phanerozoic/Coq-HoTT · Datasets at Hugging Face

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} {P Q : A -> Type} (f : forall a, P a -> Q a) (C : sig Q -> Type) (ef : forall a, ExtendableAlong 1 (f a) (fun v => C (a;v))) (s : forall z, C (functor_sigma idmap f z)) : ExtensionAlong (functor_sigma idmap f) C s. Proof. srefine (_;_). - intros [a v]; revert v. refine ((fst (ef a) _).1). intros u. exact (s (a;u)). - intros [a u]; cbn. exact ((fst (ef a) _).2 u). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	extension_functor_sigma_id	100
n {A} {P Q : A -> Type} (f : forall a, P a -> Q a) (C : sig Q -> Type) (ef : forall a, ExtendableAlong n (f a) (fun v => C (a;v))) : ExtendableAlong n (functor_sigma idmap f) C. Proof. revert C ef; simple_induction n n IH; intros C ef; [ exact tt \| split ]. - apply extension_functor_sigma_id. intros a; exact (fst (ef a) , fun _ _ => tt). - intros h k; apply IH. intros a; apply (snd (ef a)). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	extendable_functor_sigma_id	101
{A} {P Q : A -> Type} (f : forall a, P a -> Q a) (C : sig Q -> Type) (ef : forall a, ooExtendableAlong (f a) (fun v => C (a;v))) : ooExtendableAlong (functor_sigma idmap f) C := fun n => extendable_functor_sigma_id n f C (fun a => ef a n).	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	ooextendable_functor_sigma_id	102
{A B} {Q : B -> Type} (f : A -> B) (C : sig Q -> Type) (p : forall (a:A) (v:Q (f a)), C (f a;v)) := { q : forall (b:B) (v:Q b), C (b;v) & forall a v, q (f a) v = p a v }.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	HomotopyExtensionAlong	103
(n : nat) {A B} {Q : B -> Type} (f : A -> B) (C : sig Q -> Type) : Type := match n with \| 0 => Unit \| S n => ((forall (p : forall (a:A) (v:Q (f a)), C (f a;v)), HomotopyExtensionAlong f C p) * (forall (h k : forall z, C z), n f (fun z => h z = k z))) end.	Fixpoint	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	HomotopyExtendableAlong	104
{A B} {Q : B -> Type} (f : A -> B) (C : sig Q -> Type) := forall n, HomotopyExtendableAlong n f C.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	ooHomotopyExtendableAlong	105
{A B} {P : A -> Type} {Q : B -> Type} (f : A -> B) (g : forall a, P a -> Q (f a)) (C : sig Q -> Type) (ef : HomotopyExtendableAlong 1 f C) (eg : forall a, ExtendableAlong 1 (g a) (fun v => C (f a ; v))) (s : forall z, C (functor_sigma f g z)) : ExtensionAlong (functor_sigma f g) C s. Proof. srefine (_;_). - intros [b v]; revert b v. refine ((fst ef _).1). intros a. refine ((fst (eg a) _).1). intros u. exact (s (a;u)). - intros [a u]; cbn. refine ((fst ef _).2 _ _ @ _). exact ((fst (eg a) _).2 u). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	extension_functor_sigma	106
(n : nat) {A B} {P : A -> Type} {Q : B -> Type} (f : A -> B) (g : forall a, P a -> Q (f a)) (C : sig Q -> Type) (ef : HomotopyExtendableAlong n f C) (eg : forall a, ExtendableAlong n (g a) (fun v => C (f a ; v))) : ExtendableAlong n (functor_sigma f g) C. Proof. revert C ef eg; simple_induction n n IH; intros C ef eg; [ exact tt \| split ]. - apply extension_functor_sigma. + exact (fst ef, fun _ _ => tt). + intros a; exact (fst (eg a) , fun _ _ => tt). - intros h k; apply IH. + exact (snd ef h k). + intros a; apply (snd (eg a)). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	extendable_functor_sigma	107
{A B} {P : A -> Type} {Q : B -> Type} (f : A -> B) (g : forall a, P a -> Q (f a)) (C : sig Q -> Type) (ef : ooHomotopyExtendableAlong f C) (eg : forall a, ooExtendableAlong (g a) (fun v => C (f a ; v))) : ooExtendableAlong (functor_sigma f g) C := fun n => extendable_functor_sigma n f g C (ef n) (fun a => eg a n).	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	ooextendable_functor_sigma	108
{A B A' B'} (f : A -> A') (g : B -> B') (P : A' + B' -> Type) (ef : ExtendableAlong 1 f (P o inl)) (eg : ExtendableAlong 1 g (P o inr)) (h : forall z, P (functor_sum f g z)) : ExtensionAlong (functor_sum f g) P h. Proof. srefine (sum_ind _ _ _ ; sum_ind _ _ _). + exact (fst ef (h o inl)).1. + exact (fst eg (h o inr)).1. + exact (fst ef (h o inl)).2. + exact (fst eg (h o inr)).2. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	extension_functor_sum	109
(n : nat) {A B A' B'} (f : A -> A') (g : B -> B') (P : A' + B' -> Type) (ef : ExtendableAlong n f (P o inl)) (eg : ExtendableAlong n g (P o inr)) : ExtendableAlong n (functor_sum f g) P. Proof. revert P ef eg; induction n as [\|n IH]; intros P ef eg; [ exact tt \| split ]. - intros h; apply extension_functor_sum. + exact (fst ef, fun _ _ => tt). + exact (fst eg, fun _ _ => tt). - intros h k. apply IH. + exact (snd ef (h o inl) (k o inl)). + exact (snd eg (h o inr) (k o inr)). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	extendable_functor_sum	110
{A B A' B'} (f : A -> A') (g : B -> B') (P : A' + B' -> Type) (ef : ooExtendableAlong f (P o inl)) (eg : ooExtendableAlong g (P o inr)) : ooExtendableAlong (functor_sum f g) P. Proof. intros n; apply extendable_functor_sum; [ apply ef \| apply eg ]. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	ooextendable_functor_sum	111
{B A f g B' A' f' g'} {h : B -> B'} {k : A -> A'} {p : k o f == f' o h} {q : k o g == g' o h} {C : Coeq f' g' -> Type} (ek : ExtendableAlong 1 k (C o coeq)) (eh : forall (u v : forall x : B', C (coeq (g' x))), ExtendableAlong 1 h (fun x => u x = v x)) (s : forall x, C (functor_coeq h k p q x)) : ExtensionAlong (functor_coeq h k p q) C s. Proof. set (C' := C o pr_cylcoeq p q). set (s' x := pr_cyl_cylcoeq p q x # s x). assert (e : ExtensionAlong (cyl_cylcoeq p q) C' s'). 2:{ pose (ex := fst (extendable_equiv 1 C (pr_cylcoeq p q)) e.1). exists (ex.1); intros x. apply (equiv_inj (transport C (pr_cyl_cylcoeq p q x))). exact (apD _ (pr_cyl_cylcoeq p q x) @ ex.2 _ @ e.2 x). } pose (ea1 := fun u => (fst (cyl_extendable' 1 _ _ ek) u).1). assert (eb'' : forall u v, ExtendableAlong 1 cyl (fun x:Cyl h => DPath C' (cglue x) (u x) (v x))). { intros u v. rapply extendable_postcompose'. 2:{ rapply (cancelL_extendable 1 _ cyl pr_cyl). - rapply extendable_equiv. - exact (eh (fun x => cglue x # u (cyr x)) (v o cyr)). } intros x; subst C'. refine ((dp_compose (pr_cylcoeq p q) C _)^-1 oE _). symmetry; srapply equiv_ds_fill_lr. 3:rapply ap_pr_cylcoeq_cglue. all:srapply (transport (fun r => DPath C r _ _)). 3:exact (dp_inverse (dp_compose _ C _ (apD u (eissect pr_cyl x) : DPath _ _ _ _))). 4:exact (dp_inverse (dp_compose _ C _ (apD v (eissect pr_cyl x) : DPath _ _ _ _))). 1:change (fun y => pr_cylcoeq p q (coeq (functor_cyl p y))) with (fun y => coeq (f := f') (g := g') (pr_cyl (functor_cyl p y))). 2:change (fun y => pr_cylcoeq p q (coeq (functor_cyl q y))) with (fun y => coeq (f := f') (g := g') (pr_cyl (functor_cyl q y))). all:refine ((ap_V _ (eissect pr_cyl x))^ @ _). all: exact (ap_compose (fun x => pr_cyl (functor_cyl _ x)) coeq _). } pose (eb1 := fun u v w => (fst (cyl_extendable _ _ _ (eb'' u v)) w).1). srefine (_;_); srapply Coeq_ind. + exact (ea1 (s' o coeq)). + apply eb1; intros b. rapply (dp_compose' _ _ (ap_cyl_cylcoeq_cglue p q b)). exact (apD s' (cglue b)). + intros a; reflexivity. + intros b. apply ds_dp. rapply ds_G1. refine (dp_apD_compose' _ _ (ap_cyl_cylcoeq_cglue p q b) _ @ _). apply moveR_equiv_V. nrapply Coeq_ind_beta_cglue. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	extension_functor_coeq	112
(n : nat) {B A f g B' A' f' g'} {h : B -> B'} {k : A -> A'} {p : k o f == f' o h} {q : k o g == g' o h} {C : Coeq f' g' -> Type} (ek : ExtendableAlong n k (C o coeq)) (eh : forall (u v : forall x : B', C (coeq (g' x))), ExtendableAlong n h (fun x => u x = v x)) : ExtendableAlong n (functor_coeq h k p q) C. Proof. revert C ek eh; simple_induction n n IH; intros C ek eh; [ exact tt \| split ]. - apply extension_functor_coeq. + exact (fst ek , fun _ _ => tt). + exact (fun u v => (fst (eh u v) , fun _ _ => tt)). - intros u v; apply IH. + exact (snd ek (u o coeq) (v o coeq)). + exact (snd (eh (u o coeq o g') (v o coeq o g'))). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	extendable_functor_coeq	113
{B A f g B' A' f' g'} {h : B -> B'} {k : A -> A'} {p : k o f == f' o h} {q : k o g == g' o h} {C : Coeq f' g' -> Type} (ek : ooExtendableAlong k (C o coeq)) (eh : forall (u v : forall x : B', C (coeq (g' x))), ooExtendableAlong h (fun x => u x = v x)) : ooExtendableAlong (functor_coeq h k p q) C := fun n => extendable_functor_coeq n (ek n) (fun u v => eh u v n).	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	ooextendable_functor_coeq	114
extendable_functor_coeq' (n : nat) {B A f g B' A' f' g'} {h : B -> B'} {k : A -> A'} {p : k o f == f' o h} {q : k o g == g' o h} {C : Coeq f' g' -> Type} (ek : ExtendableAlong n k (C o coeq)) (eh : ExtendableAlong n.+1 h (C o coeq o g')) : ExtendableAlong n (functor_coeq h k p q) C. Proof. apply extendable_functor_coeq. 1:assumption. exact (snd eh). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	extendable_functor_coeq'	115
ooextendable_functor_coeq' {B A f g B' A' f' g'} {h : B -> B'} {k : A -> A'} {p : k o f == f' o h} {q : k o g == g' o h} {C : Coeq f' g' -> Type} (ek : ooExtendableAlong k (C o coeq)) (eh : ooExtendableAlong h (C o coeq o g')) : ooExtendableAlong (functor_coeq h k p q) C := fun n => extendable_functor_coeq' n (ek n) (eh n.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.	Extensions.v	ooextendable_functor_coeq'	116
	Record	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	Factorization	117
{ I : Type & { g : A -> I & { h : I -> B & { p : h o g == f & { gin1 : class1 _ _ g & class2 _ _ h }}}}} <~> Factorization. Proof. issig. Defined.	Lemma	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	issig_Factorization	118
{fact fact' : Factorization} :=	Record	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	PathFactorization	119
(fact fact' : Factorization) : { path_intermediate : intermediate fact <~> intermediate fact' & { path_factor1 : path_intermediate o factor1 fact == factor1 fact' & { path_factor2 : factor2 fact == factor2 fact' o path_intermediate & forall a, path_factor2 (factor1 fact a) @ ap (factor2 fact') (path_factor1 a) @ fact_factors fact' a = fact_factors fact a }}} <~> PathFactorization fact fact'. Proof. issig. Defined.	Lemma	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	issig_PathFactorization	120
`{Univalence} (fact fact' : Factorization) : PathFactorization fact fact' <~> fact = fact'. Proof. refine (_ oE (issig_PathFactorization fact fact')^-1). revert fact fact'; apply (equiv_path_issig_contr issig_Factorization). { intros [I [f1 [f2 [ff [oc1 oc2]]]]]. exists (equiv_idmap I); cbn. exists (fun x:A => 1%path); cbn. exists (fun x:I => 1%path); cbn. intros; apply concat_1p. } intros [I [f1 [f2 [ff [oc1 oc2]]]]]. contr_sigsig I (equiv_idmap I); cbn. contr_sigsig f1 (fun x:A => idpath (f1 x)); cbn. contr_sigsig f2 (fun x:I => idpath (f2 x)); cbn. refine (contr_equiv' {ff' : f2 o f1 == f & ff == ff'} _). symmetry; srefine (equiv_functor_sigma' (equiv_sigma_contr _) _). { intros h; cbn. srefine (@istrunc_sigma _ _ _ _ _); [ \| intros a]; apply contr_inhabited_hprop; try exact _; assumption. } intros [ff' [oc1' oc2']]; cbn. refine (equiv_functor_forall' (equiv_idmap _) _); intros a. refine (equiv_path_inverse _ _ oE _). apply equiv_concat_l; symmetry; apply concat_1p. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	equiv_path_factorization	121
`{Univalence} (fact fact' : Factorization) : PathFactorization fact fact' -> fact = fact' := equiv_path_factorization fact fact'.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	path_factorization	122
FactorizationSystem@{i j k} :=	Record	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	FactorizationSystem@	123
`{Univalence} (factsys : FactorizationSystem) {X Y : Type} (f : X -> Y) : Contr (Factorization (@class1 factsys) (@class2 factsys) f). Proof. apply contr_inhabited_hprop. - apply hprop_allpath. intros fact fact'. apply path_factorization; try exact _. apply path_factor. - apply factor. Defined.	Theorem	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	contr_factor	124
Build_Factorization' {X Y} := @Build_Factorization (@class1 factsys) (@class2 factsys) X Y.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	Build_Factorization'	125
Build_PathFactorization' {X Y} := @Build_PathFactorization (@class1 factsys) (@class2 factsys) X Y.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	Build_PathFactorization'	126
`{Funext} {X Y Z} (f : X -> Y) (g : Y -> Z) : class1 factsys f -> class1 factsys (g o f) -> class1 factsys g. Proof. intros c1f c1gf. destruct (factor factsys g) as [I g1 g2 gf c1g1 c2g2]. pose (fact := Build_Factorization' (g o f) Z (g o f) (idmap) (fun x => 1) c1gf (class2_isequiv factsys idmap)). pose (fact' := Build_Factorization' (g o f) I (g1 o f) g2 (fun x => gf (f x)) (class1_compose factsys f g1 c1f c1g1) c2g2). destruct (path_factor factsys (g o f) fact' fact) as [q q1 q2 qf]; simpl in *. refine (transport (class1 factsys) (path_arrow _ _ gf) _). refine (class1_compose factsys g1 g2 c1g1 _). apply class1_isequiv. apply (isequiv_homotopic _ (fun i => (q2 i)^)). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	cancelR_class1	127
`{Funext} {X Y Z} (f : X -> Y) (g : Y -> Z) : class2 factsys g -> class2 factsys (g o f) -> class2 factsys f. Proof. intros c2g c2gf. destruct (factor factsys f) as [I f1 f2 ff c1f1 c2f2]. pose (fact := Build_Factorization' (g o f) X (idmap) (g o f) (fun x => 1) (class1_isequiv factsys idmap) c2gf). pose (fact' := Build_Factorization' (g o f) I f1 (g o f2) (fun x => ap g (ff x)) c1f1 (class2_compose factsys f2 g c2f2 c2g)). destruct (path_factor factsys (g o f) fact fact') as [q q1 q2 qf]; simpl in *. refine (transport (class2 factsys) (path_arrow _ _ ff) _). refine (class2_compose factsys f1 f2 _ c2f2). apply class2_isequiv. apply (isequiv_homotopic _ q1). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	cancelL_class2	128
B -> X := f2 o q^-1 o g1.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	lift_factsys	129
lift_factsys o i == f. Proof. intros x. refine (ap (f2 o q^-1) (q1 x)^ @ _). transitivity (f2 (f1 x)). + apply ap, eissect. + apply ff. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	lift_factsys_tri1	130
p o lift_factsys == g. Proof. intros x. refine (q2 _ @ _). transitivity (g2 (g1 x)). + apply ap, eisretr. + apply gf. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	lift_factsys_tri2	131
(x : A) : ap p (lift_factsys_tri1 x)^ @ lift_factsys_tri2 (i x) = h x. Proof. unfold lift_factsys_tri1, lift_factsys_tri2. Open Scope long_path_scope. pose (r := path_fact_factors (path_factor factsys (p o f) fact fact') x : q2 (f1 x) @ ap g2 (q1 x) @ (gf (i x) @ (h x)^) = ap p (ff x)). rewrite concat_p_pp in r. apply moveL_pM, moveR_Vp in r. refine (_ @ r); clear r. repeat rewrite inv_pp; repeat rewrite ap_pp; rewrite ap_V. repeat rewrite concat_pp_p; apply whiskerL. repeat rewrite concat_p_pp; apply whiskerR. rewrite (ap_compose q^-1 f2), <- ap_pp, <- inv_pp. unfold lift_factsys. rewrite <- ap_pp. rewrite <- ap_V, <- ap_compose. rewrite (concat_Ap q2). rewrite concat_pp_p; apply whiskerL. rewrite ap_compose. rewrite <- ap_pp. apply ap. rewrite inv_pp, ap_pp, ap_V. rewrite <- eisadj. rewrite concat_pp_p; apply moveR_Vp. rewrite <- ap_V, inv_V, <- ap_compose. exact (concat_A1p (eisretr q) (q1 x)). Close Scope long_path_scope. Qed.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	lift_factsys_square	132
{A B : Type} (f : A -> B) {c1f : class1 factsys f} (P : B -> Type) (c2P : class2 factsys (@pr1 B P)) (d : forall a:A, P (f a)) : ExtensionAlong f P d. Proof. pose (e := lift_factsys factsys f c1f pr1 c2P (fun a => (f a ; d a)) idmap (fun a => 1)). pose (e2 := lift_factsys_tri2 factsys f c1f pr1 c2P (fun a => (f a ; d a)) idmap (fun a => 1)). exists (fun a => (e2 a) # (e a).2). intros a. pose (e1 := lift_factsys_tri1 factsys f c1f pr1 c2P (fun a => (f a ; d a)) idmap (fun a => 1) a : e (f a) = (f a ; d a)). pose (e3 := moveL_M1 _ _ (((ap_V _ _)^ @@ 1) @ lift_factsys_square factsys f c1f pr1 c2P (fun a => (f a ; d a)) idmap (fun a => 1) a) : e2 (f a) = pr1_path e1). refine (ap (fun p => transport P p (e (f a)).2) e3 @ _). exact (pr2_path e1). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.	Factorization.v	extension_factsys	133
{A B} (f : A -> B) `{IsEquiv _ _ f} : IsEquiv (fmap F f). Proof. refine (equiv_induction (fun A' e => IsEquiv (fmap F e)) _ _ (Build_Equiv _ _ f _)). refine (transport _ (fmap_idmap F)^ _); try apply isequiv_idmap. Defined.	Proposition	Require Import HoTT.Basics Types.Universe.	Functorish.v	isequiv_fmap	134
{A B} (f : A -> B) `{IsEquiv _ _ f} : fmap F f = equiv_path _ _ (ap F (path_universe f)). Proof. refine (equiv_induction (fun A' e => fmap F e = equiv_path _ _ (ap F (path_universe e))) _ _ (Build_Equiv _ _ f _)). transitivity (idmap : F A -> F A). - apply fmap_idmap. - change (equiv_idmap A) with (equiv_path A A 1). rewrite (@eta_path_universe _ A A 1). exact 1. Defined.	Proposition	Require Import HoTT.Basics Types.Universe.	Functorish.v	fmap_agrees_with_univalence	135
{A B : Type} {f : A -> B} {y : B} (x1 x2 : hfiber f y) : { q : x1.1 = x2.1 & x1.2 = ap f q @ x2.2 } <~> (x1 = x2). Proof. refine (equiv_path_sigma _ _ _ oE _). apply equiv_functor_sigma_id. intros p; simpl. refine (_ oE equiv_moveR_Vp _ _ _). exact (equiv_concat_l (transport_paths_Fl _ _) _). Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	equiv_path_hfiber	136
{A B : Type} {f : A -> B} {y : B} {x1 x2 : hfiber f y} : { q : x1.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	path_hfiber_uncurried	137
{A B : Type} {f : A -> B} {y : B} {x1 x2 : hfiber f y} (q : x1.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	path_hfiber	138
{A B : Type} {f : A -> B} {x1 x2 : A} (p : f x1 = f x2) : hfiber (ap f) p <~> ((x1 ; p) = (x2 ; 1) :> hfiber f (f x2)). Proof. refine (equiv_path_hfiber (x1;p) (x2;1%path) oE _). unfold hfiber; simpl. apply equiv_functor_sigma_id; intros q. refine (_ oE equiv_path_inverse _ _). exact (equiv_concat_r (concat_p1 _)^ _). Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	hfiber_ap	139
{A B : Type} (f g : A -> B) (h : f == g) (b : B) : hfiber f b <~> hfiber g b. Proof. refine (Build_Equiv _ _ (fun fx => (fx.1 ; (h fx.1)^ @ fx.2)) _). refine (isequiv_adjointify _ (fun gx => (gx.1 ; (h gx.1) @ gx.2)) _ _); intros [a p]; simpl; apply ap. - apply concat_V_pp. - apply concat_p_Vp. Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	equiv_hfiber_homotopic	140
{A B C D} {f : A -> B} {g : C -> D} {h : A -> C} {k : B -> D} (p : k o f == g o h) (b : B) : hfiber f b -> hfiber g (k b). Proof. snrapply @functor_sigma. - exact h. - intros a e; exact ((p a)^ @ ap k e). Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	functor_hfiber	141
{A B C D} {f : A -> B} {g : C -> D} {h : A -> C} {k : B -> D} `{IsEquiv A C h} `{IsEquiv B D k} (p : k o f == g o h) (b : B) : IsEquiv (functor_hfiber p b) := _.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	isequiv_functor_hfiber	142
{A B C D} {f : A -> B} {g : C -> D} {h : A <~> C} {k : B <~> D} (p : k o f == g o h) (b : B) : hfiber f b <~> hfiber g (k b) := Build_Equiv _ _ (functor_hfiber p b) _.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	equiv_functor_hfiber	143
{A B C D} {f : A -> B} {g : C -> D} {h : A -> C} {k : B -> D} (p : k o f == g o h) {b : B} {b' : D} (q : k b = b') : hfiber f b -> hfiber g b'. Proof. srapply functor_sigma. - exact h. - intros a e. exact ((p a)^ @ ap k e @ q). Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	functor_hfiber2	144
{A B C D} {f : A -> B} {g : C -> D} {h : A <~> C} {k : B <~> D} (p : k o f == g o h) {b : B} {b' : D} (q : k b = b') : hfiber f b <~> hfiber g b' := Build_Equiv _ _ (functor_hfiber2 p q) _.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	equiv_functor_hfiber2	145
{A B C X Y Z : Type} {k : A -> B} {l : B -> C} {f : A -> X} {g : B -> Y} {h : C -> Z} {i : X -> Y} {j : Y -> Z} (H : i o f == g o k) (K : j o g == h o l) : forall x, functor_hfiber (comm_square_comp' H K) x == (functor_hfiber K (i x)) o (functor_hfiber H x : hfiber f x -> _). Proof. intros x [y p]. destruct p. apply (path_sigma' _ idpath). refine (concat_p1 _ @ _). refine (inv_pp _ _ @ ap _ _). refine ((ap_V _ _)^ @ ap _ _^). apply concat_p1. Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	functor_hfiber_compose	146
{A B C D} {f : A -> B} {g : C -> D} {h : A -> C} {k : B -> D} (p : k o f == g o h) (b : B) (c : C) (q : g c = k b) : hfiber (functor_hfiber p b) (c;q) <~> hfiber (functor_hfiber (fun x => (p x)^) c) (b;q^). Proof. rapply (equiv_functor_sigma_id _ oE _ oE (equiv_functor_sigma_id _)^-1). 1,3:intros; rapply equiv_path_sigma. refine (equiv_sigma_assoc _ _ oE _ oE (equiv_sigma_assoc _ _)^-1). apply equiv_functor_sigma_id; intros a; cbn. refine (equiv_sigma_symm _ oE _). do 2 (apply equiv_functor_sigma_id; intro). refine ((equiv_ap inverse _ _)^-1 oE _). refine (equiv_concat_r (inv_V q)^ _ oE _). apply equiv_concat_l. abstract (rewrite !transport_paths_Fl, !inv_pp, !inv_V, concat_pp_p; reflexivity). Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	hfiber_functor_hfiber	147
{B C} (f:C ->B) : C <~> {y:B & hfiber f y}. Proof. snrefine (Build_Equiv _ _ _ ( Build_IsEquiv C {y:B & {x:C & f x = y}} (fun c => (f c; (c; idpath))) (fun c => c.2.1) _ (fun c => idpath) _)). - intros [? [? []]]; reflexivity. - reflexivity. Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	equiv_fibration_replacement	148
{X} (x : X) (P:X->Type) : P x <~> @hfiber (sig P) X pr1 x. Proof. snrefine (Build_Equiv _ _ _ (Build_IsEquiv (P x) { z : sig P & z.1 = x } (fun Px => ((x; Px); idpath)) (fun Px => transport P Px.2 Px.1.2) _ (fun Px => idpath) _)). - intros [[] []]; reflexivity. - reflexivity. Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	hfiber_fibration	149
(c : C) : hfiber (g o f) c -> hfiber g c := fun x => (f x.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	hfiber_compose_map	150
(b : B) : hfiber (hfiber_compose_map (g b)) (b;1) <~> hfiber f b. Proof. unfold hfiber, hfiber_compose_map. refine (_ oE equiv_functor_sigma_id (fun x => (equiv_path_sigma _ _ _)^-1)); cbn. make_equiv_contr_basedpaths. Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	hfiber_hfiber_compose_map	151
(c : C) : hfiber (g o f) c <~> { w : hfiber g c & hfiber f w.1 }. Proof. make_equiv_contr_basedpaths. Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	hfiber_compose	152
`{Funext} {A : Type} {P Q : A -> Type} (h : forall a, P a -> Q a) (g : forall a, Q a) : hfiber (functor_forall_id h) g <~> (forall a, hfiber (h a) (g a)). Proof. unfold hfiber, functor_forall_id, functor_forall. nrefine (equiv_sig_coind _ _ oE _). apply equiv_functor_sigma_id; intro f. apply equiv_apD10. Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	hfiber_functor_forall_id	153
A {B} (y y' : B) : hfiber (fun _ : A => y) y' <~> A * (y = y') := equiv_sigma_prod0 A (y = y').	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	hfiber_const	154
{A B} n (f:A -> B) `{!forall x y, IsTruncMap n (@ap _ _ f x y)} : IsTruncMap n.+1 f. Proof. intro y; apply istrunc_S. intros [a p] [b q]; destruct q; exact (istrunc_equiv_istrunc _ (hfiber_ap p)). Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	istruncmap_from_ap	155
`{Funext} {A B} n (f:A -> B) : IsTruncMap n.+1 f <~> (forall x y, IsTruncMap n (@ap _ _ f x y)) := equiv_iff_hprop (@istruncmap_ap _ _ n f) (@istruncmap_from_ap _ _ n f). Global Instance isequiv_ap_isembedding {A B} (f : A -> B) `{!IsEmbedding f} : forall x y, IsEquiv (@ap _ _ f x y). Proof. intros x y. apply isequiv_contr_map,_. Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	equiv_istruncmap_ap	156
{A B} (f : A -> B) `{!IsEmbedding f} (x y : A) : (x = y) <~> (f x = f y) := Build_Equiv _ _ (ap f) _.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	equiv_ap_isembedding	157
{A B} (f : A -> B) `{!forall x y, IsEquiv (@ap _ _ f x y)} : IsEmbedding f. Proof. rapply istruncmap_from_ap. Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	isembedding_isequiv_ap	158
`{Funext} {A B} (f : A -> B) : IsEmbedding f <~> (forall x y, IsEquiv (@ap _ _ f x y)). Proof. exact (equiv_iff_hprop (@isequiv_ap_isembedding _ _ f) (@isembedding_isequiv_ap _ _ f)). Defined.	Definition	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	equiv_isequiv_ap_isembedding	159
{X Y : Type} (f : X -> Y) {I : IsEmbedding f} {x0 x1 : X} : forall (p : f x0 = f x1), ap f (isinj_embedding f I x0 x1 p) = p. Proof. equiv_intro (equiv_ap_isembedding f x0 x1) q. induction q. cbn. exact (ap _ (isinj_embedding_beta f)). Defined.	Lemma	Require Import Basics Types Diagrams.CommutativeSquares HSet.	HFiber.v	ap_isinj_embedding_beta	160
{X : Type} :=	Record	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	RetractOf	161
(X : Type) : RetractOf X := Build_RetractOf X X idmap idmap (fun _ => 1).	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	idmap_retractof	162
{X Y : Type} (f : X -> Y) `{feq : IsEquiv _ _ f} : RetractOf X -> RetractOf Y. Proof. intros [A r s H]; refine (Build_RetractOf Y A (r o f^-1) (f o s) _); intros x. exact (ap r (eissect f (s x)) @ H x). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	retractof_equiv	163
retractof_equiv' {X Y : Type} (f : X <~> Y) : RetractOf X -> RetractOf Y := retractof_equiv f.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	retractof_equiv'	164
{X : Type} (R : RetractOf X) {B : Type} (f : retract_type R -> B) `{feq : IsEquiv _ _ f} : RetractOf X. Proof. destruct R as [A r s H]; refine (Build_RetractOf X B (f o r) (s o f^-1) _); intros x. exact (ap f (H (f^-1 x)) @ eisretr f x). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	equiv_retractof	165
equiv_retractof' {X : Type} (R : RetractOf X) {B : Type} (f : retract_type R <~> B) : RetractOf X := equiv_retractof R f.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	equiv_retractof'	166
{X Y : Type} (R : RetractOf X) (f : X -> Y) (g : retract_type R -> Y) (p : g o retract_retr R == f) (y : Y) : RetractOf (hfiber f y). Proof. destruct R as [A r s H]; simpl in *. simple refine (Build_RetractOf (hfiber f y) (hfiber g y) _ _ _). - intros [x q]. exists (r x). exact (p x @ q). - intros [a q]. exists (s a). exact ((p (s a))^ @ ap g (H a) @ q). - intros [a q]. simple refine (path_sigma' _ _ _). + exact (H a). + abstract ( rewrite transport_paths_Fl, !concat_p_pp, concat_pp_V, concat_Vp, concat_1p; reflexivity). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	retractof_hfiber	167
{X : Type} (R : RetractOf X ) (contra : Contr X) : Contr (retract_type R ) := contr_retract (retract_retr R) (retract_sect R) (retract_issect R).	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	contr_retracttype	168
(X : Type) : { A : Type & {r : X -> A & {s : A -> X & r o s == idmap }}} <~> RetractOf X. Proof. issig. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	issig_retractof	169
(X : Type) (R' R : RetractOf X) := { Ap : retract_type R' <~> retract_type R & { rp : Ap o retract_retr R' == retract_retr R & { sp : retract_sect R' o Ap^-1 == retract_sect R & forall a, ap Ap (retract_issect R' (Ap^-1 a)) @ eisretr Ap a = rp (retract_sect R' (Ap^-1 a)) @ ap (retract_retr R) (sp a) @ retract_issect R a } } }.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	PathRetractOf	170
`{ua : Univalence} {X : Type} (R' R : RetractOf X) : PathRetractOf X R' R <~> R' = R. Proof. revert R' R; apply (equiv_path_issig_contr (issig_retractof X)). { intros [A [r [s H]]]; cbn. exists equiv_idmap. exists (fun x => 1%path). exists (fun x => 1%path). cbn. exact (fun a => equiv_p1_1q (ap_idmap (H a))). } intros [A [r [s H]]]; cbn. unfold PathRetractOf. contr_sigsig A (equiv_idmap A); cbn. contr_sigsig r (fun x:X => idpath (r x)); cbn. contr_sigsig s (fun x:A => idpath (s x)); cbn. refine (contr_equiv' {K : r o s == idmap & H == K} _). apply equiv_functor_sigma_id; intros K. apply equiv_functor_forall_id; intros a; cbn. apply equiv_concat_lr. - refine (concat_p1 _ @ ap_idmap (H a)). - symmetry; apply concat_1p. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	equiv_path_retractof	171
`{ua : Univalence} {X : Type} {R' R : RetractOf X} Ap rp sp Hp : R' = R := equiv_path_retractof R' R (Ap;rp;sp;Hp).	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	path_retractof	172
{X : Type} (R : RetractOf X) : (X -> X) := retract_sect R o retract_retr R.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	retract_idem	173
{X : Type} (f : X -> X) := { R : RetractOf X & retract_idem R == f}.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	Splitting	174
(X : Type) : @Splitting X idmap := (idmap_retractof X ; fun _ => 1).	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	splitting_idmap	175
{X : Type} (f : X -> X) `{IsPreIdempotent _ f} {g : X -> X} (p : f == g) : IsPreIdempotent g. Proof. intros x; refine (_ @ isidem f x @ p x). refine (_ @ (p (f x))^). apply ap; symmetry; apply p. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	ispreidem_homotopic	176
(X : Type) := { f : X -> X & IsPreIdempotent f }.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	PreIdempotent	177
{X : Type} : PreIdempotent X -> X -> X := pr1.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	preidempotent_pr1	178
(X : Type) : PreIdempotent X. Proof. exists idmap; exact _. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	preidem_idmap	179
(X : Type) `{IsHSet X} (f : PreIdempotent X) : Splitting f. Proof. simple refine (Build_RetractOf X { x : X & f x = x } (fun x => (f x ; isidem f x)) pr1 _ ; _). - intros [x p]; simpl. apply path_sigma with p; simpl. apply path_ishprop. - simpl. intros x; reflexivity. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	split_preidem_set	180
(X : Type) (f : PreIdempotent X) `{WeaklyConstant _ _ f} : Splitting f. Proof. simple refine (Build_RetractOf X (FixedBy f) (fun x => (f x ; isidem f x)) pr1 _ ; _). - intros x; apply path_ishprop. - simpl. intros x; reflexivity. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	split_preidem_wconst	181
(X : Type) (f : PreIdempotent X) (ss : forall x, Collapsible (f x = x)) : Splitting f. Proof. simple refine (Build_RetractOf X { x : X & FixedBy (@collapse (f x = x) _) } _ pr1 _ ; _). - intros x; exists (f x); unfold FixedBy. exists (collapse (isidem f x)). apply wconst. - intros [x [p q]]; simpl. apply path_sigma with p. apply path_ishprop. - simpl. intros x; reflexivity. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	split_preidem_splitsupp	182
(X : Type) (f : PreIdempotent X) (ss : forall x, Collapsible (f x = x)) : IsEmbedding (retract_sect (split_preidem_splitsupp X f ss).1). Proof. apply istruncmap_mapinO_tr; exact _. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	isemb_split_preidem_splitsupp	183
(X : Type) (f : X -> X) (S : Splitting f) `{IsEmbedding (retract_sect S.1)} : forall x, Collapsible (f x = x). Proof. intros x. destruct S as [[A r s H] K]; simpl in *. assert (c1 : f x = x -> { a : A & s a = x }). { intros p; exists (r x). exact (K x @ p). } assert (c2 : { a : A & s a = x } -> f x = x). { intros [a q]. exact ((K x)^ @ ap (s o r) q^ @ ap s (H a) @ q). } exists (c2 o c1). apply wconst_through_hprop. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	splitsupp_split_isemb	184
{X : Type} (f : X -> X) `{IsQuasiIdempotent _ f} {g : X -> X} (p : f == g) : @IsQuasiIdempotent X g (ispreidem_homotopic f p). Proof. intros x; unfold isidem; simpl. Open Scope long_path_scope. rewrite (concat_Ap (fun x => (p x)^) (p x)^). rewrite !ap_pp, !concat_pp_p; apply whiskerL. rewrite !concat_p_pp; apply moveL_pM. rewrite (concat_pA_p (fun x => (p x)^) (p x)). rewrite (concat_pA_p (fun x => (p x)^) (isidem _ x)). rewrite (concat_Ap (fun x => (p x)^) (ap f (p x)^)). rewrite !concat_pp_p; apply whiskerL. rewrite !ap_V; apply moveR_Vp. rewrite <- ap_compose. rewrite isidem2; try exact _. symmetry; refine (concat_Ap (isidem f) (p x)). Close Scope long_path_scope. Qed.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	isqidem_homotopic	185
(X : Type) := { f : PreIdempotent X & IsQuasiIdempotent f }.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	QuasiIdempotent	186
{X : Type} : QuasiIdempotent X -> X -> X := pr1.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	quasiidempotent_pr1	187
(X : Type) : QuasiIdempotent X. Proof. exists (preidem_idmap X); exact _. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	qidem_idmap	188
{X : Type} (R : RetractOf X) : PreIdempotent X := (retract_idem R ; ispreidem_retract R). Arguments ispreidem_retract / . Arguments / . Global Instance isqidem_retract {X : Type} (R : RetractOf X) : IsQuasiIdempotent (retract_idem R). Proof. destruct R as [A r s H]; intros x; unfold isidem; simpl. refine ((ap_compose _ _ _) @ _). apply ap. refine ((ap_compose _ _ _)^ @ _). refine (cancelR _ _ (H (r x)) _). refine (concat_A1p H (H (r x))). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	preidem_retract	189
{X : Type} (R : RetractOf X) : QuasiIdempotent X := (preidem_retract R ; isqidem_retract R). Global Instance ispreidem_split {X : Type} (f : X -> X) (S : Splitting f) : IsPreIdempotent f. Proof. destruct S as [R p]. refine (ispreidem_homotopic _ p); exact _. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	qidem_retract	190
Type := { a : nat -> X & forall n, f (a n.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	split_idem	191
split_idem -> (nat -> X) := pr1.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	split_idem_pr1	192
split_idem -> X := fun a => a 0.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	split_idem_sect	193
X -> split_idem. Proof. intros x. exists (fun n => f x). exact (fun n => I x). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	split_idem_retr	194
(x : X) : split_idem_sect (split_idem_retr x) = f x := 1.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	split_idem_splits	195
{a a' : split_idem} (p : a.1 == a'.1) (q : forall n, a.2 n @ p n = ap f (p n.+1) @ a'.2 n) : a = a'. Proof. simple refine (path_sigma' _ _ _). - apply path_arrow; intros n. exact (p n). - apply path_forall; intros n. abstract ( rewrite transport_forall_constant; rewrite transport_paths_FlFr; rewrite ap_apply_l, ap10_path_arrow; rewrite (ap_compose (fun b => b n.+1) (fun x => f x) _); rewrite ap_apply_l, ap10_path_arrow; rewrite concat_pp_p; apply moveR_Vp; by symmetry ). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	path_split_idem	196
{a a' : split_idem} (p : a.1 == a'.1) (q : forall n, a.2 n @ p n = ap f (p n.+1) @ a'.2 n) : ap split_idem_sect (path_split_idem p q) = p 0. Proof. change (ap ((fun b => b 0) o pr1) (path_split_idem p q) = p 0). refine (ap_compose pr1 (fun b => b 0) _ @ _). refine (ap (ap (fun b => b 0)) (pr1_path_sigma _ _) @ _). refine (ap_apply_l _ 0 @ _). apply ap10_path_arrow. Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	sect_path_split_idem	197
(a : split_idem) : split_idem_retr (split_idem_sect a) = a. Proof. refine (_ @ nudge_eq a); symmetry. simple refine (path_split_idem _ _). - exact (split_idem_issect_part1 a). - exact (split_idem_issect_part2 a). Defined.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	split_idem_issect	198
RetractOf X := Build_RetractOf X split_idem split_idem_retr split_idem_sect split_idem_issect.	Definition	Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.	Idempotents.v	split_idem_retract	199

{A} {P Q : A -> Type} (f : forall a, P a -> Q a) (C : sig Q -> Type) (ef : forall a, ExtendableAlong 1 (f a) (fun v => C (a;v))) (s : forall z, C (functor_sigma idmap f z)) : ExtensionAlong (functor_sigma idmap f) C s. Proof. srefine (_;_). - intros [a v]; revert v. refine ((fst (ef a) _).1). intros u. exact (s (a;u)). - intros [a u]; cbn. exact ((fst (ef a) _).2 u). Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

extension_functor_sigma_id

100

n {A} {P Q : A -> Type} (f : forall a, P a -> Q a) (C : sig Q -> Type) (ef : forall a, ExtendableAlong n (f a) (fun v => C (a;v))) : ExtendableAlong n (functor_sigma idmap f) C. Proof. revert C ef; simple_induction n n IH; intros C ef; [ exact tt | split ]. - apply extension_functor_sigma_id. intros a; exact (fst (ef a) , fun _ _ => tt). - intros h k; apply IH. intros a; apply (snd (ef a)). Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

extendable_functor_sigma_id

101

{A} {P Q : A -> Type} (f : forall a, P a -> Q a) (C : sig Q -> Type) (ef : forall a, ooExtendableAlong (f a) (fun v => C (a;v))) : ooExtendableAlong (functor_sigma idmap f) C := fun n => extendable_functor_sigma_id n f C (fun a => ef a n).

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

ooextendable_functor_sigma_id

102

{A B} {Q : B -> Type} (f : A -> B) (C : sig Q -> Type) (p : forall (a:A) (v:Q (f a)), C (f a;v)) := { q : forall (b:B) (v:Q b), C (b;v) & forall a v, q (f a) v = p a v }.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

HomotopyExtensionAlong

103

(n : nat) {A B} {Q : B -> Type} (f : A -> B) (C : sig Q -> Type) : Type := match n with | 0 => Unit | S n => ((forall (p : forall (a:A) (v:Q (f a)), C (f a;v)), HomotopyExtensionAlong f C p) * (forall (h k : forall z, C z), n f (fun z => h z = k z))) end.

Fixpoint

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

HomotopyExtendableAlong

104

{A B} {Q : B -> Type} (f : A -> B) (C : sig Q -> Type) := forall n, HomotopyExtendableAlong n f C.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

ooHomotopyExtendableAlong

105

{A B} {P : A -> Type} {Q : B -> Type} (f : A -> B) (g : forall a, P a -> Q (f a)) (C : sig Q -> Type) (ef : HomotopyExtendableAlong 1 f C) (eg : forall a, ExtendableAlong 1 (g a) (fun v => C (f a ; v))) (s : forall z, C (functor_sigma f g z)) : ExtensionAlong (functor_sigma f g) C s. Proof. srefine (_;_). - intros [b v]; revert b v. refine ((fst ef _).1). intros a. refine ((fst (eg a) _).1). intros u. exact (s (a;u)). - intros [a u]; cbn. refine ((fst ef _).2 _ _ @ _). exact ((fst (eg a) _).2 u). Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

extension_functor_sigma

106

(n : nat) {A B} {P : A -> Type} {Q : B -> Type} (f : A -> B) (g : forall a, P a -> Q (f a)) (C : sig Q -> Type) (ef : HomotopyExtendableAlong n f C) (eg : forall a, ExtendableAlong n (g a) (fun v => C (f a ; v))) : ExtendableAlong n (functor_sigma f g) C. Proof. revert C ef eg; simple_induction n n IH; intros C ef eg; [ exact tt | split ]. - apply extension_functor_sigma. + exact (fst ef, fun _ _ => tt). + intros a; exact (fst (eg a) , fun _ _ => tt). - intros h k; apply IH. + exact (snd ef h k). + intros a; apply (snd (eg a)). Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

extendable_functor_sigma

107

{A B} {P : A -> Type} {Q : B -> Type} (f : A -> B) (g : forall a, P a -> Q (f a)) (C : sig Q -> Type) (ef : ooHomotopyExtendableAlong f C) (eg : forall a, ooExtendableAlong (g a) (fun v => C (f a ; v))) : ooExtendableAlong (functor_sigma f g) C := fun n => extendable_functor_sigma n f g C (ef n) (fun a => eg a n).

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

ooextendable_functor_sigma

108

{A B A' B'} (f : A -> A') (g : B -> B') (P : A' + B' -> Type) (ef : ExtendableAlong 1 f (P o inl)) (eg : ExtendableAlong 1 g (P o inr)) (h : forall z, P (functor_sum f g z)) : ExtensionAlong (functor_sum f g) P h. Proof. srefine (sum_ind _ _ _ ; sum_ind _ _ _). + exact (fst ef (h o inl)).1. + exact (fst eg (h o inr)).1. + exact (fst ef (h o inl)).2. + exact (fst eg (h o inr)).2. Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

extension_functor_sum

109

(n : nat) {A B A' B'} (f : A -> A') (g : B -> B') (P : A' + B' -> Type) (ef : ExtendableAlong n f (P o inl)) (eg : ExtendableAlong n g (P o inr)) : ExtendableAlong n (functor_sum f g) P. Proof. revert P ef eg; induction n as [|n IH]; intros P ef eg; [ exact tt | split ]. - intros h; apply extension_functor_sum. + exact (fst ef, fun _ _ => tt). + exact (fst eg, fun _ _ => tt). - intros h k. apply IH. + exact (snd ef (h o inl) (k o inl)). + exact (snd eg (h o inr) (k o inr)). Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

extendable_functor_sum

110

{A B A' B'} (f : A -> A') (g : B -> B') (P : A' + B' -> Type) (ef : ooExtendableAlong f (P o inl)) (eg : ooExtendableAlong g (P o inr)) : ooExtendableAlong (functor_sum f g) P. Proof. intros n; apply extendable_functor_sum; [ apply ef | apply eg ]. Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

ooextendable_functor_sum

111

{B A f g B' A' f' g'} {h : B -> B'} {k : A -> A'} {p : k o f == f' o h} {q : k o g == g' o h} {C : Coeq f' g' -> Type} (ek : ExtendableAlong 1 k (C o coeq)) (eh : forall (u v : forall x : B', C (coeq (g' x))), ExtendableAlong 1 h (fun x => u x = v x)) (s : forall x, C (functor_coeq h k p q x)) : ExtensionAlong (functor_coeq h k p q) C s. Proof. set (C' := C o pr_cylcoeq p q). set (s' x := pr_cyl_cylcoeq p q x # s x). assert (e : ExtensionAlong (cyl_cylcoeq p q) C' s'). 2:{ pose (ex := fst (extendable_equiv 1 C (pr_cylcoeq p q)) e.1). exists (ex.1); intros x. apply (equiv_inj (transport C (pr_cyl_cylcoeq p q x))). exact (apD _ (pr_cyl_cylcoeq p q x) @ ex.2 _ @ e.2 x). } pose (ea1 := fun u => (fst (cyl_extendable' 1 _ _ ek) u).1). assert (eb'' : forall u v, ExtendableAlong 1 cyl (fun x:Cyl h => DPath C' (cglue x) (u x) (v x))). { intros u v. rapply extendable_postcompose'. 2:{ rapply (cancelL_extendable 1 _ cyl pr_cyl). - rapply extendable_equiv. - exact (eh (fun x => cglue x # u (cyr x)) (v o cyr)). } intros x; subst C'. refine ((dp_compose (pr_cylcoeq p q) C _)^-1 oE _). symmetry; srapply equiv_ds_fill_lr. 3:rapply ap_pr_cylcoeq_cglue. all:srapply (transport (fun r => DPath C r _ _)). 3:exact (dp_inverse (dp_compose _ C _ (apD u (eissect pr_cyl x) : DPath _ _ _ _))). 4:exact (dp_inverse (dp_compose _ C _ (apD v (eissect pr_cyl x) : DPath _ _ _ _))). 1:change (fun y => pr_cylcoeq p q (coeq (functor_cyl p y))) with (fun y => coeq (f := f') (g := g') (pr_cyl (functor_cyl p y))). 2:change (fun y => pr_cylcoeq p q (coeq (functor_cyl q y))) with (fun y => coeq (f := f') (g := g') (pr_cyl (functor_cyl q y))). all:refine ((ap_V _ (eissect pr_cyl x))^ @ _). all: exact (ap_compose (fun x => pr_cyl (functor_cyl _ x)) coeq _). } pose (eb1 := fun u v w => (fst (cyl_extendable _ _ _ (eb'' u v)) w).1). srefine (_;_); srapply Coeq_ind. + exact (ea1 (s' o coeq)). + apply eb1; intros b. rapply (dp_compose' _ _ (ap_cyl_cylcoeq_cglue p q b)). exact (apD s' (cglue b)). + intros a; reflexivity. + intros b. apply ds_dp. rapply ds_G1. refine (dp_apD_compose' _ _ (ap_cyl_cylcoeq_cglue p q b) _ @ _). apply moveR_equiv_V. nrapply Coeq_ind_beta_cglue. Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

extension_functor_coeq

112

(n : nat) {B A f g B' A' f' g'} {h : B -> B'} {k : A -> A'} {p : k o f == f' o h} {q : k o g == g' o h} {C : Coeq f' g' -> Type} (ek : ExtendableAlong n k (C o coeq)) (eh : forall (u v : forall x : B', C (coeq (g' x))), ExtendableAlong n h (fun x => u x = v x)) : ExtendableAlong n (functor_coeq h k p q) C. Proof. revert C ek eh; simple_induction n n IH; intros C ek eh; [ exact tt | split ]. - apply extension_functor_coeq. + exact (fst ek , fun _ _ => tt). + exact (fun u v => (fst (eh u v) , fun _ _ => tt)). - intros u v; apply IH. + exact (snd ek (u o coeq) (v o coeq)). + exact (snd (eh (u o coeq o g') (v o coeq o g'))). Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

extendable_functor_coeq

113

{B A f g B' A' f' g'} {h : B -> B'} {k : A -> A'} {p : k o f == f' o h} {q : k o g == g' o h} {C : Coeq f' g' -> Type} (ek : ooExtendableAlong k (C o coeq)) (eh : forall (u v : forall x : B', C (coeq (g' x))), ooExtendableAlong h (fun x => u x = v x)) : ooExtendableAlong (functor_coeq h k p q) C := fun n => extendable_functor_coeq n (ek n) (fun u v => eh u v n).

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

ooextendable_functor_coeq

114

extendable_functor_coeq' (n : nat) {B A f g B' A' f' g'} {h : B -> B'} {k : A -> A'} {p : k o f == f' o h} {q : k o g == g' o h} {C : Coeq f' g' -> Type} (ek : ExtendableAlong n k (C o coeq)) (eh : ExtendableAlong n.+1 h (C o coeq o g')) : ExtendableAlong n (functor_coeq h k p q) C. Proof. apply extendable_functor_coeq. 1:assumption. exact (snd eh). Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

extendable_functor_coeq'

115

ooextendable_functor_coeq' {B A f g B' A' f' g'} {h : B -> B'} {k : A -> A'} {p : k o f == f' o h} {q : k o g == g' o h} {C : Coeq f' g' -> Type} (ek : ooExtendableAlong k (C o coeq)) (eh : ooExtendableAlong h (C o coeq o g')) : ooExtendableAlong (functor_coeq h k p q) C := fun n => extendable_functor_coeq' n (ek n) (eh n.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.

Extensions.v

ooextendable_functor_coeq'

116

Record

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

Factorization

117

{ I : Type & { g : A -> I & { h : I -> B & { p : h o g == f & { gin1 : class1 _ _ g & class2 _ _ h }}}}} <~> Factorization. Proof. issig. Defined.

Lemma

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

issig_Factorization

118

{fact fact' : Factorization} :=

Record

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

PathFactorization

119

(fact fact' : Factorization) : { path_intermediate : intermediate fact <~> intermediate fact' & { path_factor1 : path_intermediate o factor1 fact == factor1 fact' & { path_factor2 : factor2 fact == factor2 fact' o path_intermediate & forall a, path_factor2 (factor1 fact a) @ ap (factor2 fact') (path_factor1 a) @ fact_factors fact' a = fact_factors fact a }}} <~> PathFactorization fact fact'. Proof. issig. Defined.

Lemma

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

issig_PathFactorization

120

`{Univalence} (fact fact' : Factorization) : PathFactorization fact fact' <~> fact = fact'. Proof. refine (_ oE (issig_PathFactorization fact fact')^-1). revert fact fact'; apply (equiv_path_issig_contr issig_Factorization). { intros [I [f1 [f2 [ff [oc1 oc2]]]]]. exists (equiv_idmap I); cbn. exists (fun x:A => 1%path); cbn. exists (fun x:I => 1%path); cbn. intros; apply concat_1p. } intros [I [f1 [f2 [ff [oc1 oc2]]]]]. contr_sigsig I (equiv_idmap I); cbn. contr_sigsig f1 (fun x:A => idpath (f1 x)); cbn. contr_sigsig f2 (fun x:I => idpath (f2 x)); cbn. refine (contr_equiv' {ff' : f2 o f1 == f & ff == ff'} _). symmetry; srefine (equiv_functor_sigma' (equiv_sigma_contr _) _). { intros h; cbn. srefine (@istrunc_sigma _ _ _ _ _); [ | intros a]; apply contr_inhabited_hprop; try exact _; assumption. } intros [ff' [oc1' oc2']]; cbn. refine (equiv_functor_forall' (equiv_idmap _) _); intros a. refine (equiv_path_inverse _ _ oE _). apply equiv_concat_l; symmetry; apply concat_1p. Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

equiv_path_factorization

121

`{Univalence} (fact fact' : Factorization) : PathFactorization fact fact' -> fact = fact' := equiv_path_factorization fact fact'.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

path_factorization

122

FactorizationSystem@{i j k} :=

Record

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

FactorizationSystem@

123

`{Univalence} (factsys : FactorizationSystem) {X Y : Type} (f : X -> Y) : Contr (Factorization (@class1 factsys) (@class2 factsys) f). Proof. apply contr_inhabited_hprop. - apply hprop_allpath. intros fact fact'. apply path_factorization; try exact _. apply path_factor. - apply factor. Defined.

Theorem

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

contr_factor

124

Build_Factorization' {X Y} := @Build_Factorization (@class1 factsys) (@class2 factsys) X Y.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

Build_Factorization'

125

Build_PathFactorization' {X Y} := @Build_PathFactorization (@class1 factsys) (@class2 factsys) X Y.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

Build_PathFactorization'

126

`{Funext} {X Y Z} (f : X -> Y) (g : Y -> Z) : class1 factsys f -> class1 factsys (g o f) -> class1 factsys g. Proof. intros c1f c1gf. destruct (factor factsys g) as [I g1 g2 gf c1g1 c2g2]. pose (fact := Build_Factorization' (g o f) Z (g o f) (idmap) (fun x => 1) c1gf (class2_isequiv factsys idmap)). pose (fact' := Build_Factorization' (g o f) I (g1 o f) g2 (fun x => gf (f x)) (class1_compose factsys f g1 c1f c1g1) c2g2). destruct (path_factor factsys (g o f) fact' fact) as [q q1 q2 qf]; simpl in *. refine (transport (class1 factsys) (path_arrow _ _ gf) _). refine (class1_compose factsys g1 g2 c1g1 _). apply class1_isequiv. apply (isequiv_homotopic _ (fun i => (q2 i)^)). Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

cancelR_class1

127

`{Funext} {X Y Z} (f : X -> Y) (g : Y -> Z) : class2 factsys g -> class2 factsys (g o f) -> class2 factsys f. Proof. intros c2g c2gf. destruct (factor factsys f) as [I f1 f2 ff c1f1 c2f2]. pose (fact := Build_Factorization' (g o f) X (idmap) (g o f) (fun x => 1) (class1_isequiv factsys idmap) c2gf). pose (fact' := Build_Factorization' (g o f) I f1 (g o f2) (fun x => ap g (ff x)) c1f1 (class2_compose factsys f2 g c2f2 c2g)). destruct (path_factor factsys (g o f) fact fact') as [q q1 q2 qf]; simpl in *. refine (transport (class2 factsys) (path_arrow _ _ ff) _). refine (class2_compose factsys f1 f2 _ c2f2). apply class2_isequiv. apply (isequiv_homotopic _ q1). Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

cancelL_class2

128

B -> X := f2 o q^-1 o g1.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

lift_factsys

129

lift_factsys o i == f. Proof. intros x. refine (ap (f2 o q^-1) (q1 x)^ @ _). transitivity (f2 (f1 x)). + apply ap, eissect. + apply ff. Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

lift_factsys_tri1

130

p o lift_factsys == g. Proof. intros x. refine (q2 _ @ _). transitivity (g2 (g1 x)). + apply ap, eisretr. + apply gf. Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

lift_factsys_tri2

131

(x : A) : ap p (lift_factsys_tri1 x)^ @ lift_factsys_tri2 (i x) = h x. Proof. unfold lift_factsys_tri1, lift_factsys_tri2. Open Scope long_path_scope. pose (r := path_fact_factors (path_factor factsys (p o f) fact fact') x : q2 (f1 x) @ ap g2 (q1 x) @ (gf (i x) @ (h x)^) = ap p (ff x)). rewrite concat_p_pp in r. apply moveL_pM, moveR_Vp in r. refine (_ @ r); clear r. repeat rewrite inv_pp; repeat rewrite ap_pp; rewrite ap_V. repeat rewrite concat_pp_p; apply whiskerL. repeat rewrite concat_p_pp; apply whiskerR. rewrite (ap_compose q^-1 f2), <- ap_pp, <- inv_pp. unfold lift_factsys. rewrite <- ap_pp. rewrite <- ap_V, <- ap_compose. rewrite (concat_Ap q2). rewrite concat_pp_p; apply whiskerL. rewrite ap_compose. rewrite <- ap_pp. apply ap. rewrite inv_pp, ap_pp, ap_V. rewrite <- eisadj. rewrite concat_pp_p; apply moveR_Vp. rewrite <- ap_V, inv_V, <- ap_compose. exact (concat_A1p (eisretr q) (q1 x)). Close Scope long_path_scope. Qed.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

lift_factsys_square

132

{A B : Type} (f : A -> B) {c1f : class1 factsys f} (P : B -> Type) (c2P : class2 factsys (@pr1 B P)) (d : forall a:A, P (f a)) : ExtensionAlong f P d. Proof. pose (e := lift_factsys factsys f c1f pr1 c2P (fun a => (f a ; d a)) idmap (fun a => 1)). pose (e2 := lift_factsys_tri2 factsys f c1f pr1 c2P (fun a => (f a ; d a)) idmap (fun a => 1)). exists (fun a => (e2 a) # (e a).2). intros a. pose (e1 := lift_factsys_tri1 factsys f c1f pr1 c2P (fun a => (f a ; d a)) idmap (fun a => 1) a : e (f a) = (f a ; d a)). pose (e3 := moveL_M1 _ _ (((ap_V _ _)^ @@ 1) @ lift_factsys_square factsys f c1f pr1 c2P (fun a => (f a ; d a)) idmap (fun a => 1) a) : e2 (f a) = pr1_path e1). refine (ap (fun p => transport P p (e (f a)).2) e3 @ _). exact (pr2_path e1). Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Extensions Homotopy.IdentitySystems.

Factorization.v

extension_factsys

133

{A B} (f : A -> B) `{IsEquiv _ _ f} : IsEquiv (fmap F f). Proof. refine (equiv_induction (fun A' e => IsEquiv (fmap F e)) _ _ (Build_Equiv _ _ f _)). refine (transport _ (fmap_idmap F)^ _); try apply isequiv_idmap. Defined.

Proposition

Require Import HoTT.Basics Types.Universe.

Functorish.v

isequiv_fmap

134

{A B} (f : A -> B) `{IsEquiv _ _ f} : fmap F f = equiv_path _ _ (ap F (path_universe f)). Proof. refine (equiv_induction (fun A' e => fmap F e = equiv_path _ _ (ap F (path_universe e))) _ _ (Build_Equiv _ _ f _)). transitivity (idmap : F A -> F A). - apply fmap_idmap. - change (equiv_idmap A) with (equiv_path A A 1). rewrite (@eta_path_universe _ A A 1). exact 1. Defined.

Proposition

Require Import HoTT.Basics Types.Universe.

Functorish.v

fmap_agrees_with_univalence

135

{A B : Type} {f : A -> B} {y : B} (x1 x2 : hfiber f y) : { q : x1.1 = x2.1 & x1.2 = ap f q @ x2.2 } <~> (x1 = x2). Proof. refine (equiv_path_sigma _ _ _ oE _). apply equiv_functor_sigma_id. intros p; simpl. refine (_ oE equiv_moveR_Vp _ _ _). exact (equiv_concat_l (transport_paths_Fl _ _) _). Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

equiv_path_hfiber

136

{A B : Type} {f : A -> B} {y : B} {x1 x2 : hfiber f y} : { q : x1.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

path_hfiber_uncurried

137

{A B : Type} {f : A -> B} {y : B} {x1 x2 : hfiber f y} (q : x1.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

path_hfiber

138

{A B : Type} {f : A -> B} {x1 x2 : A} (p : f x1 = f x2) : hfiber (ap f) p <~> ((x1 ; p) = (x2 ; 1) :> hfiber f (f x2)). Proof. refine (equiv_path_hfiber (x1;p) (x2;1%path) oE _). unfold hfiber; simpl. apply equiv_functor_sigma_id; intros q. refine (_ oE equiv_path_inverse _ _). exact (equiv_concat_r (concat_p1 _)^ _). Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

hfiber_ap

139

{A B : Type} (f g : A -> B) (h : f == g) (b : B) : hfiber f b <~> hfiber g b. Proof. refine (Build_Equiv _ _ (fun fx => (fx.1 ; (h fx.1)^ @ fx.2)) _). refine (isequiv_adjointify _ (fun gx => (gx.1 ; (h gx.1) @ gx.2)) _ _); intros [a p]; simpl; apply ap. - apply concat_V_pp. - apply concat_p_Vp. Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

equiv_hfiber_homotopic

140

{A B C D} {f : A -> B} {g : C -> D} {h : A -> C} {k : B -> D} (p : k o f == g o h) (b : B) : hfiber f b -> hfiber g (k b). Proof. snrapply @functor_sigma. - exact h. - intros a e; exact ((p a)^ @ ap k e). Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

functor_hfiber

141

{A B C D} {f : A -> B} {g : C -> D} {h : A -> C} {k : B -> D} `{IsEquiv A C h} `{IsEquiv B D k} (p : k o f == g o h) (b : B) : IsEquiv (functor_hfiber p b) := _.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

isequiv_functor_hfiber

142

{A B C D} {f : A -> B} {g : C -> D} {h : A <~> C} {k : B <~> D} (p : k o f == g o h) (b : B) : hfiber f b <~> hfiber g (k b) := Build_Equiv _ _ (functor_hfiber p b) _.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

equiv_functor_hfiber

143

{A B C D} {f : A -> B} {g : C -> D} {h : A -> C} {k : B -> D} (p : k o f == g o h) {b : B} {b' : D} (q : k b = b') : hfiber f b -> hfiber g b'. Proof. srapply functor_sigma. - exact h. - intros a e. exact ((p a)^ @ ap k e @ q). Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

functor_hfiber2

144

{A B C D} {f : A -> B} {g : C -> D} {h : A <~> C} {k : B <~> D} (p : k o f == g o h) {b : B} {b' : D} (q : k b = b') : hfiber f b <~> hfiber g b' := Build_Equiv _ _ (functor_hfiber2 p q) _.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

equiv_functor_hfiber2

145

{A B C X Y Z : Type} {k : A -> B} {l : B -> C} {f : A -> X} {g : B -> Y} {h : C -> Z} {i : X -> Y} {j : Y -> Z} (H : i o f == g o k) (K : j o g == h o l) : forall x, functor_hfiber (comm_square_comp' H K) x == (functor_hfiber K (i x)) o (functor_hfiber H x : hfiber f x -> _). Proof. intros x [y p]. destruct p. apply (path_sigma' _ idpath). refine (concat_p1 _ @ _). refine (inv_pp _ _ @ ap _ _). refine ((ap_V _ _)^ @ ap _ _^). apply concat_p1. Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

functor_hfiber_compose

146

{A B C D} {f : A -> B} {g : C -> D} {h : A -> C} {k : B -> D} (p : k o f == g o h) (b : B) (c : C) (q : g c = k b) : hfiber (functor_hfiber p b) (c;q) <~> hfiber (functor_hfiber (fun x => (p x)^) c) (b;q^). Proof. rapply (equiv_functor_sigma_id _ oE _ oE (equiv_functor_sigma_id _)^-1). 1,3:intros; rapply equiv_path_sigma. refine (equiv_sigma_assoc _ _ oE _ oE (equiv_sigma_assoc _ _)^-1). apply equiv_functor_sigma_id; intros a; cbn. refine (equiv_sigma_symm _ oE _). do 2 (apply equiv_functor_sigma_id; intro). refine ((equiv_ap inverse _ _)^-1 oE _). refine (equiv_concat_r (inv_V q)^ _ oE _). apply equiv_concat_l. abstract (rewrite !transport_paths_Fl, !inv_pp, !inv_V, concat_pp_p; reflexivity). Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

hfiber_functor_hfiber

147

{B C} (f:C ->B) : C <~> {y:B & hfiber f y}. Proof. snrefine (Build_Equiv _ _ _ ( Build_IsEquiv C {y:B & {x:C & f x = y}} (fun c => (f c; (c; idpath))) (fun c => c.2.1) _ (fun c => idpath) _)). - intros [? [? []]]; reflexivity. - reflexivity. Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

equiv_fibration_replacement

148

{X} (x : X) (P:X->Type) : P x <~> @hfiber (sig P) X pr1 x. Proof. snrefine (Build_Equiv _ _ _ (Build_IsEquiv (P x) { z : sig P & z.1 = x } (fun Px => ((x; Px); idpath)) (fun Px => transport P Px.2 Px.1.2) _ (fun Px => idpath) _)). - intros [[] []]; reflexivity. - reflexivity. Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

hfiber_fibration

149

(c : C) : hfiber (g o f) c -> hfiber g c := fun x => (f x.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

hfiber_compose_map

150

(b : B) : hfiber (hfiber_compose_map (g b)) (b;1) <~> hfiber f b. Proof. unfold hfiber, hfiber_compose_map. refine (_ oE equiv_functor_sigma_id (fun x => (equiv_path_sigma _ _ _)^-1)); cbn. make_equiv_contr_basedpaths. Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

hfiber_hfiber_compose_map

151

(c : C) : hfiber (g o f) c <~> { w : hfiber g c & hfiber f w.1 }. Proof. make_equiv_contr_basedpaths. Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

hfiber_compose

152

`{Funext} {A : Type} {P Q : A -> Type} (h : forall a, P a -> Q a) (g : forall a, Q a) : hfiber (functor_forall_id h) g <~> (forall a, hfiber (h a) (g a)). Proof. unfold hfiber, functor_forall_id, functor_forall. nrefine (equiv_sig_coind _ _ oE _). apply equiv_functor_sigma_id; intro f. apply equiv_apD10. Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

hfiber_functor_forall_id

153

A {B} (y y' : B) : hfiber (fun _ : A => y) y' <~> A * (y = y') := equiv_sigma_prod0 A (y = y').

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

hfiber_const

154

{A B} n (f:A -> B) `{!forall x y, IsTruncMap n (@ap _ _ f x y)} : IsTruncMap n.+1 f. Proof. intro y; apply istrunc_S. intros [a p] [b q]; destruct q; exact (istrunc_equiv_istrunc _ (hfiber_ap p)). Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

istruncmap_from_ap

155

`{Funext} {A B} n (f:A -> B) : IsTruncMap n.+1 f <~> (forall x y, IsTruncMap n (@ap _ _ f x y)) := equiv_iff_hprop (@istruncmap_ap _ _ n f) (@istruncmap_from_ap _ _ n f). Global Instance isequiv_ap_isembedding {A B} (f : A -> B) `{!IsEmbedding f} : forall x y, IsEquiv (@ap _ _ f x y). Proof. intros x y. apply isequiv_contr_map,_. Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

equiv_istruncmap_ap

156

{A B} (f : A -> B) `{!IsEmbedding f} (x y : A) : (x = y) <~> (f x = f y) := Build_Equiv _ _ (ap f) _.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

equiv_ap_isembedding

157

{A B} (f : A -> B) `{!forall x y, IsEquiv (@ap _ _ f x y)} : IsEmbedding f. Proof. rapply istruncmap_from_ap. Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

isembedding_isequiv_ap

158

`{Funext} {A B} (f : A -> B) : IsEmbedding f <~> (forall x y, IsEquiv (@ap _ _ f x y)). Proof. exact (equiv_iff_hprop (@isequiv_ap_isembedding _ _ f) (@isembedding_isequiv_ap _ _ f)). Defined.

Definition

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

equiv_isequiv_ap_isembedding

159

{X Y : Type} (f : X -> Y) {I : IsEmbedding f} {x0 x1 : X} : forall (p : f x0 = f x1), ap f (isinj_embedding f I x0 x1 p) = p. Proof. equiv_intro (equiv_ap_isembedding f x0 x1) q. induction q. cbn. exact (ap _ (isinj_embedding_beta f)). Defined.

Lemma

Require Import Basics Types Diagrams.CommutativeSquares HSet.

HFiber.v

ap_isinj_embedding_beta

160

{X : Type} :=

Record