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{z : A} : prod_0gpd I (fun i => yon_0gpd (x i) z) $<~> (yon_0gpd (cat_prod I x) z) := cate_cat_prod_corec_inv^-1$. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cate_cat_prod_corec | 7,700 |
{z : A} : (forall i, z $-> x i) -> (z $-> cat_prod I x). Proof. apply cate_cat_prod_corec. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_prod_corec | 7,701 |
{z : A} (f : forall i, z $-> x i) : forall i, cat_pr i $o cat_prod_corec f $== f i. Proof. exact (cate_isretr cate_cat_prod_corec_inv f). Defined. | Lemma | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_prod_beta | 7,702 |
{z : A} (f : z $-> cat_prod I x) : cat_prod_corec (fun i => cat_pr i $o f) $== f. Proof. exact (cate_issect cate_cat_prod_corec_inv f). Defined. | Lemma | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_prod_eta | 7,703 |
NatEquiv (yon_0gpd (cat_prod I x)) (fun z : A^op => prod_0gpd I (fun i => yon_0gpd (x i) z)). Proof. snrapply Build_NatEquiv. 1: intro; nrapply cate_cat_prod_corec_inv. exact (is1natural_yoneda_0gpd (cat_prod I x) (fun z => prod_0gpd I (fun i => yon_0gpd (x i) z)) cat_pr). Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | natequiv_cat_prod_corec_inv | 7,704 |
{z : A} {f f' : forall i, z $-> x i} : (forall i, f i $== f' i) -> cat_prod_corec f $== cat_prod_corec f'. Proof. intros p. unfold cat_prod_corec. nrapply (moveL_equiv_V_0gpd cate_cat_prod_corec_inv). nrefine (cate_isretr cate_cat_prod_corec_inv _ $@ _). exact p. Defined. | Lemma | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_prod_corec_eta | 7,705 |
{z : A} {f f' : z $-> cat_prod I x} : (forall i, cat_pr i $o f $== cat_pr i $o f') -> f $== f'. Proof. intros p. refine ((cat_prod_eta _)^$ $@ _ $@ cat_prod_eta _). by nrapply cat_prod_corec_eta. Defined. | Lemma | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_prod_pr_eta | 7,706 |
{I : Type} {A : Type} (x : A) `{Product I _ (fun _ => x)} : x $-> cat_prod I (fun _ => x) := cat_prod_corec I (fun _ => Id x). | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_prod_diag | 7,707 |
{I J : Type} (ie : I <~> J) {A : Type} `{HasEquivs A} (x : I -> A) `{!Product I x} (y : J -> A) `{!Product J y} (e : forall i : I, x i $<~> y (ie i)) : cat_prod I x $<~> cat_prod J y. Proof. nrapply yon_equiv_0gpd. nrefine (natequiv_compose _ (natequiv_cat_prod_corec_inv _)). nrefine (natequiv_compose (natequiv_inverse (natequiv_cat_prod_corec_inv _)) _). snrapply Build_NatEquiv. - intros z. nrapply (cate_prod_0gpd ie). intros i. exact (natequiv_yon_equiv_0gpd (e i) _). - snrapply Build_Is1Natural. intros a b f g j. cbn. destruct (eisretr ie j). exact (cat_assoc_opp _ _ _). Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cate_cat_prod | 7,708 |
{I A : Type} `{HasEquivs A} (x : I -> A) `{!Product I x} (y : I -> A) `{!Product I y} (e : forall i : I, x i $<~> y i) : cat_prod I x $<~> cat_prod I y. Proof. exact (cate_cat_prod 1 x y e). Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_prod_unique | 7,709 |
{A : Type} {z : A} `{Product Empty A (fun _ => z)} : IsTerminal (cat_prod Empty (fun _ => z)). Proof. intros a. snrefine (cat_prod_corec _ _; fun f => cat_prod_pr_eta _ _); intros []. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | isterminal_prodempty | 7,710 |
A := cat_prod Bool (fun b : Bool => if b then x else y). | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod | 7,711 |
cat_binprod $-> x := cat_pr true. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_pr1 | 7,712 |
cat_binprod $-> y := cat_pr false. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_pr2 | 7,713 |
{z : A} (f : z $-> x) (g : z $-> y) : z $-> cat_binprod. Proof. nrapply (cat_prod_corec Bool). intros [|]. - exact f. - exact g. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_corec | 7,714 |
{z : A} (f : z $-> x) (g : z $-> y) : cat_pr1 $o cat_binprod_corec f g $== f := cat_prod_beta _ _ true. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_beta_pr1 | 7,715 |
{z : A} (f : z $-> x) (g : z $-> y) : cat_pr2 $o cat_binprod_corec f g $== g := cat_prod_beta _ _ false. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_beta_pr2 | 7,716 |
{z : A} (f : z $-> cat_binprod) : cat_binprod_corec (cat_pr1 $o f) (cat_pr2 $o f) $== f. Proof. unfold cat_binprod_corec. nrapply cat_prod_pr_eta. intros [|]. - exact (cat_binprod_beta_pr1 _ _). - exact (cat_binprod_beta_pr2 _ _). Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_eta | 7,717 |
{z : A} (f g : z $-> cat_binprod) : cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g. Proof. intros p q. rapply cat_prod_pr_eta. intros [|]. - exact p. - exact q. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_eta_pr | 7,718 |
{z : A} (f f' : z $-> x) (g g' : z $-> y) : f $== f' -> g $== g' -> cat_binprod_corec f g $== cat_binprod_corec f' g'. Proof. intros p q. rapply cat_prod_corec_eta. intros [|]. - exact p. - exact q. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_corec_eta | 7,719 |
{A : Type} `{Is1Cat A} {x y : A} (cat_binprod : A) (cat_pr1 : cat_binprod $-> x) (cat_pr2 : cat_binprod $-> y) (cat_binprod_corec : forall z : A, z $-> x -> z $-> y -> z $-> cat_binprod) (cat_binprod_beta_pr1 : forall (z : A) (f : z $-> x) (g : z $-> y), cat_pr1 $o cat_binprod_corec z f g $== f) (cat_binprod_beta_pr2 : forall (z : A) (f : z $-> x) (g : z $-> y), cat_pr2 $o cat_binprod_corec z f g $== g) (cat_binprod_eta_pr : forall (z : A) (f g : z $-> cat_binprod), cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g) : Product Bool (fun b => if b then x else y). Proof. snrapply (Build_Product _ cat_binprod). - intros [|]. + exact cat_pr1. + exact cat_pr2. - intros z f. nrapply cat_binprod_corec. + exact (f true). + exact (f false). - intros z f [|]. + nrapply cat_binprod_beta_pr1. + nrapply cat_binprod_beta_pr2. - intros z f g p. nrapply cat_binprod_eta_pr. + exact (p true). + exact (p false). Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | Build_BinaryProduct | 7,720 |
{A : Type} `{HasBinaryProducts A} {w x y z : A} (f g : w $-> cat_binprod x (cat_binprod y z)) : cat_pr1 $o f $== cat_pr1 $o g -> cat_pr1 $o cat_pr2 $o f $== cat_pr1 $o cat_pr2 $o g -> cat_pr2 $o cat_pr2 $o f $== cat_pr2 $o cat_pr2 $o g -> f $== g. Proof. intros p q r. snrapply cat_binprod_eta_pr. - exact p. - snrapply cat_binprod_eta_pr. + exact (cat_assoc_opp _ _ _ $@ q $@ cat_assoc _ _ _). + exact (cat_assoc_opp _ _ _ $@ r $@ cat_assoc _ _ _). Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_eta_pr_x_xx | 7,721 |
{A : Type} `{HasBinaryProducts A} {w x y z : A} (f g : w $-> cat_binprod (cat_binprod x y) z) : cat_pr1 $o cat_pr1 $o f $== cat_pr1 $o cat_pr1 $o g -> cat_pr2 $o cat_pr1 $o f $== cat_pr2 $o cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g. Proof. intros p q r. snrapply cat_binprod_eta_pr. 2: exact r. snrapply cat_binprod_eta_pr. 1,2: refine (cat_assoc_opp _ _ _ $@ _ $@ cat_assoc _ _ _). - exact p. - exact q. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_eta_pr_xx_x | 7,722 |
{A : Type} `{HasBinaryProducts A} {x y z : A} (f : cat_binprod x (cat_binprod y z) $-> cat_binprod x (cat_binprod y z)) : cat_pr1 $o f $== cat_pr1 -> cat_pr1 $o cat_pr2 $o f $== cat_pr1 $o cat_pr2 -> cat_pr2 $o cat_pr2 $o f $== cat_pr2 $o cat_pr2 -> f $== Id _. Proof. intros p q r. snrapply cat_binprod_eta_pr_x_xx. - exact (p $@ (cat_idr _)^$). - exact (q $@ (cat_idr _)^$). - exact (r $@ (cat_idr _)^$). Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_eta_pr_x_xx_id | 7,723 |
{A : Type} `{HasBinaryProducts A} : HasProducts Bool A. Proof. intros x. snrapply Build_Product. - exact (cat_binprod (x true) (x false)). - intros [|]. + exact cat_pr1. + exact cat_pr2. - intros z f. exact (cat_binprod_corec (f true) (f false)). - intros z f [|]. + exact (cat_binprod_beta_pr1 (f true) (f false)). + exact (cat_binprod_beta_pr2 (f true) (f false)). - intros z f g p. nrapply cat_binprod_eta_pr. + exact (p true). + exact (p false). Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | hasproductsbool_hasbinaryproducts | 7,724 |
{I J : Type} {A : Type} `{HasBinaryProducts A} (x : I -> A) (y : J -> A) : Product I x -> Product J y -> Product (I + J) (sum_ind _ x y). Proof. intros p q. snrapply Build_Product. - exact (cat_binprod (cat_prod I x) (cat_prod J y)). - intros [i | j]. + exact (cat_pr _ $o cat_pr1). + exact (cat_pr _ $o cat_pr2). - intros z f. nrapply cat_binprod_corec. + nrapply cat_prod_corec. exact (f o inl). + nrapply cat_prod_corec. exact (f o inr). - intros z f [i | j]. + nrefine (cat_assoc _ _ _ $@ _). nrefine ((_ $@L cat_binprod_beta_pr1 _ _) $@ _). rapply cat_prod_beta. + nrefine (cat_assoc _ _ _ $@ _). nrefine ((_ $@L cat_binprod_beta_pr2 _ _) $@ _). rapply cat_prod_beta. - intros z f g r. rapply cat_binprod_eta_pr. + rapply cat_prod_pr_eta. intros i. exact ((cat_assoc _ _ _)^$ $@ r (inl i) $@ cat_assoc _ _ _). + rapply cat_prod_pr_eta. intros j. exact ((cat_assoc _ _ _)^$ $@ r (inr j) $@ cat_assoc _ _ _). Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_prod_index_sum | 7,725 |
{A : Type} `{HasBinaryProducts A} (a : A) {x y : A} (g : x $-> y) : cat_pr1 $o fmap01 (fun x y => cat_binprod x y) a g $== cat_pr1 := cat_binprod_beta_pr1 _ _ $@ cat_idl _. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_pr1_fmap01_binprod | 7,726 |
{A : Type} `{HasBinaryProducts A} {x y : A} (f : x $-> y) (a : A) : cat_pr1 $o fmap10 (fun x y => cat_binprod x y) f a $== f $o cat_pr1 := cat_binprod_beta_pr1 _ _. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_pr1_fmap10_binprod | 7,727 |
{A : Type} `{HasBinaryProducts A} {w x y z : A} (f : w $-> y) (g : x $-> z) : cat_pr1 $o fmap11 (fun x y => cat_binprod x y) f g $== f $o cat_pr1 := cat_binprod_beta_pr1 _ _. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_pr1_fmap11_binprod | 7,728 |
{A : Type} `{HasBinaryProducts A} (a : A) {x y : A} (g : x $-> y) : cat_pr2 $o fmap01 (fun x y => cat_binprod x y) a g $== g $o cat_pr2 := cat_binprod_beta_pr2 _ _. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_pr2_fmap01_binprod | 7,729 |
{A : Type} `{HasBinaryProducts A} {x y : A} (f : x $-> y) (a : A) : cat_pr2 $o fmap10 (fun x y => cat_binprod x y) f a $== cat_pr2 := cat_binprod_beta_pr2 _ _ $@ cat_idl _. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_pr2_fmap10_binprod | 7,730 |
{A : Type} `{HasBinaryProducts A} {w x y z : A} (f : w $-> y) (g : x $-> z) : cat_pr2 $o fmap11 (fun x y => cat_binprod x y) f g $== g $o cat_pr2 := cat_binprod_beta_pr2 _ _. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_pr2_fmap11_binprod | 7,731 |
{A : Type} `{Is1Cat A} (x : A) `{!BinaryProduct x x} : x $-> cat_binprod x x. Proof. snrapply cat_binprod_corec; exact (Id _). Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_diag | 7,732 |
{A : Type} `{Is1Cat A, !HasBinaryProducts A} {w x y z : A} (f : w $-> z) (g : x $-> y) (h : w $-> x) : fmap01 (fun x y => cat_binprod x y) z g $o cat_binprod_corec f h $== cat_binprod_corec f (g $o h). Proof. snrapply cat_binprod_eta_pr. - nrefine (cat_assoc_opp _ _ _ $@ _). refine ((_ $@R _) $@ cat_assoc _ _ _ $@ cat_idl _ $@ _ $@ _^$). 1-3: rapply cat_binprod_beta_pr1. - nrefine (cat_assoc_opp _ _ _ $@ _). refine ((_ $@R _) $@ cat_assoc _ _ _ $@ (_ $@L _) $@ _^$). 1-3: rapply cat_binprod_beta_pr2. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_fmap01_corec | 7,733 |
{A : Type} `{Is1Cat A, !HasBinaryProducts A} {w x y z : A} (f : x $-> y) (g : w $-> x) (h : w $-> z) : fmap10 (fun x y => cat_binprod x y) f z $o cat_binprod_corec g h $== cat_binprod_corec (f $o g) h. Proof. snrapply cat_binprod_eta_pr. - refine (cat_assoc_opp _ _ _ $@ _). refine ((_ $@R _) $@ cat_assoc _ _ _ $@ (_ $@L _) $@ _^$). 1-3: nrapply cat_binprod_beta_pr1. - refine (cat_assoc_opp _ _ _ $@ _). refine ((_ $@R _) $@ cat_assoc _ _ _ $@ cat_idl _ $@ _ $@ _^$). 1-3: nrapply cat_binprod_beta_pr2. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_fmap10_corec | 7,734 |
{A : Type} `{Is1Cat A, !HasBinaryProducts A} {v w x y z : A} (f : w $-> y) (g : x $-> z) (h : v $-> w) (i : v $-> x) : fmap11 (fun x y => cat_binprod x y) f g $o cat_binprod_corec h i $== cat_binprod_corec (f $o h) (g $o i). Proof. snrapply cat_binprod_eta_pr. - refine (cat_assoc_opp _ _ _ $@ _). refine ((_ $@R _) $@ cat_assoc _ _ _ $@ (_ $@L _) $@ _^$). 1-3: nrapply cat_binprod_beta_pr1. - nrefine (cat_assoc_opp _ _ _ $@ _). refine ((_ $@R _) $@ cat_assoc _ _ _ $@ (_ $@L _) $@ _^$). 1-3: rapply cat_binprod_beta_pr2. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_fmap11_corec | 7,735 |
(x y : A) : cat_binprod x y $-> cat_binprod y x := cat_binprod_corec cat_pr2 cat_pr1. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_swap | 7,736 |
(x y : A) : cat_binprod_swap x y $o cat_binprod_swap y x $== Id _. Proof. nrapply cat_binprod_eta_pr. - refine ((cat_assoc _ _ _)^$ $@ _). nrefine (cat_binprod_beta_pr1 _ _ $@R _ $@ _). exact (cat_binprod_beta_pr2 _ _ $@ (cat_idr _)^$). - refine ((cat_assoc _ _ _)^$ $@ _). nrefine (cat_binprod_beta_pr2 _ _ $@R _ $@ _). exact (cat_binprod_beta_pr1 _ _ $@ (cat_idr _)^$). Defined. | Lemma | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_swap_cat_binprod_swap | 7,737 |
(x y : A) : cat_binprod x y $<~> cat_binprod y x. Proof. snrapply cate_adjointify. 1,2: nrapply cat_binprod_swap. all: nrapply cat_binprod_swap_cat_binprod_swap. Defined. | Lemma | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cate_binprod_swap | 7,738 |
{a b c : A} (f : a $-> b) (g : a $-> c) : cat_binprod_swap b c $o cat_binprod_corec f g $== cat_binprod_corec g f. Proof. nrapply cat_binprod_eta_pr. - refine (cat_assoc_opp _ _ _ $@ (_ $@R _) $@ (_ $@ _^$)). 1,3: nrapply cat_binprod_beta_pr1. nrapply cat_binprod_beta_pr2. - refine (cat_assoc_opp _ _ _ $@ (_ $@R _) $@ (_ $@ _^$)). 1,3: nrapply cat_binprod_beta_pr2. nrapply cat_binprod_beta_pr1. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_swap_corec | 7,739 |
{a b c d : A} (f : a $-> c) (g : b $-> d) : cat_binprod_swap c d $o fmap11 (fun x y : A => cat_binprod x y) f g $== fmap11 (fun x y : A => cat_binprod x y) g f $o cat_binprod_swap a b := cat_binprod_swap_corec _ _ $@ (cat_binprod_fmap11_corec _ _ _ _)^$. Local Instance symmetricbraiding_binprod : SymmetricBraiding (fun x y => cat_binprod x y). Proof. snrapply Build_SymmetricBraiding. - snrapply Build_NatTrans. + intros [x y]. exact (cat_binprod_swap x y). + snrapply Build_Is1Natural. intros [a b] [c d] [f g]; cbn in f, g. exact( f g). - exact cat_binprod_swap_cat_binprod_swap. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_swap_nat | 7,740 |
(x y z : A) : cat_binprod x (cat_binprod y z) $-> cat_binprod y (cat_binprod x z). Proof. nrapply cat_binprod_corec. - exact (cat_pr1 $o cat_pr2). - exact (fmap01 (fun x y => cat_binprod x y) x cat_pr2). Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_twist | 7,741 |
(x y z : A) : cat_pr1 $o cat_binprod_twist x y z $== cat_pr1 $o cat_pr2 := cat_binprod_beta_pr1 _ _. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_pr1_twist | 7,742 |
(x y z : A) : cat_pr1 $o cat_pr2 $o cat_binprod_twist x y z $== cat_pr1. Proof. nrefine (cat_assoc _ _ _ $@ _). nrefine ((_ $@L cat_binprod_beta_pr2 _ _) $@ _). nrapply cat_pr1_fmap01_binprod. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_pr1_pr2_twist | 7,743 |
(x y z : A) : cat_pr2 $o cat_pr2 $o cat_binprod_twist x y z $== cat_pr2 $o cat_pr2. Proof. nrefine (cat_assoc _ _ _ $@ _). nrefine ((_ $@L cat_binprod_beta_pr2 _ _) $@ _). nrapply cat_pr2_fmap01_binprod. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_pr2_pr2_twist | 7,744 |
{w x y z : A} (f : w $-> x) (g : w $-> y) (h : w $-> z) : cat_binprod_twist x y z $o cat_binprod_corec f (cat_binprod_corec g h) $== cat_binprod_corec g (cat_binprod_corec f h). Proof. nrapply cat_binprod_eta_pr. - nrefine (cat_assoc_opp _ _ _ $@ _). refine ((_ $@R _) $@ cat_assoc _ _ _ $@ (_ $@L _) $@ (_ $@ _^$)). 1: nrapply cat_binprod_pr1_twist. 1: nrapply cat_binprod_beta_pr2. 1,2: nrapply cat_binprod_beta_pr1. - refine (cat_assoc_opp _ _ _ $@ (_ $@R _) $@ _ $@ (cat_binprod_beta_pr2 _ _)^$). 1: nrapply cat_binprod_beta_pr2. nrefine (cat_binprod_fmap01_corec _ _ _ $@ _). nrapply cat_binprod_corec_eta. 1: exact (Id _). nrapply cat_binprod_beta_pr2. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_twist_corec | 7,745 |
(x y z : A) : cat_binprod_twist x y z $o cat_binprod_twist y x z $== Id _. Proof. nrapply cat_binprod_eta_pr_x_xx_id. - nrefine (cat_assoc_opp _ _ _ $@ (cat_binprod_pr1_twist _ _ _ $@R _) $@ _). nrapply cat_binprod_pr1_pr2_twist. - nrefine (cat_assoc_opp _ _ _ $@ (cat_binprod_pr1_pr2_twist _ _ _ $@R _) $@ _). nrapply cat_binprod_pr1_twist. - nrefine (cat_assoc_opp _ _ _ $@ (cat_binprod_pr2_pr2_twist _ _ _ $@R _) $@ _). nrapply cat_binprod_pr2_pr2_twist. Defined. | Lemma | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_twist_cat_binprod_twist | 7,746 |
(x y z : A) : cat_binprod x (cat_binprod y z) $<~> cat_binprod y (cat_binprod x z). Proof. snrapply cate_adjointify. 1,2: nrapply cat_binprod_twist. 1,2: nrapply cat_binprod_twist_cat_binprod_twist. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cate_binprod_twist | 7,747 |
{a a' b b' c c' : A} (f : a $-> a') (g : b $-> b') (h : c $-> c') : cat_binprod_twist a' b' c' $o fmap11 (fun x y => cat_binprod x y) f (fmap11 (fun x y => cat_binprod x y) g h) $== fmap11 (fun x y => cat_binprod x y) g (fmap11 (fun x y => cat_binprod x y) f h) $o cat_binprod_twist a b c. Proof. nrapply cat_binprod_eta_pr. - refine (cat_assoc_opp _ _ _ $@ _). nrefine ((cat_binprod_beta_pr1 _ _ $@R _) $@ _). nrefine (cat_assoc _ _ _ $@ _). nrefine ((_ $@L _) $@ _). 1: nrapply cat_pr2_fmap11_binprod. nrefine (cat_assoc_opp _ _ _ $@ _). nrefine ((_ $@R _) $@ _). 1: nrapply cat_pr1_fmap11_binprod. nrefine (_ $@ cat_assoc _ _ _). refine (_ $@ (_^$ $@R _)). 2: nrapply cat_pr1_fmap11_binprod. refine (cat_assoc _ _ _ $@ (_ $@L _^$) $@ (cat_assoc _ _ _)^$). nrapply cat_binprod_beta_pr1. - nrefine (cat_assoc_opp _ _ _ $@ (cat_binprod_beta_pr2 _ _ $@R _) $@ _). nrefine (_ $@ cat_assoc _ _ _). refine (_ $@ (_^$ $@R _)). 2: nrapply cat_pr2_fmap11_binprod. refine (_ $@ (_ $@L _^$) $@ (cat_assoc _ _ _)^$). 2: nrapply cat_binprod_beta_pr2. refine (_^$ $@ _ $@ _). 1,3: rapply fmap11_comp. rapply fmap22. 1: exact (cat_idl _ $@ (cat_idr _)^$). nrapply cat_binprod_beta_pr2. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_twist_nat | 7,748 |
x y z : cat_pr1 $o cat_pr1 $o associator_cat_binprod x y z $== cat_pr1. Proof. nrefine ((_ $@L associator_twist'_unfold _ _ _ _ _ _ _ _) $@ _). nrefine (cat_assoc _ _ _ $@ (_ $@L (cat_assoc_opp _ _ _ $@ (_ $@R _))) $@ _). 1: nrapply cat_binprod_beta_pr1. do 2 nrefine (cat_assoc_opp _ _ _ $@ _). nrefine ((cat_binprod_pr1_pr2_twist _ _ _ $@R _) $@ _). nrapply cat_pr1_fmap01_binprod. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_pr1_pr1_associator_binprod | 7,749 |
x y z : cat_pr2 $o cat_pr1 $o associator_cat_binprod x y z $== cat_pr1 $o cat_pr2. Proof. nrefine ((_ $@L associator_twist'_unfold _ _ _ _ _ _ _ _) $@ _). nrefine (cat_assoc _ _ _ $@ (_ $@L (cat_assoc_opp _ _ _ $@ (_ $@R _))) $@ _). 1: nrapply cat_binprod_beta_pr1. do 2 nrefine (cat_assoc_opp _ _ _ $@ _). nrefine ((cat_binprod_pr2_pr2_twist _ _ _ $@R _) $@ _). nrefine (cat_assoc _ _ _ $@ (_ $@L cat_pr2_fmap01_binprod _ _) $@ _). exact (cat_assoc_opp _ _ _ $@ (cat_binprod_beta_pr2 _ _ $@R _)). Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_pr2_pr1_associator_binprod | 7,750 |
x y z : cat_pr2 $o associator_cat_binprod x y z $== cat_pr2 $o cat_pr2. Proof. nrefine ((_ $@L associator_twist'_unfold _ _ _ _ _ _ _ _) $@ _). nrefine (cat_assoc_opp _ _ _ $@ (cat_binprod_beta_pr2 _ _ $@R _) $@ _). nrefine (cat_assoc_opp _ _ _ $@ (cat_binprod_pr1_twist _ _ _ $@R _) $@ _). nrefine (cat_assoc _ _ _ $@ (_ $@L cat_pr2_fmap01_binprod _ _) $@ _). exact (cat_assoc_opp _ _ _ $@ (cat_binprod_beta_pr1 _ _ $@R _)). Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_pr2_associator_binprod | 7,751 |
{w x y z} (f : w $-> x) (g : w $-> y) (h : w $-> z) : associator_cat_binprod x y z $o cat_binprod_corec f (cat_binprod_corec g h) $== cat_binprod_corec (cat_binprod_corec f g) h. Proof. nrefine ((associator_twist'_unfold _ _ _ _ _ _ _ _ $@R _) $@ _). nrefine ((cat_assoc_opp _ _ _ $@R _) $@ cat_assoc _ _ _ $@ (_ $@L (_ $@ _)) $@ _). 1: nrapply cat_binprod_fmap01_corec. 1: rapply (cat_binprod_corec_eta _ _ _ _ (Id _)). 1: nrapply cat_binprod_swap_corec. nrefine (cat_assoc _ _ _ $@ (_ $@L _) $@ _). 1: nrapply cat_binprod_twist_corec. nrapply cat_binprod_swap_corec. Defined. | Definition | Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\Products.v | cat_binprod_associator_corec | 7,752 |
Square@{u v w} {A : Type@{u}} `{Is1Cat@{u w v} A} {x00 x20 x02 x22 : A} (f01 : x00 $-> x02) (f21 : x20 $-> x22) (f10 : x00 $-> x20) (f12 : x02 $-> x22) : Type@{w} := f21 $o f10 $== f12 $o f01. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | Square@ | 7,753 |
(p : f21 $o f10 $== f12 $o f01) : Square f01 f21 f10 f12 := p. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | Build_Square | 7,754 |
(s : Square f01 f21 f10 f12) : f21 $o f10 $== f12 $o f01 := s. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | gpdhom_square | 7,755 |
{f f' : x $-> x'} (p : f $== f') : Square f f' (Id x) (Id x') := cat_idr f' $@ p^$ $@ (cat_idl f)^$. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | hdeg_square | 7,756 |
{f f' : x $-> x'} (p : f $== f') : Square (Id x) (Id x') f f' := cat_idl f $@ p $@ (cat_idr f')^$. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | vdeg_square | 7,757 |
(f : x $-> x') : Square f f (Id x) (Id x') := hdeg_square (Id f). | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | hrefl | 7,758 |
(f : x $-> x') : Square (Id x) (Id x') f f := vdeg_square (Id f). | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | vrefl | 7,759 |
(s : Square f01 f21 f10 f12) : Square f10 f12 f01 f21 := s^$. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | transpose | 7,760 |
(s : Square f01 f21 f10 f12) (t : Square f21 f41 f30 f32) : Square f01 f41 (f30 $o f10) (f32 $o f12) := (cat_assoc _ _ _)^$ $@ (t $@R f10) $@ cat_assoc _ _ _ $@ (f32 $@L s) $@ (cat_assoc _ _ _)^$. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | hconcat | 7,761 |
(s : Square f01 f21 f10 f12) (t : Square f03 f23 f12 f14) : Square (f03 $o f01) (f23 $o f21) f10 f14 := cat_assoc _ _ _ $@ (f23 $@L s) $@ (cat_assoc _ _ _)^$ $@ (t $@R f01) $@ cat_assoc _ _ _. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | vconcat | 7,762 |
{HE : HasEquivs A} (f10 : x00 $<~> x20) (f12 : x02 $<~> x22) (s : Square f01 f21 f10 f12) : Square f21 f01 f10^-1$ f12^-1$ := (cat_idl _)^$ $@ ((cate_issect f12)^$ $@R _) $@ cat_assoc _ _ _ $@ (_ $@L ((cat_assoc _ _ _)^$ $@ (s^$ $@R _) $@ cat_assoc _ _ _ $@ (_ $@L cate_isretr f10) $@ cat_idr _)). | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | hinverse | 7,763 |
(p : f01' $== f01) (s : Square f01 f21 f10 f12) : Square f01' f21 f10 f12 := s $@ (f12 $@L p^$). | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | hconcatL | 7,764 |
(s : Square f01 f21 f10 f12) (p : f21' $== f21) : Square f01 f21' f10 f12 := (p $@R f10) $@ s. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | hconcatR | 7,765 |
(p : f10' $== f10) (s : Square f01 f21 f10 f12) : Square f01 f21 f10' f12 := (f21 $@L p) $@ s. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | vconcatL | 7,766 |
(s : Square f01 f21 f10 f12) (p : f12' $== f12) : Square f01 f21 f10 f12' := s $@ (p^$ $@R f01). | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | vconcatR | 7,767 |
(f01 : x00 $<~> x02) (f21 : x20 $<~> x22) (s : Square f01 f21 f10 f12) : Square (f01^-1$) (f21^-1$) f12 f10 := transpose (hinverse _ _ (transpose s)). | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | vinverse | 7,768 |
{f : x $-> x00} (s : Square f01 f21 f10 f12) : Square (f01 $o f) f21 (f10 $o f) f12 := (cat_assoc _ _ _)^$ $@ (s $@R f) $@ cat_assoc _ _ _. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | whiskerTL | 7,769 |
{f : x22 $-> x} (s : Square f01 f21 f10 f12) : Square f01 (f $o f21) f10 (f $o f12) := cat_assoc _ _ _ $@ (f $@L s) $@ (cat_assoc _ _ _)^$. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | whiskerBR | 7,770 |
{f : x $<~> x02} (s : Square f01 f21 f10 f12) : Square (f^-1$ $o f01) f21 f10 (f12 $o f) := s $@ ((compose_hh_V _ _)^$ $@R f01) $@ cat_assoc _ _ _. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | whiskerBL | 7,771 |
{f : x02 $<~> x} (s : Square f01 f21 f10 f12) : Square (f $o f01) f21 f10 (f12 $o f^-1$) := s $@ ((compose_hV_h _ _)^$ $@R f01) $@ cat_assoc _ _ _. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | whiskerLB | 7,772 |
{f : x20 $<~> x} (s : Square f01 f21 f10 f12) : Square f01 (f21 $o f^-1$) (f $o f10) f12 := cat_assoc _ _ _ $@ (f21 $@L compose_V_hh _ _) $@ s. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | whiskerTR | 7,773 |
{f : x $<~> x20} (s : Square f01 f21 f10 f12) : Square f01 (f21 $o f) (f^-1$ $o f10) f12 := cat_assoc _ _ _ $@ (f21 $@L compose_h_Vh _ _) $@ s. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | whiskerRT | 7,774 |
{f01 : x00 $-> x} {f01' : x $-> x02} (s : Square (f01' $o f01) f21 f10 f12) : Square f01 f21 f10 (f12 $o f01') := s $@ (cat_assoc _ _ _)^$. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | move_bottom_left | 7,775 |
{f12 : x02 $-> x} {f12' : x $-> x22} (s : Square f01 f21 f10 (f12' $o f12)) : Square (f12 $o f01) f21 f10 f12' := s $@ cat_assoc _ _ _. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | move_left_bottom | 7,776 |
{f10 : x00 $-> x} {f10' : x $-> x20} (s : Square f01 f21 (f10' $o f10) f12) : Square f01 (f21 $o f10') f10 f12 := cat_assoc _ _ _ $@ s. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | move_right_top | 7,777 |
{f21 : x20 $-> x} {f21' : x $-> x22} (s : Square f01 (f21' $o f21) f10 f12) : Square f01 f21' (f21 $o f10) f12 := (cat_assoc _ _ _)^$ $@ s. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | move_top_right | 7,778 |
{B : Type} `{Is1Cat B} (f : A -> B) `{!Is0Functor f} `{!Is1Functor f} (s : Square f01 f21 f10 f12) : Square (fmap f f01) (fmap f f21) (fmap f f10) (fmap f f12) := (fmap_comp f _ _)^$ $@ fmap2 f s $@ fmap_comp f _ _. | Definition | Require Import Basics.Overture. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Square.v | fmap_square | 7,779 |
{A} `{Is21Cat A} {a b c : A} {f g h : a $-> b} (k : b $-> c) (p : f $== g) (q : g $== h) : k $@L (p $@ q) $== (k $@L p) $@ (k $@L q). Proof. rapply fmap_comp. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.NatTrans. | WildCat\TwoOneCat.v | cat_postwhisker_pp | 7,780 |
{A} `{Is21Cat A} {a b c : A} {f g h : b $-> c} (k : a $-> b) (p : f $== g) (q : g $== h) : (p $@ q) $@R k $== (p $@R k) $@ (q $@R k). Proof. rapply fmap_comp. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.NatTrans. | WildCat\TwoOneCat.v | cat_prewhisker_pp | 7,781 |
{A : Type} `{Is21Cat A} {a b c : A} {f f' f'' : a $-> b} {g g' g'' : b $-> c} (p : f $== f') (q : f' $== f'') (r : g $== g') (s : g' $== g'') : (p $@ q) $@@ (r $@ s) $== (p $@@ r) $@ (q $@@ s). Proof. unfold "$@@". nrefine ((_ $@L cat_prewhisker_pp _ _ _ ) $@ _). nrefine ((cat_postwhisker_pp _ _ _ $@R _) $@ _). nrefine (cat_assoc _ _ _ $@ _). refine (_ $@ (cat_assoc _ _ _)^$). nrefine (_ $@L _). refine (_ $@ cat_assoc _ _ _). refine ((cat_assoc _ _ _)^$ $@ _). nrefine (_ $@R _). apply bifunctor_coh_comp. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.NatTrans. | WildCat\TwoOneCat.v | cat_exchange | 7,782 |
{A B : Type} {f : A $-> B} `{IsEquiv A B f} : CatIsEquiv f. Proof. assumption. Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Equivalences Basics.PathGroupoids. Require Import Types.Equiv. Require Import WildCat.Core WildCat.Equiv WildCat.NatTrans WildCat.TwoOneCat. | WildCat\Universe.v | catie_isequiv | 7,783 |
{A} `{Is01Cat A} : Fun01 (A^op * A) Type := @Build_Fun01 _ _ _ _ _ is0functor_hom. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | fun01_hom | 7,784 |
{A : Type} `{IsGraph A} (a : A) : A -> Type := fun b => (a $-> b). Global Instance is0functor_opyon {A : Type} `{Is01Cat A} (a : A) : Is0Functor ( a). Proof. apply Build_Is0Functor. unfold ; intros b c f g; cbn in *. exact (f $o g). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyon | 7,785 |
{A : Type} `{HasEquivs A} `{!HasMorExt A} {x y z : A} (f : y $<~> z) : (x $-> y) <~> (x $-> z) := emap (opyon x) f. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | equiv_postcompose_cat_equiv | 7,786 |
{A : Type} `{HasEquivs A} `{!HasMorExt A} {x y z : A} (f : x $<~> y) : (y $-> z) <~> (x $-> z) := @equiv_postcompose_cat_equiv A^op _ _ _ _ _ _ z y x f. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | equiv_precompose_cat_equiv | 7,787 |
{A : Type} `{HasEquivs A} `{!HasMorExt (core A)} {x y z : A} (f : y $<~> z) : (x $<~> y) <~> (x $<~> z). Proof. change ((Build_core x $-> Build_core y) <~> (Build_core x $-> Build_core z)). refine (equiv_postcompose_cat_equiv (A := core A) _). exact f. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | equiv_postcompose_core_cat_equiv | 7,788 |
{A : Type} `{HasEquivs A} `{!HasMorExt (core A)} {x y z : A} (f : x $<~> y) : (y $<~> z) <~> (x $<~> z). Proof. change ((Build_core y $-> Build_core z) <~> (Build_core x $-> Build_core z)). refine (equiv_precompose_cat_equiv (A := core A) _). exact f. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | equiv_precompose_core_cat_equiv | 7,789 |
{A : Type} `{Is01Cat A} (a : A) (F : A -> Type) {ff : Is0Functor F} : F a -> (opyon a $=> F). Proof. intros x b f. exact (fmap F f x). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyoneda | 7,790 |
{A : Type} `{Is01Cat A} (a : A) (F : A -> Type) {ff : Is0Functor F} : (opyon a $=> F) -> F a := fun alpha => alpha a (Id a). Global Instance is1natural_opyoneda {A : Type} `{Is1Cat A} (a : A) (F : A -> Type) `{!Is0Functor F, !Is1Functor F} (x : F a) : Is1Natural (opyon a) F (opyoneda a F x). Proof. snrapply Build_Is1Natural. unfold opyon, opyoneda; intros b c f g; cbn in *. exact (fmap_comp F g f x). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | un_opyoneda | 7,791 |
{A : Type} `{Is1Cat A} (a : A) (F : A -> Type) `{!Is0Functor F, !Is1Functor F} (x x' : F a) (p : forall b, opyoneda a F x b == opyoneda a F x' b) : x = x'. Proof. refine ((fmap_id F a x)^ @ _ @ fmap_id F a x'). cbn in p. exact (p a (Id a)). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyoneda_isinj | 7,792 |
{A : Type} `{Is1Cat_Strong A} (a b : A) (f g : b $-> a) (p : forall (c : A) (h : a $-> c), h $o f = h $o g) : f = g := (cat_idl_strong f)^ @ p a (Id a) @ cat_idl_strong g. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyon_faithful | 7,793 |
{A : Type} `{Is1Cat A} (a : A) (F : A -> Type) `{!Is0Functor F, !Is1Functor F} (x : F a) : un_opyoneda a F (opyoneda a F x) = x := fmap_id F a x. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyoneda_issect | 7,794 |
{A : Type} `{Is1Cat_Strong A} (a : A) (F : A -> Type) `{!Is0Functor F, !Is1Functor F} (alpha : opyon a $=> F) {alnat : Is1Natural (opyon a) F alpha} (b : A) : opyoneda a F (un_opyoneda a F alpha) b $== alpha b. Proof. unfold opyoneda, un_opyoneda, opyon; intros f. refine ((isnat alpha f (Id a))^ @ _). cbn. apply ap. exact (cat_idr_strong f). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyoneda_isretr | 7,795 |
{A : Type} `{Is01Cat A} (a b : A) : (opyon a $=> opyon b) -> (b $-> a) := un_opyoneda a (opyon b). | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyon_cancel | 7,796 |
{A : Type} `{Is01Cat A} (a : A) : Fun01 A Type. Proof. rapply (Build_Fun01 _ _ (opyon a)). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyon1 | 7,797 |
{A : Type} `{Is1Cat A} `{!HasMorExt A} (a : A) : Fun11 A Type. Proof. rapply (Build_Fun11 _ _ (opyon a)). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyon11 | 7,798 |
{A : Type} `{HasEquivs A} `{!Is1Cat_Strong A} {a b : A} : (opyon1 a $<~> opyon1 b) -> (b $<~> a). Proof. intros f. refine (cate_adjointify (f a (Id a)) (f^-1$ b (Id b)) _ _); apply GpdHom_path; cbn in *. - refine ((isnat_natequiv (natequiv_inverse f) (f a (Id a)) (Id b))^ @ _); cbn. refine (_ @ cate_issect (f a) (Id a)); cbn. apply ap. srapply cat_idr_strong. - refine ((isnat_natequiv f (f^-1$ b (Id b)) (Id a))^ @ _); cbn. refine (_ @ cate_isretr (f b) (Id b)); cbn. apply ap. srapply cat_idr_strong. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyon_equiv | 7,799 |
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