fact
stringlengths 0
6.66k
| type
stringclasses 10
values | imports
stringclasses 399
values | filename
stringclasses 465
values | symbolic_name
stringlengths 1
75
| index_level
int64 0
7.85k
|
|---|---|---|---|---|---|
{a b} := @cat_precomp Group _ _ a b. | Definition | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | group_precomp | 300 |
`{Funext} {G : Group} {G_ab : AbGroup} (eta : GroupHomomorphism G G_ab) `{!IsAbelianization G_ab eta} (A : AbGroup) : IsEquiv (group_precomp A eta). Proof. snrapply isequiv_adjointify. - intros g. rapply (surjinj_inv (group_precomp A eta) g). - intros f. snrapply equiv_path_grouphomomorphism. exact (eisretr0gpd_inv (group_precomp A eta) f). - intros f. snrapply equiv_path_grouphomomorphism. exact (eissect0gpd_inv (group_precomp A eta) f). Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | isequiv_group_precomp_isabelianization | 301 |
`{Funext} {G : Group} {G_ab : AbGroup} (eta : GroupHomomorphism G G_ab) `{!IsAbelianization G_ab eta} (A : AbGroup) : (G_ab $-> A) <~> (G $-> A) := Build_Equiv _ _ _ (isequiv_group_precomp_isabelianization eta A). Section Abel. Context (G : Group). Local uncurry2 {A B C D : Type} : (A -> B -> C -> D) -> A * B * C -> D. Proof. intros f [[a b] c]. by apply f. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | equiv_group_precomp_isabelianization | 302 |
Tr 0 (Coeq (uncurry2 (fun a b c : G => a * (b * c))) (uncurry2 (fun a b c : G => a * (c * b)))). | Definition | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | Abel | 303 |
G -> Abel. Proof. intro g. apply tr, coeq, g. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | abel_in | 304 |
a b c : abel_in (a * (b * c)) = abel_in (a * (c * b)). Proof. apply (ap tr). exact (cglue (a, b, c)). Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | abel_in_comm | 305 |
(P : Abel -> Type) `{forall x, IsHSet (P x)} (a : forall x, P (abel_in x)) (c : forall x y z, DPath P (abel_in_comm x y z) (a (x * (y * z))) (a (x * (z * y)))) : forall (x : Abel), P x. Proof. srapply Trunc_ind. srapply Coeq_ind. 1: apply a. intros [[x y] z]. refine (transport_compose _ _ _ _ @ _). apply c. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | Abel_ind | 306 |
(P : Abel -> Type) `{forall x, IsHSet (P x)} (a : forall x, P (abel_in x)) (c : forall x y z, DPath P (abel_in_comm x y z) (a (x * (y * z))) (a (x * (z * y)))) (x : G) : Abel_ind P a c (abel_in x) = a x := idpath. | Definition | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | Abel_ind_beta_abel_in | 307 |
(P : Abel -> Type) `{forall x, IsHSet (P x)} (a : forall x, P (abel_in x)) (c : forall x y z, DPath P (abel_in_comm x y z) (a (x * (y * z))) (a (x * (z * y)))) (x y z : G) : apD (Abel_ind P a c) (abel_in_comm x y z) = c x y z. Proof. refine (apD_compose' tr _ _ @ ap _ _ @ concat_V_pp _ _). rapply Coeq_ind_beta_cglue. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | Abel_ind_beta_abel_in_comm | 308 |
(P : Type) `{IsHSet P} (a : G -> P) (c : forall x y z, a (x * (y * z)) = a (x * (z * y))) : Abel -> P. Proof. apply (Abel_ind _ a). intros; apply dp_const, c. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | Abel_rec | 309 |
(P : Abel -> Type) `{forall x, IsHProp (P x)} (a : forall x, P (abel_in x)) : forall (x : Abel), P x. Proof. srapply Trunc_ind. srapply Coeq_ind_hprop. exact a. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | Abel_ind_hprop | 310 |
RightInverse abel_sgop abel_negate abel_mon_unit. Proof. Abel_ind_hprop x; simpl. apply ap; apply right_inverse. Defined. | Instance | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | abel_rightinverse | 311 |
Group -> AbGroup. Proof. intro G. snrapply Build_AbGroup. - srapply (Build_Group (Abel G)). - exact _. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | abel | 312 |
{G : Group} : G $-> (abel G) := @Build_GroupHomomorphism G (abel G) abel_in _. | Definition | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | abel_unit | 313 |
{G : Group} {A : AbGroup} (f : G $-> A) : abel G $-> A. Proof. snrapply Build_GroupHomomorphism. { srapply (Abel_rec _ _ f). intros x y z. nrapply grp_homo_op_agree; trivial. refine (grp_homo_op _ _ _ @ _ @ (grp_homo_op _ _ _)^). apply commutativity. } intros y. Abel_ind_hprop x; revert y. Abel_ind_hprop y. apply grp_homo_op. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | grp_homo_abel_rec | 314 |
{G H : Group} {f g : Hom (A:=Group) (abel G) H} (p : f $o abel_unit $== g $o abel_unit) : f $== g. Proof. rapply Abel_ind_hprop. rapply p. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | abel_ind_homotopy | 315 |
{G : Group} (A B : AbGroup) (eta1 : GroupHomomorphism G A) (eta2 : GroupHomomorphism G B) {isab1 : IsAbelianization A eta1} {isab2 : IsAbelianization B eta2} : A ≅ B. Proof. destruct (esssurj (group_precomp B eta1) eta2) as [a ac]. destruct (esssurj (group_precomp A eta2) eta1) as [b bc]. srapply (Build_GroupIsomorphism _ _ a). srapply (isequiv_adjointify _ b). { refine (essinj (group_precomp B eta2) (x := a $o b) (y := Id (A := Group) B) _). intros x; cbn in *. refine (_ @ ac x). apply ap, bc. } { refine (essinj (group_precomp A eta1) (x := b $o a) (y := Id (A := Group) A) _). intros x; cbn in *. refine (_ @ bc x). apply ap, ac. } Defined. | Theorem | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | groupiso_isabelianization | 316 |
{G : Group} (A B : AbGroup) (eta1 : GroupHomomorphism G A) (eta2 : GroupHomomorphism G B) {isab1 : IsAbelianization A eta1} {isab2 : IsAbelianization B eta2} : eta2 == grp_homo_compose (groupiso_isabelianization A B eta1 eta2) eta1. Proof. intros x. exact (((esssurj (group_precomp B eta1) eta2).2 x)^). Defined. | Theorem | Require Import Basics Types Truncations.Core. Require Import Cubical.DPath WildCat. Require Import Colimits.Coeq. Require Import Algebra.AbGroups.AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Abelianization.v | homotopic_isabelianization | 317 |
{A : Group} {B : AbGroup} (f g : A $-> B) : A $-> B. Proof. refine (grp_homo_compose ab_codiagonal _). exact (grp_prod_corec f g). Defined. | Definition | Require Import Basics Types. Require Import WildCat HSet Truncations.Core Modalities.ReflectiveSubuniverse. Require Import Groups.QuotientGroup AbelianGroup Biproduct. | Algebra\AbGroups\AbHom.v | ab_homo_add | 318 |
`{Funext} (A : Group) (B : AbGroup) : Group. Proof. nrefine (Build_Group (GroupHomomorphism A B) ab_homo_add grp_homo_const negate_hom _). repeat split. 1: exact _. all: hnf; intros; apply equiv_path_grouphomomorphism; intro; cbn. + apply associativity. + apply left_identity. + apply right_identity. + apply left_inverse. + apply right_inverse. Defined. | Definition | Require Import Basics Types. Require Import WildCat HSet Truncations.Core Modalities.ReflectiveSubuniverse. Require Import Groups.QuotientGroup AbelianGroup Biproduct. | Algebra\AbGroups\AbHom.v | grp_hom | 319 |
`{Funext} (A : Group) (B : AbGroup) : AbGroup. Proof. snrapply (Build_AbGroup (grp_hom A B)). intros f g; cbn. apply equiv_path_grouphomomorphism; intro x; cbn. apply commutativity. Defined. | Definition | Require Import Basics Types. Require Import WildCat HSet Truncations.Core Modalities.ReflectiveSubuniverse. Require Import Groups.QuotientGroup AbelianGroup Biproduct. | Algebra\AbGroups\AbHom.v | ab_hom | 320 |
{A B : AbGroup} (f g : GroupHomomorphism A B) := ab_cokernel (ab_homo_add (negate_hom f) g). | Definition | Require Import Basics Types. Require Import WildCat HSet Truncations.Core Modalities.ReflectiveSubuniverse. Require Import Groups.QuotientGroup AbelianGroup Biproduct. | Algebra\AbGroups\AbHom.v | ab_coeq | 321 |
{A B} {f g : A $-> B} : B $-> ab_coeq f g. Proof. snrapply grp_quotient_map. Defined. | Definition | Require Import Basics Types. Require Import WildCat HSet Truncations.Core Modalities.ReflectiveSubuniverse. Require Import Groups.QuotientGroup AbelianGroup Biproduct. | Algebra\AbGroups\AbHom.v | ab_coeq_in | 322 |
{A B} {f g : A $-> B} : ab_coeq_in (f:=f) (g:=g) $o f $== ab_coeq_in $o g. Proof. intros x. nrapply qglue. apply tr. by exists x. Defined. | Definition | Require Import Basics Types. Require Import WildCat HSet Truncations.Core Modalities.ReflectiveSubuniverse. Require Import Groups.QuotientGroup AbelianGroup Biproduct. | Algebra\AbGroups\AbHom.v | ab_coeq_glue | 323 |
{A B : AbGroup} {f g : A $-> B} {C : AbGroup} (i : B $-> C) (p : i $o f $== i $o g) : ab_coeq f g $-> C. Proof. snrapply (grp_quotient_rec _ _ i). cbn. intros b H. strip_truncations. destruct H as [a q]. destruct q; simpl. lhs nrapply grp_homo_op. lhs nrapply (ap (.* _)). 1: apply grp_homo_inv. apply grp_moveL_M1^-1. exact (p a)^. Defined. | Definition | Require Import Basics Types. Require Import WildCat HSet Truncations.Core Modalities.ReflectiveSubuniverse. Require Import Groups.QuotientGroup AbelianGroup Biproduct. | Algebra\AbGroups\AbHom.v | ab_coeq_rec | 324 |
{A B : AbGroup} {f g : A $-> B} {C : AbGroup} (i : B $-> C) (p : i $o f $== i $o g) : ab_coeq_rec i p $o ab_coeq_in $== i := fun _ => idpath. | Definition | Require Import Basics Types. Require Import WildCat HSet Truncations.Core Modalities.ReflectiveSubuniverse. Require Import Groups.QuotientGroup AbelianGroup Biproduct. | Algebra\AbGroups\AbHom.v | ab_coeq_rec_beta_in | 325 |
{A B f g} (P : @ab_coeq A B f g -> Type) `{forall x, IsHProp (P x)} (i : forall b, P (ab_coeq_in b)) : forall x, P x. Proof. rapply Quotient_ind_hprop. exact i. Defined. | Definition | Require Import Basics Types. Require Import WildCat HSet Truncations.Core Modalities.ReflectiveSubuniverse. Require Import Groups.QuotientGroup AbelianGroup Biproduct. | Algebra\AbGroups\AbHom.v | ab_coeq_ind_hprop | 326 |
{A B C f g} {l r : @ab_coeq A B f g $-> C} (p : l $o ab_coeq_in $== r $o ab_coeq_in) : l $== r. Proof. srapply ab_coeq_ind_hprop. exact p. Defined. | Definition | Require Import Basics Types. Require Import WildCat HSet Truncations.Core Modalities.ReflectiveSubuniverse. Require Import Groups.QuotientGroup AbelianGroup Biproduct. | Algebra\AbGroups\AbHom.v | ab_coeq_ind_homotopy | 327 |
{A B} {f g : A $-> B} {A' B'} {f' g' : A' $-> B'} (a : A $-> A') (b : B $-> B') (p : f' $o a $== b $o f) (q : g' $o a $== b $o g) : ab_coeq f g $-> ab_coeq f' g'. Proof. snrapply ab_coeq_rec. 1: exact (ab_coeq_in $o b). refine (cat_assoc _ _ _ $@ _ $@ cat_assoc_opp _ _ _). refine ((_ $@L p^$) $@ _ $@ (_ $@L q)). refine (cat_assoc_opp _ _ _ $@ (_ $@R a) $@ cat_assoc _ _ _). nrapply ab_coeq_glue. Defined. | Definition | Require Import Basics Types. Require Import WildCat HSet Truncations.Core Modalities.ReflectiveSubuniverse. Require Import Groups.QuotientGroup AbelianGroup Biproduct. | Algebra\AbGroups\AbHom.v | functor_ab_coeq | 328 |
{A B} {f g : A $-> B} {A' B'} {f' g' : A' $-> B'} {a a' : A $-> A'} {b b' : B $-> B'} (p : f' $o a $== b $o f) (q : g' $o a $== b $o g) (p' : f' $o a' $== b' $o f) (q' : g' $o a' $== b' $o g) (s : b $== b') : functor_ab_coeq a b p q $== functor_ab_coeq a' b' p' q'. Proof. snrapply ab_coeq_ind_homotopy. intros x. exact (ap ab_coeq_in (s x)). Defined. | Definition | Require Import Basics Types. Require Import WildCat HSet Truncations.Core Modalities.ReflectiveSubuniverse. Require Import Groups.QuotientGroup AbelianGroup Biproduct. | Algebra\AbGroups\AbHom.v | functor2_ab_coeq | 329 |
{A B} {f g : A $-> B} {A' B'} {f' g' : A' $-> B'} (a : A $-> A') (b : B $-> B') (p : f' $o a $== b $o f) (q : g' $o a $== b $o g) {A'' B''} {f'' g'' : A'' $-> B''} (a' : A' $-> A'') (b' : B' $-> B'') (p' : f'' $o a' $== b' $o f') (q' : g'' $o a' $== b' $o g') : functor_ab_coeq a' b' p' q' $o functor_ab_coeq a b p q $== functor_ab_coeq (a' $o a) (b' $o b) (hconcat p p') (hconcat q q'). Proof. snrapply ab_coeq_ind_homotopy. simpl; reflexivity. Defined. | Definition | Require Import Basics Types. Require Import WildCat HSet Truncations.Core Modalities.ReflectiveSubuniverse. Require Import Groups.QuotientGroup AbelianGroup Biproduct. | Algebra\AbGroups\AbHom.v | functor_ab_coeq_compose | 330 |
{A B} (f g : A $-> B) : functor_ab_coeq (f:=f) (g:=g) (Id _) (Id _) (hrefl _) (hrefl _) $== Id _. Proof. snrapply ab_coeq_ind_homotopy. reflexivity. Defined. | Definition | Require Import Basics Types. Require Import WildCat HSet Truncations.Core Modalities.ReflectiveSubuniverse. Require Import Groups.QuotientGroup AbelianGroup Biproduct. | Algebra\AbGroups\AbHom.v | functor_ab_coeq_id | 331 |
{A B} {f g : A $-> B} {A' B'} {f' g' : A' $-> B'} (a : A $<~> A') (b : B $<~> B') (p : f' $o a $== b $o f) (q : g' $o a $== b $o g) : ab_coeq f g $<~> ab_coeq f' g'. Proof. snrapply cate_adjointify. - exact (functor_ab_coeq a b p q). - exact (functor_ab_coeq a^-1$ b^-1$ (hinverse _ _ p) (hinverse _ _ q)). - nrefine (functor_ab_coeq_compose _ _ _ _ _ _ _ _ $@ functor2_ab_coeq _ _ _ _ _ $@ functor_ab_coeq_id _ _). rapply cate_isretr. - nrefine (functor_ab_coeq_compose _ _ _ _ _ _ _ _ $@ functor2_ab_coeq _ _ _ _ _ $@ functor_ab_coeq_id _ _). rapply cate_issect. Defined. | Definition | Require Import Basics Types. Require Import WildCat HSet Truncations.Core Modalities.ReflectiveSubuniverse. Require Import Groups.QuotientGroup AbelianGroup Biproduct. | Algebra\AbGroups\AbHom.v | grp_iso_ab_coeq | 332 |
(P : AbGroup) : IsAbProjective P <-> (forall A, forall p : A $-> P, IsSurjection p -> merely (exists s : P $-> A, p $o s == grp_homo_id)). Proof. split. - intros H A B. apply H. - intros H A B e f H1. pose proof (s := H (ab_pullback f e) (grp_pullback_pr1 f e) (conn_map_pullback _ f e)). strip_truncations. destruct s as [s h]. refine (tr ((grp_pullback_pr2 f e) $o s; _)); intro x. refine ((pullback_commsq f e _)^ @ _). exact (ap f (h x)). Defined. | Proposition | Require Import Basics Types AbelianGroup AbPullback | Algebra\AbGroups\AbProjective.v | iff_isabprojective_surjections_split | 333 |
AbGroup := Build_AbGroup (grp_pullback f g) _. | Definition | Require Import Basics. Require Import Limits.Pullback Cubical.PathSquare. Require Import Algebra.AbGroups.AbelianGroup. Require Import WildCat.Core. | Algebra\AbGroups\AbPullback.v | ab_pullback | 334 |
{A B C : AbGroup} (f : A $-> B) (g : A $-> C) : Subgroup (ab_biprod B C) := grp_image (ab_biprod_corec (ab_homo_negation $o f) g). | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat.Core HSet. Require Import AbGroups.AbelianGroup AbGroups.Biproduct. | Algebra\AbGroups\AbPushout.v | ab_pushout_subgroup | 335 |
{A B C : AbGroup} (f : A $-> B) (g : A $-> C) : AbGroup := QuotientAbGroup (ab_biprod B C) (ab_pushout_subgroup f g). | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat.Core HSet. Require Import AbGroups.AbelianGroup AbGroups.Biproduct. | Algebra\AbGroups\AbPushout.v | ab_pushout | 336 |
{A B C Y : AbGroup} {f : A $-> B} {g : A $-> C} (b : B $-> Y) (c : C $-> Y) (p : b o f == c o g) : ab_pushout f g $-> Y. Proof. srapply grp_quotient_rec. - exact (ab_biprod_rec b c). - intros [x y] q; strip_truncations; simpl. destruct q as [a q]. cbn in q. refine (ap (uncurry (fun x y => b x + c y)) q^ @ _). cbn. refine (ap011 sg_op (preserves_negate _) (p a)^ @ _). exact (left_inverse _). Defined. | Theorem | Require Import Basics Types Truncations.Core. Require Import WildCat.Core HSet. Require Import AbGroups.AbelianGroup AbGroups.Biproduct. | Algebra\AbGroups\AbPushout.v | ab_pushout_rec | 337 |
{A B C : AbGroup} (f : A $-> B) (g : A $-> C) (Y : AbGroup) : {b : B $-> Y & {c : C $-> Y & b o f == c o g}} -> (ab_pushout f g $-> Y). Proof. intros [b [c p]]; exact (ab_pushout_rec b c p). Defined. | Corollary | Require Import Basics Types Truncations.Core. Require Import WildCat.Core HSet. Require Import AbGroups.AbelianGroup AbGroups.Biproduct. | Algebra\AbGroups\AbPushout.v | ab_pushout_rec_uncurried | 338 |
{A B C : AbGroup} {f : A $-> B} {g : A $-> C} : B $-> ab_pushout f g := grp_quotient_map $o grp_prod_inl. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat.Core HSet. Require Import AbGroups.AbelianGroup AbGroups.Biproduct. | Algebra\AbGroups\AbPushout.v | ab_pushout_inl | 339 |
{A B C : AbGroup} {f : A $-> B} {g : A $-> C} : C $-> ab_pushout f g := grp_quotient_map $o grp_prod_inr. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat.Core HSet. Require Import AbGroups.AbelianGroup AbGroups.Biproduct. | Algebra\AbGroups\AbPushout.v | ab_pushout_inr | 340 |
{A B C : AbGroup} {f : A $-> B} {g : A $-> C} : (@ab_pushout_inl A B C f g) $o f == ab_pushout_inr $o g. Proof. intro a. apply qglue; cbn. apply tr. exists a. apply path_prod; simpl. - exact (right_identity _)^. - rewrite negate_mon_unit. exact (left_identity _)^. Defined. | Proposition | Require Import Basics Types Truncations.Core. Require Import WildCat.Core HSet. Require Import AbGroups.AbelianGroup AbGroups.Biproduct. | Algebra\AbGroups\AbPushout.v | ab_pushout_commsq | 341 |
`{Funext} {A B C Y : AbGroup} {f : A $-> B} {g : A $-> C} (phi : ab_pushout f g $-> Y) : ab_pushout_rec (phi $o ab_pushout_inl) (phi $o ab_pushout_inr) (fun a:A => ap phi (ab_pushout_commsq a)) = phi. Proof. rapply (equiv_ap' (equiv_quotient_abgroup_ump (G:=ab_biprod B C) _ Y)^-1%equiv _ _)^-1. srapply path_sigma_hprop. refine (grp_quotient_rec_beta _ Y _ _ @ _). apply equiv_path_grouphomomorphism; intro bc. exact (ab_biprod_rec_eta (phi $o grp_quotient_map) bc). Defined. | Proposition | Require Import Basics Types Truncations.Core. Require Import WildCat.Core HSet. Require Import AbGroups.AbelianGroup AbGroups.Biproduct. | Algebra\AbGroups\AbPushout.v | ab_pushout_rec_beta | 342 |
{A B C Y : AbGroup} (f : A $-> B) (g : A $-> C) (l : B $-> Y) (r : C $-> Y) (p : l o f == r o g) : ab_pushout_rec l r p $o ab_pushout_inl == l. Proof. intro x; simpl. rewrite (grp_homo_unit r). apply right_identity. Defined. | Lemma | Require Import Basics Types Truncations.Core. Require Import WildCat.Core HSet. Require Import AbGroups.AbelianGroup AbGroups.Biproduct. | Algebra\AbGroups\AbPushout.v | ab_pushout_rec_beta_left | 343 |
{A B C Y : AbGroup} (f : A $-> B) (g : A $-> C) (l : B $-> Y) (r : C $-> Y) (p : l o f == r o g) : ab_pushout_rec l r p $o ab_pushout_inr == r. Proof. intro x; simpl. rewrite (grp_homo_unit l). apply left_identity. Defined. | Lemma | Require Import Basics Types Truncations.Core. Require Import WildCat.Core HSet. Require Import AbGroups.AbelianGroup AbGroups.Biproduct. | Algebra\AbGroups\AbPushout.v | ab_pushout_rec_beta_right | 344 |
`{Funext} {A B C Y : AbGroup} {f : A $-> B} {g : A $-> C} : IsEquiv (ab_pushout_rec_uncurried f g Y). Proof. srapply isequiv_adjointify. - intro phi. refine (phi $o ab_pushout_inl; phi $o ab_pushout_inr; _). intro a. apply (ap phi). exact (ab_pushout_commsq a). - intro phi. exact (ab_pushout_rec_beta phi). - intros [b [c p]]. srapply path_sigma. + apply equiv_path_grouphomomorphism. intro x; simpl. refine (ap (fun k => b x + k) (grp_homo_unit c) @ _). apply right_identity. + refine (transport_sigma' _ _ @ _). apply path_sigma_hprop; simpl. apply equiv_path_grouphomomorphism. intro y; simpl. refine (ap (fun k => k + c y) (grp_homo_unit b) @ _). apply left_identity. Defined. | Theorem | Require Import Basics Types Truncations.Core. Require Import WildCat.Core HSet. Require Import AbGroups.AbelianGroup AbGroups.Biproduct. | Algebra\AbGroups\AbPushout.v | isequiv_ab_pushout_rec | 345 |
`{Univalence} {A B C : AbGroup} (f : A $-> B) (g : A $-> C) (bc0 bc1 : ab_biprod B C) : @in_cosetL (ab_biprod B C) (ab_pushout_subgroup f g) bc0 bc1 <~> (grp_quotient_map bc0 = grp_quotient_map bc1 :> ab_pushout f g). Proof. rapply path_quotient. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat.Core HSet. Require Import AbGroups.AbelianGroup AbGroups.Biproduct. | Algebra\AbGroups\AbPushout.v | path_ab_pushout | 346 |
`{Univalence} {A B C : AbGroup} (f : A $-> B) (g : A $-> C) `{IsEmbedding g} : IsEmbedding (ab_pushout_inl (f:=f) (g:=g)). Proof. apply isembedding_isinj_hset. intros c0 c1. refine (_ o (path_ab_pushout f g (grp_prod_inl c0) (grp_prod_inl c1))^-1). rapply Trunc_ind. cbn; intros [a p]. assert (z : a = mon_unit). - rapply (isinj_embedding g). refine (ap snd p @ _); cbn. exact (left_inverse mon_unit @ (grp_homo_unit g)^). - apply (grp_moveR_M1). refine (_ @ ap fst p); cbn; symmetry. refine (_ @ negate_mon_unit). refine (ap _ _). exact (ap f z @ grp_homo_unit f). Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat.Core HSet. Require Import AbGroups.AbelianGroup AbGroups.Biproduct. | Algebra\AbGroups\AbPushout.v | ab_pushout_embedding_inl | 347 |
{A A' B B' C C' : AbGroup} (f : A $-> B) (f' : A' $-> B') (g : A $-> C) (g' : A' $-> C') (alpha : A $-> A') (beta : B $-> B') (gamma : C $-> C') (h : beta $o f == f' $o alpha) (k : g' $o alpha == gamma $o g) : ab_pushout f g $-> ab_pushout f' g'. Proof. srapply ab_pushout_rec. - exact (ab_pushout_inl $o beta). - exact (ab_pushout_inr $o gamma). - intro a. refine (ap ab_pushout_inl (h a) @ _ @ ap ab_pushout_inr (k a)). exact (ab_pushout_commsq (alpha a)). Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat.Core HSet. Require Import AbGroups.AbelianGroup AbGroups.Biproduct. | Algebra\AbGroups\AbPushout.v | functor_ab_pushout | 348 |
ab_biprod@{u} (A B : AbGroup@{u}) : AbGroup@{u}. Proof. rapply (Build_AbGroup (grp_prod A B)). intros [a b] [a' b']. apply path_prod; simpl; apply commutativity. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod@ | 349 |
{A B : AbGroup} : A $-> ab_biprod A B := grp_prod_inl. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_inl | 350 |
{A B : AbGroup} : B $-> ab_biprod A B := grp_prod_inr. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_inr | 351 |
{A B : AbGroup} : ab_biprod A B $-> A := grp_prod_pr1. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_pr1 | 352 |
{A B : AbGroup} : ab_biprod A B $-> B := grp_prod_pr2. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_pr2 | 353 |
{A B : AbGroup} (P : ab_biprod A B -> Type) (Hinl : forall a, P (ab_biprod_inl a)) (Hinr : forall b, P (ab_biprod_inr b)) (Hop : forall x y, P x -> P y -> P (x + y)) : forall x, P x. Proof. intros [a b]. snrapply ((grp_prod_decompose a b)^ # _). apply Hop. - exact (Hinl a). - exact (Hinr b). Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_ind | 354 |
{A B C : AbGroup} {f g : ab_biprod A B $-> C} (Hinl : f $o ab_biprod_inl $== g $o ab_biprod_inl) (Hinr : f $o ab_biprod_inr $== g $o ab_biprod_inr) : f $== g. Proof. rapply ab_biprod_ind. - exact Hinl. - exact Hinr. - intros x y p q. lhs nrapply grp_homo_op. rhs nrapply grp_homo_op. f_ap. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_ind_homotopy | 355 |
`{Funext} {A B X : AbGroup} (phi psi : ab_biprod A B $-> X) : (phi $o ab_biprod_inl == psi $o ab_biprod_inl) * (phi $o ab_biprod_inr == psi $o ab_biprod_inr) <~> phi == psi. Proof. apply equiv_iff_hprop. - exact (uncurry ab_biprod_ind_homotopy). - exact (fun h => (fun a => h _, fun b => h _)). Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | equiv_ab_biprod_ind_homotopy | 356 |
{A B Y : AbGroup} (f : A $-> Y) (g : B $-> Y) : (ab_biprod A B) $-> Y. Proof. snrapply Build_GroupHomomorphism. - intros [a b]; exact (f a + g b). - intros [a b] [a' b']; simpl. rewrite (grp_homo_op f). rewrite (grp_homo_op g). rewrite (associativity _ (g b) _). rewrite <- (associativity _ (f a') _). rewrite (commutativity (f a') _). rewrite (associativity _ (g b) _). exact (associativity _ (f a') _)^. Defined. | Proposition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_rec | 357 |
{A B Y : AbGroup} : (A $-> Y) * (B $-> Y) -> (ab_biprod A B) $-> Y. Proof. intros [f g]. exact (ab_biprod_rec f g). Defined. | Corollary | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_rec_uncurried | 358 |
{A B Y : AbGroup} (u : ab_biprod A B $-> Y) : ab_biprod_rec (u $o ab_biprod_inl) (u $o ab_biprod_inr) == u. Proof. intros [a b]; simpl. refine ((grp_homo_op u _ _)^ @ ap u _). apply path_prod. - exact (right_identity a). - exact (left_identity b). Defined. | Proposition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_rec_eta | 359 |
{A B Y : AbGroup} (a : A $-> Y) (b : B $-> Y) : (ab_biprod_rec a b) $o ab_biprod_inl == a. Proof. intro x; simpl. rewrite (grp_homo_unit b). exact (right_identity (a x)). Defined. | Proposition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_rec_beta_inl | 360 |
{A B Y : AbGroup} (a : A $-> Y) (b : B $-> Y) : (ab_biprod_rec a b) $o ab_biprod_inr == b. Proof. intro y; simpl. rewrite (grp_homo_unit a). exact (left_identity (b y)). Defined. | Proposition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_rec_beta_inr | 361 |
`{Funext} {A B Y : AbGroup} : IsEquiv (@ab_biprod_rec_uncurried A B Y). Proof. srapply isequiv_adjointify. - intro phi. exact (phi $o ab_biprod_inl, phi $o ab_biprod_inr). - intro phi. apply equiv_path_grouphomomorphism. exact (ab_biprod_rec_eta phi). - intros [a b]. apply path_prod. + apply equiv_path_grouphomomorphism. apply ab_biprod_rec_beta_inl. + apply equiv_path_grouphomomorphism. apply ab_biprod_rec_beta_inr. Defined. | Theorem | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | isequiv_ab_biprod_rec | 362 |
{A B X : AbGroup} (f : X $-> A) (g : X $-> B) : X $-> ab_biprod A B := grp_prod_corec f g. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_corec | 363 |
{A A' B B' : AbGroup} (f : A $-> A') (g: B $-> B') : ab_biprod A B $-> ab_biprod A' B' := (ab_biprod_corec (f $o ab_biprod_pr1) (g $o ab_biprod_pr2)). | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | functor_ab_biprod | 364 |
{Z X Y A B : AbGroup} (f0 : Z $-> X) (f1 : Z $-> Y) (g0 : X $-> A) (g1 : Y $-> B) : functor_ab_biprod g0 g1 $o ab_biprod_corec f0 f1 $== ab_biprod_corec (g0 $o f0) (g1 $o f1) := fun _ => idpath. Global Instance is0bifunctor_ab_biprod : Is0Bifunctor ab_biprod. Proof. srapply Build_Is0Bifunctor'. snrapply Build_Is0Functor. intros [A B] [A' B'] [f g]. exact (functor_ab_biprod f g). Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_functor_beta | 365 |
{A A' B B' : AbGroup} (f : A $-> A') (g : B $-> B') `{IsEquiv _ _ f} `{IsEquiv _ _ g} : IsEquiv (functor_ab_biprod f g). Proof. srapply isequiv_adjointify. 1: { rapply functor_ab_biprod; apply grp_iso_inverse. + exact (Build_GroupIsomorphism _ _ f _). + exact (Build_GroupIsomorphism _ _ g _). } all: intros [a b]; simpl. all: apply path_prod'. 1,2: apply eisretr. all: apply eissect. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | isequiv_functor_ab_biprod | 366 |
{A A' B B' : AbGroup} (f : A $-> A') (g : B $-> B') `{IsEquiv _ _ f} `{IsEquiv _ _ g} : GroupIsomorphism (ab_biprod A B) (ab_biprod A' B') := Build_GroupIsomorphism _ _ _ (isequiv_functor_ab_biprod f g). | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | equiv_functor_ab_biprod | 367 |
{A A' B B' : AbGroup} (i : A $-> B) (i' : A' $-> B') `{IsEmbedding i} `{IsEmbedding i'} : IsEmbedding (functor_ab_biprod i i'). Proof. intros [b b']. apply hprop_allpath. intros [[a0 a0'] p] [[a1 a1'] p']; cbn in p, p'. rapply path_sigma_hprop; cbn. pose (q := (equiv_path_prod _ _)^-1 p); cbn in q. pose (q' := (equiv_path_prod _ _)^-1 p'); cbn in q'. destruct q as [q0 q1], q' as [q0' q1']. apply path_prod; rapply isinj_embedding; cbn. - exact (q0 @ q0'^). - exact (q1 @ q1'^). Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | functor_ab_biprod_embedding | 368 |
`{Funext} {A A' B B' : AbGroup} (p : A $-> B) (p' : A' $-> B') `{S : IsSurjection p} `{S' : IsSurjection p'} : IsSurjection (functor_ab_biprod p p'). Proof. intros [b b']. pose proof (a := S b); pose proof (a' := S' b'). apply center in a, a'. strip_truncations. rapply contr_inhabited_hprop. apply tr. exists (ab_biprod_inl a.1 + ab_biprod_inr a'.1); cbn. apply path_prod; refine (grp_homo_op _ _ _ @ _); rewrite (grp_homo_unit _); cbn. - exact (right_identity _ @ a.2). - exact (left_identity _ @ a'.2). Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | functor_ab_biprod_surjection | 369 |
{A B : AbGroup} : ab_biprod A B $<~> ab_biprod B A. Proof. snrapply Build_GroupIsomorphism'. - apply equiv_prod_symm. - intro; reflexivity. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | direct_sum_swap | 370 |
{A : AbGroup} : ab_biprod A A $-> A := ab_biprod_rec grp_homo_id grp_homo_id. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_codiagonal | 371 |
{A B : AbGroup} (f : A $-> B) : f $o ab_codiagonal $== ab_codiagonal $o functor_ab_biprod f f := fun a => grp_homo_op f _ _. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_codiagonal_natural | 372 |
{A : AbGroup} : A $-> ab_biprod A A := ab_biprod_corec grp_homo_id grp_homo_id. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_diagonal | 373 |
`{Funext} {A B : AbGroup} (f g : B $-> A) : ab_biprod_corec f g = (functor_ab_biprod f g) $o ab_diagonal. Proof. apply equiv_path_grouphomomorphism; reflexivity. Defined. | Lemma | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_corec_diagonal | 374 |
`{Funext} {A : AbGroup} : (@ab_codiagonal A) $o direct_sum_swap = ab_codiagonal. Proof. apply equiv_path_grouphomomorphism. intro a; cbn. exact (abgroup_commutative _ _ _). Defined. | Lemma | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_codiagonal_swap | 375 |
{A : AbGroup} : direct_sum_swap $o (@ab_diagonal A) = ab_diagonal := idpath. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_diagonal_swap | 376 |
{A B C : AbGroup} : ab_biprod A (ab_biprod B C) $<~> ab_biprod (ab_biprod A B) C. Proof. snrapply Build_GroupIsomorphism'. - apply equiv_prod_assoc. - unfold IsSemiGroupPreserving; reflexivity. Defined. | Lemma | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_assoc | 377 |
{A : AbGroup} : (functor_ab_biprod (@ab_diagonal A) grp_homo_id) $o ab_diagonal = ab_biprod_assoc $o (functor_ab_biprod grp_homo_id ab_diagonal) $o ab_diagonal := idpath. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_commute_id_diagonal | 378 |
`{Funext} {A : AbGroup} : (@ab_codiagonal A) $o (functor_ab_biprod ab_codiagonal grp_homo_id) $o ab_biprod_assoc = ab_codiagonal $o (functor_ab_biprod grp_homo_id ab_codiagonal). Proof. apply equiv_path_grouphomomorphism. intro a; cbn. exact (grp_assoc _ _ _)^. Defined. | Lemma | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_commute_id_codiagonal | 379 |
{A B C : AbGroup@{u}} : ab_biprod (ab_biprod A B) C $<~> ab_biprod (ab_biprod C B) A. Proof. snrapply Build_GroupIsomorphism. - snrapply Build_GroupHomomorphism. + intros [[a b] c]. exact ((c,b),a). + unfold IsSemiGroupPreserving. reflexivity. - snrapply isequiv_adjointify. + intros [[c b] a]. exact ((a,b),c). + reflexivity. + reflexivity. Defined. | Lemma | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_biprod_twist | 380 |
{A : AbGroup} : A $-> ab_biprod (ab_biprod A A) A := (functor_ab_biprod ab_diagonal grp_homo_id) $o ab_diagonal. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_triagonal | 381 |
{A : AbGroup} : ab_biprod (ab_biprod A A) A $-> A := ab_codiagonal $o (functor_ab_biprod ab_codiagonal grp_homo_id). | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_cotriagonal | 382 |
{A : AbGroup} : ab_biprod_twist $o @ab_triagonal A = ab_triagonal := idpath. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_triagonal_twist | 383 |
`{Funext} {A : AbGroup} : @ab_cotriagonal A $o ab_biprod_twist = ab_cotriagonal. Proof. apply equiv_path_grouphomomorphism. intro x. cbn. refine ((grp_assoc _ _ _)^ @ _). refine (abgroup_commutative _ _ _ @ _). exact (ap (fun a => a + snd x) (abgroup_commutative _ _ _)). Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import WildCat. Require Import HSet. Require Import AbelianGroup. Require Import Modalities.ReflectiveSubuniverse. | Algebra\AbGroups\Biproduct.v | ab_cotriagonal_twist | 384 |
{G : Group} (g : G) := fun h => g * h = h * g. | Definition | Require Import Basics Types Truncations.Core. Require Import HFiber AbelianGroup. | Algebra\AbGroups\Centralizer.v | centralizer | 385 |
{G : Group} (g : G) : centralizer g mon_unit. Proof. exact (grp_unit_r _ @ (grp_unit_l _)^). Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import HFiber AbelianGroup. | Algebra\AbGroups\Centralizer.v | centralizer_unit | 386 |
{G : Group} (g h k : G) (p : centralizer g h) (q : centralizer g k) : centralizer g (h * k). Proof. refine (grp_assoc _ _ _ @ _). refine (ap (fun x => x * k) p @ _). refine ((grp_assoc _ _ _)^ @ _). refine (ap (fun x => h * x) q @ _). apply grp_assoc. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import HFiber AbelianGroup. | Algebra\AbGroups\Centralizer.v | centralizer_sgop | 387 |
{G : Group} (g h : G) (p : centralizer g h) : centralizer g (-h). Proof. unfold centralizer in *. symmetry. refine ((grp_unit_r _)^ @ _ @ grp_unit_l _). refine (ap (fun x => (-h * g * x)) (grp_inv_r h)^ @ _ @ ap (fun x => x * (g * -h)) (grp_inv_l h)). refine (grp_assoc _ _ _ @ _ @ (grp_assoc _ _ _)^). refine (ap (fun x => x * (-h)) _). refine ((grp_assoc _ _ _)^ @ _ @ grp_assoc _ _ _). exact (ap (fun x => (-h) * x) p). Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import HFiber AbelianGroup. | Algebra\AbGroups\Centralizer.v | centralizer_inverse | 388 |
{G : Group} (g : G) := Build_Subgroup G (centralizer g) _. | Definition | Require Import Basics Types Truncations.Core. Require Import HFiber AbelianGroup. | Algebra\AbGroups\Centralizer.v | centralizer_subgroup | 389 |
{G : Group} (gen : Unit -> G) := subgroup_generated (hfiber gen). | Definition | Require Import Basics Types Truncations.Core. Require Import HFiber AbelianGroup. | Algebra\AbGroups\Centralizer.v | cyclic_subgroup_from_unit | 390 |
{G : Group} (g : G) := cyclic_subgroup_from_unit (unit_name g). Global Instance commutative_cyclic_subgroup {G : Group} (gen : Unit -> G) : Commutative (@group_sgop (cyclic_subgroup_from_unit gen)). Proof. intros h k. destruct h as [h H]; cbn in H. destruct k as [k K]; cbn in K. strip_truncations. apply (equiv_ap_isembedding (subgroup_incl _) _ _)^-1. cbn. induction H as [h [[] p]| |h1 h2 H1 H2 IHH1 IHH2]. - induction p. induction K as [k [[] q]| |k1 k2 K1 K2 IHK1 IHK2]. + induction q. reflexivity. + apply centralizer_unit. + srapply (issubgroup_in_op_inv (H:=centralizer (gen tt))); assumption. - symmetry; apply centralizer_unit. - symmetry. srapply (issubgroup_in_op_inv (H:=centralizer k)); unfold centralizer; symmetry; assumption. Defined. | Definition | Require Import Basics Types Truncations.Core. Require Import HFiber AbelianGroup. | Algebra\AbGroups\Centralizer.v | cyclic_subgroup | 391 |
{G : Group} (g : G) : AbGroup := Build_AbGroup (cyclic_subgroup g) _. | Definition | Require Import Basics Types Truncations.Core. Require Import HFiber AbelianGroup. | Algebra\AbGroups\Centralizer.v | abgroup_cyclic_subgroup | 392 |
(n : nat) : AbGroup := ab_cokernel (ab_mul (A:=abgroup_Z) n). | Definition | Require Import Basics.Overture Basics.Tactics WildCat.Core AbelianGroup | Algebra\AbGroups\Cyclic.v | cyclic | 393 |
(n : nat) : abgroup_Z $-> cyclic n := grp_quotient_map. | Definition | Require Import Basics.Overture Basics.Tactics WildCat.Core AbelianGroup | Algebra\AbGroups\Cyclic.v | cyclic_in | 394 |
(n : nat) (x y : abgroup_Z) : ab_mul y (cyclic_in n x) = cyclic_in n (y * x)%int. Proof. lhs_V nrapply ab_mul_natural. apply ap, abgroup_Z_ab_mul. Defined. | Definition | Require Import Basics.Overture Basics.Tactics WildCat.Core AbelianGroup | Algebra\AbGroups\Cyclic.v | ab_mul_cyclic_in | 395 |
{A : AbGroup} (n : nat) (f : forall k, (k < n)%nat -> A) : A. Proof. induction n as [|n IHn]. - exact zero. - refine (f n _ + IHn _). intros k Hk. exact (f k _). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import Spaces.Nat.Core Spaces.Int. Require Import AbelianGroup. | Algebra\AbGroups\FiniteSum.v | ab_sum | 396 |
{A : AbGroup} (n : nat) (a : A) (f : forall k, (k < n)%nat -> A) (p : forall k Hk, f k Hk = a) : ab_sum n f = ab_mul n a. Proof. induction n as [|n IHn] in f, p |- *. - reflexivity. - rhs_V nrapply (ap@{Set _} _ (int_nat_succ n)). rhs nrapply grp_pow_succ. simpl. f_ap. apply IHn. intros. apply p. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import Spaces.Nat.Core Spaces.Int. Require Import AbelianGroup. | Algebra\AbGroups\FiniteSum.v | ab_sum_const | 397 |
{A : AbGroup} (n : nat) (f : forall k, (k < n)%nat -> A) (p : forall k Hk, f k Hk = 0) : ab_sum n f = 0. Proof. lhs nrapply (ab_sum_const _ 0 f p). apply grp_pow_unit. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import Spaces.Nat.Core Spaces.Int. Require Import AbelianGroup. | Algebra\AbGroups\FiniteSum.v | ab_sum_zero | 398 |
{A : AbGroup} (n : nat) (f g : forall k, (k < n)%nat -> A) : ab_sum n (fun k Hk => f k Hk + g k Hk) = ab_sum n (fun k Hk => f k Hk) + ab_sum n (fun k Hk => g k Hk). Proof. induction n as [|n IHn]. 1: by rewrite grp_unit_l. simpl. rewrite <- !grp_assoc; f_ap. rewrite IHn, ab_comm, <- grp_assoc; f_ap. by rewrite ab_comm. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import Spaces.Nat.Core Spaces.Int. Require Import AbelianGroup. | Algebra\AbGroups\FiniteSum.v | ab_sum_plus | 399 |