type). *)
Class IsFreeGroupOn (S : Type) (F_S : Group) (i : S -> F_S)
  := contr_isfreegroupon : forall (A : Group) (g : S -> A),
      Contr (FactorsThroughFreeGroup S F_S i A g).
Global Existing Instance contr_isfreegroupon.

(** A group is free if there exists a generating type on which it is a free group. *)
Class IsFreeGroup (F_S : Group)
  := isfreegroup : {S : _ & {i : _ & IsFreeGroupOn S F_S i}}.

Global Instance isfreegroup_isfreegroupon (S : Type) (F_S : Group) (i : S -> F_S)
  {H : IsFreeGroupOn S F_S i}
  : IsFreeGroup F_S
  := (S; i; H).


(** ** Further properties of group homomorphisms. *)

(** Characterisation of injective group homomorphisms. *)
Lemma isembedding_grouphomomorphism {A B : Group} (f : A $-> B)
  : (forall a, f a = group_unit -> a = group_unit) <-> IsEmbedding f.
A, B: Group
f: A $-> B
(forall a : A, f a = group_unit -> a = group_unit) <->
IsEmbedding f
Proof.
A, B: Group
f: A $-> B
(forall a : A, f a = group_unit -> a = group_unit) <->
IsEmbedding f
  split.
A, B: Group
f: A $-> B
(forall a : A, f a = group_unit -> a = group_unit) ->
IsEmbedding f
A, B: Group
f: A $-> B
IsEmbedding f ->
forall a : A, f a = group_unit -> a = group_unit
  -
A, B: Group
f: A $-> B
(forall a : A, f a = group_unit -> a = group_unit) ->
IsEmbedding f intros h b.
A, B: Group
f: A $-> B
h: forall a : A, f a = group_unit -> a = group_unit
b: B
IsHProp (hfiber f b)
    apply hprop_allpath.
A, B: Group
f: A $-> B
h: forall a : A, f a = group_unit -> a = group_unit
b: B
forall x y : hfiber f b, x = y
    intros [a0 p0] [a1 p1].
A, B: Group
f: A $-> B
h: forall a : A, f a = group_unit -> a = group_unit
b: B
a0: A
p0: f a0 = b
a1: A
p1: f a1 = b
(a0; p0) = (a1; p1)
    srapply path_sigma_hprop; simpl.
A, B: Group
f: A $-> B
h: forall a : A, f a = group_unit -> a = group_unit
b: B
a0: A
p0: f a0 = b
a1: A
p1: f a1 = b
a0 = a1
    apply grp_moveL_1M.
A, B: Group
f: A $-> B
h: forall a : A, f a = group_unit -> a = group_unit
b: B
a0: A
p0: f a0 = b
a1: A
p1: f a1 = b
a0 * - a1 = mon_unit
    apply h.
A, B: Group
f: A $-> B
h: forall a : A, f a = group_unit -> a = group_unit
b: B
a0: A
p0: f a0 = b
a1: A
p1: f a1 = b
f (a0 * - a1) = group_unit
    rewrite grp_homo_op, grp_homo_inv.
A, B: Group
f: A $-> B
h: forall a : A, f a = group_unit -> a = group_unit
b: B
a0: A
p0: f a0 = b
a1: A
p1: f a1 = b
f a0 * - f a1 = group_unit
    rewrite p0, p1.
A, B: Group
f: A $-> B
h: forall a : A, f a = group_unit -> a = group_unit
b: B
a0: A
p0: f a0 = b
a1: A
p1: f a1 = b
b * - b = group_unit
    apply right_inverse.
  -
A, B: Group
f: A $-> B
IsEmbedding f ->
forall a : A, f a = group_unit -> a = group_unit intros E a p.
A, B: Group
f: A $-> B
E: IsEmbedding f
a: A
p: f a = group_unit
a = group_unit
    rapply (isinj_embedding f).
A, B: Group
f: A $-> B
E: IsEmbedding f
a: A
p: f a = group_unit
f a = f group_unit
    exact (p @ (grp_homo_unit f)^).
Defined.

(** Commutativity can be transferred across isomorphisms. *)
Definition commutative_iso_commutative {G H : Group}
  {C : Commutative (@group_sgop G)} (f : GroupIsomorphism G H)
  : Commutative (@group_sgop H).
G, H: Group
C: Commutative group_sgop
f: GroupIsomorphism G H
Commutative group_sgop
Proof.
G, H: Group
C: Commutative group_sgop
f: GroupIsomorphism G H
Commutative group_sgop
  unfold Commutative.
G, H: Group
C: Commutative group_sgop
f: GroupIsomorphism G H
forall x y : H, group_sgop x y = group_sgop y x
  rapply (equiv_ind f); intro g1.
G, H: Group
C: Commutative group_sgop
f: GroupIsomorphism G H
g1: G
forall y : H,
group_sgop (f g1) y = group_sgop y (f g1)
  rapply (equiv_ind f); intro g2.
G, H: Group
C: Commutative group_sgop
f: GroupIsomorphism G H
g1, g2: G
group_sgop (f g1) (f g2) = group_sgop (f g2) (f g1)
  refine ((preserves_sg_op _ _)^ @ _ @ (preserves_sg_op _ _)).
G, H: Group
C: Commutative group_sgop
f: GroupIsomorphism G H
g1, g2: G
f (g1 * g2) = f (g2 * g1)
  refine (ap f _).
G, H: Group
C: Commutative group_sgop
f: GroupIsomorphism G H
g1, g2: G
g1 * g2 = g2 * g1
  apply C.
Defined.