Datasets:

phanerozoic

/

Coq-HoTT

like 2

`{funext : Funext} (G H K : Group) : (GroupHomomorphism G K) * (GroupHomomorphism H K) <~> GroupHomomorphism (FreeProduct G H) K. Proof. refine (equiv_amalgamatedfreeproduct_rec _ _ _ _ _ K oE _^-1). refine (equiv_sigma_prod0 _ _ oE equiv_functor_sigma_id (fun _ => equiv_sigma_contr _)). intros f. rapply contr_forall. intros []; apply contr_inhab_prop. apply tr. refine (grp_homo_unit _ @ (grp_homo_unit _)^). Defined.

Definition

Require Import Basics Types. Require Import Cubical.DPath. Require Import Spaces.List.Core Spaces.List.Theory. Require Import Colimits.Pushout. Require Import Truncations.Core Truncations.SeparatedTrunc. Require Import Algebra.Groups.Group. Require Import WildCat.

Algebra\Groups\FreeProduct.v

equiv_freeproduct_rec

{ group_type :> Type; group_sgop :: SgOp group_type; group_unit :: MonUnit group_type; group_inverse :: Negate group_type; group_isgroup :: IsGroup group_type; }.

Record

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

Group

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

issig_group

associativity x y z.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_assoc

left_identity x.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_unit_l

right_identity x.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_unit_r

left_inverse x.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_inv_l

right_inverse x.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_inv_r

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

ptype_group

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

identity_unique

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

identity_unique'

`{IsMonoid A} (a x y : A) {p : x * a = mon_unit} {q : a * y = mon_unit} : x = y. Proof. refine ((right_identity x)^ @ ap _ q^ @ _). refine (associativity _ _ _ @ _). refine (ap (fun x => x * y) p @ _). apply left_identity. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

inverse_unique

(G H : Group) := Build_GroupHomomorphism { grp_homo_map :> group_type G -> group_type H; issemigrouppreserving_grp_homo :: IsSemiGroupPreserving grp_homo_map; }.

Record

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

GroupHomomorphism

forall {G H : Group} (f : GroupHomomorphism G H) (x y : G), f (x * y) = f x * f y := @issemigrouppreserving_grp_homo. #[export] Hint Immediate : group_db. Global Instance isunitpreserving_grp_homo {G H : Group} (f : GroupHomomorphism G H) : IsUnitPreserving f. Proof. unfold IsUnitPreserving. apply (group_cancelL (f mon_unit)). rhs nrapply grp_unit_r. rhs_V rapply (ap _ (monoid_left_id _ mon_unit)). symmetry. nrapply issemigrouppreserving_grp_homo. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_homo_op

forall {G H : Group} (f : GroupHomomorphism G H), f mon_unit = mon_unit := @isunitpreserving_grp_homo.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_homo_unit

{G H : Group} (f : GroupHomomorphism G H) : G ->* H := Build_pMap G H f (isunitpreserving_grp_homo f).

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

pmap_GroupHomomorphism

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

issig_GroupHomomorphism

{F : Funext} {G H : Group} {g h : GroupHomomorphism G H} : g == h <~> g = h. Proof. refine ((equiv_ap (issig_GroupHomomorphism G H)^-1 _ _)^-1 oE _). refine (equiv_path_sigma_hprop _ _ oE _). apply equiv_path_forall. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

equiv_path_grouphomomorphism

{G H} (f : GroupHomomorphism G H) : forall x, f (- x) = -(f x). Proof. intro x. apply (inverse_unique (f x)). + refine (_ @ grp_homo_unit f). refine ((grp_homo_op f (-x) x)^ @ _). apply ap. apply grp_inv_l. + apply grp_inv_r. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_homo_inv

{G : Group} : GroupHomomorphism G G := Build_GroupHomomorphism idmap _.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_homo_id

{G H K : Group} : GroupHomomorphism H K -> GroupHomomorphism G H -> GroupHomomorphism G K. Proof. intros f g. srapply (Build_GroupHomomorphism (f o g)). Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_homo_compose

(G H : Group) := Build_GroupIsomorphism { grp_iso_homo :> GroupHomomorphism G H; isequiv_group_iso :: IsEquiv grp_iso_homo; }.

Record

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

GroupIsomorphism

Build_GroupIsomorphism' {G H : Group} (f : G <~> H) (h : IsSemiGroupPreserving f) : GroupIsomorphism G H. Proof. srapply Build_GroupIsomorphism. 1: srapply Build_GroupHomomorphism. exact _. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

Build_GroupIsomorphism'

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

issig_GroupIsomorphism

{G H : Group} : GroupIsomorphism G H -> G <~> H := fun f => Build_Equiv G H f _.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

equiv_groupisomorphism

{A B : Group} : GroupIsomorphism A B -> (A <~>* B) := fun f => Build_pEquiv _ _ f _.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

pequiv_groupisomorphism

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

equiv_path_groupisomorphism

`{F : Funext} {G H : Group} : IsHSet (GroupIsomorphism G H). Proof. apply istrunc_S. intros f g; apply (istrunc_equiv_istrunc _ (equiv_path_groupisomorphism _ _)). Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

ishset_groupisomorphism

{G : Group} : GroupIsomorphism G G := Build_GroupIsomorphism _ _ grp_homo_id _.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_iso_id

{G H K : Group} (g : GroupIsomorphism H K) (f : GroupIsomorphism G H) : GroupIsomorphism G K := Build_GroupIsomorphism _ _ (grp_homo_compose g f) _.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_iso_compose

{G H : Group} : GroupIsomorphism G H -> GroupIsomorphism H G. Proof. intros [f e]. srapply Build_GroupIsomorphism. - srapply (Build_GroupHomomorphism f^-1). - exact _. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_iso_inverse

equiv_path_group' {U : Univalence} {G H : Group} : GroupIsomorphism G H <~> G = H. Proof. equiv_via {f : G <~> H & IsSemiGroupPreserving f}. 1: make_equiv. revert G H; apply (equiv_path_issig_contr issig_group). - intros [G [? [? [? ?]]]]. exists 1%equiv. exact _. - intros [G [op [unit [neg ax]]]]; cbn. contr_sigsig G (equiv_idmap G). srefine (Build_Contr _ ((_;(_;(_;_)));_) _); cbn. 1: assumption. 1: exact _. intros [[op' [unit' [neg' ax']]] eq]. apply path_sigma_hprop; cbn. 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') _). apply path_sigma_hprop; cbn. srefine (path_sigma' _ _ _). 1: funext x y; apply eq. rewrite transport_const. pose (f := Build_GroupHomomorphism (G:=Build_Group G op unit neg ax) (H:=Build_Group G op' unit' neg' ax') idmap eq). srefine (path_sigma' _ _ _). 1: exact (grp_homo_unit f). lhs nrapply transport_const. funext x. exact (grp_homo_inv f x). Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

equiv_path_group'

equiv_path_group@{u v | u < v} {U : Univalence} {G H : Group@{u}} : GroupIsomorphism G H <~> (paths@{v} G H) := equiv_path_group'. Global Instance isequiv_group_left_op {G : Group} : forall (x : G), IsEquiv (x *.). Proof. intro x. srapply isequiv_adjointify. 1: exact (-x *.). all: intro y. all: refine (grp_assoc _ _ _ @ _ @ grp_unit_l y). all: refine (ap (fun x => x * y) _). 1: apply grp_inv_r. apply grp_inv_l. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

equiv_path_group@

--x = x := negate_involutive x.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_inv_inv

- (x * y) = -y * -x := negate_sg_op x y.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_inv_op

-mon_unit = mon_unit := negate_mon_unit (G :=G).

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_inv_unit

{G : Group} {x y : G} z : x = y <~> z * x = z * y := equiv_ap (fun x => z * x) _ _.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_cancelL

{G : Group} {x y : G} z : x = y <~> x * z = y * z := equiv_ap (fun x => x * z) _ _.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_cancelR

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_moveL_gM

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_moveL_Mg

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_moveR_gM

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_moveR_Mg

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_moveR_gV

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_moveR_Vg

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_moveL_gV

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_moveL_Vg

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_moveL_1M

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_moveL_1V

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_moveL_M1

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_moveR_1M

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_moveR_M1

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_cancelL1

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_cancelR1

{G : Group} (g h : G) (p : g * h = h * g) : g * (-h) = (-h) * g. Proof. apply grp_moveR_gV. rhs_V apply simple_associativity. by apply grp_moveL_Vg. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_commutes_inv

{G : Group} (g h h' : G) (p : g * h = h * g) (p' : g * h' = h' * g) : g * (h * h') = (h * h') * g. Proof. lhs apply simple_associativity. lhs nrapply (ap (.* h') p). lhs_V apply simple_associativity. lhs nrapply (ap (h *.) p'). by apply simple_associativity. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_commutes_op

{G : Group} (g : G) (n : Int) : G := int_iter (g *.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_pow

{G H : Group} (f : GroupHomomorphism G H) (n : Int) (g : G) : f (grp_pow g n) = grp_pow (f g) n. Proof. lhs snrapply (int_iter_commute_map _ ((f g) *.)). 1: nrapply grp_homo_op. apply (ap (int_iter _ n)), grp_homo_unit. Defined.

Lemma

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_pow_natural

{G : Group} (n : Int) : grp_pow (G:=G) mon_unit n = mon_unit. Proof. snrapply (int_iter_invariant n _ (fun g => g = mon_unit)); cbn. 1, 2: apply paths_ind_r. - apply grp_unit_r. - lhs nrapply grp_unit_r. apply grp_inv_unit. - reflexivity. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_pow_unit

{G : Group} (n : Int) (g : G) : grp_pow g (n.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_pow_succ

{G : Group} (n : Int) (g : G) : grp_pow g (n.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_pow_pred

{G : Group} (m n : Int) (g : G) : grp_pow g (n + m)%int = grp_pow g n * grp_pow g m. Proof. lhs nrapply int_iter_add. induction n; cbn. 1: exact (grp_unit_l _)^. 1: rewrite int_iter_succ_l, grp_pow_succ. 2: rewrite int_iter_pred_l, grp_pow_pred; cbn. 1,2 : rhs_V srapply associativity; apply ap, IHn. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_pow_add

{G : Group} (n : Int) (g : G) : grp_pow g (int_neg n) = grp_pow (- g) n. Proof. lhs nrapply int_iter_neg. cbn; unfold grp_pow. apply int_iter_agree. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_pow_neg

{G: Group} (m : Int) (g : G) : grp_pow g (- m)%int = - grp_pow g m. Proof. apply grp_moveL_1V. lhs_V nrapply grp_pow_add. by rewrite int_add_neg_l. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_pow_neg_inv

grp_pow_neg_inv' {G: Group} (n: Int) (g : G) : grp_pow (- g) n = - grp_pow g n. Proof. lhs_V nrapply grp_pow_neg. apply grp_pow_neg_inv. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_pow_neg_inv'

{G : Group} (m n : Int) (g : G) : grp_pow g (m * n)%int = grp_pow (grp_pow g m) n. Proof. induction n. - simpl. by rewrite int_mul_0_r. - rewrite int_mul_succ_r. rewrite grp_pow_add. rewrite grp_pow_succ. apply grp_cancelL, IHn. - rewrite int_mul_pred_r. rewrite grp_pow_add. rewrite grp_pow_neg_inv. rewrite grp_pow_pred. apply grp_cancelL, IHn. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_pow_int_mul

{G : Group} (n : Int) (g h : G) (p : h * g = g * h) : h * (grp_pow g n) = (grp_pow g n) * h. Proof. induction n. - exact (grp_unit_r _ @ (grp_unit_l _)^). - rewrite grp_pow_succ. nrapply grp_commutes_op; assumption. - rewrite grp_pow_pred. nrapply grp_commutes_op. 2: assumption. apply grp_commutes_inv, p. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_pow_commutes

grp_pow_commutes' {G : Group} (n : Int) (g : G) : g * grp_pow g n = grp_pow g n * g. Proof. by apply grp_pow_commutes. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_pow_commutes'

{G : Group} (n : Int) (g h : G) (c : g * h = h * g) : grp_pow (g * h) n = (grp_pow g n) * (grp_pow h n). Proof. induction n. - simpl. symmetry; nrapply grp_unit_r. - rewrite 3 grp_pow_succ. rewrite IHn. rewrite 2 grp_assoc. apply grp_cancelR. rewrite <- 2 grp_assoc. apply grp_cancelL. apply grp_pow_commutes. exact c^. - simpl. rewrite 3 grp_pow_pred. rewrite IHn. rewrite 2 grp_assoc. apply grp_cancelR. rewrite c. rewrite grp_inv_op. rewrite <- 2 grp_assoc. apply grp_cancelL. apply grp_pow_commutes. symmetry; apply grp_commutes_inv, c. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_pow_mul

{A B : Group} (f : GroupHomomorphism A B) (b : B) : forall (a0 : hfiber f b), hfiber f b <~> hfiber f mon_unit. Proof. intros [a0 p]. refine (equiv_transport (hfiber f) (right_inverse b) oE _). snrapply Build_Equiv. { srapply (functor_hfiber (h := fun t => t * -a0) (k := fun t => t * -b)). intro a; cbn; symmetry. refine (_ @ ap (fun x => f a * (- x)) p). exact (grp_homo_op f _ _ @ ap (fun x => f a * x) (grp_homo_inv f a0)). } srapply isequiv_functor_hfiber. Defined.

Lemma

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

equiv_grp_hfiber

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_trivial

(G : Group) : GroupHomomorphism grp_trivial G. Proof. snrapply Build_GroupHomomorphism. 1: exact (fun _ => group_unit). intros ??; symmetry; apply grp_unit_l. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_trivial_rec

(G : Group) : GroupHomomorphism G grp_trivial. Proof. snrapply Build_GroupHomomorphism. 1: exact (fun _ => tt). intros ??; symmetry; exact (grp_unit_l _). Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_trivial_corec

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_homo_const

Group -> Group -> Group. Proof. intros G H. srapply (Build_Group (G * H)). { intros [g1 h1] [g2 h2]. exact (g1 * g2, h1 * h2). } 1: exact (mon_unit, mon_unit). { intros [g h]. exact (-g, -h). } repeat split. 1: exact _. all: grp_auto. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_prod

{G H K : Group} (f : K $-> G) (g : K $-> H) : K $-> (grp_prod G H). Proof. snrapply Build_GroupHomomorphism. - exact (fun x : K => (f x, g x)). - intros x y. apply path_prod'; apply grp_homo_op. Defined.

Proposition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_prod_corec

{X Y A B : Group} (f : X $-> Y) (g0 : Y $-> A) (g1 : Y $-> B) : grp_prod_corec g0 g1 $o f $== grp_prod_corec (g0 $o f) (g1 $o f) := fun _ => idpath.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_prod_corec_natural

{H K : Group} : H $-> (grp_prod H K) := grp_prod_corec grp_homo_id grp_homo_const. Global Instance isembedding_grp_prod_inl {H K : Group} : IsEmbedding (@ H K). Proof. apply isembedding_isinj_hset. intros h0 h1 p; cbn in p. exact (fst ((equiv_path_prod _ _)^-1 p)). Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_prod_inl

{H K : Group} : K $-> (grp_prod H K) := grp_prod_corec grp_homo_const grp_homo_id. Global Instance isembedding_grp_prod_inr {H K : Group} : IsEmbedding (@ H K). Proof. apply isembedding_isinj_hset. intros k0 k1 q; cbn in q. exact (snd ((equiv_path_prod _ _)^-1 q)). Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_prod_inr

{A B C D : Group} : A ≅ B -> C ≅ D -> (grp_prod A C) ≅ (grp_prod B D). Proof. intros f g. srapply Build_GroupIsomorphism'. 1: srapply (equiv_functor_prod (f:=f) (g:=g)). simpl. unfold functor_prod. intros x y. apply path_prod. 1,2: apply grp_homo_op. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_iso_prod

{G H : Group} : GroupHomomorphism (grp_prod G H) G. Proof. snrapply Build_GroupHomomorphism. 1: exact fst. intros ? ?; reflexivity. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_prod_pr1

{G H : Group} : GroupHomomorphism (grp_prod G H) H. Proof. snrapply Build_GroupHomomorphism. 1: exact snd. intros ? ?; reflexivity. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_prod_pr2

{G H : Group} (g : G) (h : H) : (g, h) = ((g, group_unit) : grp_prod G H) * (group_unit, h). Proof. snrapply path_prod; symmetry. - snrapply grp_unit_r. - snrapply grp_unit_l. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_prod_decompose

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

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

FactorsThroughFreeGroup

{A B : Group} (f : A $-> B) : (forall a, f a = group_unit -> a = group_unit) <-> IsEmbedding f. Proof. split. - intros h b. apply hprop_allpath. intros [a0 p0] [a1 p1]. srapply path_sigma_hprop; simpl. apply grp_moveL_1M. apply h. rewrite grp_homo_op, grp_homo_inv. rewrite p0, p1. apply right_inverse. - intros E a p. rapply (isinj_embedding f). exact (p @ (grp_homo_unit f)^). Defined.

Lemma

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

isembedding_grouphomomorphism

{G H : Group} {C : Commutative (@group_sgop G)} (f : GroupIsomorphism G H) : Commutative (@group_sgop H). Proof. unfold Commutative. rapply (equiv_ind f); intro g1. rapply (equiv_ind f); intro g2. refine ((preserves_sg_op _ _)^ @ _ @ (preserves_sg_op _ _)). refine (ap f _). apply C. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

commutative_iso_commutative

{G G' H : Group} (f : G $-> H) (f' : G' $-> H) {x y : G} {x' y' : G'} (p : f x = f' x') (q : f y = f' y') : f (x * y) = f' (x' * y'). Proof. lhs nrapply grp_homo_op. rhs nrapply grp_homo_op. exact (ap011 _ p q). Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_homo_op_agree

{A B : Group} (f : GroupHomomorphism A B) (x y : A) : f (x * y) = group_unit <~> (f x = - f y) := grp_moveL_1V oE equiv_concat_l (grp_homo_op f x y)^ _.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_homo_moveL_1V

{A B : Group} (f : GroupHomomorphism A B) (x y : A) : f (x * -y) = group_unit <~> (f x = f y). Proof. refine (grp_moveL_1M oE equiv_concat_l _^ _). lhs nrapply grp_homo_op. apply ap, grp_homo_inv. Defined.

Definition

Require Import Basics Types HProp HFiber HSet. Require Import Homotopy.IdentitySystems. Require Import (notations) Classes.interfaces.canonical_names. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Int. Require Import Truncations.Core.

Algebra\Groups\Group.v

grp_homo_moveL_1M

{A B : Group} (f g : A $-> B) : Group. Proof. rapply (AmalgamatedFreeProduct (FreeProduct A A) A B). 1,2: apply FreeProduct_rec. + exact grp_homo_id. + exact grp_homo_id. + exact f. + exact g. Defined.

Definition

Require Import Basics Types. Require Import WildCat.Core. Require Import Truncations.Core. Require Import Algebra.Groups.Group. Require Import Colimits.Coeq. Require Import Algebra.Groups.FreeProduct.

Algebra\Groups\GroupCoeq.v

GroupCoeq

`{Funext} {A B C : Group} (f g : GroupHomomorphism A B) : {h : B $-> C & h o f == h o g} <~> (GroupCoeq f g $-> C). Proof. refine (equiv_amalgamatedfreeproduct_rec _ _ _ _ _ _ oE _). refine (equiv_sigma_symm _ oE _). apply equiv_functor_sigma_id. intros h. snrapply equiv_adjointify. { intros p. exists (grp_homo_compose h f). hnf; intro x. refine (p _ @ _). revert x. rapply Trunc_ind. srapply Coeq_ind_hprop. intros w. hnf. induction w. 1: apply ap, grp_homo_unit. simpl. destruct a as [a|a]. 1,2: refine (ap _ (grp_homo_op _ _ _) @ _). 1,2: nrapply grp_homo_op_agree; trivial. symmetry. apply p. } { intros [k p] x. assert (q1 := p (freeproduct_inl x)). assert (q2 := p (freeproduct_inr x)). simpl in q1, q2. rewrite 2 right_identity in q1, q2. refine (q1^ @ q2). } { hnf. intros [k p]. apply path_sigma_hprop. simpl. apply equiv_path_grouphomomorphism. intro y. pose (q1 := p (freeproduct_inl y)). simpl in q1. rewrite 2 right_identity in q1. exact q1^. } hnf; intros; apply path_ishprop. Defined.

Definition

Require Import Basics Types. Require Import WildCat.Core. Require Import Truncations.Core. Require Import Algebra.Groups.Group. Require Import Colimits.Coeq. Require Import Algebra.Groups.FreeProduct.

Algebra\Groups\GroupCoeq.v

equiv_groupcoeq_rec

Group := Build_Group (Pullback f g) _ _ _ _.

Definition

Require Import Basics Types Limits.Pullback Cubical.PathSquare. Require Import Algebra.Groups.Group. Require Import WildCat.Core.

Algebra\Groups\GrpPullback.v

grp_pullback

grp_pullback $-> B. Proof. snrapply Build_GroupHomomorphism. - apply pullback_pr1. - intros x y. reflexivity. Defined.

Definition

Require Import Basics Types Limits.Pullback Cubical.PathSquare. Require Import Algebra.Groups.Group. Require Import WildCat.Core.

Algebra\Groups\GrpPullback.v

grp_pullback_pr1

grp_pullback $-> C. Proof. snrapply Build_GroupHomomorphism. - apply pullback_pr2. - intros x y. reflexivity. Defined.

Definition

Require Import Basics Types Limits.Pullback Cubical.PathSquare. Require Import Algebra.Groups.Group. Require Import WildCat.Core.

Algebra\Groups\GrpPullback.v

grp_pullback_pr2

{X : Group} (b : X $-> B) (c : X $-> C) (p : f o b == g o c) : X $-> grp_pullback. Proof. snrapply Build_GroupHomomorphism. - exact (fun x => (b x; c x; p x)). - intros x y. srapply path_sigma. + simpl. apply (grp_homo_op b). + unfold pr2. refine (transport_sigma' _ _ @ _). unfold pr1. apply path_sigma_hprop. simpl. apply (grp_homo_op c). Defined.

Proposition

Require Import Basics Types Limits.Pullback Cubical.PathSquare. Require Import Algebra.Groups.Group. Require Import WildCat.Core.

Algebra\Groups\GrpPullback.v

grp_pullback_corec

grp_pullback_corec' (X : Group) : {b : X $-> B & { c : X $-> C & f o b == g o c}} -> (X $-> grp_pullback). Proof. intros [b [c p]]; exact (grp_pullback_corec b c p). Defined.

Corollary

Require Import Basics Types Limits.Pullback Cubical.PathSquare. Require Import Algebra.Groups.Group. Require Import WildCat.Core.

Algebra\Groups\GrpPullback.v

grp_pullback_corec'

{A A' B B' C C' : Group} (f : B $-> A) (f' : B' $-> A') (g : C $-> A) (g' : C' $-> A') (alpha : A $-> A') (beta : B $-> B') (gamma : C $-> C') (h : f' o beta == alpha o f) (k : alpha o g == g' o gamma) : grp_pullback f g $-> grp_pullback f' g'. Proof. srapply grp_pullback_corec. - exact (beta $o grp_pullback_pr1 f g). - exact (gamma $o grp_pullback_pr2 f g). - intro x; cbn. refine (h _ @ ap alpha _ @ k _). apply pullback_commsq. Defined.

Definition

Require Import Basics Types Limits.Pullback Cubical.PathSquare. Require Import Algebra.Groups.Group. Require Import WildCat.Core.

Algebra\Groups\GrpPullback.v

functor_grp_pullback

{A A' B B' C C' : Group} (f : B $-> A) (f' : B' $-> A') (g : C $-> A) (g' : C' $-> A') (alpha : GroupIsomorphism A A') (beta : GroupIsomorphism B B') (gamma : GroupIsomorphism C C') (h : f' o beta == alpha o f) (k : alpha o g == g' o gamma) : GroupIsomorphism (grp_pullback f g) (grp_pullback f' g'). Proof. srapply Build_GroupIsomorphism. 1: exact (functor_grp_pullback f f' g g' _ _ _ h k). srapply isequiv_adjointify. { srapply (functor_grp_pullback f' f g' g). 1-3: rapply grp_iso_inverse; assumption. + rapply (equiv_ind beta); intro b. refine (ap f (eissect _ _) @ _). apply (equiv_ap' alpha _ _)^-1. exact ((h b)^ @ (eisretr _ _)^). + rapply (equiv_ind gamma); intro c. refine (_ @ ap g (eissect _ _)^). apply (equiv_ap' alpha _ _)^-1. exact (eisretr _ _ @ (k c)^). } all: intro x; apply equiv_path_pullback_hset; split; cbn. 1-2: apply eisretr. 1-2: apply eissect. Defined.

Definition

Require Import Basics Types Limits.Pullback Cubical.PathSquare. Require Import Algebra.Groups.Group. Require Import WildCat.Core.

Algebra\Groups\GrpPullback.v

equiv_functor_grp_pullback

{X Z Y Y' : Group} (f : X $-> Z) (g' : Y' $-> Y) (g : Y $-> Z) : GroupIsomorphism (grp_pullback (grp_pullback_pr2 f g) g') (grp_pullback f (g $o g')). Proof. srapply Build_GroupIsomorphism. - srapply grp_pullback_corec. + exact (grp_pullback_pr1 _ _ $o grp_pullback_pr1 _ _). + apply grp_pullback_pr2. + intro x; cbn. exact (pullback_commsq _ _ _ @ ap g (pullback_commsq _ _ _)). - srapply isequiv_adjointify. + srapply grp_pullback_corec. * srapply functor_grp_pullback. 1,2: exact grp_homo_id. 1: exact g'. all: reflexivity. * apply grp_pullback_pr2. * reflexivity. + intro x; cbn. by srapply equiv_path_pullback_hset. + intros [[x [y z0]] [y' z1]]; srapply equiv_path_pullback_hset; split; cbn. 2: reflexivity. srapply equiv_path_pullback_hset; split; cbn. 1: reflexivity. exact z1^. Defined.

Definition

Require Import Basics Types Limits.Pullback Cubical.PathSquare. Require Import Algebra.Groups.Group. Require Import WildCat.Core.

Algebra\Groups\GrpPullback.v

equiv_grp_pullback_compose_r

{X : Group} (b : X $-> B) (c : X $-> C) (p : f o b == g o c) : grp_pullback_pr1 f g $o grp_pullback_corec f g b c p = b. Proof. apply equiv_path_grouphomomorphism; reflexivity. Defined.

Lemma

Require Import Basics Types Limits.Pullback Cubical.PathSquare. Require Import Algebra.Groups.Group. Require Import WildCat.Core.

Algebra\Groups\GrpPullback.v

grp_pullback_corec_pr1

{X : Group} (b : X $-> B) (c : X $-> C) (p : f o b == g o c) : grp_pullback_pr2 f g $o grp_pullback_corec f g b c p = c. Proof. apply equiv_path_grouphomomorphism; reflexivity. Defined.

Lemma

Require Import Basics Types Limits.Pullback Cubical.PathSquare. Require Import Algebra.Groups.Group. Require Import WildCat.Core.

Algebra\Groups\GrpPullback.v

grp_pullback_corec_pr2