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`{P : A -> Type} `{Q : A -> Type} (g : forall a, P a <~> Q a) : (forall a, P a) <~> (forall a, Q a) := equiv_functor_forall (equiv_idmap A) g. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids | Types\Forall.v | equiv_functor_forall_id | 6,800 |
{A B : Type} {P : A -> Type} (f : B <~> A) : (forall a, P a) <~> (forall b, P (f b)) := equiv_functor_forall' (Q := P o f) f (fun b => equiv_idmap). | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids | Types\Forall.v | equiv_functor_forall_pb | 6,801 |
`{P : A -> Type} `{Q : B -> Type} (f : A <~> B) (g : forall a, P a <~> Q (f a)) : (forall a, P a) <~> (forall b, Q b) := (equiv_functor_forall' f (fun a => (g a)^-1%equiv))^-1. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids | Types\Forall.v | equiv_functor_forall_covariant | 6,802 |
`{P : A -> Type} `{Q : B -> Type} `{R : C -> Type} (f0 : A <~> B) (f1 : forall a, P a <~> Q (f0 a)) (g0 : B <~> C) (g1 : forall b, Q b <~> R (g0 b)) (h : forall a, P a) : equiv_functor_forall_covariant g0 g1 (equiv_functor_forall_covariant f0 f1 h) == equiv_functor_forall_covariant (g0 oE f0) (fun a => g1 (f0 a) oE f1 a) h. Proof. apply apD10. refine ((equiv_inverse_compose (equiv_functor_forall' g0 (fun a : B => (g1 a)^-1%equiv)) (equiv_functor_forall' f0 (fun a : A => (f1 a)^-1%equiv)) h)^ @ _). revert h; apply equiv_inverse_homotopy; intros h. apply path_forall; intros c. symmetry; rapply functor_forall_compose. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids | Types\Forall.v | equiv_functor_forall_covariant_compose | 6,803 |
{A : Type} {P Q : A -> Type} (f : forall a, P a <-> Q a) : (forall a, P a) <-> (forall a, Q a) := (functor_forall idmap (fun a => fst (f a)), functor_forall idmap (fun a => snd (f a))). | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids | Types\Forall.v | iff_functor_forall | 6,804 |
{A : Type} {B : A -> Type} {P : forall a : A, B a -> Type} (f g : forall a b, P a b) : (forall (a : A) (b : B a), f a b = g a b) <~> f = g := (equiv_path_forall f g) oE (equiv_functor_forall_id (fun a => equiv_path_forall (f a) (g a))). | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids | Types\Forall.v | equiv_path_forall11 | 6,805 |
{A : Type} {B : A -> Type} {P : forall a : A, B a -> Type} (f g : forall a b, P a b) : (forall x y, f x y = g x y) -> f = g := equiv_path_forall11 f g. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids | Types\Forall.v | path_forall11 | 6,806 |
`{Contr A} `(P : A -> Type) : (forall a, P a) <~> P (center A). Proof. simple refine (equiv_adjointify (fun (f:forall a, P a) => f (center A)) _ _ _). - intros p a; exact (transport P (path_contr _ _) p). - intros p. refine (transport2 P (q := 1) _ p). apply path_contr. - intros f; apply path_forall; intros a. apply apD. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids | Types\Forall.v | equiv_contr_forall | 6,807 |
`{P : A -> B -> Type} : (forall a b, P a b) -> (forall b a, P a b) := fun f b a => f a b. Arguments {A B P} f b a /. Global Instance isequiv_flip `{P : A -> B -> Type} : IsEquiv (@ _ _ P) | 0. Proof. set (flip_P := @ _ _ P). set (flip_P_inv := @ _ _ ( P)). set (flip_P_is_sect := (fun f => 1) : flip_P_inv o flip_P == idmap). set (flip_P_is_retr := (fun g => 1) : flip_P o flip_P_inv == idmap). exists flip_P_inv flip_P_is_retr flip_P_is_sect. intro g. exact 1. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids | Types\Forall.v | flip | 6,808 |
`(P : A -> B -> Type) : (forall a b, P a b) <~> (forall b a, P a b) := Build_Equiv _ _ (@flip _ _ P) _. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids | Types\Forall.v | equiv_flip | 6,809 |
(I : Type) (A : Type) (B : A -> Type) (i : A -> I) (j : forall x, B x -> I) : I -> Type := | iw_sup (x : A) : (forall (y : B x), I A B i j (j x y)) -> I A B i j (i x). | Inductive | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | IW | 6,810 |
{A B I i j} {l : I} (w : IW I A B i j l) : A := match w with | iw_sup x y => x end. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | iw_label | 6,811 |
{A B I i j} (l : I) (w : IW I A B i j l) : forall (y : B (iw_label w)), IW I A B i j (j (iw_label w) y) := match w with | iw_sup x y => y end. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | iw_arity | 6,812 |
{A B I i j} (l : I) (w : IW I A B i j l) : i (iw_label w) = l. Proof. by destruct w. Defined. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | path_index_iw_label | 6,813 |
{A B I i j} (l : I) (w : IW I A B i j l) : path_index_iw_label l w # iw_sup I A B i j (iw_label w) (iw_arity l w) = w. Proof. by destruct w. Defined. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | iw_eta | 6,814 |
{A B I i j} (l : I) : IW I A B i j l -> hfiber i l. Proof. intros w. exists (iw_label w). apply path_index_iw_label. Defined. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | iw_to_hfiber_index | 6,815 |
`{Funext} {A : Type} {B : A -> Type} (a : forall x, B x) (P : forall b, a == b -> Type) (f : P a (fun _ => 1%path)) {b : forall x, B x} (p : a == b) : P b p. Proof. refine (equiv_ind apD10 (P b) _ p). intros []. exact f. Defined. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | pointwise_paths_ind | 6,816 |
(x : I) (w : W A B) : Type := match w with | w_sup a b => (i a = x) * (forall c, (j a c) (b c)) end. | Fixpoint | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | IsIndexedBy | 6,817 |
IW' (x : I) := sig (IsIndexedBy x). | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | IW' | 6,818 |
iw_sup' (x : A) (y : forall z : B x, IW' (j x z)) : IW' (i x) := (w_sup A B x (fun a => pr1 (y a)); (idpath, (fun a => pr2 (y a)))). | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | iw_sup' | 6,819 |
(P : forall i, IW' i -> Type) (S : forall x y, (forall c, P _ (y c)) -> P _ (iw_sup' x y)) : forall x w, P x w. Proof. intros x [w r]. revert w x r. induction w as [a b k]. intros x [p IH]. destruct p. refine (S a (fun c => (b c; IH c)) _). intros c. apply k. Defined. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | IW'_ind | 6,820 |
IW'_ind_beta_iw_sup' (P : forall i, IW' i -> Type) (S : forall x y, (forall c, P _ (y c)) -> P _ (iw_sup' x y)) x y : IW'_ind P S _ (iw_sup' x y) = S x y (fun c => IW'_ind P S _ (y c)) := idpath. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | IW'_ind_beta_iw_sup' | 6,821 |
`{Funext} (x : I) : IW' x <~> IW I A B i j x. Proof. snrapply equiv_adjointify. { rapply (IW'_ind (fun l _ => IW I A B i j l)). intros a b c. apply iw_sup. intros y. apply c. } { rapply (IW_rect I A B i j (fun l _ => IW' l)). intros a b c. apply iw_sup'. intros y. apply c. } { rapply (IW_rect I A B i j (fun x y => IW'_ind _ _ x _ = y)). cbn; intros a b c. apply ap. funext y. apply c. } simpl. intro y. rapply (IW'_ind (fun x y => IW_rect I A B i j _ _ x _ = y)). cbn; intros a b c. apply ap. funext d. apply c. Defined. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | equiv_wtype_iwtype | 6,822 |
(x : A') : (forall y : B' x, IWPath (j' x y)) -> IWPath (i' x). Proof. destruct x as [x [c1 c2]]. intros y. unfold IWPath. cbn; apply ap. funext l. apply y. Defined. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | iwpath_sup | 6,823 |
(x : A) (a : forall c : B x, IW I A B i j (j x c)) : iwpath_sup (x; (a, a)) (apD10 1) = idpath. Proof. unfold iwpath_sup. rewrite path_forall_1. reflexivity. Defined. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | iwpath_sup_refl | 6,824 |
forall l a, P (l ; (a, a)) idpath. Proof. rapply (IW_rect I A B i j (fun l a => P (l; (a, a)) idpath)). intros x a q. pose (S (x; (a, a)) _ q) as p. unfold iwpath_sup in p. refine (transport (P (i x; (iw_sup I A B i j x a, iw_sup I A B i j x a))) _ p). change (ap (iw_sup I A B i j x) (path_forall a a (apD10 idpath)) = ap (iw_sup I A B i j x) 1%path). refine (ap _ _). apply eissect. Defined. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | IWPath_ind_refl | 6,825 |
forall x p, P x p. Proof. intros [x [a b]]. unfold IWPath; cbn. destruct p. apply IWPath_ind_refl. Defined. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | IWPath_ind | 6,826 |
(x : A') (h : forall y : B' x, IWPath (j' x y)) : IWPath_ind _ (iwpath_sup x h) = S x h (fun c => IWPath_ind _ (h c)). Proof. destruct x as [x [a b]]. cbv in h. refine (_ @ _). { refine (_ @ ap _ (eisadj (path_forall _ _) h)). refine (paths_ind _ (fun b p' => paths_ind _ (fun r p'' => P (i x ; (iw_sup I A B i j x a , r)) p'') (IWPath_ind_refl (i x) (iw_sup I A B i j x a)) _ (ap (iw_sup I A B i j x) p') = paths_rec (path_forall _ _ (apD10 p')) (fun p'' => P (_ ; (_, _)) (ap (iw_sup I A B i j x) p'')) (S (x ; (a, b)) (apD10 p') (fun c => IWPath_ind (_ ; (_, _)) (apD10 p' c))) p' (eissect apD10 p')) _ _ _). exact (transport_compose _ _ _ _)^. } by cbn; destruct (eisretr apD10 h). Defined. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | IWPath_ind_beta_iwpath_sup | 6,827 |
(x : I) (a b : IW I A B i j x) : IW I' A' B' i' j' (x; (a, b)) <~> a = b. Proof. change (IW I' A' B' i' j' (x; (a, b)) <~> IWPath (x; (a,b))). snrapply equiv_adjointify. { intros y. induction y as [e f g]. apply iwpath_sup. intros y. apply g. } { intros y. induction y as [e f g] using IWPath_ind. apply iw_sup. intros y. apply g. } { intros y; cbn. induction y as [a' b' IH] using IWPath_ind. rewrite IWPath_ind_beta_iwpath_sup. simpl; f_ap. funext c. apply IH. } intros y; cbn. induction y as [e f IH]. rewrite IWPath_ind_beta_iwpath_sup. f_ap; funext c. apply IH. Defined. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | equiv_path_iwtype | 6,828 |
(l : I) (a b : IW I A B i j l) : iw_to_hfiber_index l a = iw_to_hfiber_index l b <~> hfiber i' (l; (a, b)). Proof. srapply equiv_adjointify. + apply path_iw_to_hfiber. + apply hfiber_to_path_iw. + rapply path_iw_to_hfiber_to_path_iw. + rapply hfiber_to_path_iw_to_hfiber. Defined. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | equiv_path_hfiber_index | 6,829 |
l (a : IW I A B i j l) : forall b, Decidable (a = b). Proof. destruct a as [x c1]. intro b. transparent assert (decide_children : (forall c2, Decidable (c1 = c2))). { intros c2. destruct (liftP x (fun c => c1 c = c2 c) (fun c => _ (c1 c) (c2 c))) as [p|p]. + left; by apply path_forall. + right; intro h; by apply p, apD10. } snrefine ( match b in (IW _ _ _ _ _ l) return forall iy : l = i x, Decidable (iw_sup I A B i j x c1 = paths_rec l (IW I A B i j) b (i x) iy) with iw_sup y c2 => fun iy : i y = i x => _ end idpath). destruct (fibers_dec (i x) (x ; idpath) (y ; iy)) as [feq|fneq]. + refine ( match feq in (_ = (y ; iy)) return forall c2, Decidable (iw_sup _ _ _ _ _ x c1 = paths_rec (i y) (IW I A B i j) (iw_sup _ _ _ _ _ y c2) (i x) iy) with idpath => _ end c2). cbn; intros c3. destruct (decide_children c3) as [ceq | cneq]. - left; exact (ap _ ceq). - right; intros r; apply cneq. exact (children_eq x c1 c3 r). + right; intros r; apply fneq. exact (ap getfib r @ getfib_computes x y c2 iy). Defined. | Fixpoint | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | decide_eq | 6,830 |
forall x, DecidablePaths (IW I A B i j x). Proof. intros x a b. apply decide_eq. Defined. | Definition | Require Import Basics. Require Import Types.Forall Types.Sigma Types.Prod Types.WType. | Types\IWType.v | decidablepaths_iwtype | 6,831 |
{A B} (f : A -> B) (x : option A) : option B := match x with | None => None | Some a => Some (f a) end. | Definition | Require Import Basics.Overture. | Types\Option.v | functor_option | 6,832 |
{A} {x y : A} (p : Some x = Some y) : x = y. Proof. injection p. exact idmap. Defined. | Definition | Require Import Basics.Overture. | Types\Option.v | isinj_some | 6,833 |
{A : Type} {x1 x2 y : A} (p : x1 = x2) (q : x1 = y) : transport (fun x => x = y) p q = p^ @ q. Proof. destruct p, q; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | transport_paths_l | 6,834 |
{A : Type} {x y1 y2 : A} (p : y1 = y2) (q : x = y1) : transport (fun y => x = y) p q = q @ p. Proof. destruct p, q; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | transport_paths_r | 6,835 |
{A : Type} {x1 x2 : A} (p : x1 = x2) (q : x1 = x1) : transport (fun x => x = x) p q = p^ @ q @ p. Proof. destruct p; simpl. exact ((concat_1p q)^ @ (concat_p1 (1 @ q))^). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | transport_paths_lr | 6,836 |
{A B : Type} {f : A -> B} {x1 x2 : A} {y : B} (p : x1 = x2) (q : f x1 = y) : transport (fun x => f x = y) p q = (ap f p)^ @ q. Proof. destruct p, q; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | transport_paths_Fl | 6,837 |
{A B : Type} {g : A -> B} {y1 y2 : A} {x : B} (p : y1 = y2) (q : x = g y1) : transport (fun y => x = g y) p q = q @ (ap g p). Proof. destruct p. symmetry; apply concat_p1. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | transport_paths_Fr | 6,838 |
{A B : Type} {f g : A -> B} {x1 x2 : A} (p : x1 = x2) (q : f x1 = g x1) : transport (fun x => f x = g x) p q = (ap f p)^ @ q @ (ap g p). Proof. destruct p; simpl. exact ((concat_1p q)^ @ (concat_p1 (1 @ q))^). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | transport_paths_FlFr | 6,839 |
{A : Type} {B : A -> Type} {f g : forall a, B a} {x1 x2 : A} (p : x1 = x2) (q : f x1 = g x1) : transport (fun x => f x = g x) p q = (apD f p)^ @ ap (transport B p) q @ (apD g p). Proof. destruct p; simpl. exact ((ap_idmap _)^ @ (concat_1p _)^ @ (concat_p1 _)^). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | transport_paths_FlFr_D | 6,840 |
{A B : Type} {f : A -> B} {g : B -> A} {x1 x2 : A} (p : x1 = x2) (q : g (f x1) = x1) : transport (fun x => g (f x) = x) p q = (ap g (ap f p))^ @ q @ p. Proof. destruct p; simpl. exact ((concat_1p q)^ @ (concat_p1 (1 @ q))^). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | transport_paths_FFlr | 6,841 |
{A B : Type} {f : A -> B} {g : B -> A} {x1 x2 : A} (p : x1 = x2) (q : x1 = g (f x1)) : transport (fun x => x = g (f x)) p q = p^ @ q @ (ap g (ap f p)). Proof. destruct p; simpl. exact ((concat_1p q)^ @ (concat_p1 (1 @ q))^). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | transport_paths_lFFr | 6,842 |
transport_paths_FlFr' {A B : Type} {f g : A -> B} {x1 x2 : A} (p : x1 = x2) (q : f x1 = g x1) (r : (f x2) = (g x2)) (h : (ap f p) @ r = q @ (ap g p)) : transport (fun x => f x = g x) p q = r. Proof. refine (transport_paths_FlFr _ _ @ _). refine (concat_pp_p _ _ _ @ _). apply moveR_Vp. exact h^. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | transport_paths_FlFr' | 6,843 |
transport_paths_FFlr' {A B : Type} {f : A -> B} {g : B -> A} {x1 x2 : A} (p : x1 = x2) (q : g (f x1) = x1) (r : g (f x2) = x2) (h : (ap g (ap f p)) @ r = q @ p) : transport (fun x => g (f x) = x) p q = r. Proof. refine (transport_paths_FFlr _ _ @ _). refine (concat_pp_p _ _ _ @ _). apply moveR_Vp. exact h^. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | transport_paths_FFlr' | 6,844 |
{A B X} (f : A -> X) (g : B -> X) {a1 a2 : A} {b1 b2 : B} (p : a1 = a2) (q : b1 = b2) (r : f a1 = g b1) : transport011 (fun a b => f a = g b) p q r = (ap f p)^ @ r @ ap g q. Proof. destruct p, q; cbn. symmetry. lhs nrapply concat_p1. apply concat_1p. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | transport011_paths | 6,845 |
{A : Type} {x y : A} (p : x = y) (q : idpath x = idpath x) : transport (fun a => idpath a = idpath a) p q = (concat_Vp p)^ @ whiskerL p^ ((concat_1p p)^ @ whiskerR q p @ concat_1p p) @ concat_Vp p. Proof. destruct p. simpl. refine (_ @ (concat_p1 _)^). refine (_ @ (concat_1p _)^). assert (H : forall (p : x = x) (q : 1 = p), (q @ (concat_p1 p)^) @ (concat_1p (p @ 1))^ = whiskerL (idpath x) (idpath 1 @ whiskerR q 1 @ idpath (p @ 1))). { intros p' q'. destruct q'. reflexivity. } transitivity (q @ (concat_p1 1)^ @ (concat_1p 1)^). { simpl; exact ((concat_p1 _)^ @ (concat_p1 _)^). } refine (H 1 q). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | transport_paths2 | 6,846 |
{A : Type} (x y : A) : (x = y) <~> (y = x) := Build_Equiv _ _ (@inverse A x y) _. Global Instance isequiv_concat_l {A : Type} `(p : x = y:>A) (z : A) : IsEquiv (@transitivity A _ _ x y z p) | 0. Proof. refine (Build_IsEquiv _ _ _ (concat p^) (concat_p_Vp p) (concat_V_pp p) _). intros q; destruct p; destruct q; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_path_inverse | 6,847 |
{A : Type} `(p : x = y) (z : A) : (y = z) <~> (x = z) := Build_Equiv _ _ (concat p) _. Global Instance isequiv_concat_r {A : Type} `(p : y = z) (x : A) : IsEquiv (fun q:x=y => q @ p) | 0. Proof. refine (Build_IsEquiv _ _ (fun q => q @ p) (fun q => q @ p^) (fun q => concat_pV_p q p) (fun q => concat_pp_V q p) _). intros q; destruct p; destruct q; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_concat_l | 6,848 |
{A : Type} `(p : y = z) (x : A) : (x = y) <~> (x = z) := Build_Equiv _ _ (fun q => q @ p) _. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_concat_r | 6,849 |
{A : Type} {x x' y y' : A} (p : x' = x) (q : y = y') : (x = y) <~> (x' = y') := Build_Equiv _ _ (fun r:x=y => p @ r @ q) _. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_concat_lr | 6,850 |
{A : Type} {x y : A} {p q : x = y} : p = q <~> p @ 1 = 1 @ q := equiv_concat_lr (concat_p1 p) (concat_1p q)^. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_p1_1q | 6,851 |
{A : Type} {x y : A} {p q : x = y} : p = q <~> 1 @ p = q @ 1 := equiv_concat_lr (concat_1p p) (concat_p1 q)^. Global Instance isequiv_whiskerL {A} {x y z : A} (p : x = y) {q r : y = z} : IsEquiv (@whiskerL A x y z p q r). Proof. simple refine (isequiv_adjointify _ _ _ _). - apply cancelL. - intros k. unfold cancelL. rewrite !whiskerL_pp. refine ((_ @@ 1 @@ _) @ whiskerL_pVL p k). + destruct p, q; reflexivity. + destruct p, r; reflexivity. - intros k. unfold cancelL. refine ((_ @@ 1 @@ _) @ whiskerL_VpL p k). + destruct p, q; reflexivity. + destruct p, r; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_1p_q1 | 6,852 |
{A} {x y z : A} (p : x = y) (q r : y = z) : (q = r) <~> (p @ q = p @ r) := Build_Equiv _ _ (whiskerL p) _. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_whiskerL | 6,853 |
{A} {x y z : A} (p : x = y) (q r : y = z) : (p @ q = p @ r) <~> (q = r) := equiv_inverse (equiv_whiskerL p q r). | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_cancelL | 6,854 |
{A} {x y z : A} (p : x = y) (q r : y = z) : IsEquiv (cancelL p q r). Proof. change (IsEquiv (equiv_cancelL p q r)); exact _. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | isequiv_cancelL | 6,855 |
{A} {x y z : A} (p q : x = y) (r : y = z) : (p = q) <~> (p @ r = q @ r) := Build_Equiv _ _ (fun h => whiskerR h r) _. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_whiskerR | 6,856 |
{A} {x y z : A} (p q : x = y) (r : y = z) : (p @ r = q @ r) <~> (p = q) := equiv_inverse (equiv_whiskerR p q r). | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_cancelR | 6,857 |
{A} {x y z : A} (p q : x = y) (r : y = z) : IsEquiv (cancelR p q r). Proof. change (IsEquiv (equiv_cancelR p q r)); exact _. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | isequiv_cancelR | 6,858 |
{A : Type} {x y z : A} (p : x = z) (q : y = z) (r : y = x) : (p = r^ @ q) <~> (r @ p = q) := Build_Equiv _ _ (moveR_Mp p q r) _. Global Instance isequiv_moveR_pM {A : Type} {x y z : A} (p : x = z) (q : y = z) (r : y = x) : IsEquiv (moveR_pM p q r). Proof. destruct p. apply (isequiv_compose' _ (isequiv_concat_l _ _) _ (isequiv_concat_r _ _)). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveR_Mp | 6,859 |
{A : Type} {x y z : A} (p : x = z) (q : y = z) (r : y = x) : (r = q @ p^) <~> (r @ p = q) := Build_Equiv _ _ (moveR_pM p q r) _. Global Instance isequiv_moveR_Vp {A : Type} {x y z : A} (p : x = z) (q : y = z) (r : x = y) : IsEquiv (moveR_Vp p q r). Proof. destruct r. apply (isequiv_compose' _ (isequiv_concat_l _ _) _ (isequiv_concat_r _ _)). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveR_pM | 6,860 |
{A : Type} {x y z : A} (p : x = z) (q : y = z) (r : x = y) : (p = r @ q) <~> (r^ @ p = q) := Build_Equiv _ _ (moveR_Vp p q r) _. Global Instance isequiv_moveR_pV {A : Type} {x y z : A} (p : z = x) (q : y = z) (r : y = x) : IsEquiv (moveR_pV p q r). Proof. destruct p. apply (isequiv_compose' _ (isequiv_concat_l _ _) _ (isequiv_concat_r _ _)). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveR_Vp | 6,861 |
{A : Type} {x y z : A} (p : z = x) (q : y = z) (r : y = x) : (r = q @ p) <~> (r @ p^ = q) := Build_Equiv _ _ (moveR_pV p q r) _. Global Instance isequiv_moveL_Mp {A : Type} {x y z : A} (p : x = z) (q : y = z) (r : y = x) : IsEquiv (moveL_Mp p q r). Proof. destruct r. apply (isequiv_compose' _ (isequiv_concat_l _ _) _ (isequiv_concat_r _ _)). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveR_pV | 6,862 |
{A : Type} {x y z : A} (p : x = z) (q : y = z) (r : y = x) : (r^ @ q = p) <~> (q = r @ p) := Build_Equiv _ _ (moveL_Mp p q r) _. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveL_Mp | 6,863 |
{A : Type} {x y z : A} (p : x = z) (q : y = z) (r : y = x) : IsEquiv (moveL_pM p q r). Proof. destruct p. apply (isequiv_compose' _ (isequiv_concat_l _ _) _ (isequiv_concat_r _ _)). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | isequiv_moveL_pM | 6,864 |
{A : Type} {x y z : A} (p : x = z) (q : y = z) (r : y = x) : q @ p^ = r <~> q = r @ p := Build_Equiv _ _ _ (isequiv_moveL_pM p q r). Global Instance isequiv_moveL_Vp {A : Type} {x y z : A} (p : x = z) (q : y = z) (r : x = y) : IsEquiv (moveL_Vp p q r). Proof. destruct r. apply (isequiv_compose' _ (isequiv_concat_l _ _) _ (isequiv_concat_r _ _)). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveL_pM | 6,865 |
{A : Type} {x y z : A} (p : x = z) (q : y = z) (r : x = y) : r @ q = p <~> q = r^ @ p := Build_Equiv _ _ (moveL_Vp p q r) _. Global Instance isequiv_moveL_pV {A : Type} {x y z : A} (p : z = x) (q : y = z) (r : y = x) : IsEquiv (moveL_pV p q r). Proof. destruct p. apply (isequiv_compose' _ (isequiv_concat_l _ _) _ (isequiv_concat_r _ _)). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveL_Vp | 6,866 |
{A : Type} {x y z : A} (p : z = x) (q : y = z) (r : y = x) : q @ p = r <~> q = r @ p^ := Build_Equiv _ _ (moveL_pV p q r) _. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveL_pV | 6,867 |
{A : Type} {x y : A} (p q : x = y) : IsEquiv (moveL_1M p q). Proof. destruct q. apply isequiv_concat_l. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | isequiv_moveL_1M | 6,868 |
{A : Type} {x y : A} (p q : x = y) : IsEquiv (moveL_M1 p q). Proof. destruct q. apply isequiv_concat_l. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | isequiv_moveL_M1 | 6,869 |
{A : Type} {x y : A} (p : x = y) (q : y = x) : IsEquiv (moveL_1V p q). Proof. destruct q. apply isequiv_concat_l. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | isequiv_moveL_1V | 6,870 |
{A : Type} {x y : A} (p : x = y) (q : y = x) : IsEquiv (moveL_V1 p q). Proof. destruct q. apply isequiv_concat_l. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | isequiv_moveL_V1 | 6,871 |
{A : Type} {x y : A} (p q : x = y) : IsEquiv (moveR_M1 p q). Proof. destruct p. apply isequiv_concat_r. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | isequiv_moveR_M1 | 6,872 |
{A : Type} {x y : A} (p q : x = y) : (1 = q @ p^) <~> (p = q) := Build_Equiv _ _ (moveR_1M p q) _. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveR_1M | 6,873 |
{A : Type} {x y : A} (p : x = y) (q : y = x) : IsEquiv (moveR_1V p q). Proof. destruct p. apply isequiv_concat_r. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | isequiv_moveR_1V | 6,874 |
{A : Type} {x y : A} (p : x = y) (q : y = x) : IsEquiv (moveR_V1 p q). Proof. destruct p. apply isequiv_concat_r. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | isequiv_moveR_V1 | 6,875 |
{A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) (v : P y) (q : transport P p u = v) : moveR_transport_p P p u v (moveL_transport_V P p u v q) = q. Proof. destruct p; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | moveR_moveL_transport_V | 6,876 |
{A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) (v : P y) (q : u = transport P p^ v) : moveL_transport_V P p u v (moveR_transport_p P p u v q) = q. Proof. destruct p; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | moveL_moveR_transport_p | 6,877 |
{A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) (v : P y) : u = transport P p^ v <~> transport P p u = v := Build_Equiv _ _ (moveR_transport_p P p u v) _. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveR_transport_p | 6,878 |
{A : Type} (P : A -> Type) {x y : A} (p : y = x) (u : P x) (v : P y) (q : transport P p^ u = v) : moveR_transport_V P p u v (moveL_transport_p P p u v q) = q. Proof. destruct p; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | moveR_moveL_transport_p | 6,879 |
{A : Type} (P : A -> Type) {x y : A} (p : y = x) (u : P x) (v : P y) (q : u = transport P p v) : moveL_transport_p P p u v (moveR_transport_V P p u v q) = q. Proof. destruct p; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | moveL_moveR_transport_V | 6,880 |
{A : Type} (P : A -> Type) {x y : A} (p : y = x) (u : P x) (v : P y) : u = transport P p v <~> transport P p^ u = v := Build_Equiv _ _ (moveR_transport_V P p u v) _. Global Instance isequiv_moveL_transport_V {A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) (v : P y) : IsEquiv (moveL_transport_V P p u v). Proof. srapply isequiv_adjointify. - apply moveR_transport_p. - intro q; apply moveL_moveR_transport_p. - intro q; apply moveR_moveL_transport_V. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveR_transport_V | 6,881 |
{A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) (v : P y) : transport P p u = v <~> u = transport P p^ v := Build_Equiv _ _ (moveL_transport_V P p u v) _. Global Instance isequiv_moveL_transport_p {A : Type} (P : A -> Type) {x y : A} (p : y = x) (u : P x) (v : P y) : IsEquiv (moveL_transport_p P p u v). Proof. srapply isequiv_adjointify. - apply moveR_transport_V. - intro q; apply moveL_moveR_transport_V. - intro q; apply moveR_moveL_transport_p. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveL_transport_V | 6,882 |
{A : Type} (P : A -> Type) {x y : A} (p : y = x) (u : P x) (v : P y) : transport P p^ u = v <~> u = transport P p v := Build_Equiv _ _ (moveL_transport_p P p u v) _. Global Instance isequiv_moveR_equiv_M `{IsEquiv A B f} (x : A) (y : B) : IsEquiv (@moveR_equiv_M A B f _ x y). Proof. unfold moveR_equiv_M. refine (@isequiv_compose _ _ (ap f) _ _ (fun q => q @ eisretr f y) _). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveL_transport_p | 6,883 |
`{IsEquiv A B f} (x : A) (y : B) : (x = f^-1 y) <~> (f x = y) := Build_Equiv _ _ (@moveR_equiv_M A B f _ x y) _. Global Instance isequiv_moveR_equiv_V `{IsEquiv A B f} (x : B) (y : A) : IsEquiv (@moveR_equiv_V A B f _ x y). Proof. unfold moveR_equiv_V. refine (@isequiv_compose _ _ (ap f^-1) _ _ (fun q => q @ eissect f y) _). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveR_equiv_M | 6,884 |
`{IsEquiv A B f} (x : B) (y : A) : (x = f y) <~> (f^-1 x = y) := Build_Equiv _ _ (@moveR_equiv_V A B f _ x y) _. Global Instance isequiv_moveL_equiv_M `{IsEquiv A B f} (x : A) (y : B) : IsEquiv (@moveL_equiv_M A B f _ x y). Proof. unfold moveL_equiv_M. refine (@isequiv_compose _ _ (ap f) _ _ (fun q => (eisretr f y)^ @ q) _). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveR_equiv_V | 6,885 |
`{IsEquiv A B f} (x : A) (y : B) : (f^-1 y = x) <~> (y = f x) := Build_Equiv _ _ (@moveL_equiv_M A B f _ x y) _. Global Instance isequiv_moveL_equiv_V `{IsEquiv A B f} (x : B) (y : A) : IsEquiv (@moveL_equiv_V A B f _ x y). Proof. unfold moveL_equiv_V. refine (@isequiv_compose _ _ (ap f^-1) _ _ (fun q => (eissect f y)^ @ q) _). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveL_equiv_M | 6,886 |
`{IsEquiv A B f} (x : B) (y : A) : (f y = x) <~> (y = f^-1 x) := Build_Equiv _ _ (@moveL_equiv_V A B f _ x y) _. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_moveL_equiv_V | 6,887 |
{A : Type} {x1 x2 y : A} (p : x1 = x2) (q : x1 = y) (r : x2 = y) : q = p @ r <~> transport (fun x => x = y) p q = r. Proof. destruct p; simpl. exact (equiv_concat_r (concat_1p r) q). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | dpath_path_l | 6,888 |
{A : Type} {x y1 y2 : A} (p : y1 = y2) (q : x = y1) (r : x = y2) : q @ p = r <~> transport (fun y => x = y) p q = r. Proof. destruct p; simpl. exact (equiv_concat_l (concat_p1 q)^ r). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | dpath_path_r | 6,889 |
{A : Type} {x1 x2 : A} (p : x1 = x2) (q : x1 = x1) (r : x2 = x2) : q @ p = p @ r <~> transport (fun x => x = x) p q = r. Proof. destruct p; simpl. transitivity (q @ 1 = r). - exact (equiv_concat_r (concat_1p r) (q @ 1)). - exact (equiv_concat_l (concat_p1 q)^ r). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | dpath_path_lr | 6,890 |
{A B : Type} {f : A -> B} {x1 x2 : A} {y : B} (p : x1 = x2) (q : f x1 = y) (r : f x2 = y) : q = ap f p @ r <~> transport (fun x => f x = y) p q = r. Proof. destruct p; simpl. exact (equiv_concat_r (concat_1p r) q). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | dpath_path_Fl | 6,891 |
{A B : Type} {g : A -> B} {x : B} {y1 y2 : A} (p : y1 = y2) (q : x = g y1) (r : x = g y2) : q @ ap g p = r <~> transport (fun y => x = g y) p q = r. Proof. destruct p; simpl. exact (equiv_concat_l (concat_p1 q)^ r). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | dpath_path_Fr | 6,892 |
{A B : Type} {f g : A -> B} {x1 x2 : A} (p : x1 = x2) (q : f x1 = g x1) (r : f x2 = g x2) : q @ ap g p = ap f p @ r <~> transport (fun x => f x = g x) p q = r. Proof. destruct p; simpl. transitivity (q @ 1 = r). - exact (equiv_concat_r (concat_1p r) (q @ 1)). - exact (equiv_concat_l (concat_p1 q)^ r). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | dpath_path_FlFr | 6,893 |
{A B : Type} {f : A -> B} {g : B -> A} {x1 x2 : A} (p : x1 = x2) (q : g (f x1) = x1) (r : g (f x2) = x2) : q @ p = ap g (ap f p) @ r <~> transport (fun x => g (f x) = x) p q = r. Proof. destruct p; simpl. transitivity (q @ 1 = r). - exact (equiv_concat_r (concat_1p r) (q @ 1)). - exact (equiv_concat_l (concat_p1 q)^ r). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | dpath_path_FFlr | 6,894 |
{A B : Type} {f : A -> B} {g : B -> A} {x1 x2 : A} (p : x1 = x2) (q : x1 = g (f x1)) (r : x2 = g (f x2)) : q @ ap g (ap f p) = p @ r <~> transport (fun x => x = g (f x)) p q = r. Proof. destruct p; simpl. transitivity (q @ 1 = r). - exact (equiv_concat_r (concat_1p r) (q @ 1)). - exact (equiv_concat_l (concat_p1 q)^ r). Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | dpath_path_lFFr | 6,895 |
{A : Type} {x y : A} (p : x = y) (q : idpath x = idpath x) (r : idpath y = idpath y) : (concat_1p p)^ @ whiskerR q p @ concat_1p p = (concat_p1 p)^ @ whiskerL p r @ concat_p1 p <~> transport (fun a => idpath a = idpath a) p q = r. Proof. destruct p. simpl. refine (_ oE (equiv_whiskerR _ _ 1)^-1). refine (_ oE (equiv_whiskerL 1 _ _)^-1). refine (equiv_concat_lr _ _). - symmetry; apply whiskerR_p1_1. - apply whiskerL_1p_1. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | dpath_paths2 | 6,896 |
`{Funext} {A : Type} (a : A) (P : forall x, (a = x) -> Type) : P a 1 <~> forall x p, P x p := Build_Equiv _ _ (paths_ind a P) _. Global Instance isequiv_paths_ind_r `{Funext} {A : Type} (a : A) (P : forall x, (x = a) -> Type) : IsEquiv (paths_ind_r a P) | 0. Proof. refine (isequiv_adjointify (paths_ind_r a P) (fun f => f a 1) _ _). - intros f. apply path_forall; intros x. apply path_forall; intros p. destruct p; reflexivity. - intros u. reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_paths_ind | 6,897 |
`{Funext} {A : Type} (a : A) (P : forall x, (x = a) -> Type) : P a 1 <~> forall x p, P x p := Build_Equiv _ _ (paths_ind_r a P) _. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Basics.Tactics. | Types\Paths.v | equiv_paths_ind_r | 6,898 |
`{P : A * B -> Type} (u : A * B) : P (fst u, snd u) -> P u := idmap. | Definition | Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids Require Import Types.Empty. | Types\Prod.v | unpack_prod | 6,899 |