Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal.
[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 WildCat.Opposite.

(** * 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.
  (** TODO: in 8.19 this notation causes issues, when updating to 8.20 we should uncomment this. *)
  (* Local Notation "f $== g :> A" := (GpdHom (A := A) f g)
    (at level 70, 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 simultaneously. We show the finer naturality of [twist] in each argument separately as this becomes more useful in practice. *)

  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.

  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.

  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] *)

  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.

  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.

  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.

  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.

  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.

  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.

  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.

  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. *)
  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. *)
  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. *)
  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
    snapply 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) snapply Build_Is1Natural.
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' : A * A * A) (f : a $-> a'),
(fun pat : A * A * A =>
 let x := pat in
 let fst := fst x in
 let c := snd x in
 let fst0 := fst in
 let a0 := Overture.fst fst0 in
 let b := snd fst0 in
 cate_fun (associator_twist' a0 b c)) a'
∘ fmap
    (uncurry (uncurry (fun x y z : A => x ⊗ (y ⊗ z))))
    f $==
fmap
  (uncurry (uncurry (fun x y z : A => (x ⊗ y) ⊗ z))) f
∘ (fun pat : A * A * A =>
   let x := pat in
   let fst := fst x in
   let c := snd x in
   let fst0 := fst in
   let a0 := Overture.fst fst0 in
   let b := snd fst0 in
   cate_fun (associator_twist' a0 b c)) a
      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'
∘ fmap11 cat_tensor f (fmap11 cat_tensor g h) $==
fmap (uncurry cat_tensor)
  (fmap (uncurry cat_tensor) (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')
  (fmap11 cat_tensor f (fmap11 cat_tensor g h))
  (fmap (uncurry cat_tensor)
     (fmap (uncurry cat_tensor) (f, g), h))
      (** First we remove all the equivalences from the equation. *)
      napply 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')
  (fmap11 cat_tensor f (fmap11 cat_tensor g h))
  (fmap (uncurry cat_tensor)
     (fmap (uncurry cat_tensor) (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')
  (fmap11 cat_tensor f (fmap11 cat_tensor g h))
  (fmap (uncurry cat_tensor)
     (fmap (uncurry cat_tensor) (f, g), h))
      napply 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 (fmap11 cat_tensor f (fmap11 cat_tensor g h))
  (fmap (uncurry cat_tensor)
     (fmap (uncurry cat_tensor) (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'))
  (fmap11 cat_tensor f (fmap11 cat_tensor g h))
  (fmap (uncurry cat_tensor)
     (fmap (uncurry cat_tensor) (f, g), h))
      (** The first square involving [braid] on its own is a naturality square. *)
      napply 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')
  (fmap11 cat_tensor f (fmap11 cat_tensor 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 cat_tensor)
     (fmap (uncurry cat_tensor) (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')
  (fmap11 cat_tensor f (fmap11 cat_tensor g h))
  (fmap11 cat_tensor h
     (fmap (uncurry cat_tensor) (f, g)))
      (** The second square is just the naturality of twist. *)
      napply 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')
  (fmap11 cat_tensor f (fmap11 cat_tensor 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 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')
  (fmap11 cat_tensor f (fmap11 cat_tensor g h))
  (fmap11 cat_tensor f (fmap11 cat_tensor h g))
      napply 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')
  (fmap11 cat_tensor f (fmap11 cat_tensor g h))
  (fmap11 cat_tensor f (fmap11 cat_tensor h g))
      2: napply 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
  (fmap11 cat_tensor f (fmap11 cat_tensor g h))
  (fmap11 cat_tensor f (fmap11 cat_tensor 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'))
  (fmap11 cat_tensor f (fmap11 cat_tensor g h))
  (fmap11 cat_tensor f (fmap11 cat_tensor 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 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 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]. *)
  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
    snapply 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 snapply 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) snapply Build_Is1Natural.
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' : A) (f : a $-> a'),
(fun a0 : A => right_unitor a0
 ∘ braid cat_tensor_unit a0) a'
∘ fmap (cat_tensor cat_tensor_unit) f $== fmap idmap f
∘ (fun a0 : A => right_unitor a0
   ∘ braid cat_tensor_unit a0) 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
(fun a : A => right_unitor a ∘ braid cat_tensor_unit a)
  b ∘ fmap (cat_tensor cat_tensor_unit) f $==
fmap idmap f
∘ (fun a : A => right_unitor a
   ∘ braid cat_tensor_unit a) 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
  ((fun a : A => right_unitor a
    ∘ braid cat_tensor_unit a) a)
  ((fun a : A => right_unitor a
    ∘ braid cat_tensor_unit a) b)
  (fmap (cat_tensor cat_tensor_unit) f) (fmap idmap f)
        napply 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: exact (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)
        napply 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 :=
       Build_Is1Natural
         (fun a0 : A => right_unitor a0
          ∘ braid cat_tensor_unit a0)
         (fun (a0 b : A) (f : a0 $-> b) =>
          braid_nat_r f $@v
          is1natural_natequiv right_unitor a0 b f)
   |} 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 :=
       Build_Is1Natural
         (fun a : A => right_unitor a
          ∘ braid cat_tensor_unit a)
         (fun (a b : A) (f : a $-> b) =>
          braid_nat_r f $@v
          is1natural_natequiv right_unitor a b f)
   |} 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)
      exact (catie_braid _ _).
  Defined.

  (** *** Triangle *)

  (** The triangle identity can easily be proven by rearranging the diagram, cancelling and using naturality of [braid]. *)
  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.

  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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' 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
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' 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
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R twist a d b ∘ c ⊗R (a ⊗R braid b d)
∘ twist a c (b ⊗ d) ∘ a ⊗R twist b c d
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
α a b (c ⊗ d) $== ?Goal
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
α (a ⊗ 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
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (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
twist_hexagon: forall a b c : A,
c ⊗R braid 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
?Goal0 ∘ ?Goal $== ?Goal9 ∘ ?Goal8 ∘ ?Goal4
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
α b c d $== ?Goal5
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
a ⊗R ?Goal5 $== ?Goal4
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
α a (b ⊗ c) d $== ?Goal8
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
α a b c $== ?Goal11
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
?Goal11 ⊗L d $== ?Goal9
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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: 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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
a
⊗R (braid d (b ⊗ c) ∘ (twist b d c ∘ b ⊗R braid 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
Is0Bifunctor0: Is0Bifunctor cat_tensor
Is1Bifunctor0: Is1Bifunctor cat_tensor
braid: SymmetricBraiding cat_tensor
twist: forall a b c : A, a ⊗ (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
twist_hexagon: forall a b c : A,
c ⊗R braid 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
(braid c (a ⊗ b) ∘ (twist a c b ∘ a ⊗R braid b c))
⊗L d $== ?Goal2
    2: exact (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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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: 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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
(braid c (a ⊗ b) ∘ (twist a c b ∘ a ⊗R braid b c))
⊗L d $== ?Goal0
    2: exact (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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)))
    (** 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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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))))) $==
braid c (a ⊗ b) ⊗L d
∘ (twist a c b ⊗L d ∘ (a ⊗R braid b c) ⊗L d)
∘ (braid d (a ⊗ (b ⊗ c))
   ∘ (twist a d (b ⊗ c) ∘ a ⊗R braid (b ⊗ c) d))
∘ (a ⊗R braid d (b ⊗ c)
   ∘ (a ⊗R twist b d c ∘ a ⊗R (b ⊗R braid c d)))
    do 4 refine ((cat_assoc _ _ _)^$ $@ _).
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)))
    (** 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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)
    (** Cancel two braids next to each other. *)
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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: 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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
a ⊗R braid (b ⊗ c) d ∘ 
a ⊗R braid d (b ⊗ c) $== 
Id (a ⊗ (d ⊗ (b ⊗ c)))
    2: refine ((fmap01_comp _ _ _ _)^$ $@ 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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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: 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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
braid (b ⊗ c) d ∘ braid d (b ⊗ c) $== Id (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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)
    (** *)
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
?Goal0 $== (a ⊗R braid b c) ⊗L d
∘ braid d (a ⊗ (b ⊗ c))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
?Goal0 $== twist a c b ⊗L d ∘ braid d (a ⊗ (c ⊗ b))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
?Goal $== braid c (a ⊗ b) ⊗L d ∘ braid d (c ⊗ (a ⊗ b))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)
    (** 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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    (** 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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
a ⊗R braid b (c ⊗ d) ∘ a ⊗R (b ⊗R braid d c) $== ?Goal
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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 $== 
braid d ((a ⊗ b) ⊗ c) ∘ 
d ⊗R braid c (a ⊗ b) ∘ 
d ⊗R twist a c b ∘ d ⊗R (a ⊗R braid b c)
∘ twist a d (b ⊗ c) ∘ a ⊗R twist b d c
    1: refine ((fmap01_comp _ _ _ _)^$ $@ 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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
braid b (c ⊗ d) ∘ b ⊗R braid d c $== ?Goal2 ∘ ?Goal3
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)) $==
braid d ((a ⊗ b) ⊗ c) ∘ d ⊗R braid c (a ⊗ b)
∘ d ⊗R twist a c b ∘ d ⊗R (a ⊗R braid b c)
∘ twist a d (b ⊗ c) ∘ a ⊗R twist b d c
    refine ((cat_assoc _ _ _)^$ $@ _).
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    (** 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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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 a (c ⊗ d) b ∘ a ⊗R (braid d c ⊗L b) $== ?Goal
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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) $==
braid d ((a ⊗ b) ⊗ c) ∘ d ⊗R braid c (a ⊗ b)
∘ d ⊗R twist a c b ∘ d ⊗R (a ⊗R braid b c)
∘ twist a d (b ⊗ c) ∘ a ⊗R twist b d c
∘ a ⊗R braid (d ⊗ c) b
    refine ((cat_assoc _ _ _)^$ $@ _).
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    (** 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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
braid (c ⊗ d) (a ⊗ b) ∘ braid d c ⊗L (a ⊗ b) $== ?Goal
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)) $==
braid d ((a ⊗ b) ⊗ c) ∘ d ⊗R braid c (a ⊗ b)
∘ d ⊗R twist a c b ∘ d ⊗R (a ⊗R braid b c)
∘ twist a d (b ⊗ c) ∘ a ⊗R twist b d c
∘ a ⊗R braid (d ⊗ c) b ∘ twist (d ⊗ c) a b
    refine ((cat_assoc _ _ _)^$ $@ _).
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
(a ⊗ b) ⊗R braid c d ∘ (a ⊗ b) ⊗R braid d c $==
Id ((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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
braid d ((a ⊗ b) ⊗ c) ∘ 
twist (a ⊗ b) d c $== braid d ((a ⊗ b) ⊗ c)
∘ d ⊗R braid c (a ⊗ b) ∘ 
d ⊗R twist a c b ∘ d ⊗R (a ⊗R braid b c)
∘ twist a d (b ⊗ c) ∘ a ⊗R twist b d c
∘ a ⊗R braid (d ⊗ c) b ∘ 
twist (d ⊗ c) a b ∘ braid (a ⊗ b) (d ⊗ c)
    1: refine ((fmap01_comp _ _ _ _)^$ $@ 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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
braid c d ∘ braid d c $== Id (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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    (** 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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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))))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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))))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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 (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)))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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))))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
braid (d ⊗ c) (a ⊗ b)
∘ (twist a (d ⊗ c) b ∘ a ⊗R braid b (d ⊗ c)) $==
twist d (a ⊗ b) c ∘ d ⊗R braid c (a ⊗ b)
∘ d ⊗R twist a c b ∘ d ⊗R (a ⊗R braid b c)
∘ twist a d (b ⊗ c) ∘ a ⊗R twist b d c
    refine ((cat_assoc _ _ _)^$ $@ _).
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    (** 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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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 (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))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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 (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))))) $==
d ⊗R braid c (a ⊗ b) ∘ d ⊗R twist a c b
∘ d ⊗R (a ⊗R braid b c)
    do 4 refine ((cat_assoc _ _ _)^$ $@ _).
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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 (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)
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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 (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)
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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 (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
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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 (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)))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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 (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)))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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))
    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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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)))) $==
d ⊗R twist a c b ∘ d ⊗R (a ⊗R braid b c)
∘ twist a d (b ⊗ c) ∘ a ⊗R twist b d c
    do 3 refine ((cat_assoc _ _ _)^$ $@ _).
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))
twist_nat: forall (a a' b b' c c' : A) 
(f : a $-> a') (g : b $-> b')
(h : c $-> c'),
twist a' b' c'
∘ fmap11 cat_tensor f
    (fmap11 cat_tensor g h) $==
fmap11 cat_tensor g
  (fmap11 cat_tensor f h) ∘ 
twist a b 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
twist_9_gon: forall a b c d : A,
c ⊗R braid (a ⊗ b) d ∘ 
twist (a ⊗ b) c d
∘ braid (c ⊗ d) (a ⊗ b)
∘ twist a (c ⊗ d) b
∘ a ⊗R braid b (c ⊗ d) $==
c ⊗R 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
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
    (** And finally, this is the 9-gon we asked for. *)
    apply twist_9_gon.
  Defined.

  Local Close Scope long_path_scope.

  (** *** Hexagon *)

  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 ∘ α b a c ∘ b ⊗R braid c a $== α a b c
∘ braid (b ⊗ c) a ∘ α 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
α 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
α 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
α 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]. *)
  Instance ismonoidal_twist
    : IsMonoidal A cat_tensor cat_tensor_unit
    := {}.

  (** There is a symmetric monoidal category on [A]. *)
  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.