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{A : Type} {x y z : A} (p : y = x) (q : y = z) : (p^ @ q)^ = q^ @ p := match q with idpath => match p with idpath => 1 end end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | inv_Vp | 1,600 |
{A : Type} {x y z : A} (p : x = y) (q : z = y) : (p @ q^)^ = q @ p^. Proof. destruct p. destruct q. reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | inv_pV | 1,601 |
{A : Type} {x y z : A} (p : y = x) (q : z = y) : (p^ @ q^)^ = q @ p. Proof. destruct p. destruct q. reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | inv_VV | 1,602 |
{A : Type} {x y : A} (p : x = y) : p^^ = p := match p with idpath => 1 end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | inv_V | 1,603 |
{A : Type} {x y z : A} (p : x = z) (q : y = z) (r : y = x) : p = r^ @ q -> r @ p = q. Proof. destruct r. intro h. exact (concat_1p _ @ h @ concat_1p _). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveR_Mp | 1,604 |
{A : Type} {x y z : A} (p : x = z) (q : y = z) (r : y = x) : r = q @ p^ -> r @ p = q. Proof. destruct p. intro h. exact (concat_p1 _ @ h @ concat_p1 _). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveR_pM | 1,605 |
{A : Type} {x y z : A} (p : x = z) (q : y = z) (r : x = y) : p = r @ q -> r^ @ p = q. Proof. destruct r. intro h. exact (concat_1p _ @ h @ concat_1p _). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveR_Vp | 1,606 |
{A : Type} {x y z : A} (p : z = x) (q : y = z) (r : y = x) : r = q @ p -> r @ p^ = q. Proof. destruct p. intro h. exact (concat_p1 _ @ h @ concat_p1 _). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveR_pV | 1,607 |
{A : Type} {x y z : A} (p : x = z) (q : y = z) (r : y = x) : r^ @ q = p -> q = r @ p. Proof. destruct r. intro h. exact ((concat_1p _)^ @ h @ (concat_1p _)^). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveL_Mp | 1,608 |
{A : Type} {x y z : A} (p : x = z) (q : y = z) (r : y = x) : q @ p^ = r -> q = r @ p. Proof. destruct p. intro h. exact ((concat_p1 _)^ @ h @ (concat_p1 _)^). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveL_pM | 1,609 |
{A : Type} {x y z : A} (p : x = z) (q : y = z) (r : x = y) : r @ q = p -> q = r^ @ p. Proof. destruct r. intro h. exact ((concat_1p _)^ @ h @ (concat_1p _)^). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveL_Vp | 1,610 |
{A : Type} {x y z : A} (p : z = x) (q : y = z) (r : y = x) : q @ p = r -> q = r @ p^. Proof. destruct p. intro h. exact ((concat_p1 _)^ @ h @ (concat_p1 _)^). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveL_pV | 1,611 |
{A : Type} {x y : A} (p q : x = y) : p @ q^ = 1 -> p = q. Proof. destruct q. intro h. exact ((concat_p1 _)^ @ h). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveL_1M | 1,612 |
{A : Type} {x y : A} (p q : x = y) : q^ @ p = 1 -> p = q. Proof. destruct q. intro h. exact ((concat_1p _)^ @ h). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveL_M1 | 1,613 |
{A : Type} {x y : A} (p : x = y) (q : y = x) : p @ q = 1 -> p = q^. Proof. destruct q. intro h. exact ((concat_p1 _)^ @ h). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveL_1V | 1,614 |
{A : Type} {x y : A} (p : x = y) (q : y = x) : q @ p = 1 -> p = q^. Proof. destruct q. intro h. exact ((concat_1p _)^ @ h). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveL_V1 | 1,615 |
{A : Type} {x y : A} (p q : x = y) : 1 = p^ @ q -> p = q. Proof. destruct p. intro h. exact (h @ (concat_1p _)). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveR_M1 | 1,616 |
{A : Type} {x y : A} (p q : x = y) : 1 = q @ p^ -> p = q. Proof. destruct p. intro h. exact (h @ (concat_p1 _)). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveR_1M | 1,617 |
{A : Type} {x y : A} (p : x = y) (q : y = x) : 1 = q @ p -> p^ = q. Proof. destruct p. intro h. exact (h @ (concat_p1 _)). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveR_1V | 1,618 |
{A : Type} {x y : A} (p : x = y) (q : y = x) : 1 = p @ q -> p^ = q. Proof. destruct p. intro h. exact (h @ (concat_1p _)). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveR_V1 | 1,619 |
{A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) (v : P y) : u = p^ # v -> p # u = v. Proof. destruct p. exact idmap. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveR_transport_p | 1,620 |
{A : Type} (P : A -> Type) {x y : A} (p : y = x) (u : P x) (v : P y) : u = p # v -> p^ # u = v. Proof. destruct p. exact idmap. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveR_transport_V | 1,621 |
{A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) (v : P y) : p # u = v -> u = p^ # v. Proof. destruct p. exact idmap. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveL_transport_V | 1,622 |
{A : Type} (P : A -> Type) {x y : A} (p : y = x) (u : P x) (v : P y) : p^ # u = v -> u = p # v. Proof. destruct p. exact idmap. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveL_transport_p | 1,623 |
{A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) (v : P y) (q : u = p^ # v) : (moveR_transport_p P p u v q)^ = moveL_transport_p P p v u q^. Proof. destruct p; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveR_transport_p_V | 1,624 |
{A : Type} (P : A -> Type) {x y : A} (p : y = x) (u : P x) (v : P y) (q : u = p # v) : (moveR_transport_V P p u v q)^ = moveL_transport_V P p v u q^. Proof. destruct p; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveR_transport_V_V | 1,625 |
{A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) (v : P y) (q : p # u = v) : (moveL_transport_V P p u v q)^ = moveR_transport_V P p v u q^. Proof. destruct p; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveL_transport_V_V | 1,626 |
{A : Type} (P : A -> Type) {x y : A} (p : y = x) (u : P x) (v : P y) (q : p^ # u = v) : (moveL_transport_p P p u v q)^ = moveR_transport_p P p v u q^. Proof. destruct p; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveL_transport_p_V | 1,627 |
{A B : Type} (x : A) (f : A -> B) : ap f 1 = 1 :> (f x = f x) := 1. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap_1 | 1,628 |
{A B} (x : A) (f : forall x : A, B x) : apD f 1 = 1 :> (f x = f x) := 1. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | apD_1 | 1,629 |
{A B : Type} (f : A -> B) {x y z : A} (p : x = y) (q : y = z) : ap f (p @ q) = (ap f p) @ (ap f q) := match q with idpath => match p with idpath => 1 end end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap_pp | 1,630 |
{A B : Type} (f : A -> B) {w : B} {x y z : A} (r : w = f x) (p : x = y) (q : y = z) : r @ (ap f (p @ q)) = (r @ ap f p) @ (ap f q). Proof. destruct p, q. simpl. exact (concat_p_pp r 1 1). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap_p_pp | 1,631 |
{A B : Type} (f : A -> B) {x y z : A} {w : B} (p : x = y) (q : y = z) (r : f z = w) : (ap f (p @ q)) @ r = (ap f p) @ (ap f q @ r). Proof. destruct p, q. simpl. exact (concat_pp_p 1 1 r). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap_pp_p | 1,632 |
{A B : Type} (f : A -> B) {x y : A} (p : x = y) : (ap f p)^ = ap f (p^) := match p with idpath => 1 end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | inverse_ap | 1,633 |
{A B : Type} (f : A -> B) {x y : A} (p : x = y) : ap f (p^) = (ap f p)^ := match p with idpath => 1 end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap_V | 1,634 |
{A : Type} {x y : A} (p : x = y) : ap idmap p = p := match p with idpath => 1 end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap_idmap | 1,635 |
{A B C : Type} (f : A -> B) (g : B -> C) {x y : A} (p : x = y) : ap (g o f) p = ap g (ap f p) := match p with idpath => 1 end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap_compose | 1,636 |
ap_compose' {A B C : Type} (f : A -> B) (g : B -> C) {x y : A} (p : x = y) : ap (fun a => g (f a)) p = ap g (ap f p) := match p with idpath => 1 end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap_compose' | 1,637 |
{A B : Type} {x y : A} (p : x = y) (z : B) : ap (fun _ => z) p = 1 := match p with idpath => idpath end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap_const | 1,638 |
{A B : Type} {f g : A -> B} (p : forall x, f x = g x) {x y : A} (q : x = y) : (ap f q) @ (p y) = (p x) @ (ap g q) := match q with | idpath => concat_1p_p1 _ end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | concat_Ap | 1,639 |
{A B : Type} {f g : A -> B} (p : forall x, f x = g x) {x y : A} (q : x = y) : (ap f q) = (p x) @ (ap g q) @ (p y)^. Proof. apply moveL_pV. apply concat_Ap. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap_homotopic | 1,640 |
{A : Type} {f : A -> A} (p : forall x, f x = x) {x y : A} (q : x = y) : (ap f q) @ (p y) = (p x) @ q := match q with | idpath => concat_1p_p1 _ end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | concat_A1p | 1,641 |
{A : Type} {f : A -> A} (p : forall x, f x = x) {x y : A} (q : x = y) : (ap f q) = (p x) @ q @ (p y)^. Proof. apply moveL_pV. apply concat_A1p. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap_homotopic_id | 1,642 |
{A : Type} {f : A -> A} (p : forall x, x = f x) {x y : A} (q : x = y) : (p x) @ (ap f q) = q @ (p y) := match q as i in (_ = y) return (p x @ ap f i = i @ p y) with | idpath => concat_p1_1p _ end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | concat_pA1 | 1,643 |
{A : Type} {B : A -> Type} {f g : forall x, B x} (p : forall x, f x = g x) {x y : A} (q : x = y) : apD f q = ap (transport B q) (p x) @ apD g q @ (p y)^. Proof. apply moveL_pV. destruct q; unfold apD, transport. symmetry. exact (concat_p1 _ @ ap_idmap _ @ (concat_1p _)^). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | apD_homotopic | 1,644 |
{A B : Type} {f g : A -> B} (p : forall x, f x = g x) {x y : A} (q : x = y) {w z : B} (r : w = f x) (s : g y = z) : (r @ ap f q) @ (p y @ s) = (r @ p x) @ (ap g q @ s). Proof. destruct q, s; simpl. induction (p x). reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | concat_pA_pp | 1,645 |
{A B : Type} {f g : A -> B} (p : forall x, f x = g x) {x y : A} (q : x = y) {w : B} (r : w = f x) : (r @ ap f q) @ p y = (r @ p x) @ ap g q. Proof. destruct q; simpl. induction (p x). reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | concat_pA_p | 1,646 |
{A B : Type} {f g : A -> B} (p : forall x, f x = g x) {x y : A} (q : x = y) {z : B} (s : g y = z) : (ap f q) @ (p y @ s) = (p x) @ (ap g q @ s). Proof. destruct q, s; cbn. apply concat_1p. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | concat_A_pp | 1,647 |
{A : Type} {f : A -> A} (p : forall x, f x = x) {x y : A} (q : x = y) {w z : A} (r : w = f x) (s : y = z) : (r @ ap f q) @ (p y @ s) = (r @ p x) @ (q @ s). Proof. destruct q, s; simpl. induction (p x). reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | concat_pA1_pp | 1,648 |
{A : Type} {g : A -> A} (p : forall x, x = g x) {x y : A} (q : x = y) {w z : A} (r : w = x) (s : g y = z) : (r @ p x) @ (ap g q @ s) = (r @ q) @ (p y @ s). Proof. destruct q, s; simpl. induction (p x). reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | concat_pp_A1p | 1,649 |
{A : Type} {f : A -> A} (p : forall x, f x = x) {x y : A} (q : x = y) {w : A} (r : w = f x) : (r @ ap f q) @ p y = (r @ p x) @ q. Proof. destruct q; simpl. induction (p x). reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | concat_pA1_p | 1,650 |
{A : Type} {f : A -> A} (p : forall x, f x = x) {x y : A} (q : x = y) {z : A} (s : y = z) : (ap f q) @ (p y @ s) = (p x) @ (q @ s). Proof. destruct q, s; cbn. apply concat_1p. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | concat_A1_pp | 1,651 |
{A : Type} {g : A -> A} (p : forall x, x = g x) {x y : A} (q : x = y) {w : A} (r : w = x) : (r @ p x) @ ap g q = (r @ q) @ p y. Proof. destruct q; simpl. induction (p x). reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | concat_pp_A1 | 1,652 |
{A : Type} {g : A -> A} (p : forall x, x = g x) {x y : A} (q : x = y) {z : A} (s : g y = z) : p x @ (ap g q @ s) = q @ (p y @ s). Proof. destruct q, s; simpl. symmetry; apply concat_1p. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | concat_p_A1p | 1,653 |
{A} {x : A} (p : x = x) (q : p = 1) : concat_1p p @ q = ap (fun p' => 1 @ p') q. Proof. rewrite <- (inv_V q). set (r := q^). clearbody r; clear q; destruct r. reflexivity. Defined. | Lemma | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | concat_1p_1 | 1,654 |
{A} {x : A} (p : x = x) (q : p = 1) : concat_p1 p @ q = ap (fun p' => p' @ 1) q. Proof. rewrite <- (inv_V q). set (r := q^). clearbody r; clear q; destruct r. reflexivity. Defined. | Lemma | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | concat_p1_1 | 1,655 |
{A} {B:A->Type} (f : forall x, B x) (x:A) : apD10 (idpath f) x = 1 := 1. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | apD10_1 | 1,656 |
{A} {B:A->Type} {f f' f'' : forall x, B x} (h:f=f') (h':f'=f'') (x:A) : apD10 (h @ h') x = apD10 h x @ apD10 h' x. Proof. case h, h'; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | apD10_pp | 1,657 |
{A} {B:A->Type} {f g : forall x, B x} (h:f=g) (x:A) : apD10 (h^) x = (apD10 h x)^ := match h with idpath => 1 end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | apD10_V | 1,658 |
{A B} {f:A->B} (x:A) : ap10 (idpath f) x = 1 := 1. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap10_1 | 1,659 |
{A B} {f f' f'':A->B} (h:f=f') (h':f'=f'') (x:A) : ap10 (h @ h') x = ap10 h x @ ap10 h' x := apD10_pp h h' x. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap10_pp | 1,660 |
{A B} {f g : A->B} (h : f = g) (x:A) : ap10 (h^) x = (ap10 h x)^ := apD10_V h x. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap10_V | 1,661 |
{A B C} (f : A -> B) {g g' : forall x:B, C x} (p : g = g') a : apD10 (ap (fun h : forall x:B, C x => h oD f) p) a = apD10 p (f a). Proof. destruct p; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | apD10_ap_precompose | 1,662 |
{A B C} (f : A -> B) {g g' : B -> C} (p : g = g') a : ap10 (ap (fun h : B -> C => h o f) p) a = ap10 p (f a) := apD10_ap_precompose f p a. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap10_ap_precompose | 1,663 |
{A B C} (f : forall x, B x -> C) {g g' : forall x:A, B x} (p : g = g') a : apD10 (ap (fun h : forall x:A, B x => fun x => f x (h x)) p) a = ap (f a) (apD10 p a). Proof. destruct p; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | apD10_ap_postcompose | 1,664 |
{A B C} (f : B -> C) {g g' : A -> B} (p : g = g') a : ap10 (ap (fun h : A -> B => f o h) p) a = ap f (ap10 p a) := apD10_ap_postcompose (fun a => f) p a. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap10_ap_postcompose | 1,665 |
{X Y Z : Type} {f g : X -> Y -> Z} (p : f = g) (x : X) (y : Y) : f x y = g x y := (ap10 (ap10 p x) y). | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap100 | 1,666 |
{A : Type} (P : A -> Type) {x : A} (u : P x) : 1 # u = u := 1. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transport_1 | 1,667 |
{A : Type} (P : A -> Type) {x y z : A} (p : x = y) (q : y = z) (u : P x) : p @ q # u = q # p # u := match q with idpath => match p with idpath => 1 end end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transport_pp | 1,668 |
{A : Type} (P : A -> Type) {x y : A} (p : x = y) (z : P y) : p # p^ # z = z := (transport_pp P p^ p z)^ @ ap (fun r => transport P r z) (concat_Vp p). | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transport_pV | 1,669 |
{A : Type} (P : A -> Type) {x y : A} (p : x = y) (z : P x) : p^ # p # z = z := (transport_pp P p p^ z)^ @ ap (fun r => transport P r z) (concat_pV p). | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transport_Vp | 1,670 |
{A : Type} (P : A -> Type) {x y z w : A} (p : x = y) (q : y = z) (r : z = w) (u : P x) : ap (fun e => e # u) (concat_p_pp p q r) @ (transport_pp P (p@q) r u) @ ap (transport P r) (transport_pp P p q u) = (transport_pp P p (q@r) u) @ (transport_pp P q r (p#u)) :> ((p @ (q @ r)) # u = r # q # p # u) . Proof. destruct p, q, r. simpl. exact 1. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transport_p_pp | 1,671 |
{A} (P : A -> Type) {x y:A} (p:x=y) (z:P x) : transport_pV P p (transport P p z) = ap (transport P p) (transport_Vp P p z). Proof. destruct p; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transport_pVp | 1,672 |
{A} (P : A -> Type) {x y : A} (p : x = y) (z : P y) : transport_Vp P p (transport P p^ z) = ap (transport P p^) (transport_pV P p z). Proof. destruct p; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transport_VpV | 1,673 |
{A} (P : A -> Type) {x y : A} (p : x = y) (u : P x) (v : P y) (e : transport P p u = v) : ap (transport P p) (moveL_transport_V P p u v e) @ transport_pV P p v = e. Proof. by destruct e, p. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap_transport_transport_pV | 1,674 |
{A} (P : A -> Type) {x y : A} (p : x = y) (u : P x) : moveL_transport_V P p u (p # u) 1 = (transport_Vp P p u)^. Proof. destruct p; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | moveL_transport_V_1 | 1,675 |
{A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) : paths_rect A x (fun a _ => P a) u y p = transport P p u := 1. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | paths_rect_transport | 1,676 |
{A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) : paths_ind x (fun a _ => P a) u y p = transport P p u := 1. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | paths_ind_transport | 1,677 |
{A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P y) : paths_ind_r y (fun b _ => P b) u x p = transport P p^ u. Proof. by destruct p. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | paths_ind_r_transport | 1,678 |
{A B} {f g:A->B} (h:f=g) {x y:A} (p:x=y) : ap11 h p = ap10 h x @ ap g p. Proof. by path_induction. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap11_is_ap10_ap01 | 1,679 |
{A : Type} (B : A -> Type) (C : forall a:A, B a -> Type) {x1 x2 : A} (p : x1 = x2) (y : B x1) (z : C x1 y) : C x2 (p # y) := match p with idpath => z end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transportD | 1,680 |
{A : Type} (B C : A -> Type) (D : forall a:A, B a -> C a -> Type) {x1 x2 : A} (p : x1 = x2) (y : B x1) (z : C x1) (w : D x1 y z) : D x2 (p # y) (p # z) := match p with idpath => w end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transportD2 | 1,681 |
{A B C} (f : A -> B -> C) {x x' y y'} (p : x = x') (q : y = y') : f x y = f x' y'. Proof. destruct p. apply ap. exact q. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap011 | 1,682 |
{A B C} (f : A -> B -> C) {x x' y y'} (p : x = x') (q : y = y') : ap011 f p^ q^ = (ap011 f p q)^. Proof. destruct p. apply ap_V. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap011_V | 1,683 |
{A B C} (f : A -> B -> C) {x x' x'' y y' y''} (p : x = x') (p' : x' = x'') (q : y = y') (q' : y' = y'') : ap011 f (p @ p') (q @ q') = ap011 f p q @ ap011 f p' q'. Proof. destruct p, p'. apply ap_pp. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap011_pp | 1,684 |
{A B C D} (f : A -> B -> C) (g : C -> D) {x x' y y'} (p : x = x') (q : y = y') : ap011 (fun x y => g (f x y)) p q = ap g (ap011 f p q). Proof. destruct p; simpl. apply ap_compose. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap011_compose | 1,685 |
ap011_compose' {A B C D E} (f : A -> B -> C) (g : D -> A) (h : E -> B) {x x' y y'} (p : x = x') (q : y = y') : ap011 (fun x y => f (g x) (h y)) p q = ap011 f (ap g p) (ap h q). Proof. destruct p; simpl. apply ap_compose. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap011_compose' | 1,686 |
{A B C} (f : A -> B -> C) {x x' : A} {y y' : B} (p : x = x') (q : y = y') : ap011 f p q = ap (fun x => f x y) p @ ap (fun y => f x' y) q. Proof. destruct p. symmetry. apply concat_1p. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap011_is_ap | 1,687 |
{A B C} (f : forall (a:A), B a -> C) {x x'} (p : x = x') {y y'} (q : p # y = y') : f x y = f x' y'. Proof. destruct p, q; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap011D | 1,688 |
{A B C} (f : forall (a:A), B a -> C a) {x x'} (p : x = x') {y y'} (q : p # y = y') : transport C p (f x y) = f x' y'. Proof. destruct p, q; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap01D1 | 1,689 |
{A B C} (f : forall (a:A) (b:B a), C a b) {x x'} (p : x = x') {y y'} (q : p # y = y') : transport (C x') q (transportD B C p y (f x y)) = f x' y'. Proof. destruct p, q; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | apD011 | 1,690 |
{A B} (P : A -> B -> Type) {x1 x2 : A} {y1 y2 : B} (p : x1 = x2) (q : y1 = y2) (z : P x1 y1) : P x2 y2 := transport (fun x => P x y2) p (transport (fun y => P x1 y) q z). | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transport011 | 1,691 |
{A B} (P : A -> B -> Type) {x1 x2 x3 : A} {y1 y2 y3 : B} (p1 : x1 = x2) (p2 : x2 = x3) (q1 : y1 = y2) (q2 : y2 = y3) (z : P x1 y1) : transport011 P (p1 @ p2) (q1 @ q2) z = transport011 P p2 q2 (transport011 P p1 q1 z). Proof. destruct p1, p2, q1, q2; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transport011_pp | 1,692 |
{A B A' B'} (P : A -> B -> Type) (f : A' -> A) (g : B' -> B) {x1 x2 : A'} {y1 y2 : B'} (p : x1 = x2) (q : y1 = y2) (z : P (f x1) (g y1)) : transport011 (fun x y => P (f x) (g y)) p q z = transport011 P (ap f p) (ap g q) z. Proof. destruct p, q; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transport011_compose | 1,693 |
{A B} {P Q : A -> B -> Type} {a1 a2 : A} {b1 b2 : B} (p : a1 = a2) (q : b1 = b2) (f : forall {a b}, P a b -> Q a b) (x : P a1 b1) : f (transport011 P p q x) = transport011 Q p q (f x). Proof. destruct p, q; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | ap_transport011 | 1,694 |
{A : Type} (P : A -> Type) {x y : A} {p q : x = y} (r : p = q) (z : P x) : p # z = q # z := ap (fun p' => p' # z) r. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transport2 | 1,695 |
{A : Type} (Q : A -> Type) {x y : A} {p q : x = y} (r : p = q) (z : Q x) : transport2 Q r z = ap10 (ap (transport Q) r) z := match r with idpath => 1 end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transport2_is_ap10 | 1,696 |
{A : Type} (P : A -> Type) {x y : A} {p1 p2 p3 : x = y} (r1 : p1 = p2) (r2 : p2 = p3) (z : P x) : transport2 P (r1 @ r2) z = transport2 P r1 z @ transport2 P r2 z. Proof. destruct r1, r2; reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transport2_p2p | 1,697 |
{A : Type} (Q : A -> Type) {x y : A} {p q : x = y} (r : p = q) (z : Q x) : transport2 Q (r^) z = (transport2 Q r z)^ := match r with idpath => 1 end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | transport2_V | 1,698 |
{A : Type} (P : A -> Type) {x y : A} {p q : x = y} {z w : P x} (r : p = q) (s : z = w) : ap (transport P p) s @ transport2 P r w = transport2 P r z @ ap (transport P q) s := match r with idpath => (concat_p1_1p _) end. | Definition | Require Import Basics.Overture Basics.Tactics. | Basics\PathGroupoids.v | concat_AT | 1,699 |