# Group actions

Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.Algebra.Monoids.
Require Import UniMath.Algebra.Groups.
Require Import UniMath.Combinatorics.OrderedSets.

Import UniMath.MoreFoundations.PartA.

## Definitions

Definition action_op G (X:hSet) : Type := (g:G) (x:X), X.

Section A.
Context (G:gr) (X:hSet).
Definition ActionStructure : Type :=
(act_mult : action_op G X)
(act_unit : x, act_mult (unel G) x = x),
g h x, act_mult (op g h) x = act_mult g (act_mult h x).
Definition make act_mult act_unit act_assoc : ActionStructure := act_mult,, act_unit,, act_assoc.
Definition act_mult (x:ActionStructure) := pr1 x.
Definition act_unit (x:ActionStructure) := pr12 x.
Definition act_assoc (x:ActionStructure) := pr22 x.
End A.
Arguments act_mult {_ _} _ _ _.

Lemma isaset_ActionStructure (G:gr) (X:hSet) : isaset (ActionStructure G X).
Proof.
intros.
apply isaset_total2.
{ apply (impred 2); intro g. apply impred; intro x. apply setproperty. }
intro op.
apply isaset_total2.
{ apply (impred 2); intro x. apply hlevelntosn. apply setproperty. }
intro un. apply (impred 2); intro g. apply (impred 2); intro h. apply (impred 2); intro x.
apply hlevelntosn. apply setproperty.
Qed.

Definition Action (G:gr) := total2 (ActionStructure G).
Definition makeAction {G:gr} (X:hSet) (ac:ActionStructure G X) :=
X,,ac : Action G.

Definition ac_set {G:gr} (X:Action G) := pr1 X.
Coercion ac_set : Action >-> hSet.
Definition ac_type {G:gr} (X:Action G) := pr1hSet (ac_set X).
Definition ac_str {G:gr} (X:Action G) := pr2 X : ActionStructure G (ac_set X).
Definition ac_mult {G:gr} (X:Action G) := act_mult (pr2 X).
Declare Scope action_scope.
Delimit Scope action_scope with action.
Local Notation "g * x" := (ac_mult _ g x) : action_scope.
Local Open Scope action_scope.
Definition ac_assoc {G:gr} (X:Action G) := act_assoc _ _ (pr2 X) : g h x, (op g h)*x = g*(h×x).

Definition right_mult {G:gr} {X:Action G} (x:X) := λ g, g×x.
Definition left_mult {G:gr} {X:Action G} (g:G) := λ x:X, g×x.

Definition is_equivariant {G:gr} {X Y:Action G} (f:XY) : hProp :=
( g x, f (g×x) = g*(f x))%set.

Definition is_equivariant_isaprop {G:gr} {X Y:Action G} (f:XY) :
isaprop (is_equivariant f).
Proof.
apply propproperty.
Qed.

The following fact is fundamental: it shows that our definition of is_equivariant captures all of the structure. The proof reduces to showing that if G acts on a set X in two ways, and the identity function is equivariant, then the two actions are equal. A similar fact will hold in other cases: groups, rings, monoids, etc. Refer to section 9.8 of the HoTT book, on the "structure identity principle", a term coined by Peter Aczel.
Local Open Scope transport.

Definition is_equivariant_identity {G:gr} {X Y:Action G}
(p:ac_set X = ac_set Y) :
p # ac_str X = ac_str Y is_equivariant (cast (maponpaths pr1hSet p)).
Proof.
revert X Y p; intros [X [Xm [Xu Xa]]] [Y [Ym [Yu Ya]]] ? .
simpl in p. destruct p; simpl. unfold transportf; simpl. unfold idfun; simpl.
simple refine (make_weq _ _).
{ intros p g x. simpl in x. simpl.
exact (eqtohomot (eqtohomot (maponpaths act_mult p) g) x). }
use isweq_iso.
{ unfold cast; simpl.
intro i.
assert (p:Xm=Ym).
{ apply funextsec; intro g. apply funextsec; intro x; simpl in x.
exact (i g x). }
destruct p. clear i. assert (p:Xu=Yu).
{ apply funextsec; intro x; simpl in x. apply setproperty. }
destruct p. assert (p:Xa=Ya).
{ apply funextsec; intro g. apply funextsec; intro h.
apply funextsec; intro x. apply setproperty. }
destruct p. apply idpath. }
{ intro p. apply isaset_ActionStructure. }
{ intro is. apply proofirrelevance.
apply impred; intros g.
apply impred; intros x.
apply setproperty. }
Defined.

Definition is_equivariant_comp {G:gr} {X Y Z:Action G}
(p:XY) (i:is_equivariant p)
(q:YZ) (j:is_equivariant q) : is_equivariant (funcomp p q).
Proof.
intros. intros g x. exact (maponpaths q (i g x) @ j g (p x)).
Defined.

Definition ActionMap {G:gr} (X Y:Action G) := total2 (@is_equivariant _ X Y).

Definition underlyingFunction {G:gr} {X Y:Action G} (f:ActionMap X Y) := pr1 f.

Coercion underlyingFunction : ActionMap >-> Funclass.

Definition equivariance {G:gr} {X Y:Action G} (f:ActionMap X Y) : is_equivariant f
:= pr2 f.

Definition composeActionMap {G:gr} (X Y Z:Action G)
(p:ActionMap X Y) (q:ActionMap Y Z) : ActionMap X Z.
Proof.
revert p q; intros [p i] [q j]. (funcomp p q).
apply is_equivariant_comp. assumption. assumption.
Defined.

Definition ActionIso {G:gr} (X Y:Action G) : Type.
Proof.
exact ( f:(ac_set X) (ac_set Y), is_equivariant f).
Defined.

Lemma ActionIso_isaset {G:gr} (X Y:Action G) : isaset (ActionIso X Y).
Proof.
apply (isofhlevelsninclb _ pr1).
{ apply isinclpr1; intro f. apply propproperty. }
apply isofhlevelsnweqtohlevelsn.
apply setproperty.
Defined.

Coercion underlyingIso {G:gr} {X Y:Action G} (e:ActionIso X Y) : X Y := pr1 e.

Lemma underlyingIso_incl {G:gr} {X Y:Action G} :
isincl (underlyingIso : ActionIso X Y X Y).
Proof.
intros. apply isinclpr1; intro f. apply propproperty.
Defined.

Local Goal G (X Y:Action G) (i : ActionIso X Y) (x:X), Y.
intros.
exact (i x).
Qed.

Lemma underlyingIso_injectivity {G:gr} {X Y:Action G}
(e f:ActionIso X Y) :
(e = f) (underlyingIso e = underlyingIso f).
Proof.
intros. apply weqonpathsincl. apply underlyingIso_incl.
Defined.

Definition underlyingActionMap {G:gr} {X Y:Action G} (e:ActionIso X Y) : ActionMap X Y :=
pr1weq (pr1 e),, pr2 e.

Definition idActionIso {G:gr} (X:Action G) : ActionIso X X.
Proof.
intros. (idweq _). intros g x. reflexivity.
Defined.

Definition composeActionIso {G:gr} {X Y Z:Action G}
(e:ActionIso X Y) (f:ActionIso Y Z) : ActionIso X Z.
Proof.
revert e f; intros [e i] [f j]. (weqcomp e f).
destruct e as [e e'], f as [f f']; simpl.
apply is_equivariant_comp. exact i. exact j.
Defined.

Lemma composeActionIsoId {G:gr} {X Y:Action G} (f : ActionIso X Y) : composeActionIso (idActionIso X) f = f.
Proof.
apply subtypePath.
{ intros g. apply propproperty. }
apply subtypePath.
{ intros g. apply isapropisweq. }
reflexivity.
Defined.

Lemma composeActionIsoId' {G:gr} {X Y:Action G} (f : ActionIso X Y) : composeActionIso f (idActionIso Y) = f.
Proof.
apply subtypePath.
{ intros g. apply propproperty. }
apply subtypePath.
{ intros g. apply isapropisweq. }
reflexivity.
Defined.

Definition path_to_ActionIso {G:gr} {X Y:Action G} (e:X = Y) : ActionIso X Y.
Proof.
intros. destruct e. exact (idActionIso X).
Defined.

Definition castAction {G:gr} {X Y:Action G} (e:X = Y) : X Y.
Proof.
intros x. exact (path_to_ActionIso e x).
Defined.

## Applications of univalence

Definition Action_univalence_prelim {G:gr} {X Y:Action G} :
(X = Y) (ActionIso X Y).
Proof.
intros.
simple refine (weqcomp (total2_paths_equiv (ActionStructure G) X Y) _).
simple refine (weqbandf _ _ _ _).
{ apply hSet_univalence. }
simpl. intro p. simple refine (weqcomp (is_equivariant_identity p) _).
exact (eqweqmap (maponpaths (λ f, hProptoType (is_equivariant f)) (pr1_eqweqmap (maponpaths pr1hSet p)))).
Defined.

Definition Action_univalence_prelim_comp {G:gr} {X Y:Action G} (p:X = Y) :
Action_univalence_prelim p = path_to_ActionIso p.
Proof.
intros. destruct p. apply (maponpaths (tpair _ _)). apply funextsec; intro g.
apply funextsec; intro x. apply setproperty.
Defined.

Lemma path_to_ActionIsweq_iso {G:gr} {X Y:Action G} :
isweq (@path_to_ActionIso G X Y).
Proof.
intros. exact (isweqhomot Action_univalence_prelim
path_to_ActionIso
Action_univalence_prelim_comp
(pr2 Action_univalence_prelim)).
Qed.

Definition Action_univalence {G:gr} {X Y:Action G} :
(X = Y) (ActionIso X Y).
Proof.
intros. path_to_ActionIso. apply path_to_ActionIsweq_iso.
Defined.

Definition Action_univalence_comp {G:gr} {X Y:Action G} (p:X = Y) :
Action_univalence p = path_to_ActionIso p.
Proof.
reflexivity.
Defined.

Definition Action_univalence_inv {G:gr} {X Y:Action G}
: (ActionIso X Y) (X=Y) := invweq Action_univalence.

Definition Action_univalence_inv_comp {G:gr} {X Y:Action G} (f:ActionIso X Y) :
path_to_ActionIso (Action_univalence_inv f) = f.
Proof.
intros.
unfold Action_univalence_inv, Action_univalence.
apply (homotweqinvweq Action_univalence f).
Defined.

Definition Action_univalence_inv_comp_eval {G:gr} {X Y:Action G} (f:ActionIso X Y) (x:X) :
castAction (Action_univalence_inv f) x = f x.
Proof.
intros. exact (eqtohomot
(maponpaths pr1weq
(maponpaths underlyingIso
(Action_univalence_inv_comp f)))
x).
Defined.

## Torsors

Definition is_torsor {G:gr} (X:Action G) :=
nonempty X × x:X, isweq (right_mult x).

Lemma is_torsor_isaprop {G:gr} (X:Action G) : isaprop (is_torsor X).
Proof.
intros. apply isapropdirprod.
{ apply propproperty. }
{ apply impred; intro x. apply isapropisweq. }
Qed.

Definition Torsor (G:gr) := total2 (@is_torsor G).

Coercion underlyingAction {G} (X:Torsor G) := pr1 X : Action G.

Definition is_torsor_prop {G} (X:Torsor G) := pr2 X.

Definition torsor_nonempty {G} (X:Torsor G) := pr1 (is_torsor_prop X).

Definition torsor_splitting {G} (X:Torsor G) := pr2 (is_torsor_prop X).

Definition torsor_mult_weq {G} (X:Torsor G) (x:X) :=
make_weq (right_mult x) (torsor_splitting X x) : G X.

Definition torsor_mult_weq' {G} (X:Torsor G) (g:G) : X X.
Proof.
(left_mult g).
use isweq_iso.
- exact (left_mult (grinv G g)).
- intros x. unfold left_mult.
intermediate_path ((grinv G g × g)%multmonoid × x).
+ apply pathsinv0,act_assoc.
+ intermediate_path (unel G × x).
× apply (maponpaths (right_mult x)). apply grlinvax.
× apply act_unit.
- intros x. unfold left_mult.
intermediate_path ((g × grinv G g)%multmonoid × x).
+ apply pathsinv0,act_assoc.
+ intermediate_path (unel G × x).
× apply (maponpaths (right_mult x)). apply grrinvax.
× apply act_unit.
Defined.

Definition left_mult_Iso {G:abgr} (X:Torsor G) (g:G) : ActionIso X X.
Proof.
(torsor_mult_weq' X g). intros h x.
change (g × (h × x) = h × (g × x)).
refine (! ac_assoc X g h x @ _ @ ac_assoc X h g x).
exact (maponpaths (right_mult x) (commax G g h)).
Defined.

Definition torsor_update_nonempty {G} (X:Torsor G) (x:nonempty X) : Torsor G.
Proof.
exact (underlyingAction X,,(x,,pr2(is_torsor_prop X))).
Defined.

Definition castTorsor {G} {T T':Torsor G} (q:T = T') : T T'.
Proof.
exact (castAction (maponpaths underlyingAction q)).
Defined.

Lemma castTorsor_transportf {G} {T T':Torsor G} (q:T = T') (t:T) :
transportf (λ S, underlyingAction S) q t = castTorsor q t.
Proof.
now induction q.
Defined.

Lemma underlyingAction_incl {G:gr} :
isincl (underlyingAction : Torsor G Action G).
Proof.
intros. refine (isinclpr1 _ _); intro X. apply is_torsor_isaprop.
Defined.

Lemma underlyingAction_injectivity {G:gr} {X Y:Torsor G} :
(X = Y) (underlyingAction X = underlyingAction Y).
Proof.
intros. apply weqonpathsincl. apply underlyingAction_incl.
Defined.

Definition underlyingAction_injectivity_comp {G:gr} {X Y:Torsor G} (p:X = Y) :
underlyingAction_injectivity p = maponpaths underlyingAction p.
Proof.
reflexivity.
Defined.

Definition underlyingAction_injectivity_comp' {G:gr} {X Y:Torsor G} :
pr1weq (@underlyingAction_injectivity G X Y)
= @maponpaths (Torsor G) (Action G) (@underlyingAction G) X Y.
Proof.
reflexivity.
Defined.

Definition underlyingAction_injectivity_inv_comp {G:gr} {X Y:Torsor G}
(f:underlyingAction X = underlyingAction Y) :
maponpaths underlyingAction (invmap underlyingAction_injectivity f) = f.
Proof.
intros. apply (homotweqinvweq underlyingAction_injectivity f).
Defined.

Definition PointedTorsor (G:gr) := X:Torsor G, X.
Definition underlyingTorsor {G} (X:PointedTorsor G) := pr1 X : Torsor G.
Coercion underlyingTorsor : PointedTorsor >-> Torsor.
Definition underlyingPoint {G} (X:PointedTorsor G) := pr2 X : X.

Lemma is_quotient {G} (X:Torsor G) (y x:X) : ∃! g, g×x = y.
Proof.
intros. exact (pr2 (is_torsor_prop X) x y).
Defined.

Definition quotient {G} (X:Torsor G) (y x:X) := pr1 (iscontrpr1 (is_quotient X y x)) : G.
Local Notation "y / x" := (quotient _ y x) : action_scope.

Lemma quotient_times {G} {X:Torsor G} (y x:X) : (y/x)*x = y.
Proof.
intros. exact (pr2 (iscontrpr1 (is_quotient _ y x))).
Defined.

Lemma quotient_uniqueness {G} {X:Torsor G} (y x:X) (g:G) : g×x = y g = y/x.
Proof.
intros e.
exact (maponpaths pr1 (uniqueness (is_quotient _ y x) (g,,e))).
Defined.

Lemma quotient_mult {G} (X:Torsor G) (g:G) (x:X) : (g×x)/x = g.
Proof.
intros. apply pathsinv0. apply quotient_uniqueness. reflexivity.
Defined.

Lemma quotient_1 {G} (X:Torsor G) (x:X) : x/x = 1%multmonoid.
Proof.
intros. apply pathsinv0. apply quotient_uniqueness. apply act_unit.
Defined.

Lemma quotient_product {G} (X:Torsor G) (z y x:X) : op (z/y) (y/x) = z/x.
Proof.
intros. apply quotient_uniqueness.
exact (ac_assoc _ _ _ _
@ maponpaths (left_mult (z/y)) (quotient_times y x)
@ quotient_times z y).
Defined.

Lemma quotient_map {G} {X Y:Torsor G} (f : ActionMap X Y) (x x':X) : f x' / f x = x' / x.
Proof.
refine (! (quotient_uniqueness (f x') (f x) (x' / x) _)).
assert (p := equivariance f (x'/x) x).
refine (!p @ _); clear p.
apply maponpaths.
apply quotient_times.
Qed.

Lemma torsorMapIsIso {G} {X Y : Torsor G} (f : ActionMap X Y) : isweq f.
Proof.
apply (squash_to_prop (torsor_nonempty X)).
- apply isapropisweq.
- intros x.
set (y := f x).
set (f' := λ y', y' / y × x).
apply (isweq_iso f f').
+ intros x'.
unfold f', y.
assert (p := quotient_times x' x).
refine (_ @ p); clear p.
apply (maponpaths (λ g, g × x)).
apply quotient_map.
+ intros y'.
unfold f'.
assert (p := equivariance f (y'/y) x).
refine (p @ _); clear p.
fold y.
apply quotient_times.
Defined.

Definition torsorMap_to_torsorIso {G} {X Y : Torsor G} (f : ActionMap X Y) : ActionIso X Y.
Proof.
use tpair.
- f. apply torsorMapIsIso.
- simpl. apply equivariance.
Defined.

Definition trivialTorsor (G:gr) : Torsor G.
Proof.
intros. (makeAction G (make G G op (lunax G) (assocax G))).
exact (hinhpr (unel G),,
λ x, isweq_iso
(λ g, op g x)
(λ g, op g (grinv _ x))
(λ g, assocax _ g x (grinv _ x) @ maponpaths (op g) (grrinvax G x) @ runax _ g)
(λ g, assocax _ g (grinv _ x) x @ maponpaths (op g) (grlinvax G x) @ runax _ g)).
Defined.

Definition toTrivialTorsor {G:gr} (g:G) : trivialTorsor G.
Proof.
exact g.
Defined.

Definition pointedTrivialTorsor (G:gr) : PointedTorsor G.
Proof.
intros. (trivialTorsor G). exact (unel G).
Defined.

Definition univ_function {G:gr} (X:Torsor G) (x:X) : trivialTorsor G X.
Proof.
apply right_mult. assumption.
Defined.

Definition univ_function_pointed {G:gr} (X:Torsor G) (x:X) :
univ_function X x (unel _) = x.
Proof.
intros. apply act_unit.
Defined.

Definition univ_function_is_equivariant {G:gr} (X:Torsor G) (x:X) :
is_equivariant (univ_function X x).
Proof.
intros. intros g h. apply act_assoc.
Defined.

Definition triviality_isomorphism {G:gr} (X:Torsor G) (x:X) :
ActionIso (trivialTorsor G) X.
Proof.
intros.
exact (torsor_mult_weq X x,, univ_function_is_equivariant X x).
Defined.

Lemma triviality_isomorphism_compute (G:gr) :
triviality_isomorphism (trivialTorsor G) (unel G) = idActionIso (trivialTorsor G).
Proof.
apply subtypePath_prop.
apply subtypePath.
{ intros X. apply isapropisweq. }
apply funextsec; intros g.
change (op g (unel _) = g).
apply runax.
Defined.

Definition trivialTorsor_weq (G:gr) (g:G) : (trivialTorsor G) (trivialTorsor G).
Proof.
intros. (λ h, op h g). apply (isweq_iso _ (λ h, op h (grinv G g))).
{ exact (λ h, assocax _ _ _ _ @ maponpaths (op _) (grrinvax _ _) @ runax _ _). }
{ exact (λ h, assocax _ _ _ _ @ maponpaths (op _) (grlinvax _ _) @ runax _ _). }
Defined.

Definition trivialTorsorAuto (G:gr) (g:G) :
ActionIso (trivialTorsor G) (trivialTorsor G).
Proof.
intros. (trivialTorsor_weq G g).
intros h x. simpl. exact (assocax _ h x g).
Defined.

Lemma pr1weq_injectivity {X Y} (f g:X Y) : (f = g) (pr1weq f = pr1weq g).
Proof.
intros. apply weqonpathsincl. apply isinclpr1weq.
Defined.

Definition trivialTorsorRightMultiplication (G:gr) : G ActionIso (trivialTorsor G) (trivialTorsor G).
Proof.
(trivialTorsorAuto G). simple refine (isweq_iso _ _ _ _).
{ intro f. exact (f (unel G)). }
{ intro g; simpl. exact (lunax _ g). }
{ intro f; simpl. apply (invweq (underlyingIso_injectivity _ _)); simpl.
apply (invweq (pr1weq_injectivity _ _)). apply funextsec; intro g.
simpl. exact ((! (pr2 f) g (unel G)) @ (maponpaths (pr1 f) (runax G g))). }
Defined.

Definition autos_comp (G:gr) (g:G) :
underlyingIso (trivialTorsorRightMultiplication G g) = trivialTorsor_weq G g.
Proof.
reflexivity. Defined.

Definition autos_comp_apply (G:gr) (g h:G) :
(trivialTorsorRightMultiplication _ g) h = (h × g)%multmonoid.
Proof.
reflexivity. Defined.

Lemma trivialTorsorAuto_unit (G:gr) :
trivialTorsorAuto G (unel _) = idActionIso _.
Proof.
intros. simple refine (subtypePath _ _).
{ intro k. apply is_equivariant_isaprop. }
{ simple refine (subtypePath _ _).
{ intro; apply isapropisweq. }
{ apply funextsec; intro x; simpl. exact (runax G x). } }
Defined.

Lemma trivialTorsorAuto_mult (G:gr) (g h:G) :
composeActionIso (trivialTorsorAuto G g) (trivialTorsorAuto G h)
= (trivialTorsorAuto G (op g h)).
Proof.
intros. simple refine (subtypePath _ _).
{ intro; apply is_equivariant_isaprop. }
{ simple refine (subtypePath _ _).
{ intro; apply isapropisweq. }
{ apply funextsec; intro x; simpl. exact (assocax _ x g h). } }
Defined.

## Applications of univalence

Definition torsor_univalence {G:gr} {X Y:Torsor G} : (X = Y) (ActionIso X Y).
Proof.
intros. simple refine (weqcomp underlyingAction_injectivity _).
apply Action_univalence.
Defined.

Definition torsor_univalence_transport {G:gr} {X Y:Torsor G} (p:X=Y) (x:X) :
torsor_univalence p x = transportf (λ X:Torsor G, X:Type) p x.
Proof.
now induction p.
Defined.

Corollary torsor_hlevel {G:gr} : isofhlevel 3 (Torsor G).
Proof.
intros X Y.
apply (isofhlevelweqb 2 torsor_univalence).
apply ActionIso_isaset.
Defined.

Definition torsor_univalence_comp {G:gr} {X Y:Torsor G} (p:X = Y) :
torsor_univalence p = path_to_ActionIso (maponpaths underlyingAction p).
Proof.
reflexivity.
Defined.

Definition torsor_univalence_inv_comp_eval {G:gr} {X Y:Torsor G}
(f:ActionIso X Y) (x:X) :
castTorsor (invmap torsor_univalence f) x = f x.
Proof.
intros. unfold torsor_univalence.
unfold castTorsor. rewrite invmapweqcomp.   unfold weqcomp; simpl.
rewrite underlyingAction_injectivity_inv_comp.
apply Action_univalence_inv_comp_eval.
Defined.

Definition torsor_eqweq_to_path {G:gr} {X Y:Torsor G} : ActionIso X Y X = Y.
Proof.
intros f. exact (invweq torsor_univalence f).
Defined.

Definition torsorMap_to_path {G:gr} {X Y:Torsor G} : ActionMap X Y X = Y.
Proof.
intros f.
apply (invweq torsor_univalence).
apply torsorMap_to_torsorIso.
exact f.
Defined.

Theorem TorsorIso_rect {G:gr} {X Y : Torsor G} (P : ActionIso X Y UU) :
( e : X = Y, P (torsor_univalence e)) f, P f.
Proof.
intros ih ?.
set (p := ih (invmap torsor_univalence f)).
set (h := homotweqinvweq torsor_univalence f).
exact (transportf P h p).
Defined.

Ltac torsor_induction f e :=
generalize f; apply TorsorIso_rect; intro e; clear f.

Theorem TorsorIso_rect' {G:gr} {X : Torsor G} (P : Y : Torsor G, ActionIso X Y Type) :
P X (idActionIso X) (Y : Torsor G) (f:ActionIso X Y), P Y f.
Proof.
intros p ? ?. torsor_induction f q. induction q. exact p.
Defined.

Ltac torsor_induction' f X :=
generalize f; generalize X; apply TorsorIso_rect'; clear f X.

Lemma torsor_univalence_id {G:gr} (X:Torsor G) : invmap torsor_univalence (idActionIso X) = idpath X.
Proof.
change (idActionIso X) with (torsor_univalence (idpath X)).
apply homotinvweqweq.
Defined.

Definition invUnivalenceCompose {G:gr} {X Y Z : Torsor G} (f : ActionIso X Y) (g : ActionIso Y Z) :
invmap torsor_univalence f @ invmap torsor_univalence g = invmap torsor_univalence (composeActionIso f g).
Proof.
torsor_induction' g Z. rewrite composeActionIsoId'. rewrite torsor_univalence_id. apply pathscomp0rid.
Defined.

Definition PointedActionIso {G:gr} (X Y:PointedTorsor G)
:= f:ActionIso X Y, f (underlyingPoint X) = underlyingPoint Y.

Definition pointed_triviality_isomorphism {G:gr} (X:PointedTorsor G) :
PointedActionIso (pointedTrivialTorsor G) X.
Proof.
revert X; intros [X x]. (triviality_isomorphism X x).
simpl. apply univ_function_pointed.
Defined.

Definition Pointedtorsor_univalence {G:gr} {X Y:PointedTorsor G} :
(X = Y) (PointedActionIso X Y).
Proof.
intros.
simple refine (weqcomp (total2_paths_equiv _ X Y) _).
simple refine (weqbandf _ _ _ _).
{ intros.
exact (weqcomp (weqonpathsincl underlyingAction underlyingAction_incl X Y)
Action_univalence). }
destruct X as [X x], Y as [Y y]; simpl. intro p. destruct p; simpl.
exact (idweq _).
Defined.

Definition ClassifyingSpace G := pointedType (Torsor G) (trivialTorsor G).
Definition E := PointedTorsor.
Definition B := ClassifyingSpace.
Definition π {G:gr} := underlyingTorsor : E G B G.

Lemma isBaseConnected_BG (G:gr) : isBaseConnected (B G).
Proof.
intros X. use (hinhfun _ (torsor_nonempty X)); intros x.
exact (torsor_eqweq_to_path (triviality_isomorphism X x)).
Defined.

Goal (G:gr), triviality_isomorphism (trivialTorsor G) (unel G) = idActionIso (trivialTorsor G).
Fail reflexivity.
Abort.

Goal (G:gr), isBaseConnected_BG G (trivialTorsor G) = hinhpr (idpath (trivialTorsor G)).
intros.
unfold isBaseConnected_BG, pr2.
change (pr1 (trivialTorsor G) : Type) with (G : Type).
change (torsor_nonempty (trivialTorsor G)) with (hinhpr (unel G)).
change (hinhpr (torsor_eqweq_to_path (triviality_isomorphism (trivialTorsor G) (unel G)))
= hinhpr (idpath (trivialTorsor G))).
apply maponpaths.
Fail reflexivity.
Abort.

Lemma isConnected_BG (G:gr) : isConnected (B G).
Proof.
apply baseConnectedness. apply isBaseConnected_BG.
Defined.

Lemma iscontr_EG (G:gr) : iscontr (E G).
Proof.
intros. (pointedTrivialTorsor G). intros [X x].
apply pathsinv0. apply (invweq Pointedtorsor_univalence).
apply pointed_triviality_isomorphism.
Defined.

Theorem loopsBG (G:gr) : G Ω (B G).
Proof.
intros.
simple refine (weqcomp _ (invweq torsor_univalence)).
apply trivialTorsorRightMultiplication.
Defined.

Definition loopsBG_comp (G:gr) (g:G) :
loopsBG G g = invmap torsor_univalence (trivialTorsorAuto G g).
Proof.
reflexivity.
Defined.

Definition loopsBG_comp' {G:gr} (p : Ω (B G)) :
invmap (loopsBG G) p = path_to_ActionIso (maponpaths underlyingAction p) (unel G).
Proof.
reflexivity.
Defined.

Definition loopsBG_comp_2 (G:gr) (g h:G)
: castTorsor (loopsBG G g) h = (h×g)%multmonoid.
Proof.
exact (torsor_univalence_inv_comp_eval (trivialTorsorAuto G g) h).
Defined.

Theorem loopsBG also follows from the Rezk Completion theorem of the CategoryTheory package. To see that, regard G as a category with one object. Consider a merely representable functor F : G^op -> Set. Let X be F of the object *. Apply F to the arrows to get an action of G on X. Try to prove that X is a torsor. Since being a torsor is a mere property, we may assume F is actually representable. There is only one object *, so F is isomorphic to h*. Apply h* to * and we get Hom, which is G, regarded as a G-set. That's a torsor. So the Rezk completion RCG is equivalent to BG, the type of G-torsors. Now the theorem also says there is an equivalence G -> RCG. So RCG is connected and its loop space is G.
A formalization of that argument should be added eventually.