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{A} `{HasEquivs A} (a b : A) := @' A _ _ _ _ _ a b. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | CatEquiv | 7,500 |
{A} `{HasEquivs A} {a b : A} (f : a $<~> b) : a $-> b := @' A _ _ _ _ _ a b f. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_fun | 7,501 |
{A} `{HasEquivs A} {a b : A} (f : a $-> b) {fe : CatIsEquiv f} : a $<~> b := cate_buildequiv' a b f fe. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | Build_CatEquiv | 7,502 |
{A} `{HasEquivs A} {a b : A} (f : a $-> b) {fe : CatIsEquiv f} : cate_fun (Build_CatEquiv f) $== f := ' a b f fe. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_buildequiv_fun | 7,503 |
{A} `{HasEquivs A} {a b : A} (f : a $-> b) (g : b $-> a) (r : f $o g $== Id b) (s : g $o f $== Id a) : CatIsEquiv f := ' a b f g r s. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | catie_adjointify | 7,504 |
{A} `{HasEquivs A} {a b : A} (f : a $-> b) (g : b $-> a) (r : f $o g $== Id b) (s : g $o f $== Id a) : a $<~> b := Build_CatEquiv f (fe:=catie_adjointify f g r s). | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_adjointify | 7,505 |
{A} `{HasEquivs A} {a b : A} (f : a $<~> b) : b $<~> a. Proof. simple refine (cate_adjointify _ _ _ _). - exact (' a b f). - exact f. - exact (cate_issect' a b f). - exact (cate_isretr' a b f). Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_inv | 7,506 |
{A} `{HasEquivs A} {a b} (f : a $<~> b) : f^-1$ $o f $== Id a. Proof. refine (_ $@ ' a b f). refine (_ $@R f). apply cate_buildequiv_fun'. Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_issect | 7,507 |
{A} `{HasEquivs A} {a b} (f : a $<~> b) : f $o f^-1$ $== Id b := cate_issect (A:=A^op) (b:=a) (a:=b) f. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_isretr | 7,508 |
{A} `{HasEquivs A} {a b} (f : a $<~> b) (g : b $-> a) (p : f $o g $== Id b) : cate_fun f^-1$ $== g. Proof. refine ((cat_idr _)^$ $@ _). refine ((_ $@L p^$) $@ _). refine (cat_assoc_opp _ _ _ $@ _). refine (cate_issect f $@R _ $@ _). apply cat_idl. Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_inverse_sect | 7,509 |
{A} `{HasEquivs A} {a b} (f : a $<~> b) (g : b $-> a) (p : g $o f $== Id a) : cate_fun f^-1$ $== g := cate_inverse_sect (A:=A^op) (a:=b) (b:=a) f g p. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_inverse_retr | 7,510 |
{A} `{HasEquivs A} {a b : A} (f : a $-> b) (g : b $-> a) (r : f $o g $== Id b) (s : g $o f $== Id a) : cate_fun (cate_adjointify f g r s)^-1$ $== g. Proof. apply cate_inverse_sect. exact ((cate_buildequiv_fun f $@R _) $@ r). Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_inv_adjointify | 7,511 |
{A} `{HasEquivs A} (a : A) : a $<~> a := Build_CatEquiv (Id a). | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | id_cate | 7,512 |
{A} `{HasEquivs A} {a b : A} (f : a $-> b) `{!CatIsEquiv f} {g : a $-> b} (p : f $== g) : CatIsEquiv g. Proof. snrapply catie_adjointify. - exact (Build_CatEquiv f)^-1$. - refine (p^$ $@R _ $@ _). refine ((cate_buildequiv_fun f)^$ $@R _ $@ _). apply cate_isretr. - refine (_ $@L p^$ $@ _). refine (_ $@L (cate_buildequiv_fun f)^$ $@ _). apply cate_issect. Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | catie_homotopic | 7,513 |
{A} `{HasEquivs A} {a b c : A} (g : b $<~> c) (f : a $<~> b) : g $o f $o (f^-1$ $o g^-1$) $== Id c. Proof. refine (cat_assoc _ _ _ $@ _). refine ((_ $@L cat_assoc_opp _ _ _) $@ _). refine ((_ $@L (cate_isretr _ $@R _)) $@ _). refine ((_ $@L cat_idl _) $@ _). apply cate_isretr. Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | compose_catie_isretr | 7,514 |
{A} `{HasEquivs A} {a b c : A} (g : b $<~> c) (f : a $<~> b) : (f^-1$ $o g^-1$ $o (g $o f) $== Id a) := compose_catie_isretr (A:=A^op) (a:=c) (b:=b) (c:=a) f g. Global Instance compose_catie {A} `{HasEquivs A} {a b c : A} (g : b $<~> c) (f : a $<~> b) : CatIsEquiv (g $o f). Proof. refine (catie_adjointify _ (f^-1$ $o g^-1$) _ _). - apply compose_catie_isretr. - apply . Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | compose_catie_issect | 7,515 |
{A} `{HasEquivs A} {a b c : A} (g : b $<~> c) (f : a $<~> b) : a $<~> c := Build_CatEquiv (g $o f). | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | compose_cate | 7,516 |
{A} `{HasEquivs A} {a b c : A} (g : b $<~> c) (f : a $<~> b) : cate_fun (g $oE f) $== g $o f. Proof. apply cate_buildequiv_fun. Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | compose_cate_fun | 7,517 |
{A} `{HasEquivs A} {a b c : A} (g : b $<~> c) (f : a $<~> b) : g $o f $== cate_fun (g $oE f). Proof. apply gpd_rev. apply cate_buildequiv_fun. Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | compose_cate_funinv | 7,518 |
{A} `{HasEquivs A} (a : A) : cate_fun (id_cate a) $== Id a. Proof. apply cate_buildequiv_fun. Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | id_cate_fun | 7,519 |
{A} `{HasEquivs A} {a b c d : A} (f : a $<~> b) (g : b $<~> c) (h : c $<~> d) : cate_fun ((h $oE g) $oE f) $== cate_fun (h $oE (g $oE f)). Proof. refine (compose_cate_fun _ f $@ _ $@ cat_assoc f g h $@ _ $@ compose_cate_funinv h _). - refine (compose_cate_fun h g $@R _). - refine (_ $@L compose_cate_funinv g f). Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | compose_cate_assoc | 7,520 |
{A} `{HasEquivs A} {a b c d : A} (f : a $<~> b) (g : b $<~> c) (h : c $<~> d) : cate_fun (h $oE (g $oE f)) $== cate_fun ((h $oE g) $oE f). Proof. Opaque compose_catie_isretr. exact_no_check (compose_cate_assoc (A:=A^op) (a:=d) (b:=c) (c:=b) (d:=a) h g f). Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | compose_cate_assoc_opp | 7,521 |
{A} `{HasEquivs A} {a b : A} (f : a $<~> b) : cate_fun (id_cate b $oE f) $== cate_fun f. Proof. refine (compose_cate_fun _ f $@ _ $@ cat_idl f). refine (cate_buildequiv_fun _ $@R _). Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | compose_cate_idl | 7,522 |
{A} `{HasEquivs A} {a b : A} (f : a $<~> b) : cate_fun (f $oE id_cate a) $== cate_fun f := compose_cate_idl (A:=A^op) (a:=b) (b:=a) f. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | compose_cate_idr | 7,523 |
{A} `{HasEquivs A} {a b c : A} (f : b $<~> c) (g : a $-> b) : f^-1$ $o (f $o g) $== g := (cat_assoc_opp _ _ _) $@ (cate_issect f $@R g) $@ cat_idl g. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | compose_V_hh | 7,524 |
{A} `{HasEquivs A} {a b c : A} (f : c $<~> b) (g : a $-> b) : f $o (f^-1$ $o g) $== g := (cat_assoc_opp _ _ _) $@ (cate_isretr f $@R g) $@ cat_idl g. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | compose_h_Vh | 7,525 |
{A} `{HasEquivs A} {a b c : A} (f : b $-> c) (g : a $<~> b) : (f $o g) $o g^-1$ $== f := cat_assoc _ _ _ $@ (f $@L cate_isretr g) $@ cat_idr f. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | compose_hh_V | 7,526 |
{A} `{HasEquivs A} {a b c : A} (f : b $-> c) (g : b $<~> a) : (f $o g^-1$) $o g $== f := cat_assoc _ _ _ $@ (f $@L cate_issect g) $@ cat_idr f. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | compose_hV_h | 7,527 |
{A} `{HasEquivs A} {a b : A} (e : a $<~> b) : Monic e. Proof. intros c f g p. refine ((compose_V_hh e _)^$ $@ _ $@ compose_V_hh e _). exact (_ $@L p). Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_monic_equiv | 7,528 |
{A} `{HasEquivs A} {a b : A} (e : a $<~> b) : Epic e := cate_monic_equiv (A:=A^op) (a:=b) (b:=a) e. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_epic_equiv | 7,529 |
{A} `{HasEquivs A} {a b c : A} (e : a $<~> b) (f : a $-> c) (g : b $-> c) (p : f $o e^-1$ $== g) : f $== g $o e. Proof. apply (cate_epic_equiv e^-1$). exact (p $@ (compose_hh_V _ _)^$). Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_moveL_eM | 7,530 |
{A} `{HasEquivs A} {a b c : A} (e : b $<~> a) (f : a $-> c) (g : b $-> c) (p : f $== g $o e^-1$) : f $o e $== g. Proof. apply (cate_epic_equiv e^-1$). exact (compose_hh_V _ _ $@ p). Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_moveR_eM | 7,531 |
{A} `{HasEquivs A} {a b c : A} (e : b $<~> c) (f : a $-> c) (g : a $-> b) (p : e^-1$ $o f $== g) : f $== e $o g := cate_moveL_eM (A:=A^op) (a:=c) (b:=b) (c:=a) e f g p. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_moveL_Me | 7,532 |
{A} `{HasEquivs A} {a b c : A} (e : c $<~> b) (f : a $-> c) (g : a $-> b) (p : f $== e^-1$ $o g) : e $o f $== g := cate_moveR_eM (A:=A^op) (a:=c) (b:=b) (c:=a) e f g p. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_moveR_Me | 7,533 |
{A} `{HasEquivs A} {a b c : A} (e : a $<~> b) (f : b $-> c) (g : a $-> c) (p : f $o e $== g) : f $== g $o e^-1$. Proof. apply (cate_epic_equiv e). exact (p $@ (compose_hV_h _ _)^$). Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_moveL_eV | 7,534 |
{A} `{HasEquivs A} {a b c : A} (e : b $<~> a) (f : b $-> c) (g : a $-> c) (p : f $== g $o e) : f $o e^-1$ $== g. Proof. apply (cate_epic_equiv e). exact (compose_hV_h _ _ $@ p). Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_moveR_eV | 7,535 |
{A} `{HasEquivs A} {a b c : A} (e : b $<~> c) (f : a $-> b) (g : a $-> c) (p : e $o f $== g) : f $== e^-1$ $o g := cate_moveL_eV (A:=A^op) (a:=c) (b:=b) (c:=a) e f g p. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_moveL_Ve | 7,536 |
{A} `{HasEquivs A} {a b c : A} (e : b $<~> c) (f : a $-> c) (g : a $-> b) (p : f $== e $o g) : e^-1$ $o f $== g := cate_moveR_eV (A:=A^op) (a:=b) (b:=c) (c:=a) e f g p. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_moveR_Ve | 7,537 |
{A} `{HasEquivs A} {a b : A} {e : a $<~> b} (f : b $-> a) (p : e $o f $== Id _) : f $== cate_fun e^-1$. Proof. apply (cate_monic_equiv e). exact (p $@ (cate_isretr e)^$). Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_moveL_V1 | 7,538 |
{A} `{HasEquivs A} {a b : A} {e : a $<~> b} (f : b $-> a) (p : f $o e $== Id _) : f $== cate_fun e^-1$ := cate_moveL_V1 (A:=A^op) (a:=b) (b:=a) f p. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_moveL_1V | 7,539 |
{A} `{HasEquivs A} {a b : A} {e : a $<~> b} (f : b $-> a) (p : Id _ $== e $o f) : cate_fun e^-1$ $== f. Proof. apply (cate_monic_equiv e). exact (cate_isretr e $@ p). Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_moveR_V1 | 7,540 |
{A} `{HasEquivs A} {a b : A} {e : a $<~> b} (f : b $-> a) (p : Id _ $== f $o e) : cate_fun e^-1$ $== f := cate_moveR_V1 (A:=A^op) (a:=b) (b:=a) f p. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_moveR_1V | 7,541 |
{A} `{HasEquivs A} {a b : A} {e f : a $<~> b} (p : cate_fun e $== cate_fun f) : cate_fun e^-1$ $== cate_fun f^-1$. Proof. apply cate_moveL_V1. exact ((p^$ $@R _) $@ cate_isretr _). Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_inv2 | 7,542 |
{A} `{HasEquivs A} {a b c : A} (e : a $<~> b) (f : b $<~> c) : cate_fun (f $oE e)^-1$ $== cate_fun (e^-1$ $oE f^-1$). Proof. refine (_ $@ (compose_cate_fun _ _)^$). apply cate_inv_adjointify. Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_inv_compose | 7,543 |
cate_inv_compose' {A} `{HasEquivs A} {a b c : A} (e : a $<~> b) (f : b $<~> c) : cate_fun (f $oE e)^-1$ $== e^-1$ $o f^-1$. Proof. nrefine (_ $@ cate_buildequiv_fun _). nrapply cate_inv_compose. Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_inv_compose' | 7,544 |
{A} `{HasEquivs A} {a b : A} (e : a $<~> b) : cate_fun (e^-1$)^-1$ $== cate_fun e. Proof. apply cate_moveR_V1. symmetry; apply cate_issect. Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_inv_V | 7,545 |
{A B : Type} `{HasEquivs A} `{HasEquivs B} (F : A -> B) `{!Is0Functor F, !Is1Functor F} {a b : A} (f : a $<~> b) : F a $<~> F b := Build_CatEquiv (fmap F f). | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | emap | 7,546 |
{A B : Type} `{HasEquivs A} `{HasEquivs B} (F : A -> B) `{!Is0Functor F, !Is1Functor F} {a : A} : cate_fun (emap F (id_cate a)) $== cate_fun (id_cate (F a)). Proof. refine (cate_buildequiv_fun _ $@ _). refine (fmap2 F (id_cate_fun a) $@ _ $@ (id_cate_fun (F a))^$). rapply fmap_id. Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | emap_id | 7,547 |
{A B : Type} `{HasEquivs A} `{HasEquivs B} (F : A -> B) `{!Is0Functor F, !Is1Functor F} {a b c : A} (f : a $<~> b) (g : b $<~> c) : cate_fun (emap F (g $oE f)) $== fmap F (cate_fun g) $o fmap F (cate_fun f). Proof. refine (cate_buildequiv_fun _ $@ _). refine (fmap2 F (compose_cate_fun _ _) $@ _). rapply fmap_comp. Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | emap_compose | 7,548 |
emap_compose' {A B : Type} `{HasEquivs A} `{HasEquivs B} (F : A -> B) `{!Is0Functor F, !Is1Functor F} {a b c : A} (f : a $<~> b) (g : b $<~> c) : cate_fun (emap F (g $oE f)) $== cate_fun ((emap F g) $oE (emap F f)). Proof. refine (emap_compose F f g $@ _). symmetry. refine (compose_cate_fun _ _ $@ _). exact (cate_buildequiv_fun _ $@@ cate_buildequiv_fun _). Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | emap_compose' | 7,549 |
{A B : Type} `{HasEquivs A} `{HasEquivs B} (F : A -> B) `{!Is0Functor F, !Is1Functor F} {a b : A} (e : a $<~> b) : cate_fun (emap F e)^-1$ $== cate_fun (emap F e^-1$). Proof. refine (cate_inv_adjointify _ _ _ _ $@ _). exact (cate_buildequiv_fun _)^$. Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | emap_inv | 7,550 |
emap_inv' {A B : Type} `{HasEquivs A} `{HasEquivs B} (F : A -> B) `{!Is0Functor F, !Is1Functor F} {a b : A} (e : a $<~> b) : cate_fun (emap F e)^-1$ $== fmap F e^-1$ := emap_inv F e $@ cate_buildequiv_fun _. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | emap_inv' | 7,551 |
{A : Type} `{HasEquivs A} (a b : A) : (a = b) -> (a $<~> b). Proof. intros []; reflexivity. Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cat_equiv_path | 7,552 |
{A : Type} `{IsUnivalent1Cat A} (a b : A) : (a $<~> b) -> (a = b) := (cat_equiv_path a b)^-1. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cat_path_equiv | 7,553 |
(A : Type) := { uncore : A }. | Record | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | core | 7,554 |
A `{HasEquivs A} (x y : A) : IsInitial x -> IsInitial y -> x $<~> y. Proof. intros inx iny. srapply (cate_adjointify (inx y).1 (iny x).1). 1: exact (((iny _).2 _)^$ $@ (iny _).2 _). 1: exact (((inx _).2 _)^$ $@ (inx _).2 _). Defined. | Lemma | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_isinitial | 7,555 |
A `{HasEquivs A} (x y : A) : IsTerminal x -> IsTerminal y -> y $<~> x := cate_isinitial A^op x y. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cate_isterminal | 7,556 |
A `{HasEquivs A} (x y : A) : x $<~> y -> IsInitial x -> IsInitial y. Proof. intros f inx z. exists ((inx z).1 $o f^-1$). intros g. refine (_ $@ compose_hh_V _ f). refine (_ $@R _). exact ((inx z).2 _). Defined. | Lemma | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | isinitial_cate | 7,557 |
A `{HasEquivs A} (x y : A) : y $<~> x -> IsTerminal x -> IsTerminal y := isinitial_cate A^op x y. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | isterminal_cate | 7,558 |
A `{Is1Cat A} (x y : A) := { cat_equiv_fun :> x $-> y; cat_equiv_isequiv : Cat_IsBiInv cat_equiv_fun; }. | Record | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | Cat_BiInv | 7,559 |
{A} `{Is1Cat A} {x y : A} (f : x $-> y) {bif : Cat_IsBiInv f} : cat_equiv_inv f $== cat_equiv_inv' f. Proof. refine ((cat_idl _)^$ $@ _). refine (cat_prewhisker (cat_eissect' f)^$ _ $@ _). refine (cat_assoc _ _ _ $@ _). refine (cat_postwhisker _ (cat_eisretr f) $@ _). apply cat_idr. Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cat_inverses_homotopic | 7,560 |
{A} `{Is1Cat A} {x y : A} (f : x $-> y) {bif : Cat_IsBiInv f} : cat_equiv_inv f $o f $== Id x := (cat_inverses_homotopic f $@R f) $@ ' f. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cat_eissect | 7,561 |
A `{Is1Cat A} : HasEquivs A. Proof. srapply Build_HasEquivs; intros x y. 1: exact (Cat_BiInv _ x y). all:intros f; cbn beta in *. - exact (Cat_IsBiInv f). - exact f. - exact _. - apply Build_Cat_BiInv. - intros; reflexivity. - exact (cat_equiv_inv f). - apply cat_eissect. - apply cat_eisretr. - intros g r s. exact (Build_Cat_IsBiInv g r g s). Defined. | Definition | Require Import Basics.Utf8 Basics.Overture Basics.Tactics Basics.Equivalences. Require Import WildCat.Core. Require Import WildCat.Opposite. | WildCat\Equiv.v | cat_hasequivs | 7,562 |
{A B : Type} (F : A -> B) `{IsSurjInj A B F} : B -> A := fun b => (esssurj F b).1. Global Instance is0functor_surjinj_inv {A B : Type} (F : A -> B) `{IsSurjInj A B F} : Is0Functor ( F). Proof. constructor; intros x y f. pose (p := (esssurj F x).2). pose (q := (esssurj F y).2). cbn in *. pose (f' := p $@ f $@ q^$). exact (essinj F f'). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | surjinj_inv | 7,563 |
{A B : Type} (F : A -> B) `{IsSurjInj A B F} : F o surjinj_inv F $=> idmap. Proof. intros b. exact ((esssurj F b).2). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | eisretr0gpd_inv | 7,564 |
{A B : Type} (F : A -> B) `{IsSurjInj A B F} : surjinj_inv F o F $=> idmap. Proof. intros a. apply (essinj F). apply eisretr0gpd_inv. Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | eissect0gpd_inv | 7,565 |
{A B : Type} {F : A -> B} {G : A -> B} `{SplEssSurj A B F} `{!Is0Functor G} (alpha : G $=> F) : SplEssSurj G. Proof. intros b. exists ((esssurj F b).1). refine (_ $@ (esssurj F b).2). apply alpha. Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | isesssurj_transf | 7,566 |
{A B : Type} {F : A -> B} {G : A -> B} `{IsSurjInj A B F} `{!Is0Functor G} (alpha : G $=> F) : IsSurjInj G. Proof. constructor. - apply (isesssurj_transf alpha). - intros x y f. apply (essinj F). refine (_ $@ f $@ _). + symmetry; apply alpha. + apply alpha. Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | issurjinj_transf | 7,567 |
`{!IsSurjInj G, !SplEssSurj (G o F)} : SplEssSurj F. Proof. intros b. exists ((esssurj (G o F) (G b)).1). apply (essinj G). exact ((esssurj (G o F) (G b)).2). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | cancelL_isesssurj | 7,568 |
`{!IsSurjInj G} : SplEssSurj (G o F) <-> SplEssSurj F. Proof. split; [ apply @cancelL_isesssurj | apply @isesssurj_compose ]; exact _. Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | iffL_isesssurj | 7,569 |
`{!IsSurjInj G, !IsSurjInj (G o F)} : IsSurjInj F. Proof. constructor. - apply cancelL_isesssurj. - intros x y f. exact (essinj (G o F) (fmap G f)). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | cancelL_issurjinj | 7,570 |
`{!IsSurjInj G} : IsSurjInj (G o F) <-> IsSurjInj F. Proof. split; [ apply @cancelL_issurjinj | apply @issurjinj_compose ]; exact _. Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | iffL_issurjinj | 7,571 |
`{!SplEssSurj (G o F)} : SplEssSurj G. Proof. intros c. exists (F (esssurj (G o F) c).1). exact ((esssurj (G o F) c).2). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | cancelR_isesssurj | 7,572 |
`{!SplEssSurj F} : SplEssSurj (G o F) <-> SplEssSurj G. Proof. split; [ apply @cancelR_isesssurj | intros; apply @isesssurj_compose ]; exact _. Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | iffR_isesssurj | 7,573 |
`{!IsSurjInj F, !IsSurjInj (G o F)} : IsSurjInj G. Proof. constructor. - apply cancelR_isesssurj. - intros x y f. pose (p := (esssurj F x).2). pose (q := (esssurj F y).2). cbn in *. refine (p^$ $@ _ $@ q). apply (fmap F). apply (essinj (G o F)). refine (_ $@ f $@ _). + exact (fmap G p). + exact (fmap G q^$). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | cancelR_issurjinj | 7,574 |
`{!IsSurjInj F} : IsSurjInj (G o F) <-> IsSurjInj G. Proof. split; [ apply @cancelR_issurjinj | intros; apply @issurjinj_compose ]; exact _. Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | iffR_issurjinj | 7,575 |
{A B C D : Type} {F : A -> B} {G : C -> D} {H : A -> C} {K : B -> D} `{IsSurjInj A C H} `{IsSurjInj B D K} `{!Is0Functor F} `{!Is0Functor G} (p : K o F $=> G o H) : SplEssSurj F <-> SplEssSurj G. Proof. split; intros ?. - srapply (cancelR_isesssurj G H); try exact _. apply (isesssurj_transf (fun a => (p a)^$)). - srapply (cancelL_isesssurj K F); try exact _. apply (isesssurj_transf p). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | isesssurj_iff_commsq | 7,576 |
{A B C D : Type} {F : A -> B} {G : C -> D} {H : A -> C} {K : B -> D} `{IsSurjInj A C H} `{IsSurjInj B D K} `{!Is0Functor F} `{!Is0Functor G} (p : K o F $=> G o H) : IsSurjInj F <-> IsSurjInj G. Proof. split; intros ?. - srapply (cancelR_issurjinj G H); try exact _. apply (issurjinj_transf (fun a => (p a)^$)). - srapply (cancelL_issurjinj K F); try exact _. apply (issurjinj_transf p). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | issurjinj_iff_commsq | 7,577 |
{A : Type} (B C : A -> Type) `{forall a, IsGraph (B a)} `{forall a, Is01Cat (B a)} `{forall a, Is0Gpd (B a)} `{forall a, IsGraph (C a)} `{forall a, Is01Cat (C a)} `{forall a, Is0Gpd (C a)} (F : forall a, B a -> C a) {ff : forall a, Is0Functor (F a)} : SplEssSurj (fun (x:sig B) => (x.1 ; F x.1 x.2)) <-> (forall a, SplEssSurj (F a)). Proof. split. - intros fs a c. pose (s := fs (a;c)). destruct s as [[a' b] [p q]]; cbn in *. destruct p; cbn in q. exists b. exact q. - intros fs [a c]. exists (a ; (esssurj (F a) c).1); cbn. exists idpath; cbn. exact ((esssurj (F a) c).2). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | isesssurj_iff_sigma | 7,578 |
{A : Type} (B C : A -> Type) `{forall a, IsGraph (B a)} `{forall a, Is01Cat (B a)} `{forall a, Is0Gpd (B a)} `{forall a, IsGraph (C a)} `{forall a, Is01Cat (C a)} `{forall a, Is0Gpd (C a)} (F : forall a, B a -> C a) `{forall a, Is0Functor (F a)} `{forall a, IsSurjInj (F a)} : IsSurjInj (fun (x:sig B) => (x.1 ; F x.1 x.2)). Proof. constructor. - apply isesssurj_iff_sigma; exact _. - intros [a1 b1] [a2 b2] [p f]; cbn in *. destruct p; cbn in *. exists idpath; cbn. exact (essinj (F a1) f). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Basics.Iff. Require Import WildCat.Core. Require Import WildCat.NatTrans. Require Import WildCat.Sigma. | WildCat\EquivGpd.v | issurjinj_sigma | 7,579 |
(A B : Type) `{IsGraph A} `{IsGraph B} := { fun01_F : A -> B; fun01_is0functor : Is0Functor fun01_F; }. | Record | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Opposite. Require Import WildCat.Equiv. Require Import WildCat.Induced. Require Import WildCat.NatTrans. | WildCat\FunctorCat.v | Fun01 | 7,580 |
(A B : Type) `{IsGraph A} `{IsGraph B} : _ <~> Fun01 A B := ltac:(issig). Global Instance isgraph_fun01 (A B : Type) `{IsGraph A} `{Is1Cat B} : IsGraph (Fun01 A B). Proof. srapply Build_IsGraph. intros [F ?] [G ?]. exact (NatTrans F G). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Opposite. Require Import WildCat.Equiv. Require Import WildCat.Induced. Require Import WildCat.NatTrans. | WildCat\FunctorCat.v | issig_Fun01 | 7,581 |
(A B : Type) `{IsGraph A} `{IsGraph B} : Fun01 A B -> Fun01 A^op B^op. Proof. intros F. rapply (Build_Fun01 A^op B^op F). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Opposite. Require Import WildCat.Equiv. Require Import WildCat.Induced. Require Import WildCat.NatTrans. | WildCat\FunctorCat.v | fun01_op | 7,582 |
(A B : Type) `{Is1Cat A} `{Is1Cat B} := | Record | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Opposite. Require Import WildCat.Equiv. Require Import WildCat.Induced. Require Import WildCat.NatTrans. | WildCat\FunctorCat.v | Fun11 | 7,583 |
{A} `{IsGraph A} : Fun01 A A := Build_Fun01 A A idmap. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Opposite. Require Import WildCat.Equiv. Require Import WildCat.Induced. Require Import WildCat.NatTrans. | WildCat\FunctorCat.v | fun01_id | 7,584 |
{A} `{Is1Cat A} : Fun11 A A := Build_Fun11 _ _ idmap. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Opposite. Require Import WildCat.Equiv. Require Import WildCat.Induced. Require Import WildCat.NatTrans. | WildCat\FunctorCat.v | fun11_id | 7,585 |
{A B C} `{IsGraph A, IsGraph B, IsGraph C} : Fun01 B C -> Fun01 A B -> Fun01 A C := fun G F => Build_Fun01 _ _ (G o F). | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Opposite. Require Import WildCat.Equiv. Require Import WildCat.Induced. Require Import WildCat.NatTrans. | WildCat\FunctorCat.v | fun01_compose | 7,586 |
{A B C} `{IsGraph A, Is1Cat B, Is1Cat C} (F : Fun11 B C) : Fun01 A B -> Fun01 A C := fun01_compose (A:=A) F. Global Instance is0functor_fun01_postcomp {A B C} `{IsGraph A, Is1Cat B, Is1Cat C} (F : Fun11 B C) : Is0Functor ( (A:=A) F). Proof. apply Build_Is0Functor. intros a b f. rapply nattrans_postwhisker. exact f. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Opposite. Require Import WildCat.Equiv. Require Import WildCat.Induced. Require Import WildCat.NatTrans. | WildCat\FunctorCat.v | fun01_postcomp | 7,587 |
{A B C} `{IsGraph A, Is1Cat B, Is1Cat C} (F : Fun11 B C) : Fun11 (Fun01 A B) (Fun01 A C) := Build_Fun11 _ _ (fun01_postcomp F). | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Opposite. Require Import WildCat.Equiv. Require Import WildCat.Induced. Require Import WildCat.NatTrans. | WildCat\FunctorCat.v | fun11_fun01_postcomp | 7,588 |
{A B C} `{Is1Cat A, Is1Cat B, Is1Cat C} : Fun11 B C -> Fun11 A B -> Fun11 A C. Proof. intros F G. nrapply Build_Fun11. rapply (is1functor_compose G F). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Opposite. Require Import WildCat.Equiv. Require Import WildCat.Induced. Require Import WildCat.NatTrans. | WildCat\FunctorCat.v | fun11_compose | 7,589 |
`{HasMorExt B} : HasMorExt A. Proof. constructor. intros_of_type A; cbn. rapply isequiv_Htpy_path. Defined. | Instance | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Induced.v | hasmorext_induced | 7,590 |
`{HasEquivs B} : HasEquivs A. Proof. srapply Build_HasEquivs; intros a b; cbn. + exact (f a $<~> f b). + apply CatIsEquiv'. + apply cate_fun. + apply cate_isequiv'. + apply cate_buildequiv'. + nrapply cate_buildequiv_fun'. + apply cate_inv'. + nrapply cate_issect'. + nrapply cate_isretr'. + nrapply catie_adjointify'. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. | WildCat\Induced.v | hasequivs_induced | 7,591 |
{A : Type} `{HasEquivs A} {F : A -> A -> A} `{!Is0Bifunctor F, !Is1Bifunctor F, !Associator F} : forall a b c, F a (F b c) $<~> F (F a b) c := fun a b c => associator_uncurried (a, b, c). | Definition | Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite. | WildCat\Monoidal.v | associator | 7,592 |
{A : Type} `{HasEquivs A} (F : A -> A -> A) `{!Is0Bifunctor F, !Is1Bifunctor F} (associator : forall a b c, F a (F b c) $<~> F (F a b) c) (isnat_assoc : Is1Natural (fun '(a, b, c) => F a (F b c)) (fun '(a, b, c) => F (F a b) c) (fun '(a, b, c) => associator a b c)) : Associator F. Proof. snrapply Build_NatEquiv. - intros [[a b] c]. exact (associator a b c). - exact isnat_assoc. Defined. | Definition | Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite. | WildCat\Monoidal.v | Build_Associator | 7,593 |
{A : Type} `{Is1Cat A} {F : A -> A -> A} `{!Is0Bifunctor F, !Is1Bifunctor F, !Braiding F} : forall a b, F a b $-> F b a := fun a b => braid_uncurried (a, b). | Definition | Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite. | WildCat\Monoidal.v | braid | 7,594 |
{x x' y y' z z'} (f : x $-> x') (g : y $-> y') (h : z $-> z') : associator x' y' z' $o fmap11 F f (fmap11 F g h) $== fmap11 F (fmap11 F f g) h $o associator x y z. Proof. destruct assoc as [asso nat]. exact (nat (x, y, z) (x', y', z') (f, g, h)). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite. | WildCat\Monoidal.v | associator_nat | 7,595 |
{x x' : A} (f : x $-> x') (y z : A) : associator x' y z $o fmap10 F f (F y z) $== fmap10 F (fmap10 F f y) z $o associator x y z. Proof. refine ((_ $@L _^$) $@ _ $@ (_ $@R _)). 2: rapply (associator_nat f (Id _) (Id _)). - exact (fmap12 _ _ (fmap11_id _ _ _) $@ fmap10_is_fmap11 _ _ _). - exact (fmap21 _ (fmap10_is_fmap11 _ _ _) _ $@ fmap10_is_fmap11 _ _ _). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite. | WildCat\Monoidal.v | associator_nat_l | 7,596 |
(x : A) {y y' : A} (g : y $-> y') (z : A) : associator x y' z $o fmap01 F x (fmap10 F g z) $== fmap10 F (fmap01 F x g) z $o associator x y z. Proof. refine ((_ $@L _^$) $@ _ $@ (_ $@R _)). 2: nrapply (associator_nat (Id _) g (Id _)). - exact (fmap12 _ _ (fmap10_is_fmap11 _ _ _) $@ fmap01_is_fmap11 _ _ _). - exact (fmap21 _ (fmap01_is_fmap11 _ _ _) _ $@ fmap10_is_fmap11 _ _ _). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite. | WildCat\Monoidal.v | associator_nat_m | 7,597 |
(x y : A) {z z' : A} (h : z $-> z') : associator x y z' $o fmap01 F x (fmap01 F y h) $== fmap01 F (F x y) h $o associator x y z. Proof. refine ((_ $@L _^$) $@ _ $@ (_ $@R _)). 2: nrapply (associator_nat (Id _) (Id _) h). - exact (fmap12 _ _ (fmap01_is_fmap11 _ _ _) $@ fmap01_is_fmap11 _ _ _). - exact (fmap21 _ (fmap11_id _ _ _) _ $@ fmap01_is_fmap11 F _ _). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite. | WildCat\Monoidal.v | associator_nat_r | 7,598 |
associator_op' {A : Type} `{HasEquivs A} {F : A -> A -> A} `{!Is0Bifunctor F, !Is1Bifunctor F, assoc : !Associator (A:=A^op) F} : Associator F := associator_op (A:=A^op) (assoc := assoc). | Definition | Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite. | WildCat\Monoidal.v | associator_op' | 7,599 |