Require Import Basics.Overture Basics.Tactics Types.Forall.
[Loading ML file number_string_notation_plugin.cmxs (using legacy method) ... done]
Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv.
Require Import WildCat.NatTrans WildCat.Square.

(** * Monoidal Categories *)

(** In this file we define monoidal categories and symmetric monoidal categories. *)

(** ** Typeclasses for common diagrams *)

(** TODO: These should eventually be moved to a separate file in WildCat and used in other places. They can be thought of as a wildcat generalization of the classes in canonical_names.v *)

(** *** Associators *)

(** A natural equivalence witnessing the associativity of a bifunctor. *)
Class Associator {A : Type} `{HasEquivs A}
  (F : A -> A -> A) `{!Is0Bifunctor F, !Is1Bifunctor F} := {
  (** An isomorphism [associator] witnessing associativity of [F]. *)
  associator a b c : F a (F b c) $<~> F (F a b) c;

  (** The [associator] is a natural isomorphism. *)
  is1natural_associator_uncurried
    :: 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);
}.
Coercion associator : Associator >-> Funclass.
Arguments associator {A _ _ _ _ _ F _ _ _} a b c.

(** *** Unitors *)

Class LeftUnitor {A : Type} `{HasEquivs A}
  (F : A -> A -> A) `{!Is0Bifunctor F, !Is1Bifunctor F} (unit : A)
  (** A natural isomorphism [left_unitor] witnessing the left unit law of [F]. *)
  := left_unitor : NatEquiv (F unit) idmap.
Coercion left_unitor : LeftUnitor >-> NatEquiv.
Arguments left_unitor {A _ _ _ _ _ F _ _ unit _}.

Class RightUnitor {A : Type} `{HasEquivs A}
  (F : A -> A -> A) `{!Is0Bifunctor F, !Is1Bifunctor F} (unit : A)
  (** A natural isomorphism [right_unitor] witnessing the right unit law of [F]. *)
  := right_unitor : NatEquiv (flip F unit) idmap.
Coercion right_unitor : RightUnitor >-> NatEquiv.
Arguments right_unitor {A _ _ _ _ _ F _ _ unit _}.

(** *** Triangle and Pentagon identities *)

Class TriangleIdentity {A : Type} `{HasEquivs A}
  (F : A -> A -> A) `{!Is0Bifunctor F, !Is1Bifunctor F, !Associator F}
  (unit : A) `{!LeftUnitor F unit, !RightUnitor F unit}
  (** The triangle identity for an associator and unitors. *)
  := triangle_identity a b
    : fmap01 F a (left_unitor b)
    $== fmap10 F (right_unitor a) b $o (associator (F := F) a unit b).
Coercion triangle_identity : TriangleIdentity >-> Funclass.
Arguments triangle_identity {A _ _ _ _ _} F {_ _ _} unit {_}.

Class PentagonIdentity {A : Type} `{HasEquivs A}
  (F : A -> A -> A) `{!Is0Bifunctor F, !Is1Bifunctor F, !Associator F}
  (** The pentagon identity for an associator. *)
  := pentagon_identity a b c d
    : associator (F a b) c d $o associator a b (F c d)
      $== fmap10 F (associator a b c) d $o associator a (F b c) d
        $o fmap01 F a (associator b c d).
Coercion pentagon_identity : PentagonIdentity >-> Funclass.
Arguments pentagon_identity {A _ _ _ _ _} F {_ _ _}.

(** *** Braiding *)

Class Braiding {A : Type} `{Is1Cat A}
  (F : A -> A -> A) `{!Is0Bifunctor F, !Is1Bifunctor F} := {
  (** A morphism [braid] witnessing the symmetry of [F]. *)
  braid a b : F a b $-> F b a;
  (** The [braid] is a natural transformation. *)
  is1natural_braiding_uncurried
    : Is1Natural
      (uncurry F)
      (uncurry (flip F))
      (fun '(a, b) => braid a b);
}.
Coercion braid : Braiding >-> Funclass.
Arguments braid {A _ _ _ _ F _ _ _} a b.

Class SymmetricBraiding {A : Type} `{Is1Cat A}
  (F : A -> A -> A) `{!Is0Bifunctor F, !Is1Bifunctor F} := {
  braiding_symmetricbraiding :: Braiding F;
  braid_braid : forall a b, braid a b $o braid b a $== Id (F b a);
}.
(** We could have used [::>] in [braiding_symmetricbraiding] instead however due to bug https://github.com/coq/coq/issues/18971 the coercion isn't registered, so we have to register it manually instead. *)
Coercion braiding_symmetricbraiding : SymmetricBraiding >-> Braiding.
Arguments braid_braid {A _ _ _ _ F _ _ _} a b.

(** *** Hexagon identity *)

Class HexagonIdentity {A : Type} `{HasEquivs A}
  (F : A -> A -> A)
  `{!Is0Bifunctor F, !Is1Bifunctor F, !Associator F, !Braiding F}
  (** The hexagon identity for an associator and a braiding. *)
  := hexagon_identity a b c
    : fmap10 F (braid b a) c $o associator b a c $o fmap01 F b (braid c a)
      $== associator a b c $o braid (F b c) a $o associator b c a.
Coercion hexagon_identity : HexagonIdentity >-> Funclass.
Arguments hexagon_identity {A _ _ _ _ _} F {_ _}.

(** ** Monoidal Categories *)

(** A monoidal 1-category is a 1-category with equivalences together with the following: *)
Class IsMonoidal (A : Type) `{HasEquivs A}
  (** It has a binary operation [cat_tensor] called the tensor product. *)
  (cat_tensor : A -> A -> A)
  (** It has a unit object [cat_tensor_unit] called the tensor unit. *)
  (cat_tensor_unit : A)
  (** These all satisfy the following properties: *)
  := {
  (** A [cat_tensor] is a 1-bifunctor. *)
  is0bifunctor_cat_tensor :: Is0Bifunctor cat_tensor;
  is1bifunctor_cat_tensor :: Is1Bifunctor cat_tensor;
  (** A natural isomorphism [associator] witnessing the associativity of the tensor product. *)
  cat_tensor_associator :: Associator cat_tensor;
  (** A natural isomorphism [left_unitor] witnessing the left unit law. *)
  cat_tensor_left_unitor :: LeftUnitor cat_tensor cat_tensor_unit;
  (** A natural isomorphism [right_unitor] witnessing the right unit law. *)
  cat_tensor_right_unitor :: RightUnitor cat_tensor cat_tensor_unit;
  (** The triangle identity. *)
  cat_tensor_triangle_identity :: TriangleIdentity cat_tensor cat_tensor_unit;
  (** The pentagon identity. *)
  cat_tensor_pentagon_identity :: PentagonIdentity cat_tensor;
}.

(** TODO: Braided monoidal categories *)

(** ** Symmetric Monoidal Categories *)

(** A symmetric monoidal 1-category is a 1-category with equivalences together with the following: *)
Class IsSymmetricMonoidal (A : Type) `{HasEquivs A}
  (** A binary operation [cat_tensor] called the tensor product. *)
  (cat_tensor : A -> A -> A)
  (** A unit object [cat_tensor_unit] called the tensor unit. *)
  (cat_tensor_unit : A)
  := {
  (** A monoidal structure with [cat_tensor] and [cat_tensor_unit]. *)
  issymmetricmonoidal_ismonoidal :: IsMonoidal A cat_tensor cat_tensor_unit;
  (** A natural transformation [braid] witnessing the symmetry of the tensor product such that [braid] is its own inverse. *)
  cat_symm_tensor_braiding :: SymmetricBraiding cat_tensor;
  (** The hexagon identity. *)
  cat_symm_tensor_hexagon :: HexagonIdentity cat_tensor;
}.

(** *** Theory about [Associator] *)

Section Associator.
  Context {A : Type} {F : A -> A -> A} `{assoc : Associator A F, !HasEquivs A}.

  Local Definition associator_nat {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.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, x', y, y', z, z': A
f: x $-> x'
g: y $-> y'
h: z $-> z'
assoc x' y' z' $o fmap11 F f (fmap11 F g h) $==
fmap11 F (fmap11 F f g) h $o assoc x y z
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, x', y, y', z, z': A
f: x $-> x'
g: y $-> y'
h: z $-> z'
assoc x' y' z' $o fmap11 F f (fmap11 F g h) $==
fmap11 F (fmap11 F f g) h $o assoc x y z
    destruct assoc as [asso nat].
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, x', y, y', z, z': A
f: x $-> x'
g: y $-> y'
h: z $-> z'
asso: forall a b c : A, F a (F b c) $<~> F (F a b) c
nat: Is1Natural
  (fun pat : A * A * A =>
   let x := pat in
   let fst := fst x in
   let c := snd x in
   let fst0 := fst in
   let a := Overture.fst fst0 in
   let b := snd fst0 in F a (F b c))
  (fun pat : A * A * A =>
   let x := pat in
   let fst := fst x in
   let c := snd x in
   let fst0 := fst in
   let a := Overture.fst fst0 in
   let b := snd fst0 in F (F a b) c)
  (fun pat : A * A * A =>
   let x := pat in
   let fst := fst x in
   let c := snd x in
   let fst0 := fst in
   let a := Overture.fst fst0 in
   let b := snd fst0 in asso a b c)
{|
  associator := asso;
  is1natural_associator_uncurried := nat
|} x' y' z' $o fmap11 F f (fmap11 F g h) $==
fmap11 F (fmap11 F f g) h $o
{|
  associator := asso;
  is1natural_associator_uncurried := nat
|} x y z
    exact (nat (x, y, z) (x', y', z') (f, g, h)).
  Defined.

  Local Definition associator_nat_l {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.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, x': A
f: x $-> x'
y, z: A
assoc x' y z $o fmap10 F f (F y z) $==
fmap10 F (fmap10 F f y) z $o assoc x y z
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, x': A
f: x $-> x'
y, z: A
assoc x' y z $o fmap10 F f (F y z) $==
fmap10 F (fmap10 F f y) z $o assoc x y z
    refine ((_ $@L _^$) $@ _ $@ (_ $@R _)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, x': A
f: x $-> x'
y, z: A
?Goal $-> fmap10 F f (F y z)
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, x': A
f: x $-> x'
y, z: A
assoc x' y z $o ?Goal $== ?Goal2 $o assoc x y z
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, x': A
f: x $-> x'
y, z: A
?Goal2 $== fmap10 F (fmap10 F f y) z
    2: rapply (associator_nat f (Id _) (Id _)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, x': A
f: x $-> x'
y, z: A
fmap11 F f (fmap11 F (Id y) (Id z)) $->
fmap10 F f (F y z)
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, x': A
f: x $-> x'
y, z: A
fmap11 F (fmap11 F f (Id y)) (Id z) $==
fmap10 F (fmap10 F f y) z
    -
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, x': A
f: x $-> x'
y, z: A
fmap11 F f (fmap11 F (Id y) (Id z)) $->
fmap10 F f (F y z) exact (fmap12 _ _ (fmap11_id _ _ _) $@ fmap10_is_fmap11 _ _ _).
    -
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, x': A
f: x $-> x'
y, z: A
fmap11 F (fmap11 F f (Id y)) (Id z) $==
fmap10 F (fmap10 F f y) z exact (fmap21 _ (fmap10_is_fmap11 _ _ _) _ $@ fmap10_is_fmap11 _ _ _).
  Defined.

  Local Definition associator_nat_m (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.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, y': A
g: y $-> y'
z: A
assoc x y' z $o fmap01 F x (fmap10 F g z) $==
fmap10 F (fmap01 F x g) z $o assoc x y z
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, y': A
g: y $-> y'
z: A
assoc x y' z $o fmap01 F x (fmap10 F g z) $==
fmap10 F (fmap01 F x g) z $o assoc x y z
    refine ((_ $@L _^$) $@ _ $@ (_ $@R _)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, y': A
g: y $-> y'
z: A
?Goal $-> fmap01 F x (fmap10 F g z)
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, y': A
g: y $-> y'
z: A
assoc x y' z $o ?Goal $== ?Goal2 $o assoc x y z
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, y': A
g: y $-> y'
z: A
?Goal2 $== fmap10 F (fmap01 F x g) z
    2: nrapply (associator_nat (Id _) g (Id _)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, y': A
g: y $-> y'
z: A
fmap11 F (Id x) (fmap11 F g (Id z)) $->
fmap01 F x (fmap10 F g z)
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, y': A
g: y $-> y'
z: A
Is01Cat A
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, y': A
g: y $-> y'
z: A
Is01Cat A
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, y': A
g: y $-> y'
z: A
fmap11 F (fmap11 F (Id x) g) (Id z) $==
fmap10 F (fmap01 F x g) z
    -
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, y': A
g: y $-> y'
z: A
fmap11 F (Id x) (fmap11 F g (Id z)) $->
fmap01 F x (fmap10 F g z) exact (fmap12 _ _ (fmap10_is_fmap11 _ _ _) $@ fmap01_is_fmap11 _ _ _).
    -
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, y': A
g: y $-> y'
z: A
fmap11 F (fmap11 F (Id x) g) (Id z) $==
fmap10 F (fmap01 F x g) z exact (fmap21 _ (fmap01_is_fmap11 _ _ _) _ $@ fmap10_is_fmap11 _ _ _).
  Defined.

  Local Definition associator_nat_r (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.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, z, z': A
h: z $-> z'
assoc x y z' $o fmap01 F x (fmap01 F y h) $==
fmap01 F (F x y) h $o assoc x y z
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, z, z': A
h: z $-> z'
assoc x y z' $o fmap01 F x (fmap01 F y h) $==
fmap01 F (F x y) h $o assoc x y z
    refine ((_ $@L _^$) $@ _ $@ (_ $@R _)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, z, z': A
h: z $-> z'
?Goal $-> fmap01 F x (fmap01 F y h)
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, z, z': A
h: z $-> z'
assoc x y z' $o ?Goal $== ?Goal2 $o assoc x y z
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, z, z': A
h: z $-> z'
?Goal2 $== fmap01 F (F x y) h
    2: nrapply (associator_nat (Id _) (Id _) h).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, z, z': A
h: z $-> z'
fmap11 F (Id x) (fmap11 F (Id y) h) $->
fmap01 F x (fmap01 F y h)
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, z, z': A
h: z $-> z'
Is01Cat A
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, z, z': A
h: z $-> z'
Is01Cat A
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, z, z': A
h: z $-> z'
fmap11 F (fmap11 F (Id x) (Id y)) h $==
fmap01 F (F x y) h
    -
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, z, z': A
h: z $-> z'
fmap11 F (Id x) (fmap11 F (Id y) h) $->
fmap01 F x (fmap01 F y h) exact (fmap12 _ _ (fmap01_is_fmap11 _ _ _) $@ fmap01_is_fmap11 _ _ _).
    -
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
assoc: Associator F
HasEquivs0: HasEquivs A
x, y, z, z': A
h: z $-> z'
fmap11 F (fmap11 F (Id x) (Id y)) h $==
fmap01 F (F x y) h exact (fmap21 _ (fmap11_id _ _ _) _ $@ fmap01_is_fmap11 F _ _).
  Defined.

End Associator.

(** ** Theory about [SymmetricBraid] *)

Section SymmetricBraid.
  Context {A : Type} {F : A -> A -> A} `{SymmetricBraiding A F, !HasEquivs A}.

  (** [braid] is its own inverse and therefore an equivalence. *)
  Local Instance catie_braid a b : CatIsEquiv (braid a b)
    := catie_adjointify (braid a b) (braid b a) (braid_braid a b) (braid_braid b a).

  (** [braide] is the bundled equivalence whose underlying map is [braid]. *)
  Local Definition braide a b
    : F a b $<~> F b a
    := Build_CatEquiv (braid a b).

  Local Definition moveL_braidL a b c f (g : c $-> _)
    : braid a b $o f $== g -> f $== braid b a $o g.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: c $-> F a b
g: c $-> F b a
H0 a b $o f $== g -> f $== H0 b a $o g
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: c $-> F a b
g: c $-> F b a
H0 a b $o f $== g -> f $== H0 b a $o g
    intros p.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: c $-> F a b
g: c $-> F b a
p: H0 a b $o f $== g
f $== H0 b a $o g
    apply (cate_monic_equiv (braide a b)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: c $-> F a b
g: c $-> F b a
p: H0 a b $o f $== g
braide a b $o f $== braide a b $o (H0 b a $o g)
    refine ((cate_buildequiv_fun _ $@R _) $@ p $@ _ $@ cat_assoc _ _ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: c $-> F a b
g: c $-> F b a
p: H0 a b $o f $== g
g $== braide a b $o H0 b a $o g
    refine ((cat_idl _)^$ $@ (_^$ $@R _)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: c $-> F a b
g: c $-> F b a
p: H0 a b $o f $== g
braide a b $o H0 b a $-> Id (F b a)
    refine ((cate_buildequiv_fun _ $@R _) $@ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: c $-> F a b
g: c $-> F b a
p: H0 a b $o f $== g
H0 a b $o H0 b a $== Id (F b a)
    apply braid_braid.
  Defined.

  Local Definition moveL_braidR a b c f (g : _ $-> c)
    : f $o braid a b $== g -> f $== g $o braid b a.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: F b a $-> c
g: F a b $-> c
f $o H0 a b $== g -> f $== g $o H0 b a
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: F b a $-> c
g: F a b $-> c
f $o H0 a b $== g -> f $== g $o H0 b a
    intros p.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: F b a $-> c
g: F a b $-> c
p: f $o H0 a b $== g
f $== g $o H0 b a
    apply (cate_epic_equiv (braide a b)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: F b a $-> c
g: F a b $-> c
p: f $o H0 a b $== g
f $o braide a b $== g $o H0 b a $o braide a b
    refine (_ $@ (cat_assoc _ _ _)^$).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: F b a $-> c
g: F a b $-> c
p: f $o H0 a b $== g
f $o braide a b $== g $o (H0 b a $o braide a b)
    refine (_ $@ (_ $@L ((_ $@L cate_buildequiv_fun _) $@ _)^$)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: F b a $-> c
g: F a b $-> c
p: f $o H0 a b $== g
f $o braide a b $== g $o ?Goal0
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: F b a $-> c
g: F a b $-> c
p: f $o H0 a b $== g
H0 b a $o H0 a b $== ?Goal0
    2: apply braid_braid.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: F b a $-> c
g: F a b $-> c
p: f $o H0 a b $== g
f $o braide a b $== g $o Id (F a b)
    refine ((_ $@L _) $@ _ $@ (cat_idr _)^$).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: F b a $-> c
g: F a b $-> c
p: f $o H0 a b $== g
braide a b $== ?Goal
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: F b a $-> c
g: F a b $-> c
p: f $o H0 a b $== g
f $o ?Goal $== g
    1: apply cate_buildequiv_fun.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: F b a $-> c
g: F a b $-> c
p: f $o H0 a b $== g
f $o H0 a b $== g
    exact p.
  Defined.

  Local Definition moveR_braidL a b c f (g : c $-> _)
    : f $== braid b a $o g -> braid a b $o f $== g.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: c $-> F a b
g: c $-> F b a
f $== H0 b a $o g -> H0 a b $o f $== g
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: c $-> F a b
g: c $-> F b a
f $== H0 b a $o g -> H0 a b $o f $== g
    intros p; symmetry; apply moveL_braidL; symmetry; exact p.
  Defined.

  Local Definition moveR_braidR a b c f (g : _ $-> c)
    : f $== g $o braid b a -> f $o braid a b $== g.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: F b a $-> c
g: F a b $-> c
f $== g $o H0 b a -> f $o H0 a b $== g
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: F b a $-> c
g: F a b $-> c
f $== g $o H0 b a -> f $o H0 a b $== g
    intros p; symmetry; apply moveL_braidR; symmetry; exact p.
  Defined.

  Local Definition moveL_fmap01_braidL a b c d f (g : d $-> _)
    : fmap01 F a (braid b c) $o f $== g
      -> f $== fmap01 F a (braid c b) $o g.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
fmap01 F a (H0 b c) $o f $== g ->
f $== fmap01 F a (H0 c b) $o g
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
fmap01 F a (H0 b c) $o f $== g ->
f $== fmap01 F a (H0 c b) $o g
    intros p.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
p: fmap01 F a (H0 b c) $o f $== g
f $== fmap01 F a (H0 c b) $o g
    apply (cate_monic_equiv (emap01 F a (braide b c))).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
p: fmap01 F a (H0 b c) $o f $== g
emap01 F a (braide b c) $o f $==
emap01 F a (braide b c) $o (fmap01 F a (H0 c b) $o g)
    refine (_ $@ cat_assoc _ _ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
p: fmap01 F a (H0 b c) $o f $== g
emap01 F a (braide b c) $o f $==
emap01 F a (braide b c) $o fmap01 F a (H0 c b) $o g
    refine (_ $@ (_ $@R _)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
p: fmap01 F a (H0 b c) $o f $== g
emap01 F a (braide b c) $o f $== ?Goal0 $o g
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
p: fmap01 F a (H0 b c) $o f $== g
?Goal0 $==
emap01 F a (braide b c) $o fmap01 F a (H0 c b)
    2: {
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
p: fmap01 F a (H0 b c) $o f $== g
?Goal0 $==
emap01 F a (braide b c) $o fmap01 F a (H0 c b) refine (_ $@ (_^$ $@R _)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
p: fmap01 F a (H0 b c) $o f $== g
?Goal0 $== ?Goal2 $o fmap01 F a (H0 c b)
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
p: fmap01 F a (H0 b c) $o f $== g
emap01 F a (braide b c) $-> ?Goal2
      2: apply cate_buildequiv_fun.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
p: fmap01 F a (H0 b c) $o f $== g
?Goal0 $==
fmap01 F a (braide b c) $o fmap01 F a (H0 c b)
      refine ((fmap_id _ _)^$ $@ fmap2 _ _ $@ fmap_comp _ _ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
p: fmap01 F a (H0 b c) $o f $== g
Id (F c b) $== braide b c $o H0 c b
      refine ((_ $@R _) $@ _)^$.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
p: fmap01 F a (H0 b c) $o f $== g
braide b c $== ?Goal0
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
p: fmap01 F a (H0 b c) $o f $== g
?Goal0 $o H0 c b $== Id (F c b)
      1: apply cate_buildequiv_fun.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
p: fmap01 F a (H0 b c) $o f $== g
H0 b c $o H0 c b $== Id (F c b)
      apply braid_braid. }
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
p: fmap01 F a (H0 b c) $o f $== g
emap01 F a (braide b c) $o f $== Id (F a (F c b)) $o g
    refine ((_ $@R _) $@ p $@ (cat_idl _)^$).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
p: fmap01 F a (H0 b c) $o f $== g
emap01 F a (braide b c) $== fmap01 F a (H0 b c)
    refine (_ $@ fmap2 _ _);
    apply cate_buildequiv_fun.
  Defined.

  Local Definition moveL_fmap01_braidR a b c d f (g : _ $-> d)
    :  f $o fmap01 F a (braid b c) $== g
      -> f $== g $o fmap01 F a (braid c b).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
f $o fmap01 F a (H0 b c) $== g ->
f $== g $o fmap01 F a (H0 c b)
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
f $o fmap01 F a (H0 b c) $== g ->
f $== g $o fmap01 F a (H0 c b)
    intros p.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
p: f $o fmap01 F a (H0 b c) $== g
f $== g $o fmap01 F a (H0 c b)
    apply (cate_epic_equiv (emap01 F a (braide b c))).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
p: f $o fmap01 F a (H0 b c) $== g
f $o emap01 F a (braide b c) $==
g $o fmap01 F a (H0 c b) $o emap01 F a (braide b c)
    refine (_ $@ (cat_assoc _ _ _)^$).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
p: f $o fmap01 F a (H0 b c) $== g
f $o emap01 F a (braide b c) $==
g $o (fmap01 F a (H0 c b) $o emap01 F a (braide b c))
    refine (_ $@ (_ $@L _)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
p: f $o fmap01 F a (H0 b c) $== g
f $o emap01 F a (braide b c) $== g $o ?Goal0
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
p: f $o fmap01 F a (H0 b c) $== g
?Goal0 $==
fmap01 F a (H0 c b) $o emap01 F a (braide b c)
    2: {
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
p: f $o fmap01 F a (H0 b c) $== g
?Goal0 $==
fmap01 F a (H0 c b) $o emap01 F a (braide b c) refine (_^$ $@ (_ $@L _^$)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
p: f $o fmap01 F a (H0 b c) $== g
fmap01 F a (H0 c b) $o ?Goal2 $-> ?Goal0
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
p: f $o fmap01 F a (H0 b c) $== g
emap01 F a (braide b c) $-> ?Goal2
      2: apply cate_buildequiv_fun.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
p: f $o fmap01 F a (H0 b c) $== g
fmap01 F a (H0 c b) $o fmap01 F a (braide b c) $->
?Goal0
      refine ((fmap_comp _ _ _)^$ $@ fmap2 _ _ $@ fmap_id _ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
p: f $o fmap01 F a (H0 b c) $== g
H0 c b $o braide b c $== Id (F b c)
      refine ((_ $@L _) $@ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
p: f $o fmap01 F a (H0 b c) $== g
braide b c $== ?Goal0
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
p: f $o fmap01 F a (H0 b c) $== g
H0 c b $o ?Goal0 $== Id (F b c)
      1: apply cate_buildequiv_fun.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
p: f $o fmap01 F a (H0 b c) $== g
H0 c b $o H0 b c $== Id (F b c)
      apply braid_braid. }
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
p: f $o fmap01 F a (H0 b c) $== g
f $o emap01 F a (braide b c) $== g $o Id (F a (F b c))
    refine ((_ $@L _) $@ p $@ (cat_idr _)^$).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
p: f $o fmap01 F a (H0 b c) $== g
emap01 F a (braide b c) $== fmap01 F a (H0 b c)
    refine (_ $@ fmap2 _ _);
    apply cate_buildequiv_fun.
  Defined.

  Local Definition moveR_fmap01_braidL a b c d f (g : d $-> _)
    : f $== fmap01 F a (braid c b) $o g
      -> fmap01 F a (braid b c) $o f $== g.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
f $== fmap01 F a (H0 c b) $o g ->
fmap01 F a (H0 b c) $o f $== g
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: d $-> F a (F b c)
g: d $-> F a (F c b)
f $== fmap01 F a (H0 c b) $o g ->
fmap01 F a (H0 b c) $o f $== g
    intros p; symmetry; apply moveL_fmap01_braidL; symmetry; exact p.
  Defined.

  Local Definition moveR_fmap01_braidR a b c d f (g : _ $-> d)
    : f $== g $o fmap01 F a (braid c b)
      -> f $o fmap01 F a (braid b c) $== g.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
f $== g $o fmap01 F a (H0 c b) ->
f $o fmap01 F a (H0 b c) $== g
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: F a (F c b) $-> d
g: F a (F b c) $-> d
f $== g $o fmap01 F a (H0 c b) ->
f $o fmap01 F a (H0 b c) $== g
    intros p; symmetry; apply moveL_fmap01_braidR; symmetry; exact p.
  Defined.

  Local Definition moveL_fmap01_fmap01_braidL a b c d e f (g : e $-> _)
    : fmap01 F a (fmap01 F b (braid c d)) $o f $== g
      -> f $== fmap01 F a (fmap01 F b (braid d c)) $o g.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
fmap01 F a (fmap01 F b (H0 c d)) $o f $== g ->
f $== fmap01 F a (fmap01 F b (H0 d c)) $o g
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
fmap01 F a (fmap01 F b (H0 c d)) $o f $== g ->
f $== fmap01 F a (fmap01 F b (H0 d c)) $o g
    intros p.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
f $== fmap01 F a (fmap01 F b (H0 d c)) $o g
    apply (cate_monic_equiv (emap01 F a (emap01 F b (braide c d)))).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
emap01 F a (emap01 F b (braide c d)) $o f $==
emap01 F a (emap01 F b (braide c d)) $o
(fmap01 F a (fmap01 F b (H0 d c)) $o g)
    refine (_ $@ cat_assoc _ _ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
emap01 F a (emap01 F b (braide c d)) $o f $==
emap01 F a (emap01 F b (braide c d)) $o
fmap01 F a (fmap01 F b (H0 d c)) $o g
    refine (_ $@ (_ $@R _)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
emap01 F a (emap01 F b (braide c d)) $o f $==
?Goal0 $o g
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
?Goal0 $==
emap01 F a (emap01 F b (braide c d)) $o
fmap01 F a (fmap01 F b (H0 d c))
    2: {
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
?Goal0 $==
emap01 F a (emap01 F b (braide c d)) $o
fmap01 F a (fmap01 F b (H0 d c)) refine (_ $@ (_^$ $@R _)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
?Goal0 $== ?Goal2 $o fmap01 F a (fmap01 F b (H0 d c))
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
emap01 F a (emap01 F b (braide c d)) $-> ?Goal2
      2: apply cate_buildequiv_fun.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
?Goal0 $==
fmap01 F a (emap01 F b (braide c d)) $o
fmap01 F a (fmap01 F b (H0 d c))
      refine ((fmap_id _ _)^$ $@ fmap2 _ _ $@ fmap_comp _ _ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
Id (F b (F d c)) $==
emap01 F b (braide c d) $o fmap01 F b (H0 d c)
      refine (_ $@ (_^$ $@R _)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
Id (F b (F d c)) $== ?Goal1 $o fmap01 F b (H0 d c)
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
emap01 F b (braide c d) $-> ?Goal1
      2: apply cate_buildequiv_fun.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
Id (F b (F d c)) $==
fmap01 F b (braide c d) $o fmap01 F b (H0 d c)
      refine ((fmap_id _ _)^$ $@ fmap2 _ _ $@ fmap_comp _ _ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
Id (F d c) $== braide c d $o H0 d c
      refine ((_ $@R _) $@ _)^$.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
braide c d $== ?Goal0
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
?Goal0 $o H0 d c $== Id (F d c)
      1: apply cate_buildequiv_fun.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
H0 c d $o H0 d c $== Id (F d c)
      apply braid_braid. }
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
emap01 F a (emap01 F b (braide c d)) $o f $==
Id (F a (F b (F d c))) $o g
    refine ((_ $@R _) $@ p $@ (cat_idl _)^$).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
emap01 F a (emap01 F b (braide c d)) $==
fmap01 F a (fmap01 F b (H0 c d))
    refine (cate_buildequiv_fun _ $@ fmap02 _ _ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
emap01 F b (braide c d) $== fmap01 F b (H0 c d)
    refine (cate_buildequiv_fun _ $@ fmap02 _ _ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
p: fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
braide c d $== H0 c d
    apply cate_buildequiv_fun.
  Defined.

  Local Definition moveL_fmap01_fmap01_braidR a b c d e f (g : _ $-> e)
    : f $o fmap01 F a (fmap01 F b (braid c d)) $== g
      -> f $== g $o fmap01 F a (fmap01 F b (braid d c)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
f $o fmap01 F a (fmap01 F b (H0 c d)) $== g ->
f $== g $o fmap01 F a (fmap01 F b (H0 d c))
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
f $o fmap01 F a (fmap01 F b (H0 c d)) $== g ->
f $== g $o fmap01 F a (fmap01 F b (H0 d c))
    intros p.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
f $== g $o fmap01 F a (fmap01 F b (H0 d c))
    apply (cate_epic_equiv (emap01 F a (emap01 F b (braide c d)))).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
f $o emap01 F a (emap01 F b (braide c d)) $==
g $o fmap01 F a (fmap01 F b (H0 d c)) $o
emap01 F a (emap01 F b (braide c d))
    refine (_ $@ (cat_assoc _ _ _)^$).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
f $o emap01 F a (emap01 F b (braide c d)) $==
g $o
(fmap01 F a (fmap01 F b (H0 d c)) $o
 emap01 F a (emap01 F b (braide c d)))
    refine (_ $@ (_ $@L _)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
f $o emap01 F a (emap01 F b (braide c d)) $==
g $o ?Goal0
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
?Goal0 $==
fmap01 F a (fmap01 F b (H0 d c)) $o
emap01 F a (emap01 F b (braide c d))
    2: {
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
?Goal0 $==
fmap01 F a (fmap01 F b (H0 d c)) $o
emap01 F a (emap01 F b (braide c d)) refine (_^$ $@ (_ $@L _^$)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
fmap01 F a (fmap01 F b (H0 d c)) $o ?Goal2 $-> ?Goal0
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
emap01 F a (emap01 F b (braide c d)) $-> ?Goal2
      2: apply cate_buildequiv_fun.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
fmap01 F a (fmap01 F b (H0 d c)) $o
fmap01 F a (emap01 F b (braide c d)) $-> ?Goal0
      refine ((fmap_comp _ _ _)^$ $@ fmap2 _ _ $@ fmap_id _ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
fmap01 F b (H0 d c) $o emap01 F b (braide c d) $==
Id (F b (F c d))
      refine ((_ $@L _) $@ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
emap01 F b (braide c d) $== ?Goal0
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
fmap01 F b (H0 d c) $o ?Goal0 $== Id (F b (F c d))
      1: apply cate_buildequiv_fun.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
fmap01 F b (H0 d c) $o fmap01 F b (braide c d) $==
Id (F b (F c d))
      refine ((fmap_comp _ _ _)^$ $@ fmap2 _ _ $@ fmap_id _ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
H0 d c $o braide c d $== Id (F c d)
      refine ((_ $@L _) $@ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
braide c d $== ?Goal0
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
H0 d c $o ?Goal0 $== Id (F c d)
      1: apply cate_buildequiv_fun.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
H0 d c $o H0 c d $== Id (F c d)
      apply braid_braid. }
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
f $o emap01 F a (emap01 F b (braide c d)) $==
g $o Id (F a (F b (F c d)))
    refine ((_ $@L _) $@ p $@ (cat_idr _)^$).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
emap01 F a (emap01 F b (braide c d)) $==
fmap01 F a (fmap01 F b (H0 c d))
    refine (cate_buildequiv_fun _ $@ fmap02 _ _ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
emap01 F b (braide c d) $== fmap01 F b (H0 c d)
    refine (cate_buildequiv_fun _ $@ fmap02 _ _ _).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
p: f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
braide c d $== H0 c d
    apply cate_buildequiv_fun.
  Defined.

  Local Definition moveR_fmap01_fmap01_braidL a b c d e f (g : e $-> _)
    : f $== fmap01 F a (fmap01 F b (braid d c)) $o g
      -> fmap01 F a (fmap01 F b (braid c d)) $o f $== g.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
f $== fmap01 F a (fmap01 F b (H0 d c)) $o g ->
fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: e $-> F a (F b (F c d))
g: e $-> F a (F b (F d c))
f $== fmap01 F a (fmap01 F b (H0 d c)) $o g ->
fmap01 F a (fmap01 F b (H0 c d)) $o f $== g
    intros p; symmetry; apply moveL_fmap01_fmap01_braidL; symmetry; exact p.
  Defined.

  Local Definition moveR_fmap01_fmap01_braidR a b c d e f (g : _ $-> e)
    : f $== g $o fmap01 F a (fmap01 F b (braid d c))
      -> f $o fmap01 F a (fmap01 F b (braid c d)) $== g.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
f $== g $o fmap01 F a (fmap01 F b (H0 d c)) ->
f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d, e: A
f: F a (F b (F d c)) $-> e
g: F a (F b (F c d)) $-> e
f $== g $o fmap01 F a (fmap01 F b (H0 d c)) ->
f $o fmap01 F a (fmap01 F b (H0 c d)) $== g
    intros p; symmetry; apply moveL_fmap01_fmap01_braidR; symmetry; exact p.
  Defined.

  Local Definition braid_nat {a b c d} (f : a $-> c) (g : b $-> d)
    : braid c d $o fmap11 F f g $== fmap11 F g f $o braid a b.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: a $-> c
g: b $-> d
H0 c d $o fmap11 F f g $== fmap11 F g f $o H0 a b
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c, d: A
f: a $-> c
g: b $-> d
H0 c d $o fmap11 F f g $== fmap11 F g f $o H0 a b
    exact (is1natural_braiding_uncurried (a, b) (c, d) (f, g)).
  Defined.

  Local Definition braid_nat_l {a b c} (f : a $-> b)
    : braid b c $o fmap10 F f c $== fmap01 F c f $o braid a c.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: a $-> b
H0 b c $o fmap10 F f c $== fmap01 F c f $o H0 a c
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: a $-> b
H0 b c $o fmap10 F f c $== fmap01 F c f $o H0 a c
    refine ((_ $@L (fmap10_is_fmap11 _ _ _)^$) $@ _ $@ (fmap01_is_fmap11 _ _ _ $@R _)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
f: a $-> b
H0 b c $o fmap11 F f (Id c) $==
fmap11 F (Id c) f $o H0 a c
    exact (is1natural_braiding_uncurried (a, c) (b, c) (f, Id _)).
  Defined.

  (** This is just the inverse of above. *)
  Local Definition braid_nat_r {a b c} (g : b $-> c)
    : braid a c $o fmap01 F a g $== fmap10 F g a $o braid a b.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
g: b $-> c
H0 a c $o fmap01 F a g $== fmap10 F g a $o H0 a b
  Proof.
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
g: b $-> c
H0 a c $o fmap01 F a g $== fmap10 F g a $o H0 a b
    refine ((_ $@L (fmap01_is_fmap11 _ _ _)^$) $@ _ $@ (fmap10_is_fmap11 _ _ _ $@R _)).
A: Type
F: A -> A -> A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor0: Is0Bifunctor F
Is1Bifunctor0: Is1Bifunctor F
H0: SymmetricBraiding F
HasEquivs0: HasEquivs A
a, b, c: A
g: b $-> c
H0 a c $o fmap11 F (Id a) g $==
fmap11 F g (Id a) $o H0 a b
    exact (is1natural_braiding_uncurried (a, b) (a, c) (Id _ , g)).
  Defined.

End SymmetricBraid.

(** ** Building Symmetric Monoidal Categories *)

(** The following construction is what we call the "twist construction". It is a way to build a symmetric monoidal category from simpler pieces than the axioms ask for.

The core observation is that the associator can be broken up into a [braid] and what we call a [twist] map. The twist map takes a right associated triple [(A, (B, C))] and swaps the first two factors [(B, (A, C)]. Together with functoriality of the tensor and the braiding, here termed [braid] we can simplify the axioms we ask for.

For instance, the hexagon identity is about associators, but if we unfold the definition and simplify the diagram, we get a diagram about only twists and braids.

This means in practice, you can show a category has a symmetric monoidal structure by proving some simpler axioms. This idea has been used in TriJoin.v to show the associativity of join for example. *)

Section TwistConstruction.
  (** The aim of this section is to build a symmetric monoidal category. We do this piecewise so that the separate steps are useful in and of themselves.

  Our basic starting assumption is that we have a category with equivalences, a bifunctor called the tensor product, and a unit object.*)
  Context (A : Type) `{HasEquivs A}
    (cat_tensor : A -> A -> A) (cat_tensor_unit : A)
    `{!Is0Bifunctor cat_tensor, !Is1Bifunctor cat_tensor}

    (** Next we postulate the existence of a [braid] map. This takes a tensor pair and swaps the factors. We also postulate that [braid] is natural in both factors and self-inverse. *)
    (braid : SymmetricBraiding cat_tensor)

    (** We postulate the existence of a [twist] map. This takes a right associated triple [(A, (B, C))] and swaps the first two factors [(B, (A, C)]. We also postulate that [twist] is natural in all three factors and self-inverse. *)
    (twist : forall a b c, cat_tensor a (cat_tensor b c) $-> cat_tensor b (cat_tensor a c))
    (twist_twist : forall a b c, twist a b c $o twist b a c $== Id _)
    (twist_nat : forall a a' b b' c c' (f : a $-> a') (g : b $-> b') (h : c $-> c'),
      twist a' b' c' $o fmap11 cat_tensor f (fmap11 cat_tensor g h)
      $== fmap11 cat_tensor g (fmap11 cat_tensor f h) $o twist a b c)

    (** We assume that there is a natural isomorphism [right_unitor] witnessing the right unit law. The left unit law will be derived from this one. We also assume a coherence called [twist_unitor] which determines how the right_unitor interacts with [braid] and [twist]. This is the basis of the triangle axiom. *)
    (right_unitor : RightUnitor cat_tensor cat_tensor_unit)
    (twist_unitor : forall a b, fmap01 cat_tensor a (right_unitor b)
      $== braid b a $o fmap01 cat_tensor b (right_unitor a) $o twist a b cat_tensor_unit)

    (** The hexagon identity is about the interaction of associators and braids. We will derive this axiom from an analogous one for twists and braids. *)
    (twist_hexagon : forall a b c,
      fmap01 cat_tensor c (braid b a) $o twist b c a $o fmap01 cat_tensor b (braid a c)
      $== twist a c b $o fmap01 cat_tensor a (braid b c) $o twist b a c)

    (** The 9-gon identity. TODO: explain this *)
    (twist_9_gon : forall a b c d,
        fmap01 cat_tensor c (braid (cat_tensor a b) d)
        $o twist (cat_tensor a b) c d
        $o braid (cat_tensor c d) (cat_tensor a b)
        $o twist a (cat_tensor c d) b
        $o fmap01 cat_tensor a (braid b (cat_tensor c d))
      $== fmap01 cat_tensor c (twist a d b)
        $o fmap01 cat_tensor c (fmap01 cat_tensor a (braid b d))
        $o twist a c (cat_tensor b d)
        $o fmap01 cat_tensor a (twist b c d))
  .

  (** *** Setup *)

  (** Before starting the proofs, we need to setup some useful definitions and helpful lemmas for working with diagrams. *)

  (** We give notations and abbreviations to the morphisms that will appear in diagrams. This helps us read the goal and understand what is happening, otherwise it is too verbose. *)
  Declare Scope monoidal_scope.
  Local Infix "⊗" := cat_tensor (at level 40) : monoidal_scope.
  Local Infix "⊗R" := (fmap01 cat_tensor) (at level 40) : monoidal_scope.
  Local Infix "⊗L" := (fmap10 cat_tensor) (at level 40) : monoidal_scope.
  Local Notation "f ∘ g" := (f $o g) (at level 61, left associativity, format "f  '/' '∘'  g") : monoidal_scope.
  Local Notation "f $== g :> A" := (GpdHom (A := A) f g)
    (at level 80, format "'[v' '[v' f ']' '/'  $== '/'  '[v' g ']'  '/' :>  '[' A ']' ']'")
    : long_path_scope.
  Local Open Scope monoidal_scope.

  (** [twist] is an equivalence which we will call [twiste]. *)
  Local Definition twiste a b c
    : cat_tensor a (cat_tensor b c) $<~> cat_tensor b (cat_tensor a c)
    := cate_adjointify (twist a b c) (twist b a c)
        (twist_twist a b c) (twist_twist b a c).

  (** *** Finer naturality *)

  (** The naturality postulates we have for [twist] are natural in all their arguments similtaneously. We show the finer naturality of [twist] in each argument separately as this becomes more useful in practice. *)

  Local Definition twist_nat_l {a a'} (f : a $-> a') b c
    : twist a' b c $o fmap10 cat_tensor f (cat_tensor b c)
      $== fmap01 cat_tensor b (fmap10 cat_tensor f c) $o twist a b c.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, a': A
f: a $-> a'
b, c: A
twist a' b c ∘ f ⊗L (b ⊗ c) $== b ⊗R (f ⊗L c)
∘ twist a b c
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, a': A
f: a $-> a'
b, c: A
twist a' b c ∘ f ⊗L (b ⊗ c) $== b ⊗R (f ⊗L c)
∘ twist a b c
    refine ((_ $@L _^$) $@ twist_nat a a' b b c c f (Id _) (Id _) $@ (_ $@R _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, a': A
f: a $-> a'
b, c: A
fmap11 cat_tensor f (fmap11 cat_tensor (Id b) (Id c)) $->
f ⊗L (b ⊗ c)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, a': A
f: a $-> a'
b, c: A
fmap11 cat_tensor (Id b) (fmap11 cat_tensor f (Id c)) $==
b ⊗R (f ⊗L c)
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, a': A
f: a $-> a'
b, c: A
fmap11 cat_tensor f (fmap11 cat_tensor (Id b) (Id c)) $->
f ⊗L (b ⊗ c) refine (fmap12 _ _ _ $@ fmap10_is_fmap11 _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, a': A
f: a $-> a'
b, c: A
fmap11 cat_tensor (Id b) (Id c) $== Id (b ⊗ c)
      rapply fmap11_id.
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, a': A
f: a $-> a'
b, c: A
fmap11 cat_tensor (Id b) (fmap11 cat_tensor f (Id c)) $==
b ⊗R (f ⊗L c) refine (fmap12 _ _ _ $@ fmap01_is_fmap11 _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, a': A
f: a $-> a'
b, c: A
fmap11 cat_tensor f (Id c) $== f ⊗L c
      rapply fmap10_is_fmap11.
  Defined.

  Local Definition twist_nat_m a {b b'} (g : b $-> b') c
    : twist a b' c $o fmap01 cat_tensor a (fmap10 cat_tensor g c)
      $== fmap10 cat_tensor g (cat_tensor a c) $o twist a b c.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, b': A
g: b $-> b'
c: A
twist a b' c ∘ a ⊗R (g ⊗L c) $== g ⊗L (a ⊗ c)
∘ twist a b c
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, b': A
g: b $-> b'
c: A
twist a b' c ∘ a ⊗R (g ⊗L c) $== g ⊗L (a ⊗ c)
∘ twist a b c
    refine ((_ $@L _^$) $@ twist_nat a a b b' c c (Id _) g (Id _) $@ (_ $@R _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, b': A
g: b $-> b'
c: A
fmap11 cat_tensor (Id a) (fmap11 cat_tensor g (Id c)) $->
a ⊗R (g ⊗L c)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, b': A
g: b $-> b'
c: A
fmap11 cat_tensor g (fmap11 cat_tensor (Id a) (Id c)) $==
g ⊗L (a ⊗ c)
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, b': A
g: b $-> b'
c: A
fmap11 cat_tensor (Id a) (fmap11 cat_tensor g (Id c)) $->
a ⊗R (g ⊗L c) refine (fmap12 _ _ _ $@ fmap01_is_fmap11 _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, b': A
g: b $-> b'
c: A
fmap11 cat_tensor g (Id c) $== g ⊗L c
      rapply fmap10_is_fmap11.
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, b': A
g: b $-> b'
c: A
fmap11 cat_tensor g (fmap11 cat_tensor (Id a) (Id c)) $==
g ⊗L (a ⊗ c) refine (fmap12 _ _ _ $@ fmap10_is_fmap11 _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, b': A
g: b $-> b'
c: A
fmap11 cat_tensor (Id a) (Id c) $== Id (a ⊗ c)
      rapply fmap11_id.
  Defined.

  Local Definition twist_nat_r a b {c c'} (h : c $-> c')
    : twist a b c' $o fmap01 cat_tensor a (fmap01 cat_tensor b h)
      $== fmap01 cat_tensor b (fmap01 cat_tensor a h) $o twist a b c.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, c': A
h: c $-> c'
twist a b c' ∘ a ⊗R (b ⊗R h) $== b ⊗R (a ⊗R h)
∘ twist a b c
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, c': A
h: c $-> c'
twist a b c' ∘ a ⊗R (b ⊗R h) $== b ⊗R (a ⊗R h)
∘ twist a b c
    refine ((_ $@L _^$) $@ twist_nat a a b b c c' (Id _) (Id _) h $@ (_ $@R _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, c': A
h: c $-> c'
fmap11 cat_tensor (Id a) (fmap11 cat_tensor (Id b) h) $->
a ⊗R (b ⊗R h)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, c': A
h: c $-> c'
fmap11 cat_tensor (Id b) (fmap11 cat_tensor (Id a) h) $==
b ⊗R (a ⊗R h)
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, c': A
h: c $-> c'
fmap11 cat_tensor (Id a) (fmap11 cat_tensor (Id b) h) $->
a ⊗R (b ⊗R h) refine (fmap12 _ _ _ $@ fmap01_is_fmap11 _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, c': A
h: c $-> c'
fmap11 cat_tensor (Id b) h $== b ⊗R h
      rapply fmap01_is_fmap11.
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, c': A
h: c $-> c'
fmap11 cat_tensor (Id b) (fmap11 cat_tensor (Id a) h) $==
b ⊗R (a ⊗R h) refine (fmap12 _ _ _ $@ fmap01_is_fmap11 _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, c': A
h: c $-> c'
fmap11 cat_tensor (Id a) h $== a ⊗R h
      rapply fmap01_is_fmap11.
  Defined.

  (** *** Movement lemmas *)

  (** Here we collect lemmas about moving morphisms around in a diagram. We could have created [cate_moveL_eM]-style lemmas for [CatIsEquiv] but this leads to a lot of unnecessary unfolding and duplication. It is typically easier to use a hand crafted lemma for each situation. *)

  (** TODO: A lot of these proofs are copy and pasted between lemmas. We need to work out an efficient way of proving them. *)

  (** **** Moving [twist] *)

  Local Definition moveL_twistL a b c d f (g : d $-> _)
    : twist a b c $o f $== g -> f $== twist b a c $o g.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: d $-> a ⊗ (b ⊗ c)
g: d $-> b ⊗ (a ⊗ c)
twist a b c ∘ f $== g -> f $== twist b a c ∘ g
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: d $-> a ⊗ (b ⊗ c)
g: d $-> b ⊗ (a ⊗ c)
twist a b c ∘ f $== g -> f $== twist b a c ∘ g
    intros p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: d $-> a ⊗ (b ⊗ c)
g: d $-> b ⊗ (a ⊗ c)
p: twist a b c ∘ f $== g
f $== twist b a c ∘ g
    apply (cate_monic_equiv (twiste a b c)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: d $-> a ⊗ (b ⊗ c)
g: d $-> b ⊗ (a ⊗ c)
p: twist a b c ∘ f $== g
twiste a b c ∘ f $== twiste a b c ∘ (twist b a c ∘ g)
    nrefine ((cate_buildequiv_fun _ $@R _) $@ p $@ _ $@ cat_assoc _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: d $-> a ⊗ (b ⊗ c)
g: d $-> b ⊗ (a ⊗ c)
p: twist a b c ∘ f $== g
g $== twiste a b c ∘ twist b a c ∘ g
    refine ((cat_idl _)^$ $@ (_^$ $@R _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: d $-> a ⊗ (b ⊗ c)
g: d $-> b ⊗ (a ⊗ c)
p: twist a b c ∘ f $== g
twiste a b c ∘ twist b a c $-> Id (b ⊗ (a ⊗ c))
    refine ((cate_buildequiv_fun _ $@R _) $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: d $-> a ⊗ (b ⊗ c)
g: d $-> b ⊗ (a ⊗ c)
p: twist a b c ∘ f $== g
twist a b c ∘ twist b a c $== Id (b ⊗ (a ⊗ c))
    apply twist_twist.
  Defined.

  Local Definition moveL_twistR a b c d f (g : _ $-> d)
    : f $o twist a b c $== g -> f $== g $o twist b a c.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: b ⊗ (a ⊗ c) $-> d
g: a ⊗ (b ⊗ c) $-> d
f ∘ twist a b c $== g -> f $== g ∘ twist b a c
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: b ⊗ (a ⊗ c) $-> d
g: a ⊗ (b ⊗ c) $-> d
f ∘ twist a b c $== g -> f $== g ∘ twist b a c
    intros p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: b ⊗ (a ⊗ c) $-> d
g: a ⊗ (b ⊗ c) $-> d
p: f ∘ twist a b c $== g
f $== g ∘ twist b a c
    apply (cate_epic_equiv (twiste a b c)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: b ⊗ (a ⊗ c) $-> d
g: a ⊗ (b ⊗ c) $-> d
p: f ∘ twist a b c $== g
f ∘ twiste a b c $== g ∘ twist b a c ∘ twiste a b c
    refine (_ $@ (cat_assoc _ _ _)^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: b ⊗ (a ⊗ c) $-> d
g: a ⊗ (b ⊗ c) $-> d
p: f ∘ twist a b c $== g
f ∘ twiste a b c $== g ∘ (twist b a c ∘ twiste a b c)
    refine (_ $@ (_ $@L ((_ $@L cate_buildequiv_fun _) $@ _)^$)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: b ⊗ (a ⊗ c) $-> d
g: a ⊗ (b ⊗ c) $-> d
p: f ∘ twist a b c $== g
f ∘ twiste a b c $== g ∘ ?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: b ⊗ (a ⊗ c) $-> d
g: a ⊗ (b ⊗ c) $-> d
p: f ∘ twist a b c $== g
twist b a c ∘ twist a b c $== ?Goal0
    2: apply twist_twist.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: b ⊗ (a ⊗ c) $-> d
g: a ⊗ (b ⊗ c) $-> d
p: f ∘ twist a b c $== g
f ∘ twiste a b c $== g ∘ Id (a ⊗ (b ⊗ c))
    refine ((_ $@L _) $@ _ $@ (cat_idr _)^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: b ⊗ (a ⊗ c) $-> d
g: a ⊗ (b ⊗ c) $-> d
p: f ∘ twist a b c $== g
twiste a b c $== ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: b ⊗ (a ⊗ c) $-> d
g: a ⊗ (b ⊗ c) $-> d
p: f ∘ twist a b c $== g
f ∘ ?Goal $== g
    1: apply cate_buildequiv_fun.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: b ⊗ (a ⊗ c) $-> d
g: a ⊗ (b ⊗ c) $-> d
p: f ∘ twist a b c $== g
f ∘ twist a b c $== g
    exact p.
  Defined.

  Local Definition moveR_twistL a b c d f (g : d $-> _)
    : f $== twist b a c $o g -> twist a b c $o f $== g.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: d $-> a ⊗ (b ⊗ c)
g: d $-> b ⊗ (a ⊗ c)
f $== twist b a c ∘ g -> twist a b c ∘ f $== g
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: d $-> a ⊗ (b ⊗ c)
g: d $-> b ⊗ (a ⊗ c)
f $== twist b a c ∘ g -> twist a b c ∘ f $== g
    intros p; symmetry; apply moveL_twistL; symmetry; exact p.
  Defined.

  Local Definition moveR_twistR a b c d f (g : _ $-> d)
    : f $== g $o twist b a c -> f $o twist a b c $== g.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: b ⊗ (a ⊗ c) $-> d
g: a ⊗ (b ⊗ c) $-> d
f $== g ∘ twist b a c -> f ∘ twist a b c $== g
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
f: b ⊗ (a ⊗ c) $-> d
g: a ⊗ (b ⊗ c) $-> d
f $== g ∘ twist b a c -> f ∘ twist a b c $== g
    intros p; symmetry; apply moveL_twistR; symmetry; exact p.
  Defined.

  Local Definition moveL_fmap01_twistL a b c d e f (g : e $-> _)
    : fmap01 cat_tensor a (twist b c d) $o f $== g
      -> f $== fmap01 cat_tensor a (twist c b d) $o g.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
a ⊗R twist b c d ∘ f $== g ->
f $== a ⊗R twist c b d ∘ g
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
a ⊗R twist b c d ∘ f $== g ->
f $== a ⊗R twist c b d ∘ g
    intros p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
f $== a ⊗R twist c b d ∘ g
    apply (cate_monic_equiv (emap01 cat_tensor a (twiste b c d))).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
emap01 cat_tensor a (twiste b c d) ∘ f $==
emap01 cat_tensor a (twiste b c d)
∘ (a ⊗R twist c b d ∘ g)
    refine (_ $@ cat_assoc _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
emap01 cat_tensor a (twiste b c d) ∘ f $==
emap01 cat_tensor a (twiste b c d) ∘ a ⊗R twist c b d
∘ g
    refine (_ $@ (_ $@R _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
emap01 cat_tensor a (twiste b c d) ∘ f $== ?Goal0 ∘ g
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
?Goal0 $== emap01 cat_tensor a (twiste b c d)
∘ a ⊗R twist c b d
    2: {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
?Goal0 $== emap01 cat_tensor a (twiste b c d)
∘ a ⊗R twist c b d refine (_ $@ (_^$ $@R _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
?Goal0 $== ?Goal2 ∘ a ⊗R twist c b d
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
emap01 cat_tensor a (twiste b c d) $-> ?Goal2
      2: apply cate_buildequiv_fun.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
?Goal0 $== a ⊗R twiste b c d ∘ a ⊗R twist c b d
      refine ((fmap_id _ _)^$ $@ fmap2 _ _ $@ fmap_comp _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
Id (c ⊗ (b ⊗ d)) $== twiste b c d ∘ twist c b d
      refine (_^$ $@ (_^$ $@R _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
?Goal1 ∘ twist c b d $-> Id (c ⊗ (b ⊗ d))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
twiste b c d $-> ?Goal1
      2: apply cate_buildequiv_fun.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
twist b c d ∘ twist c b d $-> Id (c ⊗ (b ⊗ d))
      apply twist_twist. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
emap01 cat_tensor a (twiste b c d) ∘ f $==
Id (a ⊗ (c ⊗ (b ⊗ d))) ∘ g
    refine ((_ $@R _) $@ p $@ (cat_idl _)^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
emap01 cat_tensor a (twiste b c d) $==
a ⊗R twist b c d
    refine (cate_buildequiv_fun _ $@ fmap02 _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
p: a ⊗R twist b c d ∘ f $== g
twiste b c d $== twist b c d
    apply cate_buildequiv_fun.
  Defined.

  Local Definition moveL_fmap01_twistR a b c d e f (g : _ $-> e)
    : f $o fmap01 cat_tensor a (twist b c d) $== g
      -> f $== g $o fmap01 cat_tensor a (twist c b d).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
f ∘ a ⊗R twist b c d $== g ->
f $== g ∘ a ⊗R twist c b d
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
f ∘ a ⊗R twist b c d $== g ->
f $== g ∘ a ⊗R twist c b d
    intros p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
f $== g ∘ a ⊗R twist c b d
    apply (cate_epic_equiv (emap01 cat_tensor a (twiste b c d))).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
f ∘ emap01 cat_tensor a (twiste b c d) $== g
∘ a ⊗R twist c b d
∘ emap01 cat_tensor a (twiste b c d)
    refine (_ $@ (cat_assoc _ _ _)^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
f ∘ emap01 cat_tensor a (twiste b c d) $== g
∘ (a ⊗R twist c b d
   ∘ emap01 cat_tensor a (twiste b c d))
    refine (_ $@ (_ $@L _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
f ∘ emap01 cat_tensor a (twiste b c d) $== g ∘ ?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
?Goal0 $== a ⊗R twist c b d
∘ emap01 cat_tensor a (twiste b c d)
    2: {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
?Goal0 $== a ⊗R twist c b d
∘ emap01 cat_tensor a (twiste b c d) refine (_^$ $@ (_ $@L _^$)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
a ⊗R twist c b d ∘ ?Goal2 $-> ?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
emap01 cat_tensor a (twiste b c d) $-> ?Goal2
      2: apply cate_buildequiv_fun.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
a ⊗R twist c b d ∘ a ⊗R twiste b c d $-> ?Goal0
      refine ((fmap_comp _ _ _)^$ $@ fmap2 _ _ $@ fmap_id _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
twist c b d ∘ twiste b c d $== Id (b ⊗ (c ⊗ d))
      refine ((_ $@L _) $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
twiste b c d $== ?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
twist c b d ∘ ?Goal0 $== Id (b ⊗ (c ⊗ d))
      1: apply cate_buildequiv_fun.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
twist c b d ∘ twist b c d $== Id (b ⊗ (c ⊗ d))
      apply twist_twist. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
f ∘ emap01 cat_tensor a (twiste b c d) $== g
∘ Id (a ⊗ (b ⊗ (c ⊗ d)))
    refine ((_ $@L _) $@ p $@ (cat_idr _)^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
emap01 cat_tensor a (twiste b c d) $==
a ⊗R twist b c d
    refine (cate_buildequiv_fun _ $@ fmap02 _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
p: f ∘ a ⊗R twist b c d $== g
twiste b c d $== twist b c d
    apply cate_buildequiv_fun.
  Defined.

  Local Definition moveR_fmap01_twistL a b c d e f (g : e $-> _)
    : f $== fmap01 cat_tensor a (twist c b d) $o g
      -> fmap01 cat_tensor a (twist b c d) $o f $== g.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
f $== a ⊗R twist c b d ∘ g ->
a ⊗R twist b c d ∘ f $== g
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: e $-> a ⊗ (b ⊗ (c ⊗ d))
g: e $-> a ⊗ (c ⊗ (b ⊗ d))
f $== a ⊗R twist c b d ∘ g ->
a ⊗R twist b c d ∘ f $== g
    intros p; symmetry; apply moveL_fmap01_twistL; symmetry; exact p.
  Defined.

  Local Definition moveR_fmap01_twistR a b c d e f (g : _ $-> e)
    : f $== g $o fmap01 cat_tensor a (twist c b d)
      -> f $o fmap01 cat_tensor a (twist b c d) $== g.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
f $== g ∘ a ⊗R twist c b d ->
f ∘ a ⊗R twist b c d $== g
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d, e: A
f: a ⊗ (c ⊗ (b ⊗ d)) $-> e
g: a ⊗ (b ⊗ (c ⊗ d)) $-> e
f $== g ∘ a ⊗R twist c b d ->
f ∘ a ⊗R twist b c d $== g
    intros p; symmetry; apply moveL_fmap01_twistR; symmetry; exact p.
  Defined.

  (** *** The associator *)

  (** Using [braide] and [twiste] we can build an associator. *)
  Local Definition associator_twist' a b c
    : cat_tensor a (cat_tensor b c) $<~> cat_tensor (cat_tensor a b) c.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
a ⊗ (b ⊗ c) $<~> (a ⊗ b) ⊗ c
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
a ⊗ (b ⊗ c) $<~> (a ⊗ b) ⊗ c
    (** We can build the associator out of [braide] and [twiste]. *)
    refine (braide _ _ $oE _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
a ⊗ (b ⊗ c) $<~> c ⊗ (a ⊗ b)
    nrefine (twiste _ _ _ $oE _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
a ⊗ (b ⊗ c) $<~> a ⊗ (c ⊗ b)
    exact (emap01 cat_tensor a (braide _ _)).
  Defined.

  (** We would like to be able to unfold [associator_twist'] to the underlying morphisms. We use this lemma to make that process easier. *)
  Local Definition associator_twist'_unfold a b c
    : cate_fun (associator_twist' a b c)
    $== braid c (cat_tensor a b) $o (twist a c b $o fmap01 cat_tensor a (braid b c)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
associator_twist' a b c $== braid c (a ⊗ b)
∘ (twist a c b ∘ a ⊗R braid b c)
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
associator_twist' a b c $== braid c (a ⊗ b)
∘ (twist a c b ∘ a ⊗R braid b c)
    refine (cate_buildequiv_fun _ $@ (_ $@@ cate_buildequiv_fun _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
twiste a c b $oE emap01 cat_tensor a (braide b c) $==
twist a c b ∘ a ⊗R braid b c
    nrefine (cate_buildequiv_fun _ $@ (_ $@@ cate_buildequiv_fun _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
emap01 cat_tensor a (braide b c) $== a ⊗R braid b c
    refine (cate_buildequiv_fun _ $@ fmap2 _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
braide b c $== braid b c
    apply cate_buildequiv_fun.
  Defined.

  (** Now we can use [associator_twist'] and show that it is a natural equivalence in each variable. *)
  Local Instance associator_twist : Associator cat_tensor.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
Associator cat_tensor
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
Associator cat_tensor
    snrapply Build_Associator.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
forall a b c : A, a ⊗ (b ⊗ c) $<~> (a ⊗ b) ⊗ c
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
Is1Natural
  (fun pat : A * A * A =>
   let x := pat in
   let fst := fst x in
   let c := snd x in
   let fst0 := fst in
   let a := Overture.fst fst0 in
   let b := snd fst0 in a ⊗ (b ⊗ c))
  (fun pat : A * A * A =>
   let x := pat in
   let fst := fst x in
   let c := snd x in
   let fst0 := fst in
   let a := Overture.fst fst0 in
   let b := snd fst0 in (a ⊗ b) ⊗ c)
  (fun pat : A * A * A =>
   let x := pat in
   let fst := fst x in
   let c := snd x in
   let fst0 := fst in
   let a := Overture.fst fst0 in
   let b := snd fst0 in ?associator a b c)
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
forall a b c : A, a ⊗ (b ⊗ c) $<~> (a ⊗ b) ⊗ c exact associator_twist'.
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
Is1Natural
  (fun pat : A * A * A =>
   let x := pat in
   let fst := fst x in
   let c := snd x in
   let fst0 := fst in
   let a := Overture.fst fst0 in
   let b := snd fst0 in a ⊗ (b ⊗ c))
  (fun pat : A * A * A =>
   let x := pat in
   let fst := fst x in
   let c := snd x in
   let fst0 := fst in
   let a := Overture.fst fst0 in
   let b := snd fst0 in (a ⊗ b) ⊗ c)
  (fun pat : A * A * A =>
   let x := pat in
   let fst := fst x in
   let c := snd x in
   let fst0 := fst in
   let a := Overture.fst fst0 in
   let b := snd fst0 in associator_twist' a b c) simpl; intros [[a b] c] [[a' b'] c'] [[f g] h]; simpl in f, g, h.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
associator_twist' a' b' c'
∘ fmap
    (uncurry (uncurry (fun x y z : A => x ⊗ (y ⊗ z))))
    (f, g, h) $==
fmap
  (uncurry (uncurry (fun x y z : A => (x ⊗ y) ⊗ z)))
  (f, g, h) ∘ associator_twist' a b c
      (** To prove naturality it will be easier to reason about squares. *)
      change (?w $o ?x $== ?y $o ?z) with (Square z w x y).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square (associator_twist' a b c)
  (associator_twist' a' b' c')
  (fmap
     (uncurry (uncurry (fun x y z : A => x ⊗ (y ⊗ z))))
     (f, g, h))
  (fmap
     (uncurry (uncurry (fun x y z : A => (x ⊗ y) ⊗ z)))
     (f, g, h))
      (** First we remove all the equivalences from the equation. *)
      nrapply hconcatL.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
associator_twist' a b c $== ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square ?Goal (associator_twist' a' b' c')
  (fmap
     (uncurry (uncurry (fun x y z : A => x ⊗ (y ⊗ z))))
     (f, g, h))
  (fmap
     (uncurry (uncurry (fun x y z : A => (x ⊗ y) ⊗ z)))
     (f, g, h))
      1: apply associator_twist'_unfold.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square
  (braid c (a ⊗ b) ∘ (twist a c b ∘ a ⊗R braid b c))
  (associator_twist' a' b' c')
  (fmap
     (uncurry (uncurry (fun x y z : A => x ⊗ (y ⊗ z))))
     (f, g, h))
  (fmap
     (uncurry (uncurry (fun x y z : A => (x ⊗ y) ⊗ z)))
     (f, g, h))
      nrapply hconcatR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square
  (braid c (a ⊗ b) ∘ (twist a c b ∘ a ⊗R braid b c))
  ?Goal
  (fmap
     (uncurry (uncurry (fun x y z : A => x ⊗ (y ⊗ z))))
     (f, g, h))
  (fmap
     (uncurry (uncurry (fun x y z : A => (x ⊗ y) ⊗ z)))
     (f, g, h))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
associator_twist' a' b' c' $== ?Goal
      2: apply associator_twist'_unfold.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square
  (braid c (a ⊗ b) ∘ (twist a c b ∘ a ⊗R braid b c))
  (braid c' (a' ⊗ b')
   ∘ (twist a' c' b' ∘ a' ⊗R braid b' c'))
  (fmap
     (uncurry (uncurry (fun x y z : A => x ⊗ (y ⊗ z))))
     (f, g, h))
  (fmap
     (uncurry (uncurry (fun x y z : A => (x ⊗ y) ⊗ z)))
     (f, g, h))
      (** The first square involving [braid] on its own is a naturality square. *)
      nrapply vconcat.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square (twist a c b ∘ a ⊗R braid b c)
  (twist a' c' b' ∘ a' ⊗R braid b' c')
  (fmap
     (uncurry (uncurry (fun x y z : A => x ⊗ (y ⊗ z))))
     (f, g, h)) ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square (braid c (a ⊗ b)) 
  (braid c' (a' ⊗ b')) 
  ?Goal
  (fmap
     (uncurry (uncurry (fun x y z : A => (x ⊗ y) ⊗ z)))
     (f, g, h))
      2: rapply braid_nat.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square (twist a c b ∘ a ⊗R braid b c)
  (twist a' c' b' ∘ a' ⊗R braid b' c')
  (fmap
     (uncurry (uncurry (fun x y z : A => x ⊗ (y ⊗ z))))
     (f, g, h))
  (fmap11 cat_tensor (fmap idmap (snd (f, g, h)))
     (fmap (uncurry cat_tensor) (fst (f, g, h))))
      (** The second square is just the naturality of twist. *)
      nrapply vconcat.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square (a ⊗R braid b c) (a' ⊗R braid b' c')
  (fmap
     (uncurry (uncurry (fun x y z : A => x ⊗ (y ⊗ z))))
     (f, g, h)) ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square (twist a c b) (twist a' c' b') 
  ?Goal
  (fmap11 cat_tensor (fmap idmap h)
     (fmap (uncurry cat_tensor) (f, g)))
      2: apply twist_nat.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square (a ⊗R braid b c) (a' ⊗R braid b' c')
  (fmap
     (uncurry (uncurry (fun x y z : A => x ⊗ (y ⊗ z))))
     (f, g, h))
  (fmap11 cat_tensor f
     (fmap11 cat_tensor (fmap idmap h) g))
      nrapply hconcatL.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
a ⊗R braid b c $== ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square ?Goal (a' ⊗R braid b' c')
  (fmap
     (uncurry (uncurry (fun x y z : A => x ⊗ (y ⊗ z))))
     (f, g, h))
  (fmap11 cat_tensor f
     (fmap11 cat_tensor (fmap idmap h) g))
      2: nrapply hconcatR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
a ⊗R braid b c $== ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square ?Goal ?Goal1
  (fmap
     (uncurry (uncurry (fun x y z : A => x ⊗ (y ⊗ z))))
     (f, g, h))
  (fmap11 cat_tensor f
     (fmap11 cat_tensor (fmap idmap h) g))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
a' ⊗R braid b' c' $== ?Goal1
      1,3: symmetry; rapply fmap01_is_fmap11.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square (fmap11 cat_tensor (Id a) (braid b c))
  (fmap11 cat_tensor (Id a') (braid b' c'))
  (fmap
     (uncurry (uncurry (fun x y z : A => x ⊗ (y ⊗ z))))
     (f, g, h))
  (fmap11 cat_tensor f
     (fmap11 cat_tensor (fmap idmap h) g))
      (** Leaving us with a square with a functor application. *)
      rapply fmap11_square.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square (Id a) (Id a') f f
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square (braid b c) (braid b' c')
  (fmap11 cat_tensor g h)
  (fmap11 cat_tensor (fmap idmap h) g)
      1: rapply vrefl.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, a', b', c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
Square (braid b c) (braid b' c')
  (fmap11 cat_tensor g h)
  (fmap11 cat_tensor (fmap idmap h) g)
      (** We are finally left with the naturality of braid. *)
      apply braid_nat.
  Defined.

  (** We abbreviate the associator to [α] for the remainder of the section. *)
  Local Notation α := associator_twist.

  (** *** Unitors *)

  (** Since we assume the [right_unitor] exists, we can derive the [left_unitor] from it together with [braid]. *)
  Local Instance left_unitor_twist : LeftUnitor cat_tensor cat_tensor_unit.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
LeftUnitor cat_tensor cat_tensor_unit
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
LeftUnitor cat_tensor cat_tensor_unit
    snrapply Build_NatEquiv'.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
NatTrans (cat_tensor cat_tensor_unit) idmap
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
forall a : A, CatIsEquiv (?alpha a)
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
NatTrans (cat_tensor cat_tensor_unit) idmap snrapply Build_NatTrans.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
cat_tensor cat_tensor_unit $=> idmap
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
Is1Natural (cat_tensor cat_tensor_unit) idmap ?alpha
      +
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
cat_tensor cat_tensor_unit $=> idmap exact (fun a => right_unitor a $o braid cat_tensor_unit a).
      +
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
Is1Natural (cat_tensor cat_tensor_unit) idmap
  (fun a : A => right_unitor a
   ∘ braid cat_tensor_unit a) intros a b f.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b: A
f: a $-> b
right_unitor b ∘ braid cat_tensor_unit b
∘ fmap (cat_tensor cat_tensor_unit) f $== fmap idmap f
∘ (right_unitor a ∘ braid cat_tensor_unit a)
        change (?w $o ?x $== ?y $o ?z) with (Square z w x y).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b: A
f: a $-> b
Square (right_unitor a ∘ braid cat_tensor_unit a)
  (right_unitor b ∘ braid cat_tensor_unit b)
  (fmap (cat_tensor cat_tensor_unit) f) (fmap idmap f)
        nrapply vconcat.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b: A
f: a $-> b
Square (braid cat_tensor_unit a)
  (braid cat_tensor_unit b)
  (fmap (cat_tensor cat_tensor_unit) f) ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b: A
f: a $-> b
Square (right_unitor a) 
  (right_unitor b) ?Goal 
  (fmap idmap f)
        2: rapply (isnat right_unitor f).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b: A
f: a $-> b
Square (braid cat_tensor_unit a)
  (braid cat_tensor_unit b)
  (fmap (cat_tensor cat_tensor_unit) f)
  (fmap (flip cat_tensor cat_tensor_unit) f)
        rapply braid_nat_r.
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
forall a : A,
CatIsEquiv
  ({|
     trans_nattrans :=
       fun a0 : A => right_unitor a0
       ∘ braid cat_tensor_unit a0;
     is1natural_nattrans :=
       (fun (a0 b : A) (f : a0 $-> b) =>
        braid_nat_r f $@v isnat right_unitor f)
       :
       Is1Natural (cat_tensor cat_tensor_unit) idmap
         (fun a0 : A => right_unitor a0
          ∘ braid cat_tensor_unit a0)
   |} a) intros a.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a: A
CatIsEquiv
  ({|
     trans_nattrans :=
       fun a : A => right_unitor a
       ∘ braid cat_tensor_unit a;
     is1natural_nattrans :=
       (fun (a b : A) (f : a $-> b) =>
        braid_nat_r f $@v isnat right_unitor f)
       :
       Is1Natural (cat_tensor cat_tensor_unit) idmap
         (fun a : A => right_unitor a
          ∘ braid cat_tensor_unit a)
   |} a)
      rapply compose_catie'.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a: A
CatIsEquiv (braid cat_tensor_unit a)
      rapply catie_braid.
  Defined.

  (** *** Triangle *)

  (** The triangle identity can easily be proven by rearranging the diagram, cancelling and using naturality of [braid]. *)
  Local Instance triangle_twist : TriangleIdentity cat_tensor cat_tensor_unit.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
TriangleIdentity cat_tensor cat_tensor_unit
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
TriangleIdentity cat_tensor cat_tensor_unit
    intros a b.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b: A
a ⊗R left_unitor_twist b $== right_unitor a ⊗L b
∘ α a cat_tensor_unit b
    refine (_ $@ (_ $@L _)^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b: A
a ⊗R left_unitor_twist b $== right_unitor a ⊗L b
∘ ?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b: A
α a cat_tensor_unit b $== ?Goal0
    2: apply associator_twist'_unfold.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b: A
a ⊗R left_unitor_twist b $== right_unitor a ⊗L b
∘ (braid b (a ⊗ cat_tensor_unit)
   ∘ (twist a b cat_tensor_unit
      ∘ a ⊗R braid cat_tensor_unit b))
    refine (fmap02 _ a (cate_buildequiv_fun _) $@ _); cbn.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b: A
a ⊗R (right_unitor b ∘ braid cat_tensor_unit b) $==
right_unitor a ⊗L b
∘ (braid b (a ⊗ cat_tensor_unit)
   ∘ (twist a b cat_tensor_unit
      ∘ a ⊗R braid cat_tensor_unit b))
    refine (fmap01_comp _ _ _ _ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b: A
a ⊗R right_unitor b ∘ a ⊗R braid cat_tensor_unit b $==
right_unitor a ⊗L b
∘ (braid b (a ⊗ cat_tensor_unit)
   ∘ (twist a b cat_tensor_unit
      ∘ a ⊗R braid cat_tensor_unit b))
    do 2 refine (_ $@ cat_assoc _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b: A
a ⊗R right_unitor b ∘ a ⊗R braid cat_tensor_unit b $==
right_unitor a ⊗L b ∘ braid b (a ⊗ cat_tensor_unit)
∘ twist a b cat_tensor_unit
∘ a ⊗R braid cat_tensor_unit b
    refine ((twist_unitor _ _ $@ (_ $@R _)) $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b: A
braid b a ∘ b ⊗R right_unitor a $==
right_unitor a ⊗L b ∘ braid b (a ⊗ cat_tensor_unit)
    apply braid_nat_r.
  Defined.

  (** *** Pentagon *)

  Local Open Scope long_path_scope.

  Local Instance pentagon_twist : PentagonIdentity cat_tensor.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b
 $==
 braid b a
 ∘ b ⊗R right_unitor a
 ∘ twist a b cat_tensor_unit
:> (a
    ⊗ flip cat_tensor cat_tensor_unit b $->
    a ⊗ b)
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
PentagonIdentity cat_tensor
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b
 $==
 braid b a
 ∘ b ⊗R right_unitor a
 ∘ twist a b cat_tensor_unit
:> (a
    ⊗ flip cat_tensor cat_tensor_unit b $->
    a ⊗ b)
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
PentagonIdentity cat_tensor
    clear twist_unitor right_unitor cat_tensor_unit.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
PentagonIdentity cat_tensor
    intros a b c d.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
α (a ⊗ b) c d
∘ α a b (c ⊗ d)
 $==
 α a b c ⊗L d
 ∘ α a (b ⊗ c) d
 ∘ a ⊗R α b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine ((_ $@@ _) $@ _ $@ ((fmap02 _ _ _ $@ _)^$ $@@ (_ $@@ (fmap20 _ _ _ $@ _))^$)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
α a b (c ⊗ d)
 $==
 ?Goal
:> (a ⊗ (b ⊗ (c ⊗ d)) $-> (a ⊗ b) ⊗ (c ⊗ d))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
α (a ⊗ b) c d
 $==
 ?Goal0
:> ((a ⊗ b) ⊗ (c ⊗ d) $-> ((a ⊗ b) ⊗ c) ⊗ d)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
?Goal0
∘ ?Goal
 $==
 ?Goal9
 ∘ ?Goal8
 ∘ ?Goal4
:> (a ⊗ (b ⊗ (c ⊗ d)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
α b c d
 $==
 ?Goal5
:> (b ⊗ (c ⊗ d) $-> (b ⊗ c) ⊗ d)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
a ⊗R ?Goal5
 $==
 ?Goal4
:> (a ⊗ (b ⊗ (c ⊗ d)) $-> a ⊗ ((b ⊗ c) ⊗ d))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
α a (b ⊗ c) d
 $==
 ?Goal8
:> (a ⊗ ((b ⊗ c) ⊗ d) $-> (a ⊗ (b ⊗ c)) ⊗ d)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
α a b c
 $==
 ?Goal11
:> (a ⊗ (b ⊗ c) $-> (a ⊗ b) ⊗ c)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
?Goal11 ⊗L d
 $==
 ?Goal9
:> ((a ⊗ (b ⊗ c)) ⊗ d $-> ((a ⊗ b) ⊗ c) ⊗ d)
    1,2,4,6,7: apply associator_twist'_unfold.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ (twist (a ⊗ b) d c ∘ (a ⊗ b) ⊗R braid c d)
∘ (braid (c ⊗ d) (a ⊗ b)
   ∘ (twist a (c ⊗ d) b ∘ a ⊗R braid b (c ⊗ d)))
 $==
 ?Goal2
 ∘ (braid d (a ⊗ (b ⊗ c))
    ∘ (twist a d (b ⊗ c) ∘ a ⊗R braid (b ⊗ c) d))
 ∘ ?Goal0
:> (a ⊗ (b ⊗ (c ⊗ d)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
a
⊗R (braid d (b ⊗ c) ∘ (twist b d c ∘ b ⊗R braid c d))
 $==
 ?Goal0
:> (a ⊗ (b ⊗ (c ⊗ d)) $-> a ⊗ ((b ⊗ c) ⊗ d))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
(braid c (a ⊗ b) ∘ (twist a c b ∘ a ⊗R braid b c))
⊗L d
 $==
 ?Goal2
:> ((a ⊗ (b ⊗ c)) ⊗ d $-> ((a ⊗ b) ⊗ c) ⊗ d)
    2: refine (fmap01_comp _ _ _ _ $@ (_ $@L (fmap01_comp _ _ _ _))).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ (twist (a ⊗ b) d c ∘ (a ⊗ b) ⊗R braid c d)
∘ (braid (c ⊗ d) (a ⊗ b)
   ∘ (twist a (c ⊗ d) b ∘ a ⊗R braid b (c ⊗ d)))
 $==
 ?Goal0
 ∘ (braid d (a ⊗ (b ⊗ c))
    ∘ (twist a d (b ⊗ c) ∘ a ⊗R braid (b ⊗ c) d))
 ∘ (a ⊗R braid d (b ⊗ c)
    ∘ (a ⊗R twist b d c ∘ a ⊗R (b ⊗R braid c d)))
:> (a ⊗ (b ⊗ (c ⊗ d)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
(braid c (a ⊗ b) ∘ (twist a c b ∘ a ⊗R braid b c))
⊗L d
 $==
 ?Goal0
:> ((a ⊗ (b ⊗ c)) ⊗ d $-> ((a ⊗ b) ⊗ c) ⊗ d)
    2: refine (fmap10_comp _ _ _ _ $@ (_ $@L (fmap10_comp _ _ _ _))).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ (twist (a ⊗ b) d c ∘ (a ⊗ b) ⊗R braid c d)
∘ (braid (c ⊗ d) (a ⊗ b)
   ∘ (twist a (c ⊗ d) b ∘ a ⊗R braid b (c ⊗ d)))
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ (twist a c b ⊗L d ∘ (a ⊗R braid b c) ⊗L d)
 ∘ (braid d (a ⊗ (b ⊗ c))
    ∘ (twist a d (b ⊗ c) ∘ a ⊗R braid (b ⊗ c) d))
 ∘ (a ⊗R braid d (b ⊗ c)
    ∘ (a ⊗R twist b d c ∘ a ⊗R (b ⊗R braid c d)))
:> (a ⊗ (b ⊗ (c ⊗ d)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    (** We use a notation defined above that shows the base type of the groupoid hom and formats the equation in a way that is easier to read. *)
    (** Normalize brackets on LHS *)
    refine (cat_assoc _ _ _ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ (twist (a ⊗ b) d c ∘ (a ⊗ b) ⊗R braid c d
   ∘ (braid (c ⊗ d) (a ⊗ b)
      ∘ (twist a (c ⊗ d) b ∘ a ⊗R braid b (c ⊗ d))))
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ (twist a c b ⊗L d ∘ (a ⊗R braid b c) ⊗L d)
 ∘ (braid d (a ⊗ (b ⊗ c))
    ∘ (twist a d (b ⊗ c) ∘ a ⊗R braid (b ⊗ c) d))
 ∘ (a ⊗R braid d (b ⊗ c)
    ∘ (a ⊗R twist b d c ∘ a ⊗R (b ⊗R braid c d)))
:> (a ⊗ (b ⊗ (c ⊗ d)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine (_ $@L (cat_assoc _ _ _) $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ (twist (a ⊗ b) d c
   ∘ ((a ⊗ b) ⊗R braid c d
      ∘ (braid (c ⊗ d) (a ⊗ b)
         ∘ (twist a (c ⊗ d) b ∘ a ⊗R braid b (c ⊗ d)))))
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ (twist a c b ⊗L d ∘ (a ⊗R braid b c) ⊗L d)
 ∘ (braid d (a ⊗ (b ⊗ c))
    ∘ (twist a d (b ⊗ c) ∘ a ⊗R braid (b ⊗ c) d))
 ∘ (a ⊗R braid d (b ⊗ c)
    ∘ (a ⊗R twist b d c ∘ a ⊗R (b ⊗R braid c d)))
:> (a ⊗ (b ⊗ (c ⊗ d)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    do 4 refine ((cat_assoc _ _ _)^$ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ (twist a c b ⊗L d ∘ (a ⊗R braid b c) ⊗L d)
 ∘ (braid d (a ⊗ (b ⊗ c))
    ∘ (twist a d (b ⊗ c) ∘ a ⊗R braid (b ⊗ c) d))
 ∘ (a ⊗R braid d (b ⊗ c)
    ∘ (a ⊗R twist b d c ∘ a ⊗R (b ⊗R braid c d)))
:> (a ⊗ (b ⊗ (c ⊗ d)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    (** Normalize brackets on RHS *)
    refine (_ $@ (((cat_assoc _ _ _) $@R _) $@R _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ twist a c b ⊗L d
 ∘ (a ⊗R braid b c) ⊗L d
 ∘ (braid d (a ⊗ (b ⊗ c))
    ∘ (twist a d (b ⊗ c) ∘ a ⊗R braid (b ⊗ c) d))
 ∘ (a ⊗R braid d (b ⊗ c)
    ∘ (a ⊗R twist b d c ∘ a ⊗R (b ⊗R braid c d)))
:> (a ⊗ (b ⊗ (c ⊗ d)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    do 2 refine (_ $@ ((cat_assoc _ _ _) $@R _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ twist a c b ⊗L d
 ∘ (a ⊗R braid b c) ⊗L d
 ∘ braid d (a ⊗ (b ⊗ c))
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R braid (b ⊗ c) d
 ∘ (a ⊗R braid d (b ⊗ c)
    ∘ (a ⊗R twist b d c ∘ a ⊗R (b ⊗R braid c d)))
:> (a ⊗ (b ⊗ (c ⊗ d)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    do 2 refine (_ $@ cat_assoc _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ twist a c b ⊗L d
 ∘ (a ⊗R braid b c) ⊗L d
 ∘ braid d (a ⊗ (b ⊗ c))
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R braid (b ⊗ c) d
 ∘ a ⊗R braid d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R (b ⊗R braid c d)
:> (a ⊗ (b ⊗ (c ⊗ d)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    (** Cancel two braids next to eachother. *)
    apply moveL_fmap01_fmap01_braidR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ twist a c b ⊗L d
 ∘ (a ⊗R braid b c) ⊗L d
 ∘ braid d (a ⊗ (b ⊗ c))
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R braid (b ⊗ c) d
 ∘ a ⊗R braid d (b ⊗ c)
 ∘ a ⊗R twist b d c
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    apply moveL_fmap01_twistR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ twist a c b ⊗L d
 ∘ (a ⊗R braid b c) ⊗L d
 ∘ braid d (a ⊗ (b ⊗ c))
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R braid (b ⊗ c) d
 ∘ a ⊗R braid d (b ⊗ c)
:> (a ⊗ (d ⊗ (b ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine (_ $@ (cat_assoc _ _ _)^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ twist a c b ⊗L d
 ∘ (a ⊗R braid b c) ⊗L d
 ∘ braid d (a ⊗ (b ⊗ c))
 ∘ twist a d (b ⊗ c)
 ∘ (a ⊗R braid (b ⊗ c) d ∘ a ⊗R braid d (b ⊗ c))
:> (a ⊗ (d ⊗ (b ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine (_ $@ ((_ $@L _) $@ cat_idr _)^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ twist a c b ⊗L d
 ∘ (a ⊗R braid b c) ⊗L d
 ∘ braid d (a ⊗ (b ⊗ c))
 ∘ twist a d (b ⊗ c)
:> (a ⊗ (d ⊗ (b ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
a ⊗R braid (b ⊗ c) d
∘ a ⊗R braid d (b ⊗ c)
 $==
 Id (a ⊗ (d ⊗ (b ⊗ c)))
:> (a ⊗ (d ⊗ (b ⊗ c)) $-> a ⊗ (d ⊗ (b ⊗ c)))
    2: refine ((fmap01_comp _ _ _ _)^$ $@ fmap02 _ _ _ $@ fmap01_id _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ twist a c b ⊗L d
 ∘ (a ⊗R braid b c) ⊗L d
 ∘ braid d (a ⊗ (b ⊗ c))
 ∘ twist a d (b ⊗ c)
:> (a ⊗ (d ⊗ (b ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid (b ⊗ c) d
∘ braid d (b ⊗ c)
 $==
 Id (d ⊗ (b ⊗ c))
:> (d ⊗ (b ⊗ c) $-> d ⊗ (b ⊗ c))
    2: apply braid_braid.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ twist a c b ⊗L d
 ∘ (a ⊗R braid b c) ⊗L d
 ∘ braid d (a ⊗ (b ⊗ c))
 ∘ twist a d (b ⊗ c)
:> (a ⊗ (d ⊗ (b ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    (** *)
    apply moveL_twistR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ twist a c b ⊗L d
 ∘ (a ⊗R braid b c) ⊗L d
 ∘ braid d (a ⊗ (b ⊗ c))
:> (d ⊗ (a ⊗ (b ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine (_ $@ (cat_assoc _ _ _)^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ twist a c b ⊗L d
 ∘ ((a ⊗R braid b c) ⊗L d ∘ braid d (a ⊗ (b ⊗ c)))
:> (d ⊗ (a ⊗ (b ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine (_ $@ (_ $@L _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ twist a c b ⊗L d
 ∘ ?Goal0
:> (d ⊗ (a ⊗ (b ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
?Goal0
 $==
 (a ⊗R braid b c) ⊗L d
 ∘ braid d (a ⊗ (b ⊗ c))
:> (d ⊗ (a ⊗ (b ⊗ c)) $-> (a ⊗ (c ⊗ b)) ⊗ d)
    2: apply braid_nat_r.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ twist a c b ⊗L d
 ∘ (braid d (a ⊗ (c ⊗ b)) ∘ d ⊗R (a ⊗R braid b c))
:> (d ⊗ (a ⊗ (b ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine (_ $@ cat_assoc _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ twist a c b ⊗L d
 ∘ braid d (a ⊗ (c ⊗ b))
 ∘ d ⊗R (a ⊗R braid b c)
:> (d ⊗ (a ⊗ (b ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    apply moveL_fmap01_fmap01_braidR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
∘ d ⊗R (a ⊗R braid c b)
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ twist a c b ⊗L d
 ∘ braid d (a ⊗ (c ⊗ b))
:> (d ⊗ (a ⊗ (c ⊗ b)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine (_ $@ (cat_assoc _ _ _)^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
∘ d ⊗R (a ⊗R braid c b)
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ (twist a c b ⊗L d ∘ braid d (a ⊗ (c ⊗ b)))
:> (d ⊗ (a ⊗ (c ⊗ b)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine (_ $@ (_ $@L _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
∘ d ⊗R (a ⊗R braid c b)
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ ?Goal0
:> (d ⊗ (a ⊗ (c ⊗ b)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
?Goal0
 $==
 twist a c b ⊗L d
 ∘ braid d (a ⊗ (c ⊗ b))
:> (d ⊗ (a ⊗ (c ⊗ b)) $-> (c ⊗ (a ⊗ b)) ⊗ d)
    2: apply braid_nat_r.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
∘ d ⊗R (a ⊗R braid c b)
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ (braid d (c ⊗ (a ⊗ b)) ∘ d ⊗R twist a c b)
:> (d ⊗ (a ⊗ (c ⊗ b)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine (_ $@ cat_assoc _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
∘ d ⊗R (a ⊗R braid c b)
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ braid d (c ⊗ (a ⊗ b))
 ∘ d ⊗R twist a c b
:> (d ⊗ (a ⊗ (c ⊗ b)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    apply moveL_fmap01_twistR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
∘ d ⊗R (a ⊗R braid c b)
∘ d ⊗R twist c a b
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ braid d (c ⊗ (a ⊗ b))
:> (d ⊗ (c ⊗ (a ⊗ b)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine (_ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
∘ d ⊗R (a ⊗R braid c b)
∘ d ⊗R twist c a b
 $==
 ?Goal
:> (d ⊗ (c ⊗ (a ⊗ b)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
?Goal
 $==
 braid c (a ⊗ b) ⊗L d
 ∘ braid d (c ⊗ (a ⊗ b))
:> (d ⊗ (c ⊗ (a ⊗ b)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    2: apply braid_nat_r.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
∘ d ⊗R (a ⊗R braid c b)
∘ d ⊗R twist c a b
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
:> (d ⊗ (c ⊗ (a ⊗ b)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    (** Putting things back. *)
    apply moveR_fmap01_twistR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
∘ d ⊗R (a ⊗R braid c b)
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
:> (d ⊗ (a ⊗ (c ⊗ b)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    apply moveR_fmap01_fmap01_braidR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
:> (d ⊗ (a ⊗ (b ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    apply moveR_twistR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
∘ a ⊗R twist d b c
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
:> (a ⊗ (d ⊗ (b ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    apply moveR_fmap01_twistR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    (** There are two braids on the RHS of the LHS that can be swapped. *)
    refine (cat_assoc _ _ _ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ (a ⊗R braid b (c ⊗ d) ∘ a ⊗R (b ⊗R braid d c))
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine ((_ $@L _) $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
a ⊗R braid b (c ⊗ d)
∘ a ⊗R (b ⊗R braid d c)
 $==
 ?Goal
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> a ⊗ ((c ⊗ d) ⊗ b))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ ?Goal
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    1: refine ((fmap01_comp _ _ _ _)^$ $@ fmap02 _ _ _ $@ fmap01_comp _ _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid b (c ⊗ d)
∘ b ⊗R braid d c
 $==
 ?Goal2
 ∘ ?Goal3
:> (b ⊗ (d ⊗ c) $-> (c ⊗ d) ⊗ b)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ (a ⊗R ?Goal2 ∘ a ⊗R ?Goal3)
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    1: apply braid_nat_r.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ (a ⊗R (braid d c ⊗L b) ∘ a ⊗R braid b (d ⊗ c))
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine ((cat_assoc _ _ _)^$ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R (braid d c ⊗L b)
∘ a ⊗R braid b (d ⊗ c)
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    apply moveR_fmap01_braidR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R (braid d c ⊗L b)
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
:> (a ⊗ ((d ⊗ c) ⊗ b) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    (** Naturality of twist on the RHS of the LHS. *)
    refine (cat_assoc _ _ _ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ (twist a (c ⊗ d) b ∘ a ⊗R (braid d c ⊗L b))
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
:> (a ⊗ ((d ⊗ c) ⊗ b) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine ((_ $@L _) $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
twist a (c ⊗ d) b
∘ a ⊗R (braid d c ⊗L b)
 $==
 ?Goal
:> (a ⊗ ((d ⊗ c) ⊗ b) $-> (c ⊗ d) ⊗ (a ⊗ b))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ ?Goal
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
:> (a ⊗ ((d ⊗ c) ⊗ b) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    1: apply twist_nat_m.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ (braid d c ⊗L (a ⊗ b) ∘ twist a (d ⊗ c) b)
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
:> (a ⊗ ((d ⊗ c) ⊗ b) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine ((cat_assoc _ _ _)^$ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ braid d c ⊗L (a ⊗ b)
∘ twist a (d ⊗ c) b
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
:> (a ⊗ ((d ⊗ c) ⊗ b) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    (** Moving some things to the RHS so that we can braid and cancel on the LHS. *)
    apply moveR_twistR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ braid d c ⊗L (a ⊗ b)
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
 ∘ twist (d ⊗ c) a b
:> ((d ⊗ c) ⊗ (a ⊗ b) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine (cat_assoc _ _ _ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ (braid (c ⊗ d) (a ⊗ b) ∘ braid d c ⊗L (a ⊗ b))
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
 ∘ twist (d ⊗ c) a b
:> ((d ⊗ c) ⊗ (a ⊗ b) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine ((_ $@L _) $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid (c ⊗ d) (a ⊗ b)
∘ braid d c ⊗L (a ⊗ b)
 $==
 ?Goal
:> ((d ⊗ c) ⊗ (a ⊗ b) $-> (a ⊗ b) ⊗ (c ⊗ d))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ ?Goal
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
 ∘ twist (d ⊗ c) a b
:> ((d ⊗ c) ⊗ (a ⊗ b) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    1: apply braid_nat_l.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ ((a ⊗ b) ⊗R braid d c ∘ braid (d ⊗ c) (a ⊗ b))
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
 ∘ twist (d ⊗ c) a b
:> ((d ⊗ c) ⊗ (a ⊗ b) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine ((cat_assoc _ _ _)^$ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ (a ⊗ b) ⊗R braid d c
∘ braid (d ⊗ c) (a ⊗ b)
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
 ∘ twist (d ⊗ c) a b
:> ((d ⊗ c) ⊗ (a ⊗ b) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    apply moveR_braidR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ (a ⊗ b) ⊗R braid c d
∘ (a ⊗ b) ⊗R braid d c
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
 ∘ twist (d ⊗ c) a b
 ∘ braid (a ⊗ b) (d ⊗ c)
:> ((a ⊗ b) ⊗ (d ⊗ c) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine (cat_assoc _ _ _ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ ((a ⊗ b) ⊗R braid c d ∘ (a ⊗ b) ⊗R braid d c)
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
 ∘ twist (d ⊗ c) a b
 ∘ braid (a ⊗ b) (d ⊗ c)
:> ((a ⊗ b) ⊗ (d ⊗ c) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine ((_ $@L _) $@ cat_idr _ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
(a ⊗ b) ⊗R braid c d
∘ (a ⊗ b) ⊗R braid d c
 $==
 Id ((a ⊗ b) ⊗ (d ⊗ c))
:> ((a ⊗ b) ⊗ (d ⊗ c) $-> (a ⊗ b) ⊗ (d ⊗ c))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
 ∘ twist (d ⊗ c) a b
 ∘ braid (a ⊗ b) (d ⊗ c)
:> ((a ⊗ b) ⊗ (d ⊗ c) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    1: refine ((fmap01_comp _ _ _ _)^$ $@ fmap02 _ _ _ $@ fmap01_id _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid c d
∘ braid d c
 $==
 Id (d ⊗ c)
:> (d ⊗ c $-> d ⊗ c)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
 ∘ twist (d ⊗ c) a b
 ∘ braid (a ⊗ b) (d ⊗ c)
:> ((a ⊗ b) ⊗ (d ⊗ c) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    1: apply braid_braid.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
 ∘ twist (d ⊗ c) a b
 ∘ braid (a ⊗ b) (d ⊗ c)
:> ((a ⊗ b) ⊗ (d ⊗ c) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    apply moveL_braidR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ braid (d ⊗ c) (a ⊗ b)
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
 ∘ twist (d ⊗ c) a b
:> ((d ⊗ c) ⊗ (a ⊗ b) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    apply moveL_twistR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ braid (d ⊗ c) (a ⊗ b)
∘ twist a (d ⊗ c) b
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
 ∘ a ⊗R braid (d ⊗ c) b
:> (a ⊗ ((d ⊗ c) ⊗ b) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    apply moveL_fmap01_braidR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ braid (d ⊗ c) (a ⊗ b)
∘ twist a (d ⊗ c) b
∘ a ⊗R braid b (d ⊗ c)
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    (** We are almost at the desired 9-gon. Now we cancel the inner braid on the LHS. *)
    do 4 refine (_ $@ (cat_assoc _ _ _)^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ twist (a ⊗ b) d c
∘ braid (d ⊗ c) (a ⊗ b)
∘ twist a (d ⊗ c) b
∘ a ⊗R braid b (d ⊗ c)
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ (d ⊗R braid c (a ⊗ b)
    ∘ (d ⊗R twist a c b
       ∘ (d ⊗R (a ⊗R braid b c)
          ∘ (twist a d (b ⊗ c) ∘ a ⊗R twist b d c))))
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    do 3 refine (cat_assoc _ _ _ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid d ((a ⊗ b) ⊗ c)
∘ (twist (a ⊗ b) d c
   ∘ (braid (d ⊗ c) (a ⊗ b)
      ∘ (twist a (d ⊗ c) b ∘ a ⊗R braid b (d ⊗ c))))
 $==
 braid d ((a ⊗ b) ⊗ c)
 ∘ (d ⊗R braid c (a ⊗ b)
    ∘ (d ⊗R twist a c b
       ∘ (d ⊗R (a ⊗R braid b c)
          ∘ (twist a d (b ⊗ c) ∘ a ⊗R twist b d c))))
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> ((a ⊗ b) ⊗ c) ⊗ d)
    refine (_ $@L _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
twist (a ⊗ b) d c
∘ (braid (d ⊗ c) (a ⊗ b)
   ∘ (twist a (d ⊗ c) b ∘ a ⊗R braid b (d ⊗ c)))
 $==
 d ⊗R braid c (a ⊗ b)
 ∘ (d ⊗R twist a c b
    ∘ (d ⊗R (a ⊗R braid b c)
       ∘ (twist a d (b ⊗ c) ∘ a ⊗R twist b d c)))
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> d ⊗ ((a ⊗ b) ⊗ c))
    apply moveR_twistL.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid (d ⊗ c) (a ⊗ b)
∘ (twist a (d ⊗ c) b ∘ a ⊗R braid b (d ⊗ c))
 $==
 twist d (a ⊗ b) c
 ∘ (d ⊗R braid c (a ⊗ b)
    ∘ (d ⊗R twist a c b
       ∘ (d ⊗R (a ⊗R braid b c)
          ∘ (twist a d (b ⊗ c) ∘ a ⊗R twist b d c))))
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> (a ⊗ b) ⊗ (d ⊗ c))
    do 4 refine (_ $@ cat_assoc _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid (d ⊗ c) (a ⊗ b)
∘ (twist a (d ⊗ c) b ∘ a ⊗R braid b (d ⊗ c))
 $==
 twist d (a ⊗ b) c
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> (a ⊗ b) ⊗ (d ⊗ c))
    refine ((cat_assoc _ _ _)^$ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid (d ⊗ c) (a ⊗ b)
∘ twist a (d ⊗ c) b
∘ a ⊗R braid b (d ⊗ c)
 $==
 twist d (a ⊗ b) c
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> (a ⊗ b) ⊗ (d ⊗ c))
    (** Now we move terms around in order to get a homotopy in [a ⊗ (b ⊗ (d ⊗ c)) $-> d ⊗ (c ⊗ (a ⊗ b))]. *)
    apply moveL_fmap01_twistR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid (d ⊗ c) (a ⊗ b)
∘ twist a (d ⊗ c) b
∘ a ⊗R braid b (d ⊗ c)
∘ a ⊗R twist d b c
 $==
 twist d (a ⊗ b) c
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
:> (a ⊗ (d ⊗ (b ⊗ c)) $-> (a ⊗ b) ⊗ (d ⊗ c))
    apply moveL_twistR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid (d ⊗ c) (a ⊗ b)
∘ twist a (d ⊗ c) b
∘ a ⊗R braid b (d ⊗ c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
 $==
 twist d (a ⊗ b) c
 ∘ d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
:> (d ⊗ (a ⊗ (b ⊗ c)) $-> (a ⊗ b) ⊗ (d ⊗ c))
    do 2 refine (_ $@ (cat_assoc _ _ _)^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid (d ⊗ c) (a ⊗ b)
∘ twist a (d ⊗ c) b
∘ a ⊗R braid b (d ⊗ c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
 $==
 twist d (a ⊗ b) c
 ∘ (d ⊗R braid c (a ⊗ b)
    ∘ (d ⊗R twist a c b ∘ d ⊗R (a ⊗R braid b c)))
:> (d ⊗ (a ⊗ (b ⊗ c)) $-> (a ⊗ b) ⊗ (d ⊗ c))
    do 3 refine (cat_assoc _ _ _ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
braid (d ⊗ c) (a ⊗ b)
∘ (twist a (d ⊗ c) b
   ∘ (a ⊗R braid b (d ⊗ c)
      ∘ (a ⊗R twist d b c ∘ twist d a (b ⊗ c))))
 $==
 twist d (a ⊗ b) c
 ∘ (d ⊗R braid c (a ⊗ b)
    ∘ (d ⊗R twist a c b ∘ d ⊗R (a ⊗R braid b c)))
:> (d ⊗ (a ⊗ (b ⊗ c)) $-> (a ⊗ b) ⊗ (d ⊗ c))
    apply moveL_twistL.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
twist (a ⊗ b) d c
∘ (braid (d ⊗ c) (a ⊗ b)
   ∘ (twist a (d ⊗ c) b
      ∘ (a ⊗R braid b (d ⊗ c)
         ∘ (a ⊗R twist d b c ∘ twist d a (b ⊗ c)))))
 $==
 d ⊗R braid c (a ⊗ b)
 ∘ (d ⊗R twist a c b ∘ d ⊗R (a ⊗R braid b c))
:> (d ⊗ (a ⊗ (b ⊗ c)) $-> d ⊗ ((a ⊗ b) ⊗ c))
    refine (_ $@ cat_assoc _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
twist (a ⊗ b) d c
∘ (braid (d ⊗ c) (a ⊗ b)
   ∘ (twist a (d ⊗ c) b
      ∘ (a ⊗R braid b (d ⊗ c)
         ∘ (a ⊗R twist d b c ∘ twist d a (b ⊗ c)))))
 $==
 d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
:> (d ⊗ (a ⊗ (b ⊗ c)) $-> d ⊗ ((a ⊗ b) ⊗ c))
    do 4 refine ((cat_assoc _ _ _)^$ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
twist (a ⊗ b) d c
∘ braid (d ⊗ c) (a ⊗ b)
∘ twist a (d ⊗ c) b
∘ a ⊗R braid b (d ⊗ c)
∘ a ⊗R twist d b c
∘ twist d a (b ⊗ c)
 $==
 d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
:> (d ⊗ (a ⊗ (b ⊗ c)) $-> d ⊗ ((a ⊗ b) ⊗ c))
    apply moveR_twistR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
twist (a ⊗ b) d c
∘ braid (d ⊗ c) (a ⊗ b)
∘ twist a (d ⊗ c) b
∘ a ⊗R braid b (d ⊗ c)
∘ a ⊗R twist d b c
 $==
 d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
:> (a ⊗ (d ⊗ (b ⊗ c)) $-> d ⊗ ((a ⊗ b) ⊗ c))
    apply moveR_fmap01_twistR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
twist (a ⊗ b) d c
∘ braid (d ⊗ c) (a ⊗ b)
∘ twist a (d ⊗ c) b
∘ a ⊗R braid b (d ⊗ c)
 $==
 d ⊗R braid c (a ⊗ b)
 ∘ d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> d ⊗ ((a ⊗ b) ⊗ c))
    do 3 refine (_ $@ (cat_assoc _ _ _)^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
twist (a ⊗ b) d c
∘ braid (d ⊗ c) (a ⊗ b)
∘ twist a (d ⊗ c) b
∘ a ⊗R braid b (d ⊗ c)
 $==
 d ⊗R braid c (a ⊗ b)
 ∘ (d ⊗R twist a c b
    ∘ (d ⊗R (a ⊗R braid b c)
       ∘ (twist a d (b ⊗ c) ∘ a ⊗R twist b d c)))
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> d ⊗ ((a ⊗ b) ⊗ c))
    do 2 refine (cat_assoc _ _ _ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
twist (a ⊗ b) d c
∘ (braid (d ⊗ c) (a ⊗ b)
   ∘ (twist a (d ⊗ c) b ∘ a ⊗R braid b (d ⊗ c)))
 $==
 d ⊗R braid c (a ⊗ b)
 ∘ (d ⊗R twist a c b
    ∘ (d ⊗R (a ⊗R braid b c)
       ∘ (twist a d (b ⊗ c) ∘ a ⊗R twist b d c)))
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> d ⊗ ((a ⊗ b) ⊗ c))
    apply moveL_fmap01_braidL.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
d ⊗R braid (a ⊗ b) c
∘ (twist (a ⊗ b) d c
   ∘ (braid (d ⊗ c) (a ⊗ b)
      ∘ (twist a (d ⊗ c) b ∘ a ⊗R braid b (d ⊗ c))))
 $==
 d ⊗R twist a c b
 ∘ (d ⊗R (a ⊗R braid b c)
    ∘ (twist a d (b ⊗ c) ∘ a ⊗R twist b d c))
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> d ⊗ (c ⊗ (a ⊗ b)))
    do 2 refine (_ $@ cat_assoc _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
d ⊗R braid (a ⊗ b) c
∘ (twist (a ⊗ b) d c
   ∘ (braid (d ⊗ c) (a ⊗ b)
      ∘ (twist a (d ⊗ c) b ∘ a ⊗R braid b (d ⊗ c))))
 $==
 d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> d ⊗ (c ⊗ (a ⊗ b)))
    do 3 refine ((cat_assoc _ _ _)^$ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c
∘ twist b a c
 $==
 Id (b ⊗ (a ⊗ c))
:> (b ⊗ (a ⊗ c) $-> b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h)
 $==
 fmap11 cat_tensor g
   (fmap11 cat_tensor f h)
 ∘ twist a b c
:> (a ⊗ (b ⊗ c) $-> b' ⊗ (a' ⊗ c'))
twist_hexagon: forall a b c : A,
c ⊗R braid b a
∘ twist b c a
∘ b ⊗R braid a c
 $==
 twist a c b
 ∘ a ⊗R braid b c
 ∘ twist b a c
:> (b ⊗ (a ⊗ c) $-> c ⊗ (a ⊗ b))
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d
∘ twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d)
 $==
 c ⊗R twist a d b
 ∘ c ⊗R (a ⊗R braid b d)
 ∘ twist a c (b ⊗ d)
 ∘ a ⊗R twist b c d
:> (a ⊗ (b ⊗ (c ⊗ d)) $->
    c ⊗ (d ⊗ (a ⊗ b)))
a, b, c, d: A
d ⊗R braid (a ⊗ b) c
∘ twist (a ⊗ b) d c
∘ braid (d ⊗ c) (a ⊗ b)
∘ twist a (d ⊗ c) b
∘ a ⊗R braid b (d ⊗ c)
 $==
 d ⊗R twist a c b
 ∘ d ⊗R (a ⊗R braid b c)
 ∘ twist a d (b ⊗ c)
 ∘ a ⊗R twist b d c
:> (a ⊗ (b ⊗ (d ⊗ c)) $-> d ⊗ (c ⊗ (a ⊗ b)))
    (** And finally, this is the 9-gon we asked for. *)
    apply twist_9_gon.
  Defined.

  Local Close Scope long_path_scope.

  (** *** Hexagon *)

  Local Instance hexagon_twist : HexagonIdentity cat_tensor.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
HexagonIdentity cat_tensor
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
HexagonIdentity cat_tensor
    intros a b c; simpl.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
braid b a ⊗L c ∘ associator_twist' b a c
∘ b ⊗R braid c a $== associator_twist' a b c
∘ braid (b ⊗ c) a ∘ associator_twist' b c a
    refine (((_ $@L _) $@R _) $@ _ $@ (_ $@@ (_ $@R _))^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
associator_twist' b a c $== ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
braid b a ⊗L c ∘ ?Goal ∘ 
b ⊗R braid c a $== ?Goal4 ∘ 
braid (b ⊗ c) a ∘ ?Goal2
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
associator_twist' b c a $== ?Goal2
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
associator_twist' a b c $== ?Goal4
    1,3,4: apply associator_twist'_unfold.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
braid b a ⊗L c
∘ (braid c (b ⊗ a) ∘ (twist b c a ∘ b ⊗R braid a c))
∘ b ⊗R braid c a $== braid c (a ⊗ b)
∘ (twist a c b ∘ a ⊗R braid b c) ∘ braid (b ⊗ c) a
∘ (braid a (b ⊗ c) ∘ (twist b a c ∘ b ⊗R braid c a))
    do 2 refine (((cat_assoc _ _ _)^$ $@R _) $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
braid b a ⊗L c ∘ braid c (b ⊗ a) ∘ twist b c a
∘ b ⊗R braid a c ∘ b ⊗R braid c a $== braid c (a ⊗ b)
∘ (twist a c b ∘ a ⊗R braid b c) ∘ braid (b ⊗ c) a
∘ (braid a (b ⊗ c) ∘ (twist b a c ∘ b ⊗R braid c a))
    refine (cat_assoc _ _ _ $@ (_ $@L _) $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
b ⊗R braid a c ∘ b ⊗R braid c a $== ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
braid b a ⊗L c ∘ braid c (b ⊗ a) ∘ 
twist b c a ∘ ?Goal $== 
braid c (a ⊗ b) ∘ (twist a c b ∘ a ⊗R braid b c)
∘ braid (b ⊗ c) a
∘ (braid a (b ⊗ c) ∘ (twist b a c ∘ b ⊗R braid c a))
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
b ⊗R braid a c ∘ b ⊗R braid c a $== ?Goal refine ((fmap_comp _ _ _)^$ $@ fmap2 _ _ $@ fmap_id _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
braid a c ∘ braid c a $== Id (c ⊗ a)
      apply braid_braid. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
braid b a ⊗L c ∘ braid c (b ⊗ a) ∘ twist b c a
∘ Id (b ⊗ (c ⊗ a)) $== braid c (a ⊗ b)
∘ (twist a c b ∘ a ⊗R braid b c) ∘ braid (b ⊗ c) a
∘ (braid a (b ⊗ c) ∘ (twist b a c ∘ b ⊗R braid c a))
    refine (cat_idr _ $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
braid b a ⊗L c ∘ braid c (b ⊗ a) ∘ twist b c a $==
braid c (a ⊗ b) ∘ (twist a c b ∘ a ⊗R braid b c)
∘ braid (b ⊗ c) a
∘ (braid a (b ⊗ c) ∘ (twist b a c ∘ b ⊗R braid c a))
    refine (_ $@ cat_assoc _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
braid b a ⊗L c ∘ braid c (b ⊗ a) ∘ twist b c a $==
braid c (a ⊗ b) ∘ (twist a c b ∘ a ⊗R braid b c)
∘ braid (b ⊗ c) a ∘ braid a (b ⊗ c)
∘ (twist b a c ∘ b ⊗R braid c a)
    refine (_ $@ ((cat_assoc _ _ _)^$ $@R _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
braid b a ⊗L c ∘ braid c (b ⊗ a) ∘ twist b c a $==
braid c (a ⊗ b) ∘ (twist a c b ∘ a ⊗R braid b c)
∘ (braid (b ⊗ c) a ∘ braid a (b ⊗ c))
∘ (twist b a c ∘ b ⊗R braid c a)
    refine (_ $@ (((cat_idr _)^$ $@ (_ $@L _^$)) $@R _)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
braid b a ⊗L c ∘ braid c (b ⊗ a) ∘ twist b c a $==
braid c (a ⊗ b) ∘ (twist a c b ∘ a ⊗R braid b c)
∘ (twist b a c ∘ b ⊗R braid c a)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
braid (b ⊗ c) a ∘ braid a (b ⊗ c) $-> Id (a ⊗ (b ⊗ c))
    2: apply braid_braid.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
braid b a ⊗L c ∘ braid c (b ⊗ a) ∘ twist b c a $==
braid c (a ⊗ b) ∘ (twist a c b ∘ a ⊗R braid b c)
∘ (twist b a c ∘ b ⊗R braid c a)
    refine (((braid_nat_r _)^$ $@R _) $@ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
braid c (a ⊗ b) ∘ c ⊗R braid b a ∘ twist b c a $==
braid c (a ⊗ b) ∘ (twist a c b ∘ a ⊗R braid b c)
∘ (twist b a c ∘ b ⊗R braid c a)
    refine (cat_assoc _ _ _ $@ (_ $@L _) $@ (cat_assoc _ _ _)^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
c ⊗R braid b a ∘ twist b c a $== twist a c b
∘ a ⊗R braid b c ∘ (twist b a c ∘ b ⊗R braid c a)
    refine (_ $@ cat_assoc _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
c ⊗R braid b a ∘ twist b c a $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c ∘ b ⊗R braid c a
    apply moveL_fmap01_braidR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c: A
c ⊗R braid b a ∘ twist b c a ∘ b ⊗R braid a c $==
twist a c b ∘ a ⊗R braid b c ∘ twist b a c
    apply twist_hexagon.
  Defined.

  (** *** Conclusion *)

  (** In conclusion, we have proven the following: *)

  (** There is a monoidal structure on [A]. *)
  Local Instance ismonoidal_twist
    : IsMonoidal A cat_tensor cat_tensor_unit
    := {}.

  (** There is a symmetric monoidal category on [A]. *)
  Local Instance issymmetricmonoidal_twist
    : IsSymmetricMonoidal A cat_tensor cat_tensor_unit
    := {}.

  (** TODO: WIP *)

  (** Here is a hexagon involving only twist *)
  Definition twist_hex' a b c d
    : fmap01 cat_tensor c (twist a b d)
      $o twist a c (cat_tensor b d)
      $o fmap01 cat_tensor a (twist b c d)
    $== twist b c (cat_tensor a d)
      $o fmap01 cat_tensor b (twist a c d)
      $o twist a b (cat_tensor c d).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
c ⊗R twist a b d ∘ twist a c (b ⊗ d)
∘ a ⊗R twist b c d $== twist b c (a ⊗ d)
∘ b ⊗R twist a c d ∘ twist a b (c ⊗ d)
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
c ⊗R twist a b d ∘ twist a c (b ⊗ d)
∘ a ⊗R twist b c d $== twist b c (a ⊗ d)
∘ b ⊗R twist a c d ∘ twist a b (c ⊗ d)
    pose proof (twist_hexagon c a d $@ cat_assoc _ _ _) as p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
p: d ⊗R braid a c ∘ twist a d c ∘ 
a ⊗R braid c d $== twist c d a
∘ (c ⊗R braid a d ∘ twist a c d)
c ⊗R twist a b d ∘ twist a c (b ⊗ d)
∘ a ⊗R twist b c d $== twist b c (a ⊗ d)
∘ b ⊗R twist a c d ∘ twist a b (c ⊗ d)
    apply moveR_twistL in p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
p: twist d c a
∘ (d ⊗R braid a c ∘ twist a d c ∘ a ⊗R braid c d) $==
c ⊗R braid a d ∘ twist a c d
c ⊗R twist a b d ∘ twist a c (b ⊗ d)
∘ a ⊗R twist b c d $== twist b c (a ⊗ d)
∘ b ⊗R twist a c d ∘ twist a b (c ⊗ d)
    apply moveR_fmap01_braidL in p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
p: c ⊗R braid d a
∘ (twist d c a
   ∘ (d ⊗R braid a c ∘ 
      twist a d c ∘ a ⊗R braid c d)) $==
twist a c d
c ⊗R twist a b d ∘ twist a c (b ⊗ d)
∘ a ⊗R twist b c d $== twist b c (a ⊗ d)
∘ b ⊗R twist a c d ∘ twist a b (c ⊗ d)
    apply (fmap02 cat_tensor b) in p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
p: b
⊗R (c ⊗R braid d a
    ∘ (twist d c a
       ∘ (d ⊗R braid a c ∘ 
          twist a d c ∘ 
          a ⊗R braid c d))) $== 
b ⊗R twist a c d
c ⊗R twist a b d ∘ twist a c (b ⊗ d)
∘ a ⊗R twist b c d $== twist b c (a ⊗ d)
∘ b ⊗R twist a c d ∘ twist a b (c ⊗ d)
    refine (_ $@ ((_ $@L p) $@R _)); clear p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
c ⊗R twist a b d ∘ twist a c (b ⊗ d)
∘ a ⊗R twist b c d $== twist b c (a ⊗ d)
∘ b
  ⊗R (c ⊗R braid d a
      ∘ (twist d c a
         ∘ (d ⊗R braid a c ∘ twist a d c
            ∘ a ⊗R braid c d))) ∘ twist a b (c ⊗ d)
    apply moveL_twistR.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
c ⊗R twist a b d ∘ twist a c (b ⊗ d)
∘ a ⊗R twist b c d ∘ twist b a (c ⊗ d) $==
twist b c (a ⊗ d)
∘ b
  ⊗R (c ⊗R braid d a
      ∘ (twist d c a
         ∘ (d ⊗R braid a c ∘ twist a d c
            ∘ a ⊗R braid c d)))
    apply moveL_twistL.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
twist c b (a ⊗ d)
∘ (c ⊗R twist a b d ∘ twist a c (b ⊗ d)
   ∘ a ⊗R twist b c d ∘ twist b a (c ⊗ d)) $==
b
⊗R (c ⊗R braid d a
    ∘ (twist d c a
       ∘ (d ⊗R braid a c ∘ twist a d c
          ∘ a ⊗R braid c d)))
    refine (_ $@ (fmap01_comp _ _ _ _)^$).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
twist c b (a ⊗ d)
∘ (c ⊗R twist a b d ∘ twist a c (b ⊗ d)
   ∘ a ⊗R twist b c d ∘ twist b a (c ⊗ d)) $==
b ⊗R (c ⊗R braid d a)
∘ b
  ⊗R (twist d c a
      ∘ (d ⊗R braid a c ∘ twist a d c
         ∘ a ⊗R braid c d))
    (** TODO simplify *)
    apply moveR_twistL.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
c ⊗R twist a b d ∘ twist a c (b ⊗ d)
∘ a ⊗R twist b c d ∘ twist b a (c ⊗ d) $==
twist b c (a ⊗ d)
∘ (b ⊗R (c ⊗R braid d a)
   ∘ b
     ⊗R (twist d c a
         ∘ (d ⊗R braid a c ∘ twist a d c
            ∘ a ⊗R braid c d)))
    refine (_ $@ cat_assoc _ _ _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
cat_tensor: A -> A -> A
cat_tensor_unit: A
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (b ⊗ c) $-> b ⊗ (a ⊗ c)
twist_twist: forall a b c : A,
twist a b c ∘ twist b a c $==
Id (b ⊗ (a ⊗ c))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b c
right_unitor: RightUnitor cat_tensor cat_tensor_unit
twist_unitor: forall a b : A,
a ⊗R right_unitor b $== 
braid b a ∘ b ⊗R right_unitor a
∘ twist a b cat_tensor_unit
twist_hexagon: forall a b c : A,
c ⊗R braid b a ∘ twist b c a
∘ b ⊗R braid a c $== twist a c b
∘ a ⊗R braid b c ∘ twist b a c
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
a, b, c, d: A
c ⊗R twist a b d ∘ twist a c (b ⊗ d)
∘ a ⊗R twist b c d ∘ twist b a (c ⊗ d) $==
twist b c (a ⊗ d) ∘ b ⊗R (c ⊗R braid d a)
∘ b
  ⊗R (twist d c a
      ∘ (d ⊗R braid a c ∘ twist a d c
         ∘ a ⊗R braid c d))
  Abort.

End TwistConstruction.