HoTT.WildCat.Products.v.alectryon.json

Require Import Basics.Overture Basics.Equivalences Basics.Tactics.Require Import Basics.Overture Basics.Equivalences Basics.Tactics.
Require Import Types.Bool Types.Prod.
Require Import WildCat.Core WildCat.Bifunctor WildCat.Equiv WildCat.EquivGpd
               WildCat.Forall WildCat.NatTrans WildCat.Opposite
               WildCat.Universe WildCat.Yoneda WildCat.Graph WildCat.ZeroGroupoid
               WildCat.Monoidal WildCat.MonoidalTwistConstruction
               WildCat.FunctorCat.

(** * Categories with products *)

(** ** Indexed products *)

(** For [A] a wild 1-category, [I] a type, and [x : I -> A] an [I]-indexed family of objects in [A], we study the categorical product of this family of objects. *)

(** When [x] is an [I]-indexed family of objects in [A] and [prod] is an object with an [I]-indexed family of projections, we get for each [z] an induced map from the 0-groupoid of morphisms [z $-> prod] to the product of the 0-groupoids [z $-> x i] over [i : I]. *)
Definition cat_prod_corec_inv {A : Type} `{Is1Cat A}
  {I : Type} (x : I -> A) (prod : A) (pr : forall i, prod $-> x i) (z : A)
  : yon_0gpd prod z $-> prod_0gpd I (fun i => yon_0gpd (x i) z).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
prod: A
pr: forall i : I, prod $-> x i
z: A
yon_0gpd prod z $-> prod_0gpd I (fun i : I => yon_0gpd (x i) z)
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
prod: A
pr: forall i : I, prod $-> x i
z: A
yon_0gpd prod z $-> prod_0gpd I (fun i : I => yon_0gpd (x i) z)
  snapply equiv_prod_0gpd_corec.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
prod: A
pr: forall i : I, prod $-> x i
z: A
forall i : I, yon_0gpd prod z $-> (fun i0 : I => yon_0gpd (x i0) z) i
  intros i.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
prod: A
pr: forall i0 : I, prod $-> x i0
z: A
i: I
yon_0gpd prod z $-> (fun i0 : I => yon_0gpd (x i0) z) i
  exact (fmap (fun x => yon_0gpd x z) (pr i)).
Defined.

(** An object is a product of an [I]-indexed family of objects of a category if there is an [I]-indexed family of projections such that the induced map defined above is an equivalence. *)
Class IsProduct {A : Type} `{Is1Cat A} {I : Type} (x : I -> A) (cat_prod : A)
  := Build_IsProduct' {
  cat_pr : forall i : I, cat_prod $-> x i;
  cat_isequiv_cat_prod_corec_inv
    :: forall z : A, CatIsEquiv (cat_prod_corec_inv x cat_prod cat_pr z);
}.

Arguments cat_pr {A _ _ _ _ _ x cat_prod isprod} : rename.
Arguments cat_isequiv_cat_prod_corec_inv {A _ _ _ _ _} x cat_prod {isprod} : rename.
Arguments Build_IsProduct' {A _ _ _ _ _} x cat_prod.

(** A product is an object together with the data that it is a product. *)
Class Product {A : Type} `{Is1Cat A} {I : Type} (x : I -> A) := Build_Product' {
  cat_prod : A;
  cat_isprod :: IsProduct x cat_prod;
}.

Arguments Build_Product' {A _ _ _ _ _} x cat_prod cat_isprod.
Arguments cat_prod {A _ _ _ _ _} x {product} : rename.
Arguments cat_isprod {A _ _ _ _ _} x {product} : rename.

Section ProductConstructors.

  Context {A : Type} `{Is1Cat A} {I : Type} (x : I -> A)
    (cat_prod : A) (cat_pr : forall i : I, cat_prod $-> x i)
    (cat_prod_corec : forall z : A,
      (forall i : I, z $-> x i) -> (z $-> cat_prod))
    (cat_prod_beta_pr : forall (z : A) (f : forall i, z $-> x i) (i : I),
      cat_pr i $o cat_prod_corec z f $== f i)
    (cat_prod_eta_pr : forall (z : A) (f g : z $-> cat_prod),
      (forall i : I, cat_pr i $o f $== cat_pr i $o g) -> f $== g).

  (** A convenience wrapper for building [IsProduct]. *)
  Definition Build_IsProduct : IsProduct x cat_prod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_pr: forall i : I, cat_prod $-> x i
cat_prod_corec: forall z : A, (forall i : I, z $-> x i) -> z $-> cat_prod
cat_prod_beta_pr: forall (z : A) (f : forall i : I, z $-> x i) (i : I),
cat_pr i $o cat_prod_corec z f $== f i
cat_prod_eta_pr: forall (z : A) (f g : z $-> cat_prod),
(forall i : I, cat_pr i $o f $== cat_pr i $o g) -> f $== g
IsProduct x cat_prod
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_pr: forall i : I, cat_prod $-> x i
cat_prod_corec: forall z : A, (forall i : I, z $-> x i) -> z $-> cat_prod
cat_prod_beta_pr: forall (z : A) (f : forall i : I, z $-> x i) (i : I),
cat_pr i $o cat_prod_corec z f $== f i
cat_prod_eta_pr: forall (z : A) (f g : z $-> cat_prod),
(forall i : I, cat_pr i $o f $== cat_pr i $o g) -> f $== g
IsProduct x cat_prod
    snapply (Build_IsProduct' x cat_prod cat_pr).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_pr: forall i : I, cat_prod $-> x i
cat_prod_corec: forall z : A, (forall i : I, z $-> x i) -> z $-> cat_prod
cat_prod_beta_pr: forall (z : A) (f : forall i : I, z $-> x i) (i : I),
cat_pr i $o cat_prod_corec z f $== f i
cat_prod_eta_pr: forall (z : A) (f g : z $-> cat_prod),
(forall i : I, cat_pr i $o f $== cat_pr i $o g) -> f $== g
forall z : A, CatIsEquiv (cat_prod_corec_inv x cat_prod cat_pr z)
    intros z.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_pr: forall i : I, cat_prod $-> x i
cat_prod_corec: forall z0 : A, (forall i : I, z0 $-> x i) -> z0 $-> cat_prod
cat_prod_beta_pr: forall (z0 : A) (f : forall i : I, z0 $-> x i) (i : I),
cat_pr i $o cat_prod_corec z0 f $== f i
cat_prod_eta_pr: forall (z0 : A) (f g : z0 $-> cat_prod),
(forall i : I, cat_pr i $o f $== cat_pr i $o g) -> f $== g
z: A
CatIsEquiv (cat_prod_corec_inv x cat_prod cat_pr z)
    napply isequiv_0gpd_issurjinj.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_pr: forall i : I, cat_prod $-> x i
cat_prod_corec: forall z0 : A, (forall i : I, z0 $-> x i) -> z0 $-> cat_prod
cat_prod_beta_pr: forall (z0 : A) (f : forall i : I, z0 $-> x i) (i : I),
cat_pr i $o cat_prod_corec z0 f $== f i
cat_prod_eta_pr: forall (z0 : A) (f g : z0 $-> cat_prod),
(forall i : I, cat_pr i $o f $== cat_pr i $o g) -> f $== g
z: A
IsSurjInj (cat_prod_corec_inv x cat_prod cat_pr z)
    napply Build_IsSurjInj.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_pr: forall i : I, cat_prod $-> x i
cat_prod_corec: forall z0 : A, (forall i : I, z0 $-> x i) -> z0 $-> cat_prod
cat_prod_beta_pr: forall (z0 : A) (f : forall i : I, z0 $-> x i) (i : I),
cat_pr i $o cat_prod_corec z0 f $== f i
cat_prod_eta_pr: forall (z0 : A) (f g : z0 $-> cat_prod),
(forall i : I, cat_pr i $o f $== cat_pr i $o g) -> f $== g
z: A
SplEssSurj (cat_prod_corec_inv x cat_prod cat_pr z)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_pr: forall i : I, cat_prod $-> x i
cat_prod_corec: forall z0 : A, (forall i : I, z0 $-> x i) -> z0 $-> cat_prod
cat_prod_beta_pr: forall (z0 : A) (f : forall i : I, z0 $-> x i) (i : I),
cat_pr i $o cat_prod_corec z0 f $== f i
cat_prod_eta_pr: forall (z0 : A) (f g : z0 $-> cat_prod),
(forall i : I, cat_pr i $o f $== cat_pr i $o g) -> f $== g
z: A
forall x0 y : zerogpd_graph (yon_0gpd cat_prod z),
cat_prod_corec_inv x cat_prod cat_pr z x0 $==
cat_prod_corec_inv x cat_prod cat_pr z y -> x0 $== y
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_pr: forall i : I, cat_prod $-> x i
cat_prod_corec: forall z0 : A, (forall i : I, z0 $-> x i) -> z0 $-> cat_prod
cat_prod_beta_pr: forall (z0 : A) (f : forall i : I, z0 $-> x i) (i : I),
cat_pr i $o cat_prod_corec z0 f $== f i
cat_prod_eta_pr: forall (z0 : A) (f g : z0 $-> cat_prod),
(forall i : I, cat_pr i $o f $== cat_pr i $o g) -> f $== g
z: A
SplEssSurj (cat_prod_corec_inv x cat_prod cat_pr z) intros f.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_pr: forall i : I, cat_prod $-> x i
cat_prod_corec: forall z0 : A, (forall i : I, z0 $-> x i) -> z0 $-> cat_prod
cat_prod_beta_pr: forall (z0 : A) (f0 : forall i : I, z0 $-> x i) (i : I),
cat_pr i $o cat_prod_corec z0 f0 $== f0 i
cat_prod_eta_pr: forall (z0 : A) (f0 g : z0 $-> cat_prod),
(forall i : I, cat_pr i $o f0 $== cat_pr i $o g) -> f0 $== g
z: A
f: zerogpd_graph (prod_0gpd I (fun i : I => yon_0gpd (x i) z))
{a : zerogpd_graph (yon_0gpd cat_prod z) &
cat_prod_corec_inv x cat_prod cat_pr z a $== f}
      exists (cat_prod_corec z f).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_pr: forall i : I, cat_prod $-> x i
cat_prod_corec: forall z0 : A, (forall i : I, z0 $-> x i) -> z0 $-> cat_prod
cat_prod_beta_pr: forall (z0 : A) (f0 : forall i : I, z0 $-> x i) (i : I),
cat_pr i $o cat_prod_corec z0 f0 $== f0 i
cat_prod_eta_pr: forall (z0 : A) (f0 g : z0 $-> cat_prod),
(forall i : I, cat_pr i $o f0 $== cat_pr i $o g) -> f0 $== g
z: A
f: zerogpd_graph (prod_0gpd I (fun i : I => yon_0gpd (x i) z))
cat_prod_corec_inv x cat_prod cat_pr z (cat_prod_corec z f) $== f
      intros i.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_pr: forall i0 : I, cat_prod $-> x i0
cat_prod_corec: forall z0 : A, (forall i0 : I, z0 $-> x i0) -> z0 $-> cat_prod
cat_prod_beta_pr: forall (z0 : A) (f0 : forall i0 : I, z0 $-> x i0) 
(i0 : I), cat_pr i0 $o cat_prod_corec z0 f0 $== f0 i0
cat_prod_eta_pr: forall (z0 : A) (f0 g : z0 $-> cat_prod),
(forall i0 : I, cat_pr i0 $o f0 $== cat_pr i0 $o g) -> f0 $== g
z: A
f: zerogpd_graph (prod_0gpd I (fun i0 : I => yon_0gpd (x i0) z))
i: I
cat_prod_corec_inv x cat_prod cat_pr z (cat_prod_corec z f) i $-> f i
      napply cat_prod_beta_pr.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_pr: forall i : I, cat_prod $-> x i
cat_prod_corec: forall z0 : A, (forall i : I, z0 $-> x i) -> z0 $-> cat_prod
cat_prod_beta_pr: forall (z0 : A) (f : forall i : I, z0 $-> x i) (i : I),
cat_pr i $o cat_prod_corec z0 f $== f i
cat_prod_eta_pr: forall (z0 : A) (f g : z0 $-> cat_prod),
(forall i : I, cat_pr i $o f $== cat_pr i $o g) -> f $== g
z: A
forall x0 y : zerogpd_graph (yon_0gpd cat_prod z),
cat_prod_corec_inv x cat_prod cat_pr z x0 $==
cat_prod_corec_inv x cat_prod cat_pr z y -> x0 $== y intros f g p.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_pr: forall i : I, cat_prod $-> x i
cat_prod_corec: forall z0 : A, (forall i : I, z0 $-> x i) -> z0 $-> cat_prod
cat_prod_beta_pr: forall (z0 : A) (f0 : forall i : I, z0 $-> x i) (i : I),
cat_pr i $o cat_prod_corec z0 f0 $== f0 i
cat_prod_eta_pr: forall (z0 : A) (f0 g0 : z0 $-> cat_prod),
(forall i : I, cat_pr i $o f0 $== cat_pr i $o g0) -> f0 $== g0
z: A
f, g: zerogpd_graph (yon_0gpd cat_prod z)
p: cat_prod_corec_inv x cat_prod cat_pr z f $==
cat_prod_corec_inv x cat_prod cat_pr z g
f $== g
      by napply cat_prod_eta_pr.
  Defined.

  (** A convenience wrapper for building products. *)
  Definition Build_Product : Product x
    := Build_Product' x cat_prod Build_IsProduct.

End ProductConstructors.

Section Lemmata.

  Context {A : Type} `{Is1Cat A} {I : Type} {x : I -> A}
    (cat_prod : A) {cat_isprod : IsProduct x cat_prod}.

  Definition cate_cat_prod_corec_inv {z : A}
    : (yon_0gpd cat_prod z) $<~> prod_0gpd I (fun i => yon_0gpd (x i) z)
    := Build_CatEquiv (cat_prod_corec_inv x cat_prod cat_pr z).

  Definition cate_cat_prod_corec {z : A}
    : prod_0gpd I (fun i => yon_0gpd (x i) z) $<~> (yon_0gpd cat_prod z)
    := cate_cat_prod_corec_inv^-1$.

  Definition cat_prod_corec {z : A}
    : (forall i, z $-> x i) -> (z $-> cat_prod)
    := cate_fun cate_cat_prod_corec.

  (** Applying the [i]th projection after a tuple of maps gives the [ith] map. *)
  Definition cat_prod_beta {z : A} (f : forall i, z $-> x i)
    : forall i, cat_pr i $o cat_prod_corec f $== f i
    := cate_isretr cate_cat_prod_corec_inv f.

  (** The pairing map is the unique map that makes the following diagram commute. *)
  Definition cat_prod_eta {z : A} (f : z $-> cat_prod)
    : cat_prod_corec (fun i => cat_pr i $o f) $== f
    := cate_issect cate_cat_prod_corec_inv f.

  Local Instance is0functor_prod_0gpd_helper
    : Is0Functor (fun z : A^op => prod_0gpd I (fun i => yon_0gpd (x i) z)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
Is0Functor (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z))
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
Is0Functor (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z))
    snapply Build_Is0Functor.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
forall a b : A^op,
(a $-> b) ->
(fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) a $->
(fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) b
    intros a b f.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
a, b: A^op
f: a $-> b
(fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) a $->
(fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) b
    snapply Build_Fun01'.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
a, b: A^op
f: a $-> b
zerogpd_graph
  ((fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) a) ->
zerogpd_graph
  ((fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) b)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
a, b: A^op
f: a $-> b
forall
a0
 b0 : zerogpd_graph
        ((fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) a),
(a0 $-> b0) -> ?F a0 $-> ?F b0
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
a, b: A^op
f: a $-> b
zerogpd_graph
  ((fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) a) ->
zerogpd_graph
  ((fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) b) intros g i.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
a, b: A^op
f: a $-> b
g: zerogpd_graph
  ((fun z : A^op => prod_0gpd I (fun i0 : I => yon_0gpd (x i0) z)) a)
i: I
yon_0gpd (x i) b
      exact (f $o g i).
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
a, b: A^op
f: a $-> b
forall
a0
 b0 : zerogpd_graph
        ((fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) a),
(a0 $-> b0) ->
(fun
   g : zerogpd_graph
         ((fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) a) =>
 (fun i : I => f $o g i)
 :
 zerogpd_graph
   ((fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) b))
  a0 $->
(fun
   g : zerogpd_graph
         ((fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) a) =>
 (fun i : I => f $o g i)
 :
 zerogpd_graph
   ((fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) b))
  b0 intros g h p i.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
a, b: A^op
f: a $-> b
g, h: zerogpd_graph
  ((fun z : A^op => prod_0gpd I (fun i0 : I => yon_0gpd (x i0) z)) a)
p: g $-> h
i: I
f $o g i $-> f $o h i
      exact (f $@L p i).
  Defined.

  Local Instance is1functor_prod_0gpd_helper
    : Is1Functor (fun z : A^op => prod_0gpd I (fun i => yon_0gpd (x i) z)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
Is1Functor (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z))
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
Is1Functor (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z))
    snapply Build_Is1Functor.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
forall (a b : A^op) (f g : a $-> b),
f $== g ->
fmap (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) f $==
fmap (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) g
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
forall a : A^op,
fmap (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) (Id a) $==
Id ((fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) a)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
forall (a b c : A^op) (f : a $-> b) (g : b $-> c),
fmap (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) (g $o f) $==
fmap (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) g $o
fmap (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) f
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
forall (a b : A^op) (f g : a $-> b),
f $== g ->
fmap (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) f $==
fmap (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) g intros a b f g p r i.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
a, b: A^op
f, g: a $-> b
p: f $== g
r: zerogpd_graph (prod_0gpd I (fun i0 : I => yon_0gpd (x i0) a))
i: I
fmap (fun z : A^op => prod_0gpd I (fun i0 : I => yon_0gpd (x i0) z)) f r i $->
fmap (fun z : A^op => prod_0gpd I (fun i0 : I => yon_0gpd (x i0) z)) g r i
      refine (_ $@L _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
a, b: A^op
f, g: a $-> b
p: f $== g
r: zerogpd_graph (prod_0gpd I (fun i0 : I => yon_0gpd (x i0) a))
i: I
f $== g
      exact p.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
forall a : A^op,
fmap (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) (Id a) $==
Id ((fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) a) intros a r i.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
a: A^op
r: zerogpd_graph (prod_0gpd I (fun i0 : I => yon_0gpd (x i0) a))
i: I
fmap (fun z : A^op => prod_0gpd I (fun i0 : I => yon_0gpd (x i0) z)) 
  (Id a) r i $->
Id (prod_0gpd I (fun i0 : I => yon_0gpd (x i0) a)) r i
      napply cat_idl; exact _.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
forall (a b c : A^op) (f : a $-> b) (g : b $-> c),
fmap (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) (g $o f) $==
fmap (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) g $o
fmap (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) f intros a b c f g r i.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
a, b, c: A^op
f: a $-> b
g: b $-> c
r: zerogpd_graph (prod_0gpd I (fun i0 : I => yon_0gpd (x i0) a))
i: I
fmap (fun z : A^op => prod_0gpd I (fun i0 : I => yon_0gpd (x i0) z)) 
  (g $o f) r i $->
(fmap (fun z : A^op => prod_0gpd I (fun i0 : I => yon_0gpd (x i0) z)) g $o
 fmap (fun z : A^op => prod_0gpd I (fun i0 : I => yon_0gpd (x i0) z)) f) r i
      napply cat_assoc; exact _.
  Defined.

  Definition natequiv_cat_prod_corec_inv
    : NatEquiv (yon_0gpd cat_prod)
      (fun z : A^op => prod_0gpd I (fun i => yon_0gpd (x i) z)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
NatEquiv (yon_0gpd cat_prod)
  (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z))
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
NatEquiv (yon_0gpd cat_prod)
  (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z))
    snapply Build_NatEquiv.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
forall a : A^op,
yon_0gpd cat_prod a $<~>
(fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) a
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
Is1Natural (yon_0gpd cat_prod)
  (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z))
  (fun a : A^op => ?e a)
    1: intro; exact cate_cat_prod_corec_inv.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
Is1Natural (yon_0gpd cat_prod)
  (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z))
  (fun a : A^op => (fun a0 : A^op => cate_cat_prod_corec_inv) a)
    exact (is1natural_yoneda_0gpd cat_prod
      (fun z => prod_0gpd I (fun i => yon_0gpd (x i) z))
      cat_pr).
  Defined.

  Lemma cat_prod_corec_eta {z : A} {f f' : forall i, z $-> x i}
    : (forall i, f i $== f' i) -> cat_prod_corec f $== cat_prod_corec f'.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
z: A
f, f': forall i : I, z $-> x i
(forall i : I, f i $== f' i) -> cat_prod_corec f $== cat_prod_corec f'
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
z: A
f, f': forall i : I, z $-> x i
(forall i : I, f i $== f' i) -> cat_prod_corec f $== cat_prod_corec f'
    intros p.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
z: A
f, f': forall i : I, z $-> x i
p: forall i : I, f i $== f' i
cat_prod_corec f $== cat_prod_corec f'
    unfold cat_prod_corec.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
z: A
f, f': forall i : I, z $-> x i
p: forall i : I, f i $== f' i
cate_cat_prod_corec f $== cate_cat_prod_corec f'
    napply (moveL_equiv_V_0gpd cate_cat_prod_corec_inv).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
z: A
f, f': forall i : I, z $-> x i
p: forall i : I, f i $== f' i
equiv_fun_0gpd cate_cat_prod_corec_inv (cate_cat_prod_corec f) $== f'
    nrefine (cate_isretr cate_cat_prod_corec_inv _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
z: A
f, f': forall i : I, z $-> x i
p: forall i : I, f i $== f' i
Id (prod_0gpd I (fun i : I => yon_0gpd (x i) z)) f $== f'
    exact p.
  Defined.

  Lemma cat_prod_pr_eta {z : A} {f f' : z $-> cat_prod}
    : (forall i, cat_pr i $o f $== cat_pr i $o f') -> f $== f'.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
z: A
f, f': z $-> cat_prod
(forall i : I, cat_pr i $o f $== cat_pr i $o f') -> f $== f'
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
z: A
f, f': z $-> cat_prod
(forall i : I, cat_pr i $o f $== cat_pr i $o f') -> f $== f'
    intros p.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
z: A
f, f': z $-> cat_prod
p: forall i : I, cat_pr i $o f $== cat_pr i $o f'
f $== f'
    refine ((cat_prod_eta _)^$ $@ _ $@ cat_prod_eta _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
I: Type
x: I -> A
cat_prod: A
cat_isprod: IsProduct x cat_prod
z: A
f, f': z $-> cat_prod
p: forall i : I, cat_pr i $o f $== cat_pr i $o f'
cat_prod_corec (fun i : I => cat_pr i $o f) $==
cat_prod_corec (fun i : I => cat_pr i $o f')
    by napply cat_prod_corec_eta.
  Defined.

End Lemmata.

Section InducedFromEquiv.

  Context {A : Type} `{he : HasEquivs A} {I : Type} {x : I -> A}
  (cat_prod : A) `{!IsProduct x cat_prod}
  (y : A) (f : y $<~> cat_prod).

  (** A categorical equivalence into a product induces a product structure on the domain. *)
  Local Instance cat_prod_equiv_prod : IsProduct x y.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
IsProduct x y
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
IsProduct x y
    snapply Build_IsProduct.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
forall i : I, y $-> x i
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
forall z : A, (forall i : I, z $-> x i) -> z $-> y
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
forall (z : A) (f0 : forall i : I, z $-> x i) (i : I),
?cat_pr i $o ?cat_prod_corec z f0 $== f0 i
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
forall (z : A) (f0 g : z $-> y),
(forall i : I, ?cat_pr i $o f0 $== ?cat_pr i $o g) -> f0 $== g
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
forall i : I, y $-> x i intro i.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
i: I
y $-> x i
      exact (cat_pr i $o f).
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
forall z : A, (forall i : I, z $-> x i) -> z $-> y intros z D.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
z: A
D: forall i : I, z $-> x i
z $-> y
      exact (f^-1$ $o cat_prod_corec _ D).
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
forall (z : A) (f0 : forall i : I, z $-> x i) (i : I),
(fun i0 : I => cat_pr i0 $o f) i $o
(fun (z0 : A) (D : forall i0 : I, z0 $-> x i0) =>
 f^-1$ $o cat_prod_corec cat_prod D) z f0 $==
f0 i intros z D i; cbn beta.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
z: A
D: forall i0 : I, z $-> x i0
i: I
cat_pr i $o f $o (f^-1$ $o cat_prod_corec cat_prod D) $== D i
      refine (_ $@ cat_prod_beta _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
z: A
D: forall i0 : I, z $-> x i0
i: I
cat_pr i $o f $o (f^-1$ $o cat_prod_corec cat_prod D) $==
cat_pr i $o cat_prod_corec cat_prod D
      refine (cat_assoc _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
z: A
D: forall i0 : I, z $-> x i0
i: I
cat_pr i $o (f $o (f^-1$ $o cat_prod_corec cat_prod D)) $==
cat_pr i $o cat_prod_corec cat_prod D
      apply cat_postwhisker.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
z: A
D: forall i0 : I, z $-> x i0
i: I
f $o (f^-1$ $o cat_prod_corec cat_prod D) $== cat_prod_corec cat_prod D
      apply compose_h_Vh.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
forall (z : A) (f0 g : z $-> y),
(forall i : I,
 (fun i0 : I => cat_pr i0 $o f) i $o f0 $==
 (fun i0 : I => cat_pr i0 $o f) i $o g) ->
f0 $== g cbn beta; intros z g g' e.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
z: A
g, g': z $-> y
e: forall i : I, cat_pr i $o f $o g $== cat_pr i $o f $o g'
g $== g'
      napply (cate_monic_equiv f).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
z: A
g, g': z $-> y
e: forall i : I, cat_pr i $o f $o g $== cat_pr i $o f $o g'
f $o g $== f $o g'
      napply cat_prod_pr_eta.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
z: A
g, g': z $-> y
e: forall i : I, cat_pr i $o f $o g $== cat_pr i $o f $o g'
forall i : ?Goal, cat_pr i $o (f $o g) $== cat_pr i $o (f $o g')
      intro i.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
z: A
g, g': z $-> y
e: forall i0 : I, cat_pr i0 $o f $o g $== cat_pr i0 $o f $o g'
i: ?Goal
cat_pr i $o (f $o g) $== cat_pr i $o (f $o g')
      refine (cat_assoc_opp _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
z: A
g, g': z $-> y
e: forall i0 : I, cat_pr i0 $o f $o g $== cat_pr i0 $o f $o g'
i: ?Goal
cat_pr i $o f $o g $== cat_pr i $o (f $o g')
      refine (_ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
he: HasEquivs A
I: Type
x: I -> A
cat_prod: A
IsProduct0: IsProduct x cat_prod
y: A
f: y $<~> cat_prod
z: A
g, g': z $-> y
e: forall i0 : I, cat_pr i0 $o f $o g $== cat_pr i0 $o f $o g'
i: ?Goal
cat_pr i $o f $o g $== cat_pr i $o f $o g'
      exact (e i).
  Defined.

  (** The induced projection is given by the equivalence. *)
  Definition cat_pr_comp (i : I)
    : cat_pr i $== cat_pr i $o f
    := Id _.

  (** The induced corecursion is given by the equivalence. *)
  Definition cat_prod_corec_comp {z : A} (D : forall i, z $-> x i)
    : f $o cat_prod_corec (cat_isprod:=cat_prod_equiv_prod) y D $== cat_prod_corec _ D
    := compose_h_Vh _ _.

End InducedFromEquiv.

(** *** Diagonal map into the product of a constant family *)

Definition cat_prod_diag {A : Type} {I : Type} (x : A) (cat_prod : A)
  `{IsProduct _ I (fun _ => x) cat_prod}
  : x $-> cat_prod
  := cat_prod_corec cat_prod (fun _ => Id x).

(** *** Uniqueness of products *)

Definition cate_cat_prod {A : Type} `{HasEquivs A} {I J : Type} (ie : I <~> J)
  (x : I -> A) (prod_x : A) `{!IsProduct x prod_x}
  (y : J -> A) (prod_y : A) `{!IsProduct y prod_y}
  (e : forall i : I, x i $<~> y (ie i))
  : prod_x $<~> prod_y.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i : I, x i $<~> y (ie i)
prod_x $<~> prod_y
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i : I, x i $<~> y (ie i)
prod_x $<~> prod_y
  napply yon_equiv_0gpd.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i : I, x i $<~> y (ie i)
yon1_0gpd prod_x $<~> yon1_0gpd prod_y
  refine (natequiv_compose _ (natequiv_cat_prod_corec_inv _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i : I, x i $<~> y (ie i)
NatEquiv (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z))
  (yon1_0gpd prod_y)
  refine (natequiv_compose
            (natequiv_inverse (natequiv_cat_prod_corec_inv _)) _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i : I, x i $<~> y (ie i)
NatEquiv (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z))
  (fun z : A^op => prod_0gpd J (fun i : J => yon_0gpd (y i) z))
  snapply Build_NatEquiv.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i : I, x i $<~> y (ie i)
forall a : A^op,
(fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) a $<~>
(fun z : A^op => prod_0gpd J (fun i : J => yon_0gpd (y i) z)) a
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i : I, x i $<~> y (ie i)
Is1Natural (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z))
  (fun z : A^op => prod_0gpd J (fun i : J => yon_0gpd (y i) z))
  (fun a : A^op => ?e a)
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i : I, x i $<~> y (ie i)
forall a : A^op,
(fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) a $<~>
(fun z : A^op => prod_0gpd J (fun i : J => yon_0gpd (y i) z)) a intros z.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i : I, x i $<~> y (ie i)
z: A^op
(fun z0 : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z0)) z $<~>
(fun z0 : A^op => prod_0gpd J (fun i : J => yon_0gpd (y i) z0)) z
    napply (cate_prod_0gpd ie).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i : I, x i $<~> y (ie i)
z: A^op
forall i : I, yon_0gpd (x i) z $<~> yon_0gpd (y (ie i)) z
    intros i.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i0 : I, x i0 $<~> y (ie i0)
z: A^op
i: I
yon_0gpd (x i) z $<~> yon_0gpd (y (ie i)) z
    exact (natequiv_yon_equiv_0gpd (e i) _).
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i : I, x i $<~> y (ie i)
Is1Natural (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z))
  (fun z : A^op => prod_0gpd J (fun i : J => yon_0gpd (y i) z))
  (fun a : A^op =>
   (fun z : A^op =>
    cate_prod_0gpd ie (fun i : I => yon_0gpd (x i) z)
      (fun i : J => yon_0gpd (y i) z)
      (fun i : I => natequiv_yon_equiv_0gpd (e i) z))
     a) snapply Build_Is1Natural.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i : I, x i $<~> y (ie i)
forall (a a' : A^op) (f : a $-> a'),
(fun a0 : A^op =>
 cate_fun
   ((fun z : A^op =>
     cate_prod_0gpd ie (fun i : I => yon_0gpd (x i) z)
       (fun i : J => yon_0gpd (y i) z)
       (fun i : I => natequiv_yon_equiv_0gpd (e i) z))
      a0))
  a' $o
fmap (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) f $==
fmap (fun z : A^op => prod_0gpd J (fun i : J => yon_0gpd (y i) z)) f $o
(fun a0 : A^op =>
 cate_fun
   ((fun z : A^op =>
     cate_prod_0gpd ie (fun i : I => yon_0gpd (x i) z)
       (fun i : J => yon_0gpd (y i) z)
       (fun i : I => natequiv_yon_equiv_0gpd (e i) z))
      a0))
  a
    intros a b f g j.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i : I, x i $<~> y (ie i)
a, b: A^op
f: a $-> b
g: zerogpd_graph (prod_0gpd I (fun i : I => yon_0gpd (x i) a))
j: J
(cate_prod_0gpd ie (fun i : I => yon_0gpd (x i) b)
   (fun i : J => yon_0gpd (y i) b)
   (fun i : I => natequiv_yon_equiv_0gpd (e i) b) $o
 fmap (fun z : A^op => prod_0gpd I (fun i : I => yon_0gpd (x i) z)) f) g j $->
(fmap (fun z : A^op => prod_0gpd J (fun i : J => yon_0gpd (y i) z)) f $o
 cate_prod_0gpd ie (fun i : I => yon_0gpd (x i) a)
   (fun i : J => yon_0gpd (y i) a)
   (fun i : I => natequiv_yon_equiv_0gpd (e i) a))
  g j
    cbn.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i : I, x i $<~> y (ie i)
a, b: A^op
f: a $-> b
g: zerogpd_graph (prod_0gpd I (fun i : I => yon_0gpd (x i) a))
j: J
transport (fun x0 : J => b $-> y x0) (eisretr ie j)
  (cate_fun' (x (ie^-1 j)) (y (ie (ie^-1 j))) (e (ie^-1 j)) $o
   (g (ie^-1 j) $o f)) $->
transport (fun x0 : J => a $-> y x0) (eisretr ie j)
  (cate_fun' (x (ie^-1 j)) (y (ie (ie^-1 j))) (e (ie^-1 j)) $o g (ie^-1 j)) $o
f
    destruct (eisretr ie j).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
I, J: Type
ie: I <~> J
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
e: forall i : I, x i $<~> y (ie i)
a, b: A^op
f: a $-> b
g: zerogpd_graph (prod_0gpd I (fun i : I => yon_0gpd (x i) a))
j: J
transport (fun x0 : J => b $-> y x0) 1
  (cate_fun' (x (ie^-1 j)) (y (ie (ie^-1 j))) (e (ie^-1 j)) $o
   (g (ie^-1 j) $o f)) $->
transport (fun x0 : J => a $-> y x0) 1
  (cate_fun' (x (ie^-1 j)) (y (ie (ie^-1 j))) (e (ie^-1 j)) $o g (ie^-1 j)) $o
f
    exact (cat_assoc_opp _ _ _).
Defined.

(** [I]-indexed products are unique. *)
Definition cat_prod_unique {A : Type} `{HasEquivs A} {I : Type}
  (x : I -> A) (prod_x : A) `{!IsProduct x prod_x}
  (y : I -> A) (prod_y : A) `{!IsProduct y prod_y}
  (e : forall i : I, x i $<~> y i)
  : prod_x $<~> prod_y
  := cate_cat_prod 1 x _ y _ e.

(** *** Existence of products *)

Class HasProducts (A : Type) `{Is1Cat A} (I : Type)
  := has_products :: forall x : I -> A, Product x.

Arguments has_products {A _ _ _ _ I hasproducts} x : rename.

Class HasAllProducts (A : Type) `{Is1Cat A}
  := has_all_products :: forall I : Type, HasProducts A I.

(** *** Product functor *)

Instance is0functor_cat_prod (A : Type) (I : Type) `{HasProducts A I}
  : Is0Functor (fun x : I -> A => cat_prod x).A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
Is0Functor (fun x : I -> A => cat_prod x)
Proof.A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
Is0Functor (fun x : I -> A => cat_prod x)
  napply Build_Is0Functor.A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
forall a b : I -> A, (a $-> b) -> cat_prod a $-> cat_prod b
  intros x y f.A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
x, y: I -> A
f: x $-> y
cat_prod x $-> cat_prod y
  exact (cat_prod_corec _ (fun i => f i $o cat_pr i)).
Defined.

Instance is1functor_cat_prod (A : Type) (I : Type) `{HasProducts A I}
  : Is1Functor (fun x : I -> A => cat_prod x).A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
Is1Functor (fun x : I -> A => cat_prod x)
Proof.A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
Is1Functor (fun x : I -> A => cat_prod x)
  napply Build_Is1Functor.A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
forall (a b : I -> A) (f g : a $-> b),
f $== g ->
fmap (fun x : I -> A => cat_prod x) f $==
fmap (fun x : I -> A => cat_prod x) g
A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
forall a : I -> A,
fmap (fun x : I -> A => cat_prod x) (Id a) $== Id (cat_prod a)
A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
forall (a b c : I -> A) (f : a $-> b) (g : b $-> c),
fmap (fun x : I -> A => cat_prod x) (g $o f) $==
fmap (fun x : I -> A => cat_prod x) g $o
fmap (fun x : I -> A => cat_prod x) f
  -A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
forall (a b : I -> A) (f g : a $-> b),
f $== g ->
fmap (fun x : I -> A => cat_prod x) f $==
fmap (fun x : I -> A => cat_prod x) g intros x y f g p.A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
x, y: I -> A
f, g: x $-> y
p: f $== g
fmap (fun x0 : I -> A => cat_prod x0) f $==
fmap (fun x0 : I -> A => cat_prod x0) g
    exact (cat_prod_corec_eta _ (fun i => p i $@R cat_pr i)).
  -A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
forall a : I -> A,
fmap (fun x : I -> A => cat_prod x) (Id a) $== Id (cat_prod a) intros x.A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
x: I -> A
fmap (fun x0 : I -> A => cat_prod x0) (Id x) $== Id (cat_prod x)
    nrefine (_ $@ (cat_prod_eta _ (Id _))).A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
x: I -> A
fmap (fun x0 : I -> A => cat_prod x0) (Id x) $==
cat_prod_corec (cat_prod x) (fun i : ?Goal1 => cat_pr i $o Id (cat_prod x))
    exact (cat_prod_corec_eta _ (fun i => cat_idl _ $@ (cat_idr _)^$)).
  -A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
forall (a b c : I -> A) (f : a $-> b) (g : b $-> c),
fmap (fun x : I -> A => cat_prod x) (g $o f) $==
fmap (fun x : I -> A => cat_prod x) g $o
fmap (fun x : I -> A => cat_prod x) f intros x y z f g.A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
x, y, z: I -> A
f: x $-> y
g: y $-> z
fmap (fun x0 : I -> A => cat_prod x0) (g $o f) $==
fmap (fun x0 : I -> A => cat_prod x0) g $o
fmap (fun x0 : I -> A => cat_prod x0) f
    napply cat_prod_pr_eta.A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
x, y, z: I -> A
f: x $-> y
g: y $-> z
forall i : ?Goal,
cat_pr i $o fmap (fun x0 : I -> A => cat_prod x0) (g $o f) $==
cat_pr i $o
(fmap (fun x0 : I -> A => cat_prod x0) g $o
 fmap (fun x0 : I -> A => cat_prod x0) f)
    intros i.A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
x, y, z: I -> A
f: x $-> y
g: y $-> z
i: ?Goal
cat_pr i $o fmap (fun x0 : I -> A => cat_prod x0) (g $o f) $==
cat_pr i $o
(fmap (fun x0 : I -> A => cat_prod x0) g $o
 fmap (fun x0 : I -> A => cat_prod x0) f)
    nrefine (cat_prod_beta _ _ _ $@ _).A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
x, y, z: I -> A
f: x $-> y
g: y $-> z
i: I
(g $o f) i $o cat_pr i $==
cat_pr i $o
(fmap (fun x0 : I -> A => cat_prod x0) g $o
 fmap (fun x0 : I -> A => cat_prod x0) f)
    nrefine (_ $@ cat_assoc _ _ _).A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
x, y, z: I -> A
f: x $-> y
g: y $-> z
i: I
(g $o f) i $o cat_pr i $==
cat_pr i $o fmap (fun x0 : I -> A => cat_prod x0) g $o
fmap (fun x0 : I -> A => cat_prod x0) f
    symmetry.A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
x, y, z: I -> A
f: x $-> y
g: y $-> z
i: I
cat_pr i $o fmap (fun x0 : I -> A => cat_prod x0) g $o
fmap (fun x0 : I -> A => cat_prod x0) f $== (g $o f) i $o cat_pr i
    nrefine (cat_prod_beta _ _ _ $@R _ $@ _).A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
x, y, z: I -> A
f: x $-> y
g: y $-> z
i: I
g i $o cat_pr i $o fmap (fun x0 : I -> A => cat_prod x0) f $==
(g $o f) i $o cat_pr i
    nrefine (cat_assoc _ _ _ $@ _).A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
x, y, z: I -> A
f: x $-> y
g: y $-> z
i: I
g i $o (cat_pr i $o fmap (fun x0 : I -> A => cat_prod x0) f) $==
(g $o f) i $o cat_pr i
    nrefine (_ $@L cat_prod_beta _ _ _ $@ _).A, I: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasProducts A I
x, y, z: I -> A
f: x $-> y
g: y $-> z
i: I
g i $o (f i $o cat_pr i) $== (g $o f) i $o cat_pr i
    napply cat_assoc_opp.
Defined.

(** *** An empty product is terminal *)

Definition isterminal_prod_empty {A : Type} {x : Empty -> A} {prod_empty : A}
  `{isprod : IsProduct _ Empty x prod_empty}
  : IsTerminal prod_empty.A: Type
x: Empty -> A
prod_empty: A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
isprod: IsProduct x prod_empty
IsTerminal prod_empty
Proof.A: Type
x: Empty -> A
prod_empty: A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
isprod: IsProduct x prod_empty
IsTerminal prod_empty
  intros a.A: Type
x: Empty -> A
prod_empty: A
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
isprod: IsProduct x prod_empty
a: A
{f : a $-> prod_empty & forall g : a $-> prod_empty, f $== g}
  srefine (cat_prod_corec _ _; fun f => cat_prod_pr_eta _ _); intros [].
Defined.

(** ** Binary products *)

Class IsBinaryProduct {A : Type} `{Is1Cat A} (x y : A) (cat_binprod : A)
  := is_binary_product :: IsProduct (Bool_rec _ x y) (cat_binprod).

Class BinaryProduct {A : Type} `{Is1Cat A} (x y : A)
  := binary_product :: Product (Bool_rec _ x y).

Instance isbinaryproduct_binaryproduct {A : Type} `{Is1Cat A}
  (x y : A) `{!BinaryProduct x y}
  : IsBinaryProduct x y (cat_prod _)
  := cat_isprod _.

(** A category with binary products is a category with a binary product for each pair of objects. *)
Class HasBinaryProducts (A : Type) `{Is1Cat A}
  := has_binary_products :: forall x y : A, BinaryProduct x y.

Instance hasbinaryproducts_hasproductsbool {A : Type} `{HasProducts A Bool}
  : HasBinaryProducts A
  := fun x y => has_products (Bool_rec _ x y).

Section BinaryProducts.

  Context {A : Type} `{Is1Cat A} {x y : A}
    (cat_binprod : A) {isbinprod : IsBinaryProduct x y cat_binprod}.

  Definition cat_pr1 : cat_binprod $-> x := cat_pr (x:=Bool_rec _ x y) true.

  Definition cat_pr2 : cat_binprod $-> y := cat_pr (x:=Bool_rec _ x y) false.

  Definition cat_binprod_corec {z : A} (f : z $-> x) (g : z $-> y)
    : z $-> cat_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f: z $-> x
g: z $-> y
z $-> cat_binprod
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f: z $-> x
g: z $-> y
z $-> cat_binprod
    apply (cat_prod_corec _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f: z $-> x
g: z $-> y
forall i : Bool, z $-> Bool_rec A x y i
    intros [|].A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f: z $-> x
g: z $-> y
z $-> Bool_rec A x y true
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f: z $-> x
g: z $-> y
z $-> Bool_rec A x y false
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f: z $-> x
g: z $-> y
z $-> Bool_rec A x y true exact f.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f: z $-> x
g: z $-> y
z $-> Bool_rec A x y false exact g.
  Defined.

  Definition cat_binprod_beta_pr1 {z : A} (f : z $-> x) (g : z $-> y)
    : cat_pr1 $o cat_binprod_corec f g $== f
    := cat_prod_beta _ _ true.

  Definition cat_binprod_beta_pr2 {z : A} (f : z $-> x) (g : z $-> y)
    : cat_pr2 $o cat_binprod_corec f g $== g
    := cat_prod_beta _ _ false.

  Definition cat_binprod_eta {z : A} (f : z $-> cat_binprod)
    : cat_binprod_corec (cat_pr1 $o f) (cat_pr2 $o f) $== f.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f: z $-> cat_binprod
cat_binprod_corec (cat_pr1 $o f) (cat_pr2 $o f) $== f
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f: z $-> cat_binprod
cat_binprod_corec (cat_pr1 $o f) (cat_pr2 $o f) $== f
    unfold cat_binprod_corec.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f: z $-> cat_binprod
cat_prod_corec cat_binprod
  (fun i : Bool =>
   match i as b return (z $-> Bool_rec A x y b) with
   | true => cat_pr1 $o f
   | false => cat_pr2 $o f
   end) $==
f
    rapply cat_prod_pr_eta.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f: z $-> cat_binprod
forall i : Bool,
cat_pr i $o
cat_prod_corec cat_binprod
  (fun i0 : Bool =>
   match i0 as b return (z $-> Bool_rec A x y b) with
   | true => cat_pr1 $o f
   | false => cat_pr2 $o f
   end) $==
cat_pr i $o f
    intros [|].A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f: z $-> cat_binprod
cat_pr true $o
cat_prod_corec cat_binprod
  (fun i : Bool =>
   match i as b return (z $-> Bool_rec A x y b) with
   | true => cat_pr1 $o f
   | false => cat_pr2 $o f
   end) $==
cat_pr true $o f
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f: z $-> cat_binprod
cat_pr false $o
cat_prod_corec cat_binprod
  (fun i : Bool =>
   match i as b return (z $-> Bool_rec A x y b) with
   | true => cat_pr1 $o f
   | false => cat_pr2 $o f
   end) $==
cat_pr false $o f
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f: z $-> cat_binprod
cat_pr true $o
cat_prod_corec cat_binprod
  (fun i : Bool =>
   match i as b return (z $-> Bool_rec A x y b) with
   | true => cat_pr1 $o f
   | false => cat_pr2 $o f
   end) $==
cat_pr true $o f exact (cat_binprod_beta_pr1 _ _).
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f: z $-> cat_binprod
cat_pr false $o
cat_prod_corec cat_binprod
  (fun i : Bool =>
   match i as b return (z $-> Bool_rec A x y b) with
   | true => cat_pr1 $o f
   | false => cat_pr2 $o f
   end) $==
cat_pr false $o f exact (cat_binprod_beta_pr2 _ _).
  Defined.

  Definition cat_binprod_eta_pr {z : A} (f g : z $-> cat_binprod)
    : cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, g: z $-> cat_binprod
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, g: z $-> cat_binprod
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
    intros p q.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, g: z $-> cat_binprod
p: cat_pr1 $o f $== cat_pr1 $o g
q: cat_pr2 $o f $== cat_pr2 $o g
f $== g
    rapply cat_prod_pr_eta.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, g: z $-> cat_binprod
p: cat_pr1 $o f $== cat_pr1 $o g
q: cat_pr2 $o f $== cat_pr2 $o g
forall i : Bool, cat_pr i $o f $== cat_pr i $o g
    intros [|].A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, g: z $-> cat_binprod
p: cat_pr1 $o f $== cat_pr1 $o g
q: cat_pr2 $o f $== cat_pr2 $o g
cat_pr true $o f $== cat_pr true $o g
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, g: z $-> cat_binprod
p: cat_pr1 $o f $== cat_pr1 $o g
q: cat_pr2 $o f $== cat_pr2 $o g
cat_pr false $o f $== cat_pr false $o g
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, g: z $-> cat_binprod
p: cat_pr1 $o f $== cat_pr1 $o g
q: cat_pr2 $o f $== cat_pr2 $o g
cat_pr true $o f $== cat_pr true $o g exact p.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, g: z $-> cat_binprod
p: cat_pr1 $o f $== cat_pr1 $o g
q: cat_pr2 $o f $== cat_pr2 $o g
cat_pr false $o f $== cat_pr false $o g exact q.
  Defined.

  Definition cat_binprod_corec_eta {z : A} (f f' : z $-> x) (g g' : z $-> y)
    : f $== f' -> g $== g' -> cat_binprod_corec f g $== cat_binprod_corec f' g'.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, f': z $-> x
g, g': z $-> y
f $== f' -> g $== g' -> cat_binprod_corec f g $== cat_binprod_corec f' g'
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, f': z $-> x
g, g': z $-> y
f $== f' -> g $== g' -> cat_binprod_corec f g $== cat_binprod_corec f' g'
    intros p q.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, f': z $-> x
g, g': z $-> y
p: f $== f'
q: g $== g'
cat_binprod_corec f g $== cat_binprod_corec f' g'
    rapply cat_prod_corec_eta.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, f': z $-> x
g, g': z $-> y
p: f $== f'
q: g $== g'
forall i : Bool,
match i as b return (z $-> Bool_rec A x y b) with
| true => f
| false => g
end $==
match i as b return (z $-> Bool_rec A x y b) with
| true => f'
| false => g'
end
    intros [|].A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, f': z $-> x
g, g': z $-> y
p: f $== f'
q: g $== g'
f $== f'
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, f': z $-> x
g, g': z $-> y
p: f $== f'
q: g $== g'
g $== g'
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, f': z $-> x
g, g': z $-> y
p: f $== f'
q: g $== g'
f $== f' exact p.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
isbinprod: IsBinaryProduct x y cat_binprod
z: A
f, f': z $-> x
g, g': z $-> y
p: f $== f'
q: g $== g'
g $== g' exact q.
  Defined.

End BinaryProducts.

Section BinaryProductConstructors.

  Context {A : Type} `{Is1Cat A} {x y : A}
    (cat_binprod : A) (cat_pr1 : cat_binprod $-> x) (cat_pr2 : cat_binprod $-> y)
    (cat_binprod_corec : forall z : A, z $-> x -> z $-> y -> z $-> cat_binprod)
    (cat_binprod_beta_pr1 : forall (z : A) (f : z $-> x) (g : z $-> y),
      cat_pr1 $o cat_binprod_corec z f g $== f)
    (cat_binprod_beta_pr2 : forall (z : A) (f : z $-> x) (g : z $-> y),
      cat_pr2 $o cat_binprod_corec z f g $== g)
    (cat_binprod_eta_pr : forall (z : A) (f g : z $-> cat_binprod),
      cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g).

  (** A convenience wrapper for building [IsBinaryProduct]. *)
  Definition Build_IsBinaryProduct : IsBinaryProduct x y cat_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z : A, (z $-> x) -> (z $-> y) -> z $-> cat_binprod
cat_binprod_beta_pr1: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr1 $o cat_binprod_corec z f g $== f
cat_binprod_beta_pr2: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr2 $o cat_binprod_corec z f g $== g
cat_binprod_eta_pr: forall (z : A) (f g : z $-> cat_binprod),
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
IsBinaryProduct x y cat_binprod
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z : A, (z $-> x) -> (z $-> y) -> z $-> cat_binprod
cat_binprod_beta_pr1: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr1 $o cat_binprod_corec z f g $== f
cat_binprod_beta_pr2: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr2 $o cat_binprod_corec z f g $== g
cat_binprod_eta_pr: forall (z : A) (f g : z $-> cat_binprod),
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
IsBinaryProduct x y cat_binprod
    snapply (Build_IsProduct _ cat_binprod).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z : A, (z $-> x) -> (z $-> y) -> z $-> cat_binprod
cat_binprod_beta_pr1: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr1 $o cat_binprod_corec z f g $== f
cat_binprod_beta_pr2: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr2 $o cat_binprod_corec z f g $== g
cat_binprod_eta_pr: forall (z : A) (f g : z $-> cat_binprod),
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
forall i : Bool, cat_binprod $-> Bool_rec A x y i
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z : A, (z $-> x) -> (z $-> y) -> z $-> cat_binprod
cat_binprod_beta_pr1: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr1 $o cat_binprod_corec z f g $== f
cat_binprod_beta_pr2: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr2 $o cat_binprod_corec z f g $== g
cat_binprod_eta_pr: forall (z : A) (f g : z $-> cat_binprod),
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
forall z : A, (forall i : Bool, z $-> Bool_rec A x y i) -> z $-> cat_binprod
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z : A, (z $-> x) -> (z $-> y) -> z $-> cat_binprod
cat_binprod_beta_pr1: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr1 $o cat_binprod_corec z f g $== f
cat_binprod_beta_pr2: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr2 $o cat_binprod_corec z f g $== g
cat_binprod_eta_pr: forall (z : A) (f g : z $-> cat_binprod),
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
forall (z : A) (f : forall i : Bool, z $-> Bool_rec A x y i) 
(i : Bool), ?cat_pr i $o ?cat_prod_corec z f $== f i
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z : A, (z $-> x) -> (z $-> y) -> z $-> cat_binprod
cat_binprod_beta_pr1: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr1 $o cat_binprod_corec z f g $== f
cat_binprod_beta_pr2: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr2 $o cat_binprod_corec z f g $== g
cat_binprod_eta_pr: forall (z : A) (f g : z $-> cat_binprod),
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
forall (z : A) (f g : z $-> cat_binprod),
(forall i : Bool, ?cat_pr i $o f $== ?cat_pr i $o g) -> f $== g
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z : A, (z $-> x) -> (z $-> y) -> z $-> cat_binprod
cat_binprod_beta_pr1: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr1 $o cat_binprod_corec z f g $== f
cat_binprod_beta_pr2: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr2 $o cat_binprod_corec z f g $== g
cat_binprod_eta_pr: forall (z : A) (f g : z $-> cat_binprod),
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
forall i : Bool, cat_binprod $-> Bool_rec A x y i intros [|].A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z : A, (z $-> x) -> (z $-> y) -> z $-> cat_binprod
cat_binprod_beta_pr1: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr1 $o cat_binprod_corec z f g $== f
cat_binprod_beta_pr2: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr2 $o cat_binprod_corec z f g $== g
cat_binprod_eta_pr: forall (z : A) (f g : z $-> cat_binprod),
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
cat_binprod $-> Bool_rec A x y true
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z : A, (z $-> x) -> (z $-> y) -> z $-> cat_binprod
cat_binprod_beta_pr1: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr1 $o cat_binprod_corec z f g $== f
cat_binprod_beta_pr2: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr2 $o cat_binprod_corec z f g $== g
cat_binprod_eta_pr: forall (z : A) (f g : z $-> cat_binprod),
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
cat_binprod $-> Bool_rec A x y false
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z : A, (z $-> x) -> (z $-> y) -> z $-> cat_binprod
cat_binprod_beta_pr1: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr1 $o cat_binprod_corec z f g $== f
cat_binprod_beta_pr2: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr2 $o cat_binprod_corec z f g $== g
cat_binprod_eta_pr: forall (z : A) (f g : z $-> cat_binprod),
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
cat_binprod $-> Bool_rec A x y true exact cat_pr1.
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z : A, (z $-> x) -> (z $-> y) -> z $-> cat_binprod
cat_binprod_beta_pr1: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr1 $o cat_binprod_corec z f g $== f
cat_binprod_beta_pr2: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr2 $o cat_binprod_corec z f g $== g
cat_binprod_eta_pr: forall (z : A) (f g : z $-> cat_binprod),
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
cat_binprod $-> Bool_rec A x y false exact cat_pr2.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z : A, (z $-> x) -> (z $-> y) -> z $-> cat_binprod
cat_binprod_beta_pr1: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr1 $o cat_binprod_corec z f g $== f
cat_binprod_beta_pr2: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr2 $o cat_binprod_corec z f g $== g
cat_binprod_eta_pr: forall (z : A) (f g : z $-> cat_binprod),
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
forall z : A, (forall i : Bool, z $-> Bool_rec A x y i) -> z $-> cat_binprod intros z f.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z0 : A, (z0 $-> x) -> (z0 $-> y) -> z0 $-> cat_binprod
cat_binprod_beta_pr1: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr1 $o cat_binprod_corec z0 f0 g $== f0
cat_binprod_beta_pr2: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr2 $o cat_binprod_corec z0 f0 g $== g
cat_binprod_eta_pr: forall (z0 : A) (f0 g : z0 $-> cat_binprod),
cat_pr1 $o f0 $== cat_pr1 $o g -> cat_pr2 $o f0 $== cat_pr2 $o g -> f0 $== g
z: A
f: forall i : Bool, z $-> Bool_rec A x y i
z $-> cat_binprod
      napply cat_binprod_corec.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z0 : A, (z0 $-> x) -> (z0 $-> y) -> z0 $-> cat_binprod
cat_binprod_beta_pr1: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr1 $o cat_binprod_corec z0 f0 g $== f0
cat_binprod_beta_pr2: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr2 $o cat_binprod_corec z0 f0 g $== g
cat_binprod_eta_pr: forall (z0 : A) (f0 g : z0 $-> cat_binprod),
cat_pr1 $o f0 $== cat_pr1 $o g -> cat_pr2 $o f0 $== cat_pr2 $o g -> f0 $== g
z: A
f: forall i : Bool, z $-> Bool_rec A x y i
z $-> x
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z0 : A, (z0 $-> x) -> (z0 $-> y) -> z0 $-> cat_binprod
cat_binprod_beta_pr1: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr1 $o cat_binprod_corec z0 f0 g $== f0
cat_binprod_beta_pr2: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr2 $o cat_binprod_corec z0 f0 g $== g
cat_binprod_eta_pr: forall (z0 : A) (f0 g : z0 $-> cat_binprod),
cat_pr1 $o f0 $== cat_pr1 $o g -> cat_pr2 $o f0 $== cat_pr2 $o g -> f0 $== g
z: A
f: forall i : Bool, z $-> Bool_rec A x y i
z $-> y
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z0 : A, (z0 $-> x) -> (z0 $-> y) -> z0 $-> cat_binprod
cat_binprod_beta_pr1: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr1 $o cat_binprod_corec z0 f0 g $== f0
cat_binprod_beta_pr2: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr2 $o cat_binprod_corec z0 f0 g $== g
cat_binprod_eta_pr: forall (z0 : A) (f0 g : z0 $-> cat_binprod),
cat_pr1 $o f0 $== cat_pr1 $o g -> cat_pr2 $o f0 $== cat_pr2 $o g -> f0 $== g
z: A
f: forall i : Bool, z $-> Bool_rec A x y i
z $-> x exact (f true).
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z0 : A, (z0 $-> x) -> (z0 $-> y) -> z0 $-> cat_binprod
cat_binprod_beta_pr1: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr1 $o cat_binprod_corec z0 f0 g $== f0
cat_binprod_beta_pr2: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr2 $o cat_binprod_corec z0 f0 g $== g
cat_binprod_eta_pr: forall (z0 : A) (f0 g : z0 $-> cat_binprod),
cat_pr1 $o f0 $== cat_pr1 $o g -> cat_pr2 $o f0 $== cat_pr2 $o g -> f0 $== g
z: A
f: forall i : Bool, z $-> Bool_rec A x y i
z $-> y exact (f false).
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z : A, (z $-> x) -> (z $-> y) -> z $-> cat_binprod
cat_binprod_beta_pr1: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr1 $o cat_binprod_corec z f g $== f
cat_binprod_beta_pr2: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr2 $o cat_binprod_corec z f g $== g
cat_binprod_eta_pr: forall (z : A) (f g : z $-> cat_binprod),
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
forall (z : A) (f : forall i : Bool, z $-> Bool_rec A x y i) 
(i : Bool),
(fun i0 : Bool =>
 match i0 as b return (cat_binprod $-> Bool_rec A x y b) with
 | true => cat_pr1
 | false => cat_pr2
 end) i $o
(fun (z0 : A) (f0 : forall i0 : Bool, z0 $-> Bool_rec A x y i0) =>
 cat_binprod_corec z0 (f0 true) (f0 false)) z f $==
f i intros z f [|].A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z0 : A, (z0 $-> x) -> (z0 $-> y) -> z0 $-> cat_binprod
cat_binprod_beta_pr1: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr1 $o cat_binprod_corec z0 f0 g $== f0
cat_binprod_beta_pr2: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr2 $o cat_binprod_corec z0 f0 g $== g
cat_binprod_eta_pr: forall (z0 : A) (f0 g : z0 $-> cat_binprod),
cat_pr1 $o f0 $== cat_pr1 $o g -> cat_pr2 $o f0 $== cat_pr2 $o g -> f0 $== g
z: A
f: forall i : Bool, z $-> Bool_rec A x y i
cat_pr1 $o cat_binprod_corec z (f true) (f false) $== f true
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z0 : A, (z0 $-> x) -> (z0 $-> y) -> z0 $-> cat_binprod
cat_binprod_beta_pr1: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr1 $o cat_binprod_corec z0 f0 g $== f0
cat_binprod_beta_pr2: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr2 $o cat_binprod_corec z0 f0 g $== g
cat_binprod_eta_pr: forall (z0 : A) (f0 g : z0 $-> cat_binprod),
cat_pr1 $o f0 $== cat_pr1 $o g -> cat_pr2 $o f0 $== cat_pr2 $o g -> f0 $== g
z: A
f: forall i : Bool, z $-> Bool_rec A x y i
cat_pr2 $o cat_binprod_corec z (f true) (f false) $== f false
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z0 : A, (z0 $-> x) -> (z0 $-> y) -> z0 $-> cat_binprod
cat_binprod_beta_pr1: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr1 $o cat_binprod_corec z0 f0 g $== f0
cat_binprod_beta_pr2: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr2 $o cat_binprod_corec z0 f0 g $== g
cat_binprod_eta_pr: forall (z0 : A) (f0 g : z0 $-> cat_binprod),
cat_pr1 $o f0 $== cat_pr1 $o g -> cat_pr2 $o f0 $== cat_pr2 $o g -> f0 $== g
z: A
f: forall i : Bool, z $-> Bool_rec A x y i
cat_pr1 $o cat_binprod_corec z (f true) (f false) $== f true napply cat_binprod_beta_pr1.
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z0 : A, (z0 $-> x) -> (z0 $-> y) -> z0 $-> cat_binprod
cat_binprod_beta_pr1: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr1 $o cat_binprod_corec z0 f0 g $== f0
cat_binprod_beta_pr2: forall (z0 : A) (f0 : z0 $-> x) (g : z0 $-> y),
cat_pr2 $o cat_binprod_corec z0 f0 g $== g
cat_binprod_eta_pr: forall (z0 : A) (f0 g : z0 $-> cat_binprod),
cat_pr1 $o f0 $== cat_pr1 $o g -> cat_pr2 $o f0 $== cat_pr2 $o g -> f0 $== g
z: A
f: forall i : Bool, z $-> Bool_rec A x y i
cat_pr2 $o cat_binprod_corec z (f true) (f false) $== f false napply cat_binprod_beta_pr2.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z : A, (z $-> x) -> (z $-> y) -> z $-> cat_binprod
cat_binprod_beta_pr1: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr1 $o cat_binprod_corec z f g $== f
cat_binprod_beta_pr2: forall (z : A) (f : z $-> x) (g : z $-> y),
cat_pr2 $o cat_binprod_corec z f g $== g
cat_binprod_eta_pr: forall (z : A) (f g : z $-> cat_binprod),
cat_pr1 $o f $== cat_pr1 $o g -> cat_pr2 $o f $== cat_pr2 $o g -> f $== g
forall (z : A) (f g : z $-> cat_binprod),
(forall i : Bool,
 (fun i0 : Bool =>
  match i0 as b return (cat_binprod $-> Bool_rec A x y b) with
  | true => cat_pr1
  | false => cat_pr2
  end) i $o
 f $==
 (fun i0 : Bool =>
  match i0 as b return (cat_binprod $-> Bool_rec A x y b) with
  | true => cat_pr1
  | false => cat_pr2
  end) i $o
 g) ->
f $== g intros z f g p.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z0 : A, (z0 $-> x) -> (z0 $-> y) -> z0 $-> cat_binprod
cat_binprod_beta_pr1: forall (z0 : A) (f0 : z0 $-> x) (g0 : z0 $-> y),
cat_pr1 $o cat_binprod_corec z0 f0 g0 $== f0
cat_binprod_beta_pr2: forall (z0 : A) (f0 : z0 $-> x) (g0 : z0 $-> y),
cat_pr2 $o cat_binprod_corec z0 f0 g0 $== g0
cat_binprod_eta_pr: forall (z0 : A) (f0 g0 : z0 $-> cat_binprod),
cat_pr1 $o f0 $== cat_pr1 $o g0 ->
cat_pr2 $o f0 $== cat_pr2 $o g0 -> f0 $== g0
z: A
f, g: z $-> cat_binprod
p: forall i : Bool,
(fun i0 : Bool =>
 match i0 as b return (cat_binprod $-> Bool_rec A x y b) with
 | true => cat_pr1
 | false => cat_pr2
 end) i $o
f $==
(fun i0 : Bool =>
 match i0 as b return (cat_binprod $-> Bool_rec A x y b) with
 | true => cat_pr1
 | false => cat_pr2
 end) i $o
g
f $== g
      napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z0 : A, (z0 $-> x) -> (z0 $-> y) -> z0 $-> cat_binprod
cat_binprod_beta_pr1: forall (z0 : A) (f0 : z0 $-> x) (g0 : z0 $-> y),
cat_pr1 $o cat_binprod_corec z0 f0 g0 $== f0
cat_binprod_beta_pr2: forall (z0 : A) (f0 : z0 $-> x) (g0 : z0 $-> y),
cat_pr2 $o cat_binprod_corec z0 f0 g0 $== g0
cat_binprod_eta_pr: forall (z0 : A) (f0 g0 : z0 $-> cat_binprod),
cat_pr1 $o f0 $== cat_pr1 $o g0 ->
cat_pr2 $o f0 $== cat_pr2 $o g0 -> f0 $== g0
z: A
f, g: z $-> cat_binprod
p: forall i : Bool,
(fun i0 : Bool =>
 match i0 as b return (cat_binprod $-> Bool_rec A x y b) with
 | true => cat_pr1
 | false => cat_pr2
 end) i $o
f $==
(fun i0 : Bool =>
 match i0 as b return (cat_binprod $-> Bool_rec A x y b) with
 | true => cat_pr1
 | false => cat_pr2
 end) i $o
g
cat_pr1 $o f $== cat_pr1 $o g
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z0 : A, (z0 $-> x) -> (z0 $-> y) -> z0 $-> cat_binprod
cat_binprod_beta_pr1: forall (z0 : A) (f0 : z0 $-> x) (g0 : z0 $-> y),
cat_pr1 $o cat_binprod_corec z0 f0 g0 $== f0
cat_binprod_beta_pr2: forall (z0 : A) (f0 : z0 $-> x) (g0 : z0 $-> y),
cat_pr2 $o cat_binprod_corec z0 f0 g0 $== g0
cat_binprod_eta_pr: forall (z0 : A) (f0 g0 : z0 $-> cat_binprod),
cat_pr1 $o f0 $== cat_pr1 $o g0 ->
cat_pr2 $o f0 $== cat_pr2 $o g0 -> f0 $== g0
z: A
f, g: z $-> cat_binprod
p: forall i : Bool,
(fun i0 : Bool =>
 match i0 as b return (cat_binprod $-> Bool_rec A x y b) with
 | true => cat_pr1
 | false => cat_pr2
 end) i $o
f $==
(fun i0 : Bool =>
 match i0 as b return (cat_binprod $-> Bool_rec A x y b) with
 | true => cat_pr1
 | false => cat_pr2
 end) i $o
g
cat_pr2 $o f $== cat_pr2 $o g
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z0 : A, (z0 $-> x) -> (z0 $-> y) -> z0 $-> cat_binprod
cat_binprod_beta_pr1: forall (z0 : A) (f0 : z0 $-> x) (g0 : z0 $-> y),
cat_pr1 $o cat_binprod_corec z0 f0 g0 $== f0
cat_binprod_beta_pr2: forall (z0 : A) (f0 : z0 $-> x) (g0 : z0 $-> y),
cat_pr2 $o cat_binprod_corec z0 f0 g0 $== g0
cat_binprod_eta_pr: forall (z0 : A) (f0 g0 : z0 $-> cat_binprod),
cat_pr1 $o f0 $== cat_pr1 $o g0 ->
cat_pr2 $o f0 $== cat_pr2 $o g0 -> f0 $== g0
z: A
f, g: z $-> cat_binprod
p: forall i : Bool,
(fun i0 : Bool =>
 match i0 as b return (cat_binprod $-> Bool_rec A x y b) with
 | true => cat_pr1
 | false => cat_pr2
 end) i $o
f $==
(fun i0 : Bool =>
 match i0 as b return (cat_binprod $-> Bool_rec A x y b) with
 | true => cat_pr1
 | false => cat_pr2
 end) i $o
g
cat_pr1 $o f $== cat_pr1 $o g exact (p true).
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
x, y, cat_binprod: A
cat_pr1: cat_binprod $-> x
cat_pr2: cat_binprod $-> y
cat_binprod_corec: forall z0 : A, (z0 $-> x) -> (z0 $-> y) -> z0 $-> cat_binprod
cat_binprod_beta_pr1: forall (z0 : A) (f0 : z0 $-> x) (g0 : z0 $-> y),
cat_pr1 $o cat_binprod_corec z0 f0 g0 $== f0
cat_binprod_beta_pr2: forall (z0 : A) (f0 : z0 $-> x) (g0 : z0 $-> y),
cat_pr2 $o cat_binprod_corec z0 f0 g0 $== g0
cat_binprod_eta_pr: forall (z0 : A) (f0 g0 : z0 $-> cat_binprod),
cat_pr1 $o f0 $== cat_pr1 $o g0 ->
cat_pr2 $o f0 $== cat_pr2 $o g0 -> f0 $== g0
z: A
f, g: z $-> cat_binprod
p: forall i : Bool,
(fun i0 : Bool =>
 match i0 as b return (cat_binprod $-> Bool_rec A x y b) with
 | true => cat_pr1
 | false => cat_pr2
 end) i $o
f $==
(fun i0 : Bool =>
 match i0 as b return (cat_binprod $-> Bool_rec A x y b) with
 | true => cat_pr1
 | false => cat_pr2
 end) i $o
g
cat_pr2 $o f $== cat_pr2 $o g exact (p false).
  Defined.

  (** A convenience wrapper for building binary products. *)
  Definition Build_BinaryProduct : BinaryProduct x y
    := Build_Product' _ cat_binprod Build_IsBinaryProduct.

End BinaryProductConstructors.

Definition cat_binprod {A: Type} `{HasBinaryProducts A} (x y : A) : A
  := cat_prod (Bool_rec _ x y).

Definition cat_binprod_eta_pr_x_xx {A : Type} `{HasBinaryProducts A}
  {w x y z : A} (f g : w $-> cat_binprod x (cat_binprod y z))
  : cat_pr1 _ $o f $== cat_pr1 _ $o g
  -> cat_pr1 _ $o cat_pr2 _ $o f $== cat_pr1 _ $o cat_pr2 _ $o g
  -> cat_pr2 _ $o cat_pr2 _ $o f $== cat_pr2 _ $o cat_pr2 _ $o g
  -> f $== g.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod x (cat_binprod y z)
cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) $o g ->
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g ->
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g -> 
f $== g
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod x (cat_binprod y z)
cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) $o g ->
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g ->
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g -> 
f $== g
  intros p q r.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) $o g
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
f $== g
  napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) $o g
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) $o g
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) $o g
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) $o g
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) $o g exact p.
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) $o g
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) $o g
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
cat_pr1 (cat_binprod y z) $o (cat_pr2 (cat_binprod x (cat_binprod y z)) $o f) $==
cat_pr1 (cat_binprod y z) $o (cat_pr2 (cat_binprod x (cat_binprod y z)) $o g)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) $o g
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
cat_pr2 (cat_binprod y z) $o (cat_pr2 (cat_binprod x (cat_binprod y z)) $o f) $==
cat_pr2 (cat_binprod y z) $o (cat_pr2 (cat_binprod x (cat_binprod y z)) $o g)
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) $o g
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
cat_pr1 (cat_binprod y z) $o (cat_pr2 (cat_binprod x (cat_binprod y z)) $o f) $==
cat_pr1 (cat_binprod y z) $o (cat_pr2 (cat_binprod x (cat_binprod y z)) $o g) exact (cat_assoc_opp _ _ _ $@ q $@ cat_assoc _ _ _).
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) $o g
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o g
cat_pr2 (cat_binprod y z) $o (cat_pr2 (cat_binprod x (cat_binprod y z)) $o f) $==
cat_pr2 (cat_binprod y z) $o (cat_pr2 (cat_binprod x (cat_binprod y z)) $o g) exact (cat_assoc_opp _ _ _ $@ r $@ cat_assoc _ _ _).
Defined.

Definition cat_binprod_eta_pr_xx_x {A : Type} `{HasBinaryProducts A} {w x y z : A}
  (f g : w $-> cat_binprod (cat_binprod x y) z)
  : cat_pr1 _ $o cat_pr1 _ $o f $== cat_pr1 _ $o cat_pr1 _ $o g
  -> cat_pr2 _ $o cat_pr1 _ $o f $== cat_pr2 _ $o cat_pr1 _ $o g
  -> cat_pr2 _ $o f $== cat_pr2 _ $o g
  -> f $== g.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod (cat_binprod x y) z
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g ->
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g ->
cat_pr2 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_binprod (cat_binprod x y) z) $o g -> 
f $== g
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod (cat_binprod x y) z
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g ->
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g ->
cat_pr2 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_binprod (cat_binprod x y) z) $o g -> 
f $== g
  intros p q r.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod (cat_binprod x y) z
p: cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
q: cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
r: cat_pr2 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_binprod (cat_binprod x y) z) $o g
f $== g
  napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod (cat_binprod x y) z
p: cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
q: cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
r: cat_pr2 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_binprod (cat_binprod x y) z) $o g
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod (cat_binprod x y) z
p: cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
q: cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
r: cat_pr2 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_binprod (cat_binprod x y) z) $o g
cat_pr2 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_binprod (cat_binprod x y) z) $o g
  2: exact r.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod (cat_binprod x y) z
p: cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
q: cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
r: cat_pr2 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_binprod (cat_binprod x y) z) $o g
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
  napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod (cat_binprod x y) z
p: cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
q: cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
r: cat_pr2 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_binprod (cat_binprod x y) z) $o g
cat_pr1 (cat_binprod x y) $o (cat_pr1 (cat_binprod (cat_binprod x y) z) $o f) $==
cat_pr1 (cat_binprod x y) $o (cat_pr1 (cat_binprod (cat_binprod x y) z) $o g)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod (cat_binprod x y) z
p: cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
q: cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
r: cat_pr2 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_binprod (cat_binprod x y) z) $o g
cat_pr2 (cat_binprod x y) $o (cat_pr1 (cat_binprod (cat_binprod x y) z) $o f) $==
cat_pr2 (cat_binprod x y) $o (cat_pr1 (cat_binprod (cat_binprod x y) z) $o g)
  1,2: refine (cat_assoc_opp _ _ _ $@ _ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod (cat_binprod x y) z
p: cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
q: cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
r: cat_pr2 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_binprod (cat_binprod x y) z) $o g
cat_pr1 (cat_binprod x y) $o cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_binprod x y) $o cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod (cat_binprod x y) z
p: cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
q: cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
r: cat_pr2 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_binprod (cat_binprod x y) z) $o g
cat_pr2 (cat_binprod x y) $o cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_binprod x y) $o cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod (cat_binprod x y) z
p: cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
q: cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
r: cat_pr2 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_binprod (cat_binprod x y) z) $o g
cat_pr1 (cat_binprod x y) $o cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_binprod x y) $o cat_pr1 (cat_binprod (cat_binprod x y) z) $o g exact p.
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
w, x, y, z: A
f, g: w $-> cat_binprod (cat_binprod x y) z
p: cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
q: cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o g
r: cat_pr2 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_binprod (cat_binprod x y) z) $o g
cat_pr2 (cat_binprod x y) $o cat_pr1 (cat_binprod (cat_binprod x y) z) $o f $==
cat_pr2 (cat_binprod x y) $o cat_pr1 (cat_binprod (cat_binprod x y) z) $o g exact q.
Defined.

Definition cat_binprod_eta_pr_x_xx_id {A : Type} `{HasBinaryProducts A} {x y z : A}
  (f : cat_binprod x (cat_binprod y z) $-> cat_binprod x (cat_binprod y z))
  : cat_pr1 _ $o f $== cat_pr1 _
  -> cat_pr1 _ $o cat_pr2 _ $o f $== cat_pr1 _ $o cat_pr2 _
  -> cat_pr2 _ $o cat_pr2 _ $o f $== cat_pr2 _ $o cat_pr2 _
  -> f $== Id _.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
f: cat_binprod x (cat_binprod y z) $-> cat_binprod x (cat_binprod y z)
cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) ->
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) ->
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) ->
f $== Id (cat_binprod x (cat_binprod y z))
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
f: cat_binprod x (cat_binprod y z) $-> cat_binprod x (cat_binprod y z)
cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) ->
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) ->
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) ->
f $== Id (cat_binprod x (cat_binprod y z))
  intros p q r.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
f: cat_binprod x (cat_binprod y z) $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z))
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
f $== Id (cat_binprod x (cat_binprod y z))
  napply cat_binprod_eta_pr_x_xx.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
f: cat_binprod x (cat_binprod y z) $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z))
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) $o
Id (cat_binprod x (cat_binprod y z))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
f: cat_binprod x (cat_binprod y z) $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z))
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o
Id (cat_binprod x (cat_binprod y z))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
f: cat_binprod x (cat_binprod y z) $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z))
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o
Id (cat_binprod x (cat_binprod y z))
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
f: cat_binprod x (cat_binprod y z) $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z))
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z)) $o
Id (cat_binprod x (cat_binprod y z)) exact (p $@ (cat_idr _)^$).
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
f: cat_binprod x (cat_binprod y z) $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z))
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o
Id (cat_binprod x (cat_binprod y z)) exact (q $@ (cat_idr _)^$).
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
f: cat_binprod x (cat_binprod y z) $-> cat_binprod x (cat_binprod y z)
p: cat_pr1 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_binprod x (cat_binprod y z))
q: cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
r: cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o f $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o
Id (cat_binprod x (cat_binprod y z)) exact (r $@ (cat_idr _)^$).
Defined.

(** From binary products, all [Bool]-shaped products can be constructed. This should not be an instance to avoid a cycle with [hasbinaryproducts_hasproductsbool]. *)
Definition hasproductsbool_hasbinaryproducts {A : Type} `{hbp : HasBinaryProducts A}
  : HasProducts A Bool.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
HasProducts A Bool
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
HasProducts A Bool
  intros x.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
Product x
  snapply Build_Product.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
A
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
forall i : Bool, ?cat_prod $-> x i
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
forall z : A, (forall i : Bool, z $-> x i) -> z $-> ?cat_prod
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
forall (z : A) (f : forall i : Bool, z $-> x i) (i : Bool),
?cat_pr i $o ?cat_prod_corec z f $== f i
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
forall (z : A) (f g : z $-> ?cat_prod),
(forall i : Bool, ?cat_pr i $o f $== ?cat_pr i $o g) -> f $== g
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
A exact (cat_binprod (x true) (x false)).
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
forall i : Bool, cat_binprod (x true) (x false) $-> x i intros [|].A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
cat_binprod (x true) (x false) $-> x true
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
cat_binprod (x true) (x false) $-> x false
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
cat_binprod (x true) (x false) $-> x true exact (cat_pr1 _).
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
cat_binprod (x true) (x false) $-> x false exact (cat_pr2 _).
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
forall z : A,
(forall i : Bool, z $-> x i) -> z $-> cat_binprod (x true) (x false) intros z f.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
z: A
f: forall i : Bool, z $-> x i
z $-> cat_binprod (x true) (x false)
    exact (cat_binprod_corec _ (f true) (f false)).
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
forall (z : A) (f : forall i : Bool, z $-> x i) (i : Bool),
(fun i0 : Bool =>
 match i0 as b return (cat_binprod (x true) (x false) $-> x b) with
 | true => cat_pr1 (cat_binprod (x true) (x false))
 | false => cat_pr2 (cat_binprod (x true) (x false))
 end) i $o
(fun (z0 : A) (f0 : forall i0 : Bool, z0 $-> x i0) =>
 cat_binprod_corec (cat_binprod (x true) (x false)) (f0 true) (f0 false)) z f $==
f i intros z f [|].A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
z: A
f: forall i : Bool, z $-> x i
cat_pr1 (cat_binprod (x true) (x false)) $o
cat_binprod_corec (cat_binprod (x true) (x false)) (f true) (f false) $==
f true
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
z: A
f: forall i : Bool, z $-> x i
cat_pr2 (cat_binprod (x true) (x false)) $o
cat_binprod_corec (cat_binprod (x true) (x false)) (f true) (f false) $==
f false
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
z: A
f: forall i : Bool, z $-> x i
cat_pr1 (cat_binprod (x true) (x false)) $o
cat_binprod_corec (cat_binprod (x true) (x false)) (f true) (f false) $==
f true exact (cat_binprod_beta_pr1 _ (f true) (f false)).
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
z: A
f: forall i : Bool, z $-> x i
cat_pr2 (cat_binprod (x true) (x false)) $o
cat_binprod_corec (cat_binprod (x true) (x false)) (f true) (f false) $==
f false exact (cat_binprod_beta_pr2 _ (f true) (f false)).
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
forall (z : A) (f g : z $-> cat_binprod (x true) (x false)),
(forall i : Bool,
 (fun i0 : Bool =>
  match i0 as b return (cat_binprod (x true) (x false) $-> x b) with
  | true => cat_pr1 (cat_binprod (x true) (x false))
  | false => cat_pr2 (cat_binprod (x true) (x false))
  end) i $o
 f $==
 (fun i0 : Bool =>
  match i0 as b return (cat_binprod (x true) (x false) $-> x b) with
  | true => cat_pr1 (cat_binprod (x true) (x false))
  | false => cat_pr2 (cat_binprod (x true) (x false))
  end) i $o
 g) ->
f $== g intros z f g p.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
z: A
f, g: z $-> cat_binprod (x true) (x false)
p: forall i : Bool,
(fun i0 : Bool =>
 match i0 as b return (cat_binprod (x true) (x false) $-> x b) with
 | true => cat_pr1 (cat_binprod (x true) (x false))
 | false => cat_pr2 (cat_binprod (x true) (x false))
 end) i $o
f $==
(fun i0 : Bool =>
 match i0 as b return (cat_binprod (x true) (x false) $-> x b) with
 | true => cat_pr1 (cat_binprod (x true) (x false))
 | false => cat_pr2 (cat_binprod (x true) (x false))
 end) i $o
g
f $== g
    napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
z: A
f, g: z $-> cat_binprod (x true) (x false)
p: forall i : Bool,
(fun i0 : Bool =>
 match i0 as b return (cat_binprod (x true) (x false) $-> x b) with
 | true => cat_pr1 (cat_binprod (x true) (x false))
 | false => cat_pr2 (cat_binprod (x true) (x false))
 end) i $o
f $==
(fun i0 : Bool =>
 match i0 as b return (cat_binprod (x true) (x false) $-> x b) with
 | true => cat_pr1 (cat_binprod (x true) (x false))
 | false => cat_pr2 (cat_binprod (x true) (x false))
 end) i $o
g
cat_pr1 (cat_binprod (x true) (x false)) $o f $==
cat_pr1 (cat_binprod (x true) (x false)) $o g
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
z: A
f, g: z $-> cat_binprod (x true) (x false)
p: forall i : Bool,
(fun i0 : Bool =>
 match i0 as b return (cat_binprod (x true) (x false) $-> x b) with
 | true => cat_pr1 (cat_binprod (x true) (x false))
 | false => cat_pr2 (cat_binprod (x true) (x false))
 end) i $o
f $==
(fun i0 : Bool =>
 match i0 as b return (cat_binprod (x true) (x false) $-> x b) with
 | true => cat_pr1 (cat_binprod (x true) (x false))
 | false => cat_pr2 (cat_binprod (x true) (x false))
 end) i $o
g
cat_pr2 (cat_binprod (x true) (x false)) $o f $==
cat_pr2 (cat_binprod (x true) (x false)) $o g
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
z: A
f, g: z $-> cat_binprod (x true) (x false)
p: forall i : Bool,
(fun i0 : Bool =>
 match i0 as b return (cat_binprod (x true) (x false) $-> x b) with
 | true => cat_pr1 (cat_binprod (x true) (x false))
 | false => cat_pr2 (cat_binprod (x true) (x false))
 end) i $o
f $==
(fun i0 : Bool =>
 match i0 as b return (cat_binprod (x true) (x false) $-> x b) with
 | true => cat_pr1 (cat_binprod (x true) (x false))
 | false => cat_pr2 (cat_binprod (x true) (x false))
 end) i $o
g
cat_pr1 (cat_binprod (x true) (x false)) $o f $==
cat_pr1 (cat_binprod (x true) (x false)) $o g exact (p true).
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
x: Bool -> A
z: A
f, g: z $-> cat_binprod (x true) (x false)
p: forall i : Bool,
(fun i0 : Bool =>
 match i0 as b return (cat_binprod (x true) (x false) $-> x b) with
 | true => cat_pr1 (cat_binprod (x true) (x false))
 | false => cat_pr2 (cat_binprod (x true) (x false))
 end) i $o
f $==
(fun i0 : Bool =>
 match i0 as b return (cat_binprod (x true) (x false) $-> x b) with
 | true => cat_pr1 (cat_binprod (x true) (x false))
 | false => cat_pr2 (cat_binprod (x true) (x false))
 end) i $o
g
cat_pr2 (cat_binprod (x true) (x false)) $o f $==
cat_pr2 (cat_binprod (x true) (x false)) $o g exact (p false).
Defined.

(** *** Operations on indexed products *)

(** We can take the disjoint union of the index set of an indexed product if we have all binary products. *)

Definition cat_product_index_sum {A : Type} `{HasBinaryProducts A} {I J : Type}
  (x : I -> A) (prod_x : A) `{!IsProduct x prod_x}
  (y : J -> A) (prod_y : A) `{!IsProduct y prod_y}
  : Product (I:=I + J) (sum_ind _ x y).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
Product (sum_ind (fun _ : I + J => A) x y)
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
Product (sum_ind (fun _ : I + J => A) x y)
  srapply Build_Product.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
A
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
forall i : I + J, ?cat_prod $-> sum_ind (fun _ : I + J => A) x y i
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
forall z : A,
(forall i : I + J, z $-> sum_ind (fun _ : I + J => A) x y i) ->
z $-> ?cat_prod
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
forall (z : A)
(f : forall i : I + J, z $-> sum_ind (fun _ : I + J => A) x y i) 
(i : I + J), ?cat_pr i $o ?cat_prod_corec z f $== f i
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
forall (z : A) (f g : z $-> ?cat_prod),
(forall i : I + J, ?cat_pr i $o f $== ?cat_pr i $o g) -> f $== g
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
A exact (cat_binprod prod_x prod_y).
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
forall i : I + J,
cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y i intros [i | j].A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
i: I
cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y (inl i)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
j: J
cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y (inr j)
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
i: I
cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y (inl i) exact (cat_pr i $o cat_pr1 _).
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
j: J
cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y (inr j) exact (cat_pr j $o cat_pr2 _).
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
forall z : A,
(forall i : I + J, z $-> sum_ind (fun _ : I + J => A) x y i) ->
z $-> cat_binprod prod_x prod_y intros z f.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f: forall i : I + J, z $-> sum_ind (fun _ : I + J => A) x y i
z $-> cat_binprod prod_x prod_y
    rapply cat_binprod_corec.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f: forall i : I + J, z $-> sum_ind (fun _ : I + J => A) x y i
z $-> prod_x
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f: forall i : I + J, z $-> sum_ind (fun _ : I + J => A) x y i
z $-> prod_y
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f: forall i : I + J, z $-> sum_ind (fun _ : I + J => A) x y i
z $-> prod_x rapply cat_prod_corec.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f: forall i : I + J, z $-> sum_ind (fun _ : I + J => A) x y i
forall i : I, z $-> x i
      exact (f o inl).
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f: forall i : I + J, z $-> sum_ind (fun _ : I + J => A) x y i
z $-> prod_y rapply cat_prod_corec.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f: forall i : I + J, z $-> sum_ind (fun _ : I + J => A) x y i
forall i : J, z $-> y i
      exact (f o inr).
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
forall (z : A)
(f : forall i : I + J, z $-> sum_ind (fun _ : I + J => A) x y i) 
(i : I + J),
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i $o
(fun (z0 : A)
   (f0 : forall i0 : I + J, z0 $-> sum_ind (fun _ : I + J => A) x y i0) =>
 cat_binprod_corec (cat_binprod prod_x prod_y)
   (cat_prod_corec prod_x (f0 o inl)) (cat_prod_corec prod_y (f0 o inr)))
  z f $==
f i intros z f [i | j].A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f: forall i0 : I + J, z $-> sum_ind (fun _ : I + J => A) x y i0
i: I
cat_pr i $o cat_pr1 (cat_binprod prod_x prod_y) $o
cat_binprod_corec (cat_binprod prod_x prod_y)
  (cat_prod_corec prod_x (fun x0 : I => f (inl x0)))
  (cat_prod_corec prod_y (fun x0 : J => f (inr x0))) $==
f (inl i)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f: forall i : I + J, z $-> sum_ind (fun _ : I + J => A) x y i
j: J
cat_pr j $o cat_pr2 (cat_binprod prod_x prod_y) $o
cat_binprod_corec (cat_binprod prod_x prod_y)
  (cat_prod_corec prod_x (fun x0 : I => f (inl x0)))
  (cat_prod_corec prod_y (fun x0 : J => f (inr x0))) $==
f (inr j)
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f: forall i0 : I + J, z $-> sum_ind (fun _ : I + J => A) x y i0
i: I
cat_pr i $o cat_pr1 (cat_binprod prod_x prod_y) $o
cat_binprod_corec (cat_binprod prod_x prod_y)
  (cat_prod_corec prod_x (fun x0 : I => f (inl x0)))
  (cat_prod_corec prod_y (fun x0 : J => f (inr x0))) $==
f (inl i) nrefine (cat_assoc _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f: forall i0 : I + J, z $-> sum_ind (fun _ : I + J => A) x y i0
i: I
cat_pr i $o
(cat_pr1 (cat_binprod prod_x prod_y) $o
 cat_binprod_corec (cat_binprod prod_x prod_y)
   (cat_prod_corec prod_x (fun x0 : I => f (inl x0)))
   (cat_prod_corec prod_y (fun x0 : J => f (inr x0)))) $==
f (inl i)
      nrefine ((_ $@L cat_binprod_beta_pr1 _ _ _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f: forall i0 : I + J, z $-> sum_ind (fun _ : I + J => A) x y i0
i: I
cat_pr i $o cat_prod_corec prod_x (fun x0 : I => f (inl x0)) $== f (inl i)
      tapply (cat_prod_beta prod_x).
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f: forall i : I + J, z $-> sum_ind (fun _ : I + J => A) x y i
j: J
cat_pr j $o cat_pr2 (cat_binprod prod_x prod_y) $o
cat_binprod_corec (cat_binprod prod_x prod_y)
  (cat_prod_corec prod_x (fun x0 : I => f (inl x0)))
  (cat_prod_corec prod_y (fun x0 : J => f (inr x0))) $==
f (inr j) nrefine (cat_assoc _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f: forall i : I + J, z $-> sum_ind (fun _ : I + J => A) x y i
j: J
cat_pr j $o
(cat_pr2 (cat_binprod prod_x prod_y) $o
 cat_binprod_corec (cat_binprod prod_x prod_y)
   (cat_prod_corec prod_x (fun x0 : I => f (inl x0)))
   (cat_prod_corec prod_y (fun x0 : J => f (inr x0)))) $==
f (inr j)
      nrefine ((_ $@L cat_binprod_beta_pr2 _ _ _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f: forall i : I + J, z $-> sum_ind (fun _ : I + J => A) x y i
j: J
cat_pr j $o cat_prod_corec prod_y (fun x0 : J => f (inr x0)) $== f (inr j)
      tapply (cat_prod_beta prod_y).
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
forall (z : A) (f g : z $-> cat_binprod prod_x prod_y),
(forall i : I + J,
 (fun i0 : I + J =>
  match
    i0 as s
    return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
  with
  | inl i1 =>
      (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
  | inr j =>
      (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
  end) i $o
 f $==
 (fun i0 : I + J =>
  match
    i0 as s
    return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
  with
  | inl i1 =>
      (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
  | inr j =>
      (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
  end) i $o
 g) ->
f $== g intros z f g r.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f, g: z $-> cat_binprod prod_x prod_y
r: forall i : I + J,
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i $o
f $==
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i $o
g
f $== g
    rapply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f, g: z $-> cat_binprod prod_x prod_y
r: forall i : I + J,
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i $o
f $==
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i $o
g
cat_pr1 (cat_binprod prod_x prod_y) $o f $==
cat_pr1 (cat_binprod prod_x prod_y) $o g
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f, g: z $-> cat_binprod prod_x prod_y
r: forall i : I + J,
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i $o
f $==
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i $o
g
cat_pr2 (cat_binprod prod_x prod_y) $o f $==
cat_pr2 (cat_binprod prod_x prod_y) $o g
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f, g: z $-> cat_binprod prod_x prod_y
r: forall i : I + J,
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i $o
f $==
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i $o
g
cat_pr1 (cat_binprod prod_x prod_y) $o f $==
cat_pr1 (cat_binprod prod_x prod_y) $o g rapply cat_prod_pr_eta.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f, g: z $-> cat_binprod prod_x prod_y
r: forall i : I + J,
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i $o
f $==
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i $o
g
forall i : I,
cat_pr i $o (cat_pr1 (cat_binprod prod_x prod_y) $o f) $==
cat_pr i $o (cat_pr1 (cat_binprod prod_x prod_y) $o g)
      intros i.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f, g: z $-> cat_binprod prod_x prod_y
r: forall i0 : I + J,
(fun i1 : I + J =>
 match
   i1 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i2 =>
     (fun i3 : I => cat_pr i3 $o cat_pr1 (cat_binprod prod_x prod_y)) i2
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i0 $o
f $==
(fun i1 : I + J =>
 match
   i1 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i2 =>
     (fun i3 : I => cat_pr i3 $o cat_pr1 (cat_binprod prod_x prod_y)) i2
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i0 $o
g
i: I
cat_pr i $o (cat_pr1 (cat_binprod prod_x prod_y) $o f) $==
cat_pr i $o (cat_pr1 (cat_binprod prod_x prod_y) $o g)
      exact ((cat_assoc _ _ _)^$ $@ r (inl i) $@ cat_assoc _ _ _).
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f, g: z $-> cat_binprod prod_x prod_y
r: forall i : I + J,
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i $o
f $==
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i $o
g
cat_pr2 (cat_binprod prod_x prod_y) $o f $==
cat_pr2 (cat_binprod prod_x prod_y) $o g rapply cat_prod_pr_eta.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f, g: z $-> cat_binprod prod_x prod_y
r: forall i : I + J,
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i $o
f $==
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j =>
     (fun j0 : J => cat_pr j0 $o cat_pr2 (cat_binprod prod_x prod_y)) j
 end) i $o
g
forall i : J,
cat_pr i $o (cat_pr2 (cat_binprod prod_x prod_y) $o f) $==
cat_pr i $o (cat_pr2 (cat_binprod prod_x prod_y) $o g)
      intros j.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
I, J: Type
x: I -> A
prod_x: A
IsProduct0: IsProduct x prod_x
y: J -> A
prod_y: A
IsProduct1: IsProduct y prod_y
z: A
f, g: z $-> cat_binprod prod_x prod_y
r: forall i : I + J,
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j0 =>
     (fun j1 : J => cat_pr j1 $o cat_pr2 (cat_binprod prod_x prod_y)) j0
 end) i $o
f $==
(fun i0 : I + J =>
 match
   i0 as s
   return (cat_binprod prod_x prod_y $-> sum_ind (fun _ : I + J => A) x y s)
 with
 | inl i1 =>
     (fun i2 : I => cat_pr i2 $o cat_pr1 (cat_binprod prod_x prod_y)) i1
 | inr j0 =>
     (fun j1 : J => cat_pr j1 $o cat_pr2 (cat_binprod prod_x prod_y)) j0
 end) i $o
g
j: J
cat_pr j $o (cat_pr2 (cat_binprod prod_x prod_y) $o f) $==
cat_pr j $o (cat_pr2 (cat_binprod prod_x prod_y) $o g)
      exact ((cat_assoc _ _ _)^$ $@ r (inr j) $@ cat_assoc _ _ _).
Defined.

(** *** Binary product functor *)

(** We prove bifunctoriality of [cat_binprod : A -> A -> A] by factoring it as [cat_prod Bool o Bool_rec A]. First, we prove that [Bool_rec A : A -> A -> (Bool -> A)] is a bifunctor. *)
Local Instance is0bifunctor_boolrec {A : Type} `{Is1Cat A}
  : Is0Bifunctor (Bool_rec A).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor (Bool_rec A)
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Bifunctor (Bool_rec A)
  snapply Build_Is0Bifunctor'.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is01Cat A
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is01Cat A
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Functor (uncurry (Bool_rec A))
  1,2: exact _.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is0Functor (uncurry (Bool_rec A))
  snapply Build_Is0Functor.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
forall a b : A * A,
(a $-> b) -> uncurry (Bool_rec A) a $-> uncurry (Bool_rec A) b
  intros [a b] [a' b'] [f g] [ | ].A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
a, b, a', b': A
f: fst (a, b) $-> fst (a', b')
g: snd (a, b) $-> snd (a', b')
uncurry (Bool_rec A) (a, b) true $-> uncurry (Bool_rec A) (a', b') true
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
a, b, a', b': A
f: fst (a, b) $-> fst (a', b')
g: snd (a, b) $-> snd (a', b')
uncurry (Bool_rec A) (a, b) false $-> uncurry (Bool_rec A) (a', b') false
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
a, b, a', b': A
f: fst (a, b) $-> fst (a', b')
g: snd (a, b) $-> snd (a', b')
uncurry (Bool_rec A) (a, b) true $-> uncurry (Bool_rec A) (a', b') true exact f.
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
a, b, a', b': A
f: fst (a, b) $-> fst (a', b')
g: snd (a, b) $-> snd (a', b')
uncurry (Bool_rec A) (a, b) false $-> uncurry (Bool_rec A) (a', b') false exact g.
Defined.

Local Instance is1bifunctor_boolrec {A : Type} `{Is1Cat A}
  : Is1Bifunctor (Bool_rec A).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is1Bifunctor (Bool_rec A)
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is1Bifunctor (Bool_rec A)
  snapply Build_Is1Bifunctor'.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
Is1Functor (uncurry (Bool_rec A))
  snapply Build_Is1Functor.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
forall (a b : A * A) (f g : a $-> b),
f $== g -> fmap (uncurry (Bool_rec A)) f $== fmap (uncurry (Bool_rec A)) g
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
forall a : A * A,
fmap (uncurry (Bool_rec A)) (Id a) $== Id (uncurry (Bool_rec A) a)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
forall (a b c : A * A) (f : a $-> b) (g : b $-> c),
fmap (uncurry (Bool_rec A)) (g $o f) $==
fmap (uncurry (Bool_rec A)) g $o fmap (uncurry (Bool_rec A)) f
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
forall (a b : A * A) (f g : a $-> b),
f $== g -> fmap (uncurry (Bool_rec A)) f $== fmap (uncurry (Bool_rec A)) g intros [a b] [a' b'] [f g] [f' g'] [p q] [ | ].A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
a, b, a', b': A
f: fst (a, b) $-> fst (a', b')
g: snd (a, b) $-> snd (a', b')
f': fst (a, b) $-> fst (a', b')
g': snd (a, b) $-> snd (a', b')
p: fst (f, g) $-> fst (f', g')
q: snd (f, g) $-> snd (f', g')
fmap (uncurry (Bool_rec A)) (f, g) true $->
fmap (uncurry (Bool_rec A)) (f', g') true
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
a, b, a', b': A
f: fst (a, b) $-> fst (a', b')
g: snd (a, b) $-> snd (a', b')
f': fst (a, b) $-> fst (a', b')
g': snd (a, b) $-> snd (a', b')
p: fst (f, g) $-> fst (f', g')
q: snd (f, g) $-> snd (f', g')
fmap (uncurry (Bool_rec A)) (f, g) false $->
fmap (uncurry (Bool_rec A)) (f', g') false
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
a, b, a', b': A
f: fst (a, b) $-> fst (a', b')
g: snd (a, b) $-> snd (a', b')
f': fst (a, b) $-> fst (a', b')
g': snd (a, b) $-> snd (a', b')
p: fst (f, g) $-> fst (f', g')
q: snd (f, g) $-> snd (f', g')
fmap (uncurry (Bool_rec A)) (f, g) true $->
fmap (uncurry (Bool_rec A)) (f', g') true exact p.
    +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
a, b, a', b': A
f: fst (a, b) $-> fst (a', b')
g: snd (a, b) $-> snd (a', b')
f': fst (a, b) $-> fst (a', b')
g': snd (a, b) $-> snd (a', b')
p: fst (f, g) $-> fst (f', g')
q: snd (f, g) $-> snd (f', g')
fmap (uncurry (Bool_rec A)) (f, g) false $->
fmap (uncurry (Bool_rec A)) (f', g') false exact q.
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
forall a : A * A,
fmap (uncurry (Bool_rec A)) (Id a) $== Id (uncurry (Bool_rec A) a) intros [a b] [ | ]; reflexivity.
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
forall (a b c : A * A) (f : a $-> b) (g : b $-> c),
fmap (uncurry (Bool_rec A)) (g $o f) $==
fmap (uncurry (Bool_rec A)) g $o fmap (uncurry (Bool_rec A)) f intros [a b] [a' b'] [a'' b''] [f f'] [g g'] [ | ]; reflexivity.
Defined.

(** As a special case of the product functor, restriction along [Bool_rec A] yields bifunctoriality of [cat_binprod]. *)
Instance is0bifunctor_cat_binprod {A : Type} `{hbp : HasBinaryProducts A}
  : Is0Bifunctor cat_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
Is0Bifunctor cat_binprod
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
Is0Bifunctor cat_binprod
  pose (p:=has_products (hasproducts:=hasproductsbool_hasbinaryproducts)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
p:= has_products: forall x : Bool -> A, Product x
Is0Bifunctor cat_binprod
  exact (is0bifunctor_postcompose
          (Bool_rec A) (fun x => cat_prod x (product:=p x))).
Defined.

Instance is1bifunctor_cat_binprod {A : Type} `{hbp : HasBinaryProducts A}
  : Is1Bifunctor cat_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
Is1Bifunctor cat_binprod
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
Is1Bifunctor cat_binprod
  pose (p:=has_products (hasproducts:=hasproductsbool_hasbinaryproducts)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
p:= has_products: forall x : Bool -> A, Product x
Is1Bifunctor cat_binprod
  exact (is1bifunctor_postcompose
          (Bool_rec A) (fun x => cat_prod x (product:=p x))).
Defined.

(** [cat_binprod_corec] is also functorial in each morphism. *)

Instance is0functor_cat_binprod_corec_l {A : Type}
  `{HasBinaryProducts A} {x y z : A} (g : z $-> y)
  : Is0Functor (fun f : z $-> x => cat_binprod_corec _ f g).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
g: z $-> y
Is0Functor
  (fun f : z $-> x => cat_binprod_corec (cat_prod (Bool_rec A x y)) f g)
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
g: z $-> y
Is0Functor
  (fun f : z $-> x => cat_binprod_corec (cat_prod (Bool_rec A x y)) f g)
  snapply Build_Is0Functor.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
g: z $-> y
forall a b : z $-> x,
(a $-> b) ->
(fun f : z $-> x => cat_binprod_corec (cat_prod (Bool_rec A x y)) f g) a $->
(fun f : z $-> x => cat_binprod_corec (cat_prod (Bool_rec A x y)) f g) b
  intros f f' p.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
g: z $-> y
f, f': z $-> x
p: f $-> f'
(fun f0 : z $-> x => cat_binprod_corec (cat_prod (Bool_rec A x y)) f0 g) f $->
(fun f0 : z $-> x => cat_binprod_corec (cat_prod (Bool_rec A x y)) f0 g) f'
  by napply cat_binprod_corec_eta.
Defined.

Instance is0functor_cat_binprod_corec_r {A : Type}
  `{HasBinaryProducts A} {x y z : A} (f : z $-> x)
  : Is0Functor (fun g : z $-> y => cat_binprod_corec _ f g).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
f: z $-> x
Is0Functor
  (fun g : z $-> y => cat_binprod_corec (cat_prod (Bool_rec A x y)) f g)
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
f: z $-> x
Is0Functor
  (fun g : z $-> y => cat_binprod_corec (cat_prod (Bool_rec A x y)) f g)
  snapply Build_Is0Functor.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
f: z $-> x
forall a b : z $-> y,
(a $-> b) ->
(fun g : z $-> y => cat_binprod_corec (cat_prod (Bool_rec A x y)) f g) a $->
(fun g : z $-> y => cat_binprod_corec (cat_prod (Bool_rec A x y)) f g) b
  intros g h p.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y, z: A
f: z $-> x
g, h: z $-> y
p: g $-> h
(fun g0 : z $-> y => cat_binprod_corec (cat_prod (Bool_rec A x y)) f g0) g $->
(fun g0 : z $-> y => cat_binprod_corec (cat_prod (Bool_rec A x y)) f g0) h
  by napply cat_binprod_corec_eta.
Defined.

Definition cat_pr1_fmap01_binprod {A : Type} `{HasBinaryProducts A}
  (a : A) {x y : A} (g : x $-> y)
  : cat_pr1 _ $o fmap01 cat_binprod a g $== cat_pr1 _
  := cat_binprod_beta_pr1 _ _ _ $@ cat_idl _.

Definition cat_pr1_fmap10_binprod {A : Type} `{HasBinaryProducts A}
  {x y : A} (f : x $-> y) (a : A)
  : cat_pr1 _ $o fmap10 cat_binprod f a $== f $o cat_pr1 _
  := cat_binprod_beta_pr1 _ _ _.

Definition cat_pr1_fmap11_binprod {A : Type} `{HasBinaryProducts A}
  {w x y z : A} (f : w $-> y) (g : x $-> z)
  : cat_pr1 _ $o fmap11 cat_binprod f g $== f $o cat_pr1 _
  := cat_binprod_beta_pr1 _ _ _.

Definition cat_pr2_fmap01_binprod {A : Type} `{HasBinaryProducts A}
  (a : A) {x y : A} (g : x $-> y)
  : cat_pr2 _ $o fmap01 cat_binprod a g $== g $o cat_pr2 _
  := cat_binprod_beta_pr2 _ _ _.

Definition cat_pr2_fmap10_binprod {A : Type} `{HasBinaryProducts A}
  {x y : A} (f : x $-> y) (a : A)
  : cat_pr2 _ $o fmap10 cat_binprod f a $== cat_pr2 _
  := cat_binprod_beta_pr2 _ _ _ $@ cat_idl _.

Definition cat_pr2_fmap11_binprod {A : Type} `{HasBinaryProducts A}
  {w x y z : A} (f : w $-> y) (g : x $-> z)
  : cat_pr2 _ $o fmap11 cat_binprod f g $== g $o cat_pr2 _
  := cat_binprod_beta_pr2 _ _ _.

(** *** Lemmas about [cat_binprod_corec] *)

Definition cat_binprod_fmap01_corec {A : Type}
  `{Is1Cat A, hbp : !HasBinaryProducts A} {w x y z : A}
  (f : w $-> z) (g : x $-> y) (h : w $-> x)
  : fmap01 cat_binprod z g $o cat_binprod_corec _ f h
    $== cat_binprod_corec _ f (g $o h).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> z
g: x $-> y
h: w $-> x
fmap01 cat_binprod z g $o cat_binprod_corec (cat_binprod z x) f h $==
cat_binprod_corec (cat_binprod z y) f (g $o h)
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> z
g: x $-> y
h: w $-> x
fmap01 cat_binprod z g $o cat_binprod_corec (cat_binprod z x) f h $==
cat_binprod_corec (cat_binprod z y) f (g $o h)
  rapply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> z
g: x $-> y
h: w $-> x
cat_pr1 (cat_binprod z y) $o
(fmap01 cat_binprod z g $o cat_binprod_corec (cat_binprod z x) f h) $==
cat_pr1 (cat_binprod z y) $o cat_binprod_corec (cat_binprod z y) f (g $o h)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> z
g: x $-> y
h: w $-> x
cat_pr2 (cat_binprod z y) $o
(fmap01 cat_binprod z g $o cat_binprod_corec (cat_binprod z x) f h) $==
cat_pr2 (cat_binprod z y) $o cat_binprod_corec (cat_binprod z y) f (g $o h)
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> z
g: x $-> y
h: w $-> x
cat_pr1 (cat_binprod z y) $o
(fmap01 cat_binprod z g $o cat_binprod_corec (cat_binprod z x) f h) $==
cat_pr1 (cat_binprod z y) $o cat_binprod_corec (cat_binprod z y) f (g $o h) nrefine (cat_assoc_opp _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> z
g: x $-> y
h: w $-> x
cat_pr1 (cat_binprod z y) $o fmap01 cat_binprod z g $o
cat_binprod_corec (cat_binprod z x) f h $==
cat_pr1 (cat_binprod z y) $o cat_binprod_corec (cat_binprod z y) f (g $o h)
    refine ((_ $@R _) $@ cat_assoc _ _ _ $@ cat_idl _ $@ _ $@ _^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> z
g: x $-> y
h: w $-> x
cat_pr1 (cat_binprod z y) $o fmap01 cat_binprod z g $== Id z $o ?Goal2
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> z
g: x $-> y
h: w $-> x
?Goal2 $o cat_binprod_corec (cat_binprod z x) f h $== ?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> z
g: x $-> y
h: w $-> x
cat_pr1 (cat_binprod z y) $o cat_binprod_corec (cat_binprod z y) f (g $o h) $->
?Goal0
    1-3: rapply cat_binprod_beta_pr1.
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> z
g: x $-> y
h: w $-> x
cat_pr2 (cat_binprod z y) $o
(fmap01 cat_binprod z g $o cat_binprod_corec (cat_binprod z x) f h) $==
cat_pr2 (cat_binprod z y) $o cat_binprod_corec (cat_binprod z y) f (g $o h) nrefine (cat_assoc_opp _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> z
g: x $-> y
h: w $-> x
cat_pr2 (cat_binprod z y) $o fmap01 cat_binprod z g $o
cat_binprod_corec (cat_binprod z x) f h $==
cat_pr2 (cat_binprod z y) $o cat_binprod_corec (cat_binprod z y) f (g $o h)
    refine ((_ $@R _) $@ cat_assoc _ _ _ $@ (_ $@L _) $@ _^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> z
g: x $-> y
h: w $-> x
cat_pr2 (cat_binprod z y) $o fmap01 cat_binprod z g $== ?Goal2 $o ?Goal1
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> z
g: x $-> y
h: w $-> x
?Goal1 $o cat_binprod_corec (cat_binprod z x) f h $== ?Goal3
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> z
g: x $-> y
h: w $-> x
cat_pr2 (cat_binprod z y) $o cat_binprod_corec (cat_binprod z y) f (g $o h) $->
?Goal2 $o ?Goal3
    1-3: rapply cat_binprod_beta_pr2.
Defined.

Definition cat_binprod_fmap10_corec {A : Type}
  `{Is1Cat A, hbp : !HasBinaryProducts A} {w x y z : A}
  (f : x $-> y) (g : w $-> x) (h : w $-> z)
  : fmap10 cat_binprod f z $o cat_binprod_corec _ g h
    $== cat_binprod_corec _ (f $o g) h.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: x $-> y
g: w $-> x
h: w $-> z
fmap10 cat_binprod f z $o cat_binprod_corec (cat_binprod x z) g h $==
cat_binprod_corec (cat_binprod y z) (f $o g) h
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: x $-> y
g: w $-> x
h: w $-> z
fmap10 cat_binprod f z $o cat_binprod_corec (cat_binprod x z) g h $==
cat_binprod_corec (cat_binprod y z) (f $o g) h
  rapply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: x $-> y
g: w $-> x
h: w $-> z
cat_pr1 (cat_binprod y z) $o
(fmap10 cat_binprod f z $o cat_binprod_corec (cat_binprod x z) g h) $==
cat_pr1 (cat_binprod y z) $o cat_binprod_corec (cat_binprod y z) (f $o g) h
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: x $-> y
g: w $-> x
h: w $-> z
cat_pr2 (cat_binprod y z) $o
(fmap10 cat_binprod f z $o cat_binprod_corec (cat_binprod x z) g h) $==
cat_pr2 (cat_binprod y z) $o cat_binprod_corec (cat_binprod y z) (f $o g) h
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: x $-> y
g: w $-> x
h: w $-> z
cat_pr1 (cat_binprod y z) $o
(fmap10 cat_binprod f z $o cat_binprod_corec (cat_binprod x z) g h) $==
cat_pr1 (cat_binprod y z) $o cat_binprod_corec (cat_binprod y z) (f $o g) h refine (cat_assoc_opp _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: x $-> y
g: w $-> x
h: w $-> z
cat_pr1 (cat_binprod y z) $o fmap10 cat_binprod f z $o
cat_binprod_corec (cat_binprod x z) g h $==
cat_pr1 (cat_binprod y z) $o cat_binprod_corec (cat_binprod y z) (f $o g) h
    refine ((_ $@R _) $@ cat_assoc _ _ _ $@ (_ $@L _) $@ _^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: x $-> y
g: w $-> x
h: w $-> z
cat_pr1 (cat_binprod y z) $o fmap10 cat_binprod f z $== ?Goal3 $o ?Goal2
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: x $-> y
g: w $-> x
h: w $-> z
?Goal2 $o cat_binprod_corec (cat_binprod x z) g h $== ?Goal4
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: x $-> y
g: w $-> x
h: w $-> z
cat_pr1 (cat_binprod y z) $o cat_binprod_corec (cat_binprod y z) (f $o g) h $->
?Goal3 $o ?Goal4
    1-3: napply cat_binprod_beta_pr1.
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: x $-> y
g: w $-> x
h: w $-> z
cat_pr2 (cat_binprod y z) $o
(fmap10 cat_binprod f z $o cat_binprod_corec (cat_binprod x z) g h) $==
cat_pr2 (cat_binprod y z) $o cat_binprod_corec (cat_binprod y z) (f $o g) h refine (cat_assoc_opp _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: x $-> y
g: w $-> x
h: w $-> z
cat_pr2 (cat_binprod y z) $o fmap10 cat_binprod f z $o
cat_binprod_corec (cat_binprod x z) g h $==
cat_pr2 (cat_binprod y z) $o cat_binprod_corec (cat_binprod y z) (f $o g) h
    refine ((_ $@R _) $@ cat_assoc _ _ _ $@ cat_idl _ $@ _ $@ _^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: x $-> y
g: w $-> x
h: w $-> z
cat_pr2 (cat_binprod y z) $o fmap10 cat_binprod f z $== Id z $o ?Goal1
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: x $-> y
g: w $-> x
h: w $-> z
?Goal1 $o cat_binprod_corec (cat_binprod x z) g h $== ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
w, x, y, z: A
f: x $-> y
g: w $-> x
h: w $-> z
cat_pr2 (cat_binprod y z) $o cat_binprod_corec (cat_binprod y z) (f $o g) h $->
?Goal
    1-3: napply cat_binprod_beta_pr2.
Defined.

Definition cat_binprod_fmap11_corec {A : Type}
  `{Is1Cat A, hbp : !HasBinaryProducts A} {v w x y z : A}
  (f : w $-> y) (g : x $-> z) (h : v $-> w) (i : v $-> x)
  : fmap11 cat_binprod f g $o cat_binprod_corec _ h i
    $== cat_binprod_corec _ (f $o h) (g $o i).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
v, w, x, y, z: A
f: w $-> y
g: x $-> z
h: v $-> w
i: v $-> x
fmap11 cat_binprod f g $o cat_binprod_corec (cat_binprod w x) h i $==
cat_binprod_corec (cat_binprod y z) (f $o h) (g $o i)
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
v, w, x, y, z: A
f: w $-> y
g: x $-> z
h: v $-> w
i: v $-> x
fmap11 cat_binprod f g $o cat_binprod_corec (cat_binprod w x) h i $==
cat_binprod_corec (cat_binprod y z) (f $o h) (g $o i)
  rapply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
v, w, x, y, z: A
f: w $-> y
g: x $-> z
h: v $-> w
i: v $-> x
cat_pr1 (cat_binprod y z) $o
(fmap11 cat_binprod f g $o cat_binprod_corec (cat_binprod w x) h i) $==
cat_pr1 (cat_binprod y z) $o
cat_binprod_corec (cat_binprod y z) (f $o h) (g $o i)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
v, w, x, y, z: A
f: w $-> y
g: x $-> z
h: v $-> w
i: v $-> x
cat_pr2 (cat_binprod y z) $o
(fmap11 cat_binprod f g $o cat_binprod_corec (cat_binprod w x) h i) $==
cat_pr2 (cat_binprod y z) $o
cat_binprod_corec (cat_binprod y z) (f $o h) (g $o i)
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
v, w, x, y, z: A
f: w $-> y
g: x $-> z
h: v $-> w
i: v $-> x
cat_pr1 (cat_binprod y z) $o
(fmap11 cat_binprod f g $o cat_binprod_corec (cat_binprod w x) h i) $==
cat_pr1 (cat_binprod y z) $o
cat_binprod_corec (cat_binprod y z) (f $o h) (g $o i) refine (cat_assoc_opp _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
v, w, x, y, z: A
f: w $-> y
g: x $-> z
h: v $-> w
i: v $-> x
cat_pr1 (cat_binprod y z) $o fmap11 cat_binprod f g $o
cat_binprod_corec (cat_binprod w x) h i $==
cat_pr1 (cat_binprod y z) $o
cat_binprod_corec (cat_binprod y z) (f $o h) (g $o i)
    refine ((_ $@R _) $@ cat_assoc _ _ _ $@ (_ $@L _) $@ _^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
v, w, x, y, z: A
f: w $-> y
g: x $-> z
h: v $-> w
i: v $-> x
cat_pr1 (cat_binprod y z) $o fmap11 cat_binprod f g $== ?Goal3 $o ?Goal2
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
v, w, x, y, z: A
f: w $-> y
g: x $-> z
h: v $-> w
i: v $-> x
?Goal2 $o cat_binprod_corec (cat_binprod w x) h i $== ?Goal4
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
v, w, x, y, z: A
f: w $-> y
g: x $-> z
h: v $-> w
i: v $-> x
cat_pr1 (cat_binprod y z) $o
cat_binprod_corec (cat_binprod y z) (f $o h) (g $o i) $-> 
?Goal3 $o ?Goal4
    1-3: napply cat_binprod_beta_pr1.
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
v, w, x, y, z: A
f: w $-> y
g: x $-> z
h: v $-> w
i: v $-> x
cat_pr2 (cat_binprod y z) $o
(fmap11 cat_binprod f g $o cat_binprod_corec (cat_binprod w x) h i) $==
cat_pr2 (cat_binprod y z) $o
cat_binprod_corec (cat_binprod y z) (f $o h) (g $o i) nrefine (cat_assoc_opp _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
v, w, x, y, z: A
f: w $-> y
g: x $-> z
h: v $-> w
i: v $-> x
cat_pr2 (cat_binprod y z) $o fmap11 cat_binprod f g $o
cat_binprod_corec (cat_binprod w x) h i $==
cat_pr2 (cat_binprod y z) $o
cat_binprod_corec (cat_binprod y z) (f $o h) (g $o i)
    refine ((_ $@R _) $@ cat_assoc _ _ _ $@ (_ $@L _) $@ _^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
v, w, x, y, z: A
f: w $-> y
g: x $-> z
h: v $-> w
i: v $-> x
cat_pr2 (cat_binprod y z) $o fmap11 cat_binprod f g $== ?Goal2 $o ?Goal1
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
v, w, x, y, z: A
f: w $-> y
g: x $-> z
h: v $-> w
i: v $-> x
?Goal1 $o cat_binprod_corec (cat_binprod w x) h i $== ?Goal3
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
hbp: HasBinaryProducts A
v, w, x, y, z: A
f: w $-> y
g: x $-> z
h: v $-> w
i: v $-> x
cat_pr2 (cat_binprod y z) $o
cat_binprod_corec (cat_binprod y z) (f $o h) (g $o i) $-> 
?Goal2 $o ?Goal3
    1-3: rapply cat_binprod_beta_pr2.
Defined.

(** *** Diagonal *)

(** Annoyingly this doesn't follow directly from the general diagonal since [Bool_rec _ x x] is not definitionally equal to [fun _ => x]. *)
Definition cat_binprod_diag {A : Type} `{Is1Cat A} (x : A)
  (cat_binprod : A) `{isbinprod : !IsBinaryProduct x x cat_binprod}
  : x $-> cat_binprod
  := cat_binprod_corec _ (Id _) (Id _).

Definition cat_binprod_fmap11_diag {A : Type}
  `{HasBinaryProducts A} {x y : A} (f : x $-> y)
  : cat_binprod_diag y _ $o f
    $== fmap11 cat_binprod f f $o cat_binprod_diag x _.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y: A
f: x $-> y
cat_binprod_diag y (cat_binprod y y) $o f $==
fmap11 cat_binprod f f $o cat_binprod_diag x (cat_binprod x x)
Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y: A
f: x $-> y
cat_binprod_diag y (cat_binprod y y) $o f $==
fmap11 cat_binprod f f $o cat_binprod_diag x (cat_binprod x x)
  refine (_ $@ _^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y: A
f: x $-> y
cat_binprod_diag y (cat_binprod y y) $o f $== ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y: A
f: x $-> y
fmap11 cat_binprod f f $o cat_binprod_diag x (cat_binprod x x) $-> ?Goal
  2: napply cat_binprod_fmap11_corec.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y: A
f: x $-> y
cat_binprod_diag y (cat_binprod y y) $o f $==
cat_binprod_corec (cat_binprod y y) (f $o Id x) (f $o Id x)
  napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y: A
f: x $-> y
cat_pr1 (cat_binprod y y) $o (cat_binprod_diag y (cat_binprod y y) $o f) $==
cat_pr1 (cat_binprod y y) $o
cat_binprod_corec (cat_binprod y y) (f $o Id x) (f $o Id x)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y: A
f: x $-> y
cat_pr2 (cat_binprod y y) $o (cat_binprod_diag y (cat_binprod y y) $o f) $==
cat_pr2 (cat_binprod y y) $o
cat_binprod_corec (cat_binprod y y) (f $o Id x) (f $o Id x)
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y: A
f: x $-> y
cat_pr1 (cat_binprod y y) $o (cat_binprod_diag y (cat_binprod y y) $o f) $==
cat_pr1 (cat_binprod y y) $o
cat_binprod_corec (cat_binprod y y) (f $o Id x) (f $o Id x) refine ((cat_assoc _ _ _)^$ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y: A
f: x $-> y
cat_pr1 (cat_binprod y y) $o cat_binprod_diag y (cat_binprod y y) $o f $==
cat_pr1 (cat_binprod y y) $o
cat_binprod_corec (cat_binprod y y) (f $o Id x) (f $o Id x)
    refine ((_ $@R _) $@ cat_idl _ $@ (cat_idr _)^$ $@ _^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y: A
f: x $-> y
cat_pr1 (cat_binprod y y) $o cat_binprod_diag y (cat_binprod y y) $== Id y
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y: A
f: x $-> y
cat_pr1 (cat_binprod y y) $o
cat_binprod_corec (cat_binprod y y) (f $o Id x) (f $o Id x) $-> 
f $o Id x
    1,2: rapply cat_binprod_beta_pr1.
  -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y: A
f: x $-> y
cat_pr2 (cat_binprod y y) $o (cat_binprod_diag y (cat_binprod y y) $o f) $==
cat_pr2 (cat_binprod y y) $o
cat_binprod_corec (cat_binprod y y) (f $o Id x) (f $o Id x) refine ((cat_assoc _ _ _)^$ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y: A
f: x $-> y
cat_pr2 (cat_binprod y y) $o cat_binprod_diag y (cat_binprod y y) $o f $==
cat_pr2 (cat_binprod y y) $o
cat_binprod_corec (cat_binprod y y) (f $o Id x) (f $o Id x)
    refine ((_ $@R _) $@ cat_idl _ $@ (cat_idr _)^$ $@ _^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y: A
f: x $-> y
cat_pr2 (cat_binprod y y) $o cat_binprod_diag y (cat_binprod y y) $== Id y
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasBinaryProducts A
x, y: A
f: x $-> y
cat_pr2 (cat_binprod y y) $o
cat_binprod_corec (cat_binprod y y) (f $o Id x) (f $o Id x) $-> 
f $o Id x
    1,2: rapply cat_binprod_beta_pr2.
Defined.

(** *** Symmetry of binary products *)

Section Symmetry.

  (** The requirement of having all binary products can be weakened further to having specific binary products, but it is not clear this is a useful generality. *)
  Context {A : Type} `{HasEquivs A} `{hbp : !HasBinaryProducts A}.

  Definition cat_binprod_swap (x y : A) : cat_binprod x y $-> cat_binprod y x
    := cat_binprod_corec _ (cat_pr2 _) (cat_pr1 _).

  Lemma cat_binprod_swap_cat_binprod_swap (x y : A)
    : cat_binprod_swap x y $o cat_binprod_swap y x $== Id _.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_binprod_swap x y $o cat_binprod_swap y x $== Id (cat_binprod y x)
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_binprod_swap x y $o cat_binprod_swap y x $== Id (cat_binprod y x)
    napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_pr1 (cat_binprod y x) $o (cat_binprod_swap x y $o cat_binprod_swap y x) $==
cat_pr1 (cat_binprod y x) $o Id (cat_binprod y x)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_pr2 (cat_binprod y x) $o (cat_binprod_swap x y $o cat_binprod_swap y x) $==
cat_pr2 (cat_binprod y x) $o Id (cat_binprod y x)
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_pr1 (cat_binprod y x) $o (cat_binprod_swap x y $o cat_binprod_swap y x) $==
cat_pr1 (cat_binprod y x) $o Id (cat_binprod y x) refine ((cat_assoc _ _ _)^$ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_pr1 (cat_binprod y x) $o cat_binprod_swap x y $o cat_binprod_swap y x $==
cat_pr1 (cat_binprod y x) $o Id (cat_binprod y x)
      nrefine (cat_binprod_beta_pr1 _ _ _ $@R _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_pr2 (cat_binprod x y) $o cat_binprod_swap y x $==
cat_pr1 (cat_binprod y x) $o Id (cat_binprod y x)
      exact (cat_binprod_beta_pr2 _ _ _ $@ (cat_idr _)^$).
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_pr2 (cat_binprod y x) $o (cat_binprod_swap x y $o cat_binprod_swap y x) $==
cat_pr2 (cat_binprod y x) $o Id (cat_binprod y x) refine ((cat_assoc _ _ _)^$ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_pr2 (cat_binprod y x) $o cat_binprod_swap x y $o cat_binprod_swap y x $==
cat_pr2 (cat_binprod y x) $o Id (cat_binprod y x)
      nrefine (cat_binprod_beta_pr2 _ _ _ $@R _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_pr1 (cat_binprod x y) $o cat_binprod_swap y x $==
cat_pr2 (cat_binprod y x) $o Id (cat_binprod y x)
      exact (cat_binprod_beta_pr1 _ _ _ $@ (cat_idr _)^$).
  Defined.

  Lemma cate_binprod_swap (x y : A)
    : cat_binprod x y $<~> cat_binprod y x.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_binprod x y $<~> cat_binprod y x
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_binprod x y $<~> cat_binprod y x
    snapply cate_adjointify.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_binprod x y $-> cat_binprod y x
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_binprod y x $-> cat_binprod x y
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
?f $o ?g $== Id (cat_binprod y x)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
?g $o ?f $== Id (cat_binprod x y)
    1,2: napply cat_binprod_swap.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_binprod_swap x y $o cat_binprod_swap y x $== Id (cat_binprod y x)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
cat_binprod_swap y x $o cat_binprod_swap x y $== Id (cat_binprod x y)
    all: napply cat_binprod_swap_cat_binprod_swap.
  Defined.

  Definition cat_binprod_swap_corec {a b c : A} (f : a $-> b) (g : a $-> c)
    : cat_binprod_swap b c $o cat_binprod_corec _ f g $== cat_binprod_corec _ g f.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, b, c: A
f: a $-> b
g: a $-> c
cat_binprod_swap b c $o cat_binprod_corec (cat_binprod b c) f g $==
cat_binprod_corec (cat_binprod c b) g f
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, b, c: A
f: a $-> b
g: a $-> c
cat_binprod_swap b c $o cat_binprod_corec (cat_binprod b c) f g $==
cat_binprod_corec (cat_binprod c b) g f
    rapply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, b, c: A
f: a $-> b
g: a $-> c
cat_pr1 (cat_binprod c b) $o
(cat_binprod_swap b c $o cat_binprod_corec (cat_binprod b c) f g) $==
cat_pr1 (cat_binprod c b) $o cat_binprod_corec (cat_binprod c b) g f
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, b, c: A
f: a $-> b
g: a $-> c
cat_pr2 (cat_binprod c b) $o
(cat_binprod_swap b c $o cat_binprod_corec (cat_binprod b c) f g) $==
cat_pr2 (cat_binprod c b) $o cat_binprod_corec (cat_binprod c b) g f
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, b, c: A
f: a $-> b
g: a $-> c
cat_pr1 (cat_binprod c b) $o
(cat_binprod_swap b c $o cat_binprod_corec (cat_binprod b c) f g) $==
cat_pr1 (cat_binprod c b) $o cat_binprod_corec (cat_binprod c b) g f refine (cat_assoc_opp _ _ _ $@ (_ $@R _) $@ (_ $@ _^$)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, b, c: A
f: a $-> b
g: a $-> c
cat_pr1 (cat_binprod c b) $o cat_binprod_swap b c $== ?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, b, c: A
f: a $-> b
g: a $-> c
?Goal0 $o cat_binprod_corec (cat_binprod b c) f g $== ?Goal2
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, b, c: A
f: a $-> b
g: a $-> c
cat_pr1 (cat_binprod c b) $o cat_binprod_corec (cat_binprod c b) g f $->
?Goal2
      1,3: napply cat_binprod_beta_pr1.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, b, c: A
f: a $-> b
g: a $-> c
cat_pr2 (cat_binprod b c) $o cat_binprod_corec (cat_binprod b c) f g $== g
      napply cat_binprod_beta_pr2.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, b, c: A
f: a $-> b
g: a $-> c
cat_pr2 (cat_binprod c b) $o
(cat_binprod_swap b c $o cat_binprod_corec (cat_binprod b c) f g) $==
cat_pr2 (cat_binprod c b) $o cat_binprod_corec (cat_binprod c b) g f refine (cat_assoc_opp _ _ _ $@ (_ $@R _) $@ (_ $@ _^$)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, b, c: A
f: a $-> b
g: a $-> c
cat_pr2 (cat_binprod c b) $o cat_binprod_swap b c $== ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, b, c: A
f: a $-> b
g: a $-> c
?Goal $o cat_binprod_corec (cat_binprod b c) f g $== ?Goal1
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, b, c: A
f: a $-> b
g: a $-> c
cat_pr2 (cat_binprod c b) $o cat_binprod_corec (cat_binprod c b) g f $->
?Goal1
      1,3: napply cat_binprod_beta_pr2.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, b, c: A
f: a $-> b
g: a $-> c
cat_pr1 (cat_binprod b c) $o cat_binprod_corec (cat_binprod b c) f g $== f
      napply cat_binprod_beta_pr1.
  Defined.

  Definition cat_binprod_swap_nat {a b c d : A} (f : a $-> c) (g : b $-> d)
    : cat_binprod_swap c d $o fmap11 cat_binprod f g
    $== fmap11 cat_binprod g f $o cat_binprod_swap a b
    := cat_binprod_swap_corec _ _ $@ (cat_binprod_fmap11_corec _ _ _ _)^$.

  Local Instance symmetricbraiding_binprod
    : SymmetricBraiding cat_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
SymmetricBraiding cat_binprod
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
SymmetricBraiding cat_binprod
    snapply Build_SymmetricBraiding.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
Braiding cat_binprod
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
forall a b : A,
?braiding_symmetricbraiding a b $o ?braiding_symmetricbraiding b a $==
Id (cat_binprod b a)
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
Braiding cat_binprod snapply Build_NatTrans.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
uncurry cat_binprod $=> uncurry (flip cat_binprod)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
Is1Natural (uncurry cat_binprod) (uncurry (flip cat_binprod)) ?alpha
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
uncurry cat_binprod $=> uncurry (flip cat_binprod) intros [x y].A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y: A
uncurry cat_binprod (x, y) $-> uncurry (flip cat_binprod) (x, y)
        exact (cat_binprod_swap x y).
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
Is1Natural (uncurry cat_binprod) (uncurry (flip cat_binprod))
  ((fun a : A * A => (fun x y : A => cat_binprod_swap x y) (fst a) (snd a))
   :
   uncurry cat_binprod $=> uncurry (flip cat_binprod)) snapply Build_Is1Natural.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
forall (a a' : A * A) (f : a $-> a'),
(fun a0 : A * A => (fun x y : A => cat_binprod_swap x y) (fst a0) (snd a0))
  a' $o
fmap (uncurry cat_binprod) f $==
fmap (uncurry (flip cat_binprod)) f $o
(fun a0 : A * A => (fun x y : A => cat_binprod_swap x y) (fst a0) (snd a0)) a
        intros [a b] [c d] [f g]; cbn in f, g.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, b, c, d: A
f: a $-> c
g: b $-> d
cat_binprod_swap c d $o fmap (uncurry cat_binprod) (f, g) $==
fmap (uncurry (flip cat_binprod)) (f, g) $o cat_binprod_swap a b
        exact(cat_binprod_swap_nat f g).
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
forall a b : A,
{|
  trans_nattrans :=
    (fun a0 : A * A =>
     (fun x y : A => cat_binprod_swap x y) (fst a0) (snd a0))
    :
    uncurry cat_binprod $=> uncurry (flip cat_binprod);
  is1natural_nattrans :=
    Build_Is1Natural
      (fun a0 : A * A =>
       (fun x y : A => cat_binprod_swap x y) (fst a0) (snd a0))
      (fun a0 : A * A =>
       (fun (a1 b0 : A) (a' : A * A) =>
        (fun (c d : A) (f0 : (a1, b0) $-> (c, d)) =>
         (fun (f : fst (a1, b0) $-> fst (c, d))
            (g : snd (a1, b0) $-> snd (c, d)) =>
          cat_binprod_swap_nat f g) (fst f0) (snd f0)) 
          (fst a') (snd a'))
         (fst a0) (snd a0))
|} a b $o
{|
  trans_nattrans :=
    (fun a0 : A * A =>
     (fun x y : A => cat_binprod_swap x y) (fst a0) (snd a0))
    :
    uncurry cat_binprod $=> uncurry (flip cat_binprod);
  is1natural_nattrans :=
    Build_Is1Natural
      (fun a0 : A * A =>
       (fun x y : A => cat_binprod_swap x y) (fst a0) (snd a0))
      (fun a0 : A * A =>
       (fun (a1 b0 : A) (a' : A * A) =>
        (fun (c d : A) (f0 : (a1, b0) $-> (c, d)) =>
         (fun (f : fst (a1, b0) $-> fst (c, d))
            (g : snd (a1, b0) $-> snd (c, d)) =>
          cat_binprod_swap_nat f g) (fst f0) (snd f0)) 
          (fst a') (snd a'))
         (fst a0) (snd a0))
|} b a $== Id (cat_binprod b a) exact cat_binprod_swap_cat_binprod_swap.
  Defined.

    (** The swap map preserves the diagonal. *)
  Definition cat_binprod_swap_diag (x : A)
    : cat_binprod_swap x x $o cat_binprod_diag x _ $== cat_binprod_diag x _.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x: A
cat_binprod_swap x x $o cat_binprod_diag x (cat_binprod x x) $==
cat_binprod_diag x (cat_binprod x x)
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x: A
cat_binprod_swap x x $o cat_binprod_diag x (cat_binprod x x) $==
cat_binprod_diag x (cat_binprod x x)
    napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x: A
cat_pr1 (cat_binprod x x) $o
(cat_binprod_swap x x $o cat_binprod_diag x (cat_binprod x x)) $==
cat_pr1 (cat_binprod x x) $o cat_binprod_diag x (cat_binprod x x)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x: A
cat_pr2 (cat_binprod x x) $o
(cat_binprod_swap x x $o cat_binprod_diag x (cat_binprod x x)) $==
cat_pr2 (cat_binprod x x) $o cat_binprod_diag x (cat_binprod x x)
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x: A
cat_pr1 (cat_binprod x x) $o
(cat_binprod_swap x x $o cat_binprod_diag x (cat_binprod x x)) $==
cat_pr1 (cat_binprod x x) $o cat_binprod_diag x (cat_binprod x x) refine ((cat_assoc _ _ _)^$ $@ (_ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x: A
cat_pr1 (cat_binprod x x) $o cat_binprod_swap x x $== ?Goal2
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x: A
?Goal2 $o cat_binprod_diag x (cat_binprod x x) $==
cat_pr1 (cat_binprod x x) $o cat_binprod_diag x (cat_binprod x x)
      1: napply cat_binprod_beta_pr1.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x: A
cat_pr2 (cat_binprod x x) $o cat_binprod_diag x (cat_binprod x x) $==
cat_pr1 (cat_binprod x x) $o cat_binprod_diag x (cat_binprod x x)
      refine (cat_binprod_beta_pr2 _ _ _ $@ _^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x: A
cat_pr1 (cat_binprod x x) $o cat_binprod_diag x (cat_binprod x x) $-> Id x
      napply cat_binprod_beta_pr1.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x: A
cat_pr2 (cat_binprod x x) $o
(cat_binprod_swap x x $o cat_binprod_diag x (cat_binprod x x)) $==
cat_pr2 (cat_binprod x x) $o cat_binprod_diag x (cat_binprod x x) refine ((cat_assoc _ _ _)^$ $@ (_ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x: A
cat_pr2 (cat_binprod x x) $o cat_binprod_swap x x $== ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x: A
?Goal $o cat_binprod_diag x (cat_binprod x x) $==
cat_pr2 (cat_binprod x x) $o cat_binprod_diag x (cat_binprod x x)
      1: napply cat_binprod_beta_pr2.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x: A
cat_pr1 (cat_binprod x x) $o cat_binprod_diag x (cat_binprod x x) $==
cat_pr2 (cat_binprod x x) $o cat_binprod_diag x (cat_binprod x x)
      refine (cat_binprod_beta_pr1 _ _ _ $@ _^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x: A
cat_pr2 (cat_binprod x x) $o cat_binprod_diag x (cat_binprod x x) $-> Id x
      napply cat_binprod_beta_pr2.
  Defined.

End Symmetry.

(** *** Binary product gives a symmetric monoidal structure *)

Section Associativity.

  Context {A : Type} `{HasEquivs A} `{hbp : !HasBinaryProducts A}.

  Definition cat_binprod_twist (x y z : A)
    : cat_binprod x (cat_binprod y z) $-> cat_binprod y (cat_binprod x z).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_binprod x (cat_binprod y z) $-> cat_binprod y (cat_binprod x z)
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_binprod x (cat_binprod y z) $-> cat_binprod y (cat_binprod x z)
    rapply cat_binprod_corec.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_binprod x (cat_binprod y z) $-> y
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_binprod x (cat_binprod y z) $-> cat_binprod x z
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_binprod x (cat_binprod y z) $-> y exact (cat_pr1 _ $o cat_pr2 _).
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_binprod x (cat_binprod y z) $-> cat_binprod x z exact (fmap01 cat_binprod x (cat_pr2 _)).
  Defined.

  Definition cat_binprod_pr1_twist (x y z : A)
    : cat_pr1 _ $o cat_binprod_twist x y z $== cat_pr1 _ $o cat_pr2 _
    := cat_binprod_beta_pr1 _ _ _.

  Definition cat_binprod_pr1_pr2_twist (x y z : A)
    : cat_pr1 _ $o cat_pr2 _ $o cat_binprod_twist x y z $== cat_pr1 _.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_prod (Bool_rec A x z)) $o
cat_pr2 (cat_binprod y (cat_binprod x z)) $o cat_binprod_twist x y z $==
cat_pr1 (cat_binprod x (cat_binprod y z))
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_prod (Bool_rec A x z)) $o
cat_pr2 (cat_binprod y (cat_binprod x z)) $o cat_binprod_twist x y z $==
cat_pr1 (cat_binprod x (cat_binprod y z))
    nrefine (cat_assoc _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_prod (Bool_rec A x z)) $o
(cat_pr2 (cat_binprod y (cat_binprod x z)) $o cat_binprod_twist x y z) $==
cat_pr1 (cat_binprod x (cat_binprod y z))
    nrefine ((_ $@L cat_binprod_beta_pr2 _ _ _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_prod (Bool_rec A x z)) $o
fmap01 cat_binprod x (cat_pr2 (cat_binprod y z)) $==
cat_pr1 (cat_binprod x (cat_binprod y z))
    napply cat_pr1_fmap01_binprod.
  Defined.

  Definition cat_binprod_pr2_pr2_twist (x y z : A)
    : cat_pr2 _ $o cat_pr2 _ $o cat_binprod_twist x y z $== cat_pr2 _ $o cat_pr2 _.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_prod (Bool_rec A x z)) $o
cat_pr2 (cat_binprod y (cat_binprod x z)) $o cat_binprod_twist x y z $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_prod (Bool_rec A x z)) $o
cat_pr2 (cat_binprod y (cat_binprod x z)) $o cat_binprod_twist x y z $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
    nrefine (cat_assoc _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_prod (Bool_rec A x z)) $o
(cat_pr2 (cat_binprod y (cat_binprod x z)) $o cat_binprod_twist x y z) $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
    nrefine ((_ $@L cat_binprod_beta_pr2 _ _ _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_prod (Bool_rec A x z)) $o
fmap01 cat_binprod x (cat_pr2 (cat_binprod y z)) $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
    napply cat_pr2_fmap01_binprod.
  Defined.

  Definition cat_binprod_twist_corec {w x y z : A}
    (f : w $-> x) (g : w $-> y) (h : w $-> z)
    : cat_binprod_twist x y z $o cat_binprod_corec _ f (cat_binprod_corec _ g h)
      $== cat_binprod_corec _ g (cat_binprod_corec _ f h).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_binprod_twist x y z $o
cat_binprod_corec (cat_binprod x (cat_binprod y z)) f
  (cat_binprod_corec (cat_binprod y z) g h) $==
cat_binprod_corec (cat_binprod y (cat_binprod x z)) g
  (cat_binprod_corec (cat_binprod x z) f h)
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_binprod_twist x y z $o
cat_binprod_corec (cat_binprod x (cat_binprod y z)) f
  (cat_binprod_corec (cat_binprod y z) g h) $==
cat_binprod_corec (cat_binprod y (cat_binprod x z)) g
  (cat_binprod_corec (cat_binprod x z) f h)
    napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_pr1 (cat_binprod y (cat_binprod x z)) $o
(cat_binprod_twist x y z $o
 cat_binprod_corec (cat_binprod x (cat_binprod y z)) f
   (cat_binprod_corec (cat_binprod y z) g h)) $==
cat_pr1 (cat_binprod y (cat_binprod x z)) $o
cat_binprod_corec (cat_binprod y (cat_binprod x z)) g
  (cat_binprod_corec (cat_binprod x z) f h)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_pr2 (cat_binprod y (cat_binprod x z)) $o
(cat_binprod_twist x y z $o
 cat_binprod_corec (cat_binprod x (cat_binprod y z)) f
   (cat_binprod_corec (cat_binprod y z) g h)) $==
cat_pr2 (cat_binprod y (cat_binprod x z)) $o
cat_binprod_corec (cat_binprod y (cat_binprod x z)) g
  (cat_binprod_corec (cat_binprod x z) f h)
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_pr1 (cat_binprod y (cat_binprod x z)) $o
(cat_binprod_twist x y z $o
 cat_binprod_corec (cat_binprod x (cat_binprod y z)) f
   (cat_binprod_corec (cat_binprod y z) g h)) $==
cat_pr1 (cat_binprod y (cat_binprod x z)) $o
cat_binprod_corec (cat_binprod y (cat_binprod x z)) g
  (cat_binprod_corec (cat_binprod x z) f h) nrefine (cat_assoc_opp _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_pr1 (cat_binprod y (cat_binprod x z)) $o cat_binprod_twist x y z $o
cat_binprod_corec (cat_binprod x (cat_binprod y z)) f
  (cat_binprod_corec (cat_binprod y z) g h) $==
cat_pr1 (cat_binprod y (cat_binprod x z)) $o
cat_binprod_corec (cat_binprod y (cat_binprod x z)) g
  (cat_binprod_corec (cat_binprod x z) f h)
      refine ((_ $@R _) $@ cat_assoc _ _ _ $@ (_ $@L _) $@ (_ $@ _^$)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_pr1 (cat_binprod y (cat_binprod x z)) $o cat_binprod_twist x y z $==
?Goal5 $o ?Goal4
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
?Goal4 $o
cat_binprod_corec (cat_binprod x (cat_binprod y z)) f
  (cat_binprod_corec (cat_binprod y z) g h) $==
?Goal6
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
?Goal5 $o ?Goal6 $== ?Goal8
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_pr1 (cat_binprod y (cat_binprod x z)) $o
cat_binprod_corec (cat_binprod y (cat_binprod x z)) g
  (cat_binprod_corec (cat_binprod x z) f h) $->
?Goal8
      1: napply cat_binprod_pr1_twist.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_pr2 (cat_binprod x (cat_binprod y z)) $o
cat_binprod_corec (cat_binprod x (cat_binprod y z)) f
  (cat_binprod_corec (cat_binprod y z) g h) $==
?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_pr1 (cat_prod (Bool_rec A y z)) $o ?Goal0 $== ?Goal2
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_pr1 (cat_binprod y (cat_binprod x z)) $o
cat_binprod_corec (cat_binprod y (cat_binprod x z)) g
  (cat_binprod_corec (cat_binprod x z) f h) $->
?Goal2
      1: napply cat_binprod_beta_pr2.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_binprod_corec (cat_binprod y z) g h $== ?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_pr1 (cat_binprod y (cat_binprod x z)) $o
cat_binprod_corec (cat_binprod y (cat_binprod x z)) g
  (cat_binprod_corec (cat_binprod x z) f h) $->
?Goal0
      1,2: napply cat_binprod_beta_pr1.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_pr2 (cat_binprod y (cat_binprod x z)) $o
(cat_binprod_twist x y z $o
 cat_binprod_corec (cat_binprod x (cat_binprod y z)) f
   (cat_binprod_corec (cat_binprod y z) g h)) $==
cat_pr2 (cat_binprod y (cat_binprod x z)) $o
cat_binprod_corec (cat_binprod y (cat_binprod x z)) g
  (cat_binprod_corec (cat_binprod x z) f h) refine (cat_assoc_opp _ _ _ $@ (_ $@R _) $@ _ $@ (cat_binprod_beta_pr2 _ _ _)^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_pr2 (cat_binprod y (cat_binprod x z)) $o cat_binprod_twist x y z $==
?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
?Goal $o
cat_binprod_corec (cat_binprod x (cat_binprod y z)) f
  (cat_binprod_corec (cat_binprod y z) g h) $==
cat_binprod_corec (cat_binprod x z) f h
      1: napply cat_binprod_beta_pr2.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
fmap01 cat_binprod x (cat_pr2 (cat_binprod y z)) $o
cat_binprod_corec (cat_binprod x (cat_binprod y z)) f
  (cat_binprod_corec (cat_binprod y z) g h) $==
cat_binprod_corec (cat_binprod x z) f h
      nrefine (cat_binprod_fmap01_corec _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_binprod_corec (cat_binprod x z) f
  (cat_pr2 (cat_binprod y z) $o cat_binprod_corec (cat_binprod y z) g h) $==
cat_binprod_corec (cat_binprod x z) f h
      napply cat_binprod_corec_eta.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
f $== f
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_pr2 (cat_binprod y z) $o cat_binprod_corec (cat_binprod y z) g h $== h
      1: exact (Id _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_pr2 (cat_binprod y z) $o cat_binprod_corec (cat_binprod y z) g h $== h
      napply cat_binprod_beta_pr2.
  Defined.

  Lemma cat_binprod_twist_cat_binprod_twist (x y z : A)
    : cat_binprod_twist x y z $o cat_binprod_twist y x z $== Id _.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_binprod_twist x y z $o cat_binprod_twist y x z $==
Id (cat_binprod y (cat_binprod x z))
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_binprod_twist x y z $o cat_binprod_twist y x z $==
Id (cat_binprod y (cat_binprod x z))
    napply cat_binprod_eta_pr_x_xx_id.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_binprod y (cat_binprod x z)) $o
(cat_binprod_twist x y z $o cat_binprod_twist y x z) $==
cat_pr1 (cat_binprod y (cat_binprod x z))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_prod (Bool_rec A x z)) $o
cat_pr2 (cat_binprod y (cat_binprod x z)) $o
(cat_binprod_twist x y z $o cat_binprod_twist y x z) $==
cat_pr1 (cat_prod (Bool_rec A x z)) $o
cat_pr2 (cat_binprod y (cat_binprod x z))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_prod (Bool_rec A x z)) $o
cat_pr2 (cat_binprod y (cat_binprod x z)) $o
(cat_binprod_twist x y z $o cat_binprod_twist y x z) $==
cat_pr2 (cat_prod (Bool_rec A x z)) $o
cat_pr2 (cat_binprod y (cat_binprod x z))
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_binprod y (cat_binprod x z)) $o
(cat_binprod_twist x y z $o cat_binprod_twist y x z) $==
cat_pr1 (cat_binprod y (cat_binprod x z)) nrefine (cat_assoc_opp _ _ _ $@ (cat_binprod_pr1_twist _ _ _ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o cat_binprod_twist y x z $==
cat_pr1 (cat_binprod y (cat_binprod x z))
      napply cat_binprod_pr1_pr2_twist.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_prod (Bool_rec A x z)) $o
cat_pr2 (cat_binprod y (cat_binprod x z)) $o
(cat_binprod_twist x y z $o cat_binprod_twist y x z) $==
cat_pr1 (cat_prod (Bool_rec A x z)) $o
cat_pr2 (cat_binprod y (cat_binprod x z)) nrefine (cat_assoc_opp _ _ _ $@ (cat_binprod_pr1_pr2_twist _ _ _ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_binprod x (cat_binprod y z)) $o cat_binprod_twist y x z $==
cat_pr1 (cat_prod (Bool_rec A x z)) $o
cat_pr2 (cat_binprod y (cat_binprod x z))
      napply cat_binprod_pr1_twist.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_prod (Bool_rec A x z)) $o
cat_pr2 (cat_binprod y (cat_binprod x z)) $o
(cat_binprod_twist x y z $o cat_binprod_twist y x z) $==
cat_pr2 (cat_prod (Bool_rec A x z)) $o
cat_pr2 (cat_binprod y (cat_binprod x z)) nrefine (cat_assoc_opp _ _ _ $@ (cat_binprod_pr2_pr2_twist _ _ _ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z)) $o cat_binprod_twist y x z $==
cat_pr2 (cat_prod (Bool_rec A x z)) $o
cat_pr2 (cat_binprod y (cat_binprod x z))
      napply cat_binprod_pr2_pr2_twist.
  Defined.

  Definition cate_binprod_twist (x y z : A)
    : cat_binprod x (cat_binprod y z) $<~> cat_binprod y (cat_binprod x z).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_binprod x (cat_binprod y z) $<~> cat_binprod y (cat_binprod x z)
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_binprod x (cat_binprod y z) $<~> cat_binprod y (cat_binprod x z)
    snapply cate_adjointify.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_binprod x (cat_binprod y z) $-> cat_binprod y (cat_binprod x z)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_binprod y (cat_binprod x z) $-> cat_binprod x (cat_binprod y z)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
?f $o ?g $== Id (cat_binprod y (cat_binprod x z))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
?g $o ?f $== Id (cat_binprod x (cat_binprod y z))
    1,2: napply cat_binprod_twist.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_binprod_twist x y z $o cat_binprod_twist y x z $==
Id (cat_binprod y (cat_binprod x z))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_binprod_twist y x z $o cat_binprod_twist x y z $==
Id (cat_binprod x (cat_binprod y z))
    1,2: napply cat_binprod_twist_cat_binprod_twist.
  Defined.

  Definition cat_binprod_twist_nat {a a' b b' c c' : A}
    (f : a $-> a') (g : b $-> b') (h : c $-> c')
    : cat_binprod_twist a' b' c'
        $o fmap11 cat_binprod f (fmap11 cat_binprod g h)
      $== fmap11 cat_binprod g (fmap11 cat_binprod f h)
        $o cat_binprod_twist a b c.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_binprod_twist a' b' c' $o fmap11 cat_binprod f (fmap11 cat_binprod g h) $==
fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_binprod_twist a' b' c' $o fmap11 cat_binprod f (fmap11 cat_binprod g h) $==
fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c
    napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr1 (cat_binprod b' (cat_binprod a' c')) $o
(cat_binprod_twist a' b' c' $o fmap11 cat_binprod f (fmap11 cat_binprod g h)) $==
cat_pr1 (cat_binprod b' (cat_binprod a' c')) $o
(fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr2 (cat_binprod b' (cat_binprod a' c')) $o
(cat_binprod_twist a' b' c' $o fmap11 cat_binprod f (fmap11 cat_binprod g h)) $==
cat_pr2 (cat_binprod b' (cat_binprod a' c')) $o
(fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c)
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr1 (cat_binprod b' (cat_binprod a' c')) $o
(cat_binprod_twist a' b' c' $o fmap11 cat_binprod f (fmap11 cat_binprod g h)) $==
cat_pr1 (cat_binprod b' (cat_binprod a' c')) $o
(fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c) refine (cat_assoc_opp _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr1 (cat_binprod b' (cat_binprod a' c')) $o cat_binprod_twist a' b' c' $o
fmap11 cat_binprod f (fmap11 cat_binprod g h) $==
cat_pr1 (cat_binprod b' (cat_binprod a' c')) $o
(fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c)
      nrefine ((cat_binprod_beta_pr1 _ _ _ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr1 (cat_prod (Bool_rec A b' c')) $o
cat_pr2 (cat_binprod a' (cat_binprod b' c')) $o
fmap11 cat_binprod f (fmap11 cat_binprod g h) $==
cat_pr1 (cat_binprod b' (cat_binprod a' c')) $o
(fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c)
      nrefine (cat_assoc _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr1 (cat_prod (Bool_rec A b' c')) $o
(cat_pr2 (cat_binprod a' (cat_binprod b' c')) $o
 fmap11 cat_binprod f (fmap11 cat_binprod g h)) $==
cat_pr1 (cat_binprod b' (cat_binprod a' c')) $o
(fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c)
      nrefine ((_ $@L _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr2 (cat_binprod a' (cat_binprod b' c')) $o
fmap11 cat_binprod f (fmap11 cat_binprod g h) $== 
?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr1 (cat_prod (Bool_rec A b' c')) $o ?Goal0 $==
cat_pr1 (cat_binprod b' (cat_binprod a' c')) $o
(fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c)
      1: napply cat_pr2_fmap11_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr1 (cat_prod (Bool_rec A b' c')) $o
(fmap11 cat_binprod g h $o cat_pr2 (cat_binprod a (cat_binprod b c))) $==
cat_pr1 (cat_binprod b' (cat_binprod a' c')) $o
(fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c)
      nrefine (cat_assoc_opp _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr1 (cat_prod (Bool_rec A b' c')) $o fmap11 cat_binprod g h $o
cat_pr2 (cat_binprod a (cat_binprod b c)) $==
cat_pr1 (cat_binprod b' (cat_binprod a' c')) $o
(fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c)
      nrefine ((_ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr1 (cat_prod (Bool_rec A b' c')) $o fmap11 cat_binprod g h $== ?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
?Goal0 $o cat_pr2 (cat_binprod a (cat_binprod b c)) $==
cat_pr1 (cat_binprod b' (cat_binprod a' c')) $o
(fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c)
      1: napply cat_pr1_fmap11_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
g $o cat_pr1 (cat_binprod b c) $o cat_pr2 (cat_binprod a (cat_binprod b c)) $==
cat_pr1 (cat_binprod b' (cat_binprod a' c')) $o
(fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c)
      nrefine (_ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
g $o cat_pr1 (cat_binprod b c) $o cat_pr2 (cat_binprod a (cat_binprod b c)) $==
cat_pr1 (cat_binprod b' (cat_binprod a' c')) $o
fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c
      refine (_ $@ (_^$ $@R _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
g $o cat_pr1 (cat_binprod b c) $o cat_pr2 (cat_binprod a (cat_binprod b c)) $==
?Goal1 $o cat_binprod_twist a b c
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr1 (cat_binprod b' (cat_binprod a' c')) $o
fmap11 cat_binprod g (fmap11 cat_binprod f h) $-> 
?Goal1
      2: napply cat_pr1_fmap11_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
g $o cat_pr1 (cat_binprod b c) $o cat_pr2 (cat_binprod a (cat_binprod b c)) $==
g $o cat_pr1 (cat_binprod b (cat_binprod a c)) $o cat_binprod_twist a b c
      refine (cat_assoc _ _ _ $@ (_ $@L _^$) $@ (cat_assoc _ _ _)^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr1 (cat_binprod b (cat_binprod a c)) $o cat_binprod_twist a b c $->
cat_pr1 (cat_binprod b c) $o cat_pr2 (cat_binprod a (cat_binprod b c))
      napply cat_binprod_beta_pr1.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr2 (cat_binprod b' (cat_binprod a' c')) $o
(cat_binprod_twist a' b' c' $o fmap11 cat_binprod f (fmap11 cat_binprod g h)) $==
cat_pr2 (cat_binprod b' (cat_binprod a' c')) $o
(fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c) nrefine (cat_assoc_opp _ _ _ $@ (cat_binprod_beta_pr2 _ _ _ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
fmap01 cat_binprod a' (cat_pr2 (cat_binprod b' c')) $o
fmap11 cat_binprod f (fmap11 cat_binprod g h) $==
cat_pr2 (cat_binprod b' (cat_binprod a' c')) $o
(fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c)
      nrefine (_ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
fmap01 cat_binprod a' (cat_pr2 (cat_binprod b' c')) $o
fmap11 cat_binprod f (fmap11 cat_binprod g h) $==
cat_pr2 (cat_binprod b' (cat_binprod a' c')) $o
fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c
      refine (_ $@ (_^$ $@R _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
fmap01 cat_binprod a' (cat_pr2 (cat_binprod b' c')) $o
fmap11 cat_binprod f (fmap11 cat_binprod g h) $==
?Goal0 $o cat_binprod_twist a b c
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr2 (cat_binprod b' (cat_binprod a' c')) $o
fmap11 cat_binprod g (fmap11 cat_binprod f h) $-> 
?Goal0
      2: napply cat_pr2_fmap11_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
fmap01 cat_binprod a' (cat_pr2 (cat_binprod b' c')) $o
fmap11 cat_binprod f (fmap11 cat_binprod g h) $==
fmap11 cat_binprod f h $o cat_pr2 (cat_binprod b (cat_binprod a c)) $o
cat_binprod_twist a b c
      refine (_ $@ (_ $@L _^$) $@ (cat_assoc _ _ _)^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
fmap01 cat_binprod a' (cat_pr2 (cat_binprod b' c')) $o
fmap11 cat_binprod f (fmap11 cat_binprod g h) $==
fmap11 cat_binprod f h $o ?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
cat_pr2 (cat_binprod b (cat_binprod a c)) $o cat_binprod_twist a b c $->
?Goal0
      2: napply cat_binprod_beta_pr2.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
fmap01 cat_binprod a' (cat_pr2 (cat_binprod b' c')) $o
fmap11 cat_binprod f (fmap11 cat_binprod g h) $==
fmap11 cat_binprod f h $o fmap01 cat_binprod a (cat_pr2 (cat_binprod b c))
      refine (_^$ $@ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
?Goal0 $->
fmap01 cat_binprod a' (cat_pr2 (cat_binprod b' c')) $o
fmap11 cat_binprod f (fmap11 cat_binprod g h)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
?Goal0 $== ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
?Goal $==
fmap11 cat_binprod f h $o fmap01 cat_binprod a (cat_pr2 (cat_binprod b c))
      1,3: tapply fmap11_comp.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
fmap11 cat_binprod
  (fmap (uncurry (Bool_rec A))
     (fmap (fun b0 : A => (a', b0)) (cat_pr2 (cat_binprod b' c'))) true $o
   f)
  (fmap (uncurry (Bool_rec A))
     (fmap (fun b0 : A => (a', b0)) (cat_pr2 (cat_binprod b' c'))) false $o
   fmap11 cat_binprod g h) $==
fmap11 cat_binprod
  (f $o
   fmap (uncurry (Bool_rec A))
     (fmap (fun b0 : A => (a, b0)) (cat_pr2 (cat_binprod b c))) true)
  (h $o
   fmap (uncurry (Bool_rec A))
     (fmap (fun b0 : A => (a, b0)) (cat_pr2 (cat_binprod b c))) false)
      rapply fmap22.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
fmap (uncurry (Bool_rec A))
  (fmap (fun b0 : A => (a', b0)) (cat_pr2 (cat_binprod b' c'))) true $o
f $==
f $o
fmap (uncurry (Bool_rec A))
  (fmap (fun b0 : A => (a, b0)) (cat_pr2 (cat_binprod b c))) true
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
fmap (uncurry (Bool_rec A))
  (fmap (fun b0 : A => (a', b0)) (cat_pr2 (cat_binprod b' c'))) false $o
fmap11 cat_binprod g h $==
h $o
fmap (uncurry (Bool_rec A))
  (fmap (fun b0 : A => (a, b0)) (cat_pr2 (cat_binprod b c))) false
      1: exact (cat_idl _ $@ (cat_idr _)^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
a, a', b, b', c, c': A
f: a $-> a'
g: b $-> b'
h: c $-> c'
fmap (uncurry (Bool_rec A))
  (fmap (fun b0 : A => (a', b0)) (cat_pr2 (cat_binprod b' c'))) false $o
fmap11 cat_binprod g h $==
h $o
fmap (uncurry (Bool_rec A))
  (fmap (fun b0 : A => (a, b0)) (cat_pr2 (cat_binprod b c))) false
      napply cat_binprod_beta_pr2.
  Defined.

  Local Existing Instance symmetricbraiding_binprod.

  #[export] Instance associator_cat_binprod : Associator cat_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
Associator cat_binprod
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
Associator cat_binprod
    snapply associator_twist.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
SymmetricBraiding cat_binprod
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
forall a b c : A,
cat_binprod a (cat_binprod b c) $-> cat_binprod b (cat_binprod a c)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
forall a b c : A,
?twist a b c $o ?twist b a c $== Id (cat_binprod b (cat_binprod a c))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
forall (a a' b b' c c' : A) (f : a $-> a') (g : b $-> b') 
(h : c $-> c'),
?twist a' b' c' $o fmap11 cat_binprod f (fmap11 cat_binprod g h) $==
fmap11 cat_binprod g (fmap11 cat_binprod f h) $o ?twist a b c
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
SymmetricBraiding cat_binprod exact _.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
forall a b c : A,
cat_binprod a (cat_binprod b c) $-> cat_binprod b (cat_binprod a c) exact cat_binprod_twist.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
forall a b c : A,
cat_binprod_twist a b c $o cat_binprod_twist b a c $==
Id (cat_binprod b (cat_binprod a c)) exact cat_binprod_twist_cat_binprod_twist.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
forall (a a' b b' c c' : A) (f : a $-> a') (g : b $-> b') 
(h : c $-> c'),
cat_binprod_twist a' b' c' $o fmap11 cat_binprod f (fmap11 cat_binprod g h) $==
fmap11 cat_binprod g (fmap11 cat_binprod f h) $o cat_binprod_twist a b c intros ? ? ? ? ? ?; exact cat_binprod_twist_nat.
  Defined.

  Definition cat_pr1_pr1_associator_binprod x y z
    : cat_pr1 _ $o cat_pr1 _ $o associator_cat_binprod x y z $== cat_pr1 _.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o associator_cat_binprod x y z $==
cat_pr1 (cat_binprod x (cat_binprod y z))
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o associator_cat_binprod x y z $==
cat_pr1 (cat_binprod x (cat_binprod y z))
    nrefine ((_ $@L associator_twist'_unfold _ _ _ _ _ _ _ _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o
(symmetricbraiding_binprod z (cat_binprod x y) $o
 (cat_binprod_twist x z y $o
  fmap01 cat_binprod x (symmetricbraiding_binprod y z))) $==
cat_pr1 (cat_binprod x (cat_binprod y z))
    nrefine (cat_assoc _ _ _ $@ (_ $@L (cat_assoc_opp _ _ _ $@ (_ $@R _))) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_binprod (cat_binprod x y) z) $o
symmetricbraiding_binprod z (cat_binprod x y) $== 
?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_prod (Bool_rec A x y)) $o
(?Goal $o
 (cat_binprod_twist x z y $o
  fmap01 cat_binprod x (symmetricbraiding_binprod y z))) $==
cat_pr1 (cat_binprod x (cat_binprod y z))
    1: napply cat_binprod_beta_pr1.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_prod (Bool_rec A x y)) $o
(cat_pr2 (cat_binprod (fst (z, cat_binprod x y)) (snd (z, cat_binprod x y))) $o
 (cat_binprod_twist x z y $o
  fmap01 cat_binprod x (symmetricbraiding_binprod y z))) $==
cat_pr1 (cat_binprod x (cat_binprod y z))
    do 2 nrefine (cat_assoc_opp _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_prod (Bool_rec A x y)) $o
cat_pr2 (cat_binprod z (cat_binprod x y)) $o cat_binprod_twist x z y $o
fmap01 cat_binprod x (symmetricbraiding_binprod y z) $==
cat_pr1 (cat_binprod x (cat_binprod y z))
    nrefine ((cat_binprod_pr1_pr2_twist _ _ _ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_binprod x (cat_binprod z y)) $o
fmap01 cat_binprod x (symmetricbraiding_binprod y z) $==
cat_pr1 (cat_binprod x (cat_binprod y z))
    napply cat_pr1_fmap01_binprod.
  Defined.

  Definition cat_pr2_pr1_associator_binprod x y z
    : cat_pr2 _ $o cat_pr1 _ $o associator_cat_binprod x y z $== cat_pr1 _ $o cat_pr2 _.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o associator_cat_binprod x y z $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o associator_cat_binprod x y z $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
    nrefine ((_ $@L associator_twist'_unfold _ _ _ _ _ _ _ _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr1 (cat_binprod (cat_binprod x y) z) $o
(symmetricbraiding_binprod z (cat_binprod x y) $o
 (cat_binprod_twist x z y $o
  fmap01 cat_binprod x (symmetricbraiding_binprod y z))) $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
    nrefine (cat_assoc _ _ _ $@ (_ $@L (cat_assoc_opp _ _ _ $@ (_ $@R _))) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_binprod (cat_binprod x y) z) $o
symmetricbraiding_binprod z (cat_binprod x y) $== 
?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_prod (Bool_rec A x y)) $o
(?Goal $o
 (cat_binprod_twist x z y $o
  fmap01 cat_binprod x (symmetricbraiding_binprod y z))) $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
    1: napply cat_binprod_beta_pr1.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_prod (Bool_rec A x y)) $o
(cat_pr2 (cat_binprod (fst (z, cat_binprod x y)) (snd (z, cat_binprod x y))) $o
 (cat_binprod_twist x z y $o
  fmap01 cat_binprod x (symmetricbraiding_binprod y z))) $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
    do 2 nrefine (cat_assoc_opp _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_prod (Bool_rec A x y)) $o
cat_pr2 (cat_binprod z (cat_binprod x y)) $o cat_binprod_twist x z y $o
fmap01 cat_binprod x (symmetricbraiding_binprod y z) $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
    nrefine ((cat_binprod_pr2_pr2_twist _ _ _ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_prod (Bool_rec A z y)) $o
cat_pr2 (cat_binprod x (cat_binprod z y)) $o
fmap01 cat_binprod x (symmetricbraiding_binprod y z) $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
    nrefine (cat_assoc _ _ _ $@ (_ $@L cat_pr2_fmap01_binprod _ _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_prod (Bool_rec A z y)) $o
(symmetricbraiding_binprod y z $o cat_pr2 (cat_binprod x (cat_binprod y z))) $==
cat_pr1 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
    exact (cat_assoc_opp _ _ _ $@ (cat_binprod_beta_pr2 _ _ _ $@R _)).
  Defined.

  Definition cat_pr2_associator_binprod x y z
    : cat_pr2 _ $o associator_cat_binprod x y z $== cat_pr2 _ $o cat_pr2 _.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_binprod (cat_binprod x y) z) $o associator_cat_binprod x y z $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_binprod (cat_binprod x y) z) $o associator_cat_binprod x y z $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
    nrefine ((_ $@L associator_twist'_unfold _ _ _ _ _ _ _ _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr2 (cat_binprod (cat_binprod x y) z) $o
(symmetricbraiding_binprod z (cat_binprod x y) $o
 (cat_binprod_twist x z y $o
  fmap01 cat_binprod x (symmetricbraiding_binprod y z))) $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
    nrefine (cat_assoc_opp _ _ _ $@ (cat_binprod_beta_pr2 _ _ _ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_binprod z (cat_binprod x y)) $o
(cat_binprod_twist x z y $o
 fmap01 cat_binprod x (symmetricbraiding_binprod y z)) $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
    nrefine (cat_assoc_opp _ _ _ $@ (cat_binprod_pr1_twist _ _ _ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_prod (Bool_rec A z y)) $o
cat_pr2 (cat_binprod x (cat_binprod z y)) $o
fmap01 cat_binprod x (symmetricbraiding_binprod y z) $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
    nrefine (cat_assoc _ _ _ $@ (_ $@L cat_pr2_fmap01_binprod _ _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
x, y, z: A
cat_pr1 (cat_prod (Bool_rec A z y)) $o
(symmetricbraiding_binprod y z $o cat_pr2 (cat_binprod x (cat_binprod y z))) $==
cat_pr2 (cat_prod (Bool_rec A y z)) $o
cat_pr2 (cat_binprod x (cat_binprod y z))
    exact (cat_assoc_opp _ _ _ $@ (cat_binprod_beta_pr1 _ _ _ $@R _)).
  Defined.

  Definition cat_binprod_associator_corec {w x y z}
    (f : w $-> x) (g : w $-> y) (h : w $-> z)
    : associator_cat_binprod x y z $o cat_binprod_corec _ f (cat_binprod_corec _ g h)
      $== cat_binprod_corec _ (cat_binprod_corec _ f g) h.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
associator_cat_binprod x y z $o
cat_binprod_corec (cat_binprod x (cat_binprod y z)) f
  (cat_binprod_corec (cat_binprod y z) g h) $==
cat_binprod_corec (cat_binprod (cat_binprod x y) z)
  (cat_binprod_corec (cat_binprod x y) f g) h
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
associator_cat_binprod x y z $o
cat_binprod_corec (cat_binprod x (cat_binprod y z)) f
  (cat_binprod_corec (cat_binprod y z) g h) $==
cat_binprod_corec (cat_binprod (cat_binprod x y) z)
  (cat_binprod_corec (cat_binprod x y) f g) h
    nrefine ((associator_twist'_unfold _ _ _ _ _ _ _ _ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
symmetricbraiding_binprod z (cat_binprod x y) $o
(cat_binprod_twist x z y $o
 fmap01 cat_binprod x (symmetricbraiding_binprod y z)) $o
cat_binprod_corec (cat_binprod x (cat_binprod y z)) f
  (cat_binprod_corec (cat_binprod y z) g h) $==
cat_binprod_corec (cat_binprod (cat_binprod x y) z)
  (cat_binprod_corec (cat_binprod x y) f g) h
    nrefine ((cat_assoc_opp _ _ _ $@R _) $@ cat_assoc _ _ _ $@ (_ $@L (_ $@ _)) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
fmap01 cat_binprod x (symmetricbraiding_binprod y z) $o
cat_binprod_corec (cat_binprod x (cat_binprod y z)) f
  (cat_binprod_corec (cat_binprod y z) g h) $==
?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
?Goal0 $== ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
symmetricbraiding_binprod z (cat_binprod x y) $o cat_binprod_twist x z y $o
?Goal $==
cat_binprod_corec (cat_binprod (cat_binprod x y) z)
  (cat_binprod_corec (cat_binprod x y) f g) h
    1: napply cat_binprod_fmap01_corec.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_binprod_corec (cat_binprod x (cat_binprod z y)) f
  (symmetricbraiding_binprod y z $o cat_binprod_corec (cat_binprod y z) g h) $==
?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
symmetricbraiding_binprod z (cat_binprod x y) $o cat_binprod_twist x z y $o
?Goal $==
cat_binprod_corec (cat_binprod (cat_binprod x y) z)
  (cat_binprod_corec (cat_binprod x y) f g) h
    1: rapply (cat_binprod_corec_eta _ _ _ _ _ (Id _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
symmetricbraiding_binprod y z $o cat_binprod_corec (cat_binprod y z) g h $==
?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
symmetricbraiding_binprod z (cat_binprod x y) $o cat_binprod_twist x z y $o
cat_binprod_corec (cat_binprod x (cat_binprod z y)) f ?Goal0 $==
cat_binprod_corec (cat_binprod (cat_binprod x y) z)
  (cat_binprod_corec (cat_binprod x y) f g) h
    1: napply cat_binprod_swap_corec.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
symmetricbraiding_binprod z (cat_binprod x y) $o cat_binprod_twist x z y $o
cat_binprod_corec (cat_binprod x (cat_binprod z y)) f
  (cat_binprod_corec (cat_binprod z y) h g) $==
cat_binprod_corec (cat_binprod (cat_binprod x y) z)
  (cat_binprod_corec (cat_binprod x y) f g) h
    nrefine (cat_assoc _ _ _ $@ (_ $@L _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
cat_binprod_twist x z y $o
cat_binprod_corec (cat_binprod x (cat_binprod z y)) f
  (cat_binprod_corec (cat_binprod z y) h g) $==
?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
symmetricbraiding_binprod z (cat_binprod x y) $o ?Goal $==
cat_binprod_corec (cat_binprod (cat_binprod x y) z)
  (cat_binprod_corec (cat_binprod x y) f g) h
    1: napply cat_binprod_twist_corec.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
w, x, y, z: A
f: w $-> x
g: w $-> y
h: w $-> z
symmetricbraiding_binprod z (cat_binprod x y) $o
cat_binprod_corec (cat_binprod z (cat_binprod x y)) h
  (cat_binprod_corec (cat_binprod x y) f g) $==
cat_binprod_corec (cat_binprod (cat_binprod x y) z)
  (cat_binprod_corec (cat_binprod x y) f g) h
    napply cat_binprod_swap_corec.
  Defined.

  Context (unit : A) `{!IsTerminal unit}.

  Local Instance right_unitor_binprod
    : RightUnitor cat_binprod unit.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
RightUnitor cat_binprod unit
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
RightUnitor cat_binprod unit
    snapply Build_NatEquiv.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
forall a : A, flip cat_binprod unit a $<~> idmap a
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
Is1Natural (flip cat_binprod unit) idmap (fun a : A => ?e a)
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
forall a : A, flip cat_binprod unit a $<~> idmap a intros a; unfold flip.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a: A
cat_binprod a unit $<~> a
      snapply cate_adjointify.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a: A
cat_binprod a unit $-> a
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a: A
a $-> cat_binprod a unit
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a: A
?f $o ?g $== Id a
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a: A
?g $o ?f $== Id (cat_binprod a unit)
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a: A
cat_binprod a unit $-> a exact (cat_pr1 _).
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a: A
a $-> cat_binprod a unit exact (cat_binprod_corec _ (Id _) (mor_terminal _ _)).
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a: A
cat_pr1 (cat_binprod a unit) $o
cat_binprod_corec (cat_binprod a unit) (Id a) (mor_terminal a unit) $== 
Id a exact (cat_binprod_beta_pr1 _ _ _).
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a: A
cat_binprod_corec (cat_binprod a unit) (Id a) (mor_terminal a unit) $o
cat_pr1 (cat_binprod a unit) $== Id (cat_binprod a unit) napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a: A
cat_pr1 (cat_binprod a unit) $o
(cat_binprod_corec (cat_binprod a unit) (Id a) (mor_terminal a unit) $o
 cat_pr1 (cat_binprod a unit)) $==
cat_pr1 (cat_binprod a unit) $o Id (cat_binprod a unit)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a: A
cat_pr2 (cat_binprod a unit) $o
(cat_binprod_corec (cat_binprod a unit) (Id a) (mor_terminal a unit) $o
 cat_pr1 (cat_binprod a unit)) $==
cat_pr2 (cat_binprod a unit) $o Id (cat_binprod a unit)
        *A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a: A
cat_pr1 (cat_binprod a unit) $o
(cat_binprod_corec (cat_binprod a unit) (Id a) (mor_terminal a unit) $o
 cat_pr1 (cat_binprod a unit)) $==
cat_pr1 (cat_binprod a unit) $o Id (cat_binprod a unit) nrefine (cat_assoc_opp _ _ _ $@ (cat_binprod_beta_pr1 _ _ _ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a: A
Id a $o cat_pr1 (cat_binprod a unit) $==
cat_pr1 (cat_binprod a unit) $o Id (cat_binprod a unit)
          exact (cat_idl _ $@ (cat_idr _)^$).
        *A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a: A
cat_pr2 (cat_binprod a unit) $o
(cat_binprod_corec (cat_binprod a unit) (Id a) (mor_terminal a unit) $o
 cat_pr1 (cat_binprod a unit)) $==
cat_pr2 (cat_binprod a unit) $o Id (cat_binprod a unit) nrefine (cat_assoc_opp _ _ _ $@ (cat_binprod_beta_pr2 _ _ _ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a: A
mor_terminal a unit $o cat_pr1 (cat_binprod a unit) $==
cat_pr2 (cat_binprod a unit) $o Id (cat_binprod a unit)
          exact ((mor_terminal_unique _ _ _)^$ $@ mor_terminal_unique _ _ _).
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
Is1Natural (flip cat_binprod unit) idmap
  (fun a : A =>
   (fun a0 : A =>
    cate_adjointify (cat_pr1 (cat_binprod a0 unit))
      (cat_binprod_corec (cat_binprod a0 unit) (Id a0) (mor_terminal a0 unit))
      (cat_binprod_beta_pr1 (cat_binprod a0 unit) 
         (Id a0) (mor_terminal a0 unit))
      (cat_binprod_eta_pr (cat_binprod a0 unit)
         (cat_binprod_corec (cat_binprod a0 unit) 
            (Id a0) (mor_terminal a0 unit) $o
          cat_pr1 (cat_binprod a0 unit))
         (Id (cat_binprod a0 unit))
         ((cat_assoc_opp (cat_pr1 (cat_binprod a0 unit))
             (cat_binprod_corec (cat_binprod a0 unit) 
                (Id a0) (mor_terminal a0 unit))
             (cat_pr1 (cat_binprod a0 unit)) $@
           (cat_binprod_beta_pr1 (cat_binprod a0 unit) 
              (Id a0) (mor_terminal a0 unit) $@R
            cat_pr1 (cat_binprod a0 unit))) $@
          (cat_idl (cat_pr1 (cat_binprod a0 unit)) $@
           (cat_idr (cat_pr1 (cat_binprod a0 unit)))^$))
         ((cat_assoc_opp (cat_pr1 (cat_binprod a0 unit))
             (cat_binprod_corec (cat_binprod a0 unit) 
                (Id a0) (mor_terminal a0 unit))
             (cat_pr2 (cat_binprod a0 unit)) $@
           (cat_binprod_beta_pr2 (cat_binprod a0 unit) 
              (Id a0) (mor_terminal a0 unit) $@R
            cat_pr1 (cat_binprod a0 unit))) $@
          ((mor_terminal_unique (cat_binprod a0 unit) unit
              (mor_terminal a0 unit $o cat_pr1 (cat_binprod a0 unit)))^$ $@
           mor_terminal_unique (cat_binprod a0 unit) unit
             (cat_pr2 (cat_binprod a0 unit) $o Id (cat_binprod a0 unit)))))
    :
    flip cat_binprod unit a0 $<~> idmap a0) a) snapply Build_Is1Natural.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
forall (a a' : A) (f : a $-> a'),
(fun a0 : A =>
 cate_fun
   ((fun a1 : A =>
     cate_adjointify (cat_pr1 (cat_binprod a1 unit))
       (cat_binprod_corec (cat_binprod a1 unit) (Id a1)
          (mor_terminal a1 unit))
       (cat_binprod_beta_pr1 (cat_binprod a1 unit) 
          (Id a1) (mor_terminal a1 unit))
       (cat_binprod_eta_pr (cat_binprod a1 unit)
          (cat_binprod_corec (cat_binprod a1 unit) 
             (Id a1) (mor_terminal a1 unit) $o
           cat_pr1 (cat_binprod a1 unit))
          (Id (cat_binprod a1 unit))
          ((cat_assoc_opp (cat_pr1 (cat_binprod a1 unit))
              (cat_binprod_corec (cat_binprod a1 unit) 
                 (Id a1) (mor_terminal a1 unit))
              (cat_pr1 (cat_binprod a1 unit)) $@
            (cat_binprod_beta_pr1 (cat_binprod a1 unit) 
               (Id a1) (mor_terminal a1 unit) $@R
             cat_pr1 (cat_binprod a1 unit))) $@
           (cat_idl (cat_pr1 (cat_binprod a1 unit)) $@
            (cat_idr (cat_pr1 (cat_binprod a1 unit)))^$))
          ((cat_assoc_opp (cat_pr1 (cat_binprod a1 unit))
              (cat_binprod_corec (cat_binprod a1 unit) 
                 (Id a1) (mor_terminal a1 unit))
              (cat_pr2 (cat_binprod a1 unit)) $@
            (cat_binprod_beta_pr2 (cat_binprod a1 unit) 
               (Id a1) (mor_terminal a1 unit) $@R
             cat_pr1 (cat_binprod a1 unit))) $@
           ((mor_terminal_unique (cat_binprod a1 unit) unit
               (mor_terminal a1 unit $o cat_pr1 (cat_binprod a1 unit)))^$ $@
            mor_terminal_unique (cat_binprod a1 unit) unit
              (cat_pr2 (cat_binprod a1 unit) $o Id (cat_binprod a1 unit)))))
     :
     flip cat_binprod unit a1 $<~> idmap a1) a0))
  a' $o
fmap (flip cat_binprod unit) f $==
fmap idmap f $o
(fun a0 : A =>
 cate_fun
   ((fun a1 : A =>
     cate_adjointify (cat_pr1 (cat_binprod a1 unit))
       (cat_binprod_corec (cat_binprod a1 unit) (Id a1)
          (mor_terminal a1 unit))
       (cat_binprod_beta_pr1 (cat_binprod a1 unit) 
          (Id a1) (mor_terminal a1 unit))
       (cat_binprod_eta_pr (cat_binprod a1 unit)
          (cat_binprod_corec (cat_binprod a1 unit) 
             (Id a1) (mor_terminal a1 unit) $o
           cat_pr1 (cat_binprod a1 unit))
          (Id (cat_binprod a1 unit))
          ((cat_assoc_opp (cat_pr1 (cat_binprod a1 unit))
              (cat_binprod_corec (cat_binprod a1 unit) 
                 (Id a1) (mor_terminal a1 unit))
              (cat_pr1 (cat_binprod a1 unit)) $@
            (cat_binprod_beta_pr1 (cat_binprod a1 unit) 
               (Id a1) (mor_terminal a1 unit) $@R
             cat_pr1 (cat_binprod a1 unit))) $@
           (cat_idl (cat_pr1 (cat_binprod a1 unit)) $@
            (cat_idr (cat_pr1 (cat_binprod a1 unit)))^$))
          ((cat_assoc_opp (cat_pr1 (cat_binprod a1 unit))
              (cat_binprod_corec (cat_binprod a1 unit) 
                 (Id a1) (mor_terminal a1 unit))
              (cat_pr2 (cat_binprod a1 unit)) $@
            (cat_binprod_beta_pr2 (cat_binprod a1 unit) 
               (Id a1) (mor_terminal a1 unit) $@R
             cat_pr1 (cat_binprod a1 unit))) $@
           ((mor_terminal_unique (cat_binprod a1 unit) unit
               (mor_terminal a1 unit $o cat_pr1 (cat_binprod a1 unit)))^$ $@
            mor_terminal_unique (cat_binprod a1 unit) unit
              (cat_pr2 (cat_binprod a1 unit) $o Id (cat_binprod a1 unit)))))
     :
     flip cat_binprod unit a1 $<~> idmap a1) a0))
  a
      intros a b f.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
f: a $-> b
(fun a0 : A =>
 cate_fun
   ((fun a1 : A =>
     cate_adjointify (cat_pr1 (cat_binprod a1 unit))
       (cat_binprod_corec (cat_binprod a1 unit) (Id a1)
          (mor_terminal a1 unit))
       (cat_binprod_beta_pr1 (cat_binprod a1 unit) 
          (Id a1) (mor_terminal a1 unit))
       (cat_binprod_eta_pr (cat_binprod a1 unit)
          (cat_binprod_corec (cat_binprod a1 unit) 
             (Id a1) (mor_terminal a1 unit) $o
           cat_pr1 (cat_binprod a1 unit))
          (Id (cat_binprod a1 unit))
          ((cat_assoc_opp (cat_pr1 (cat_binprod a1 unit))
              (cat_binprod_corec (cat_binprod a1 unit) 
                 (Id a1) (mor_terminal a1 unit))
              (cat_pr1 (cat_binprod a1 unit)) $@
            (cat_binprod_beta_pr1 (cat_binprod a1 unit) 
               (Id a1) (mor_terminal a1 unit) $@R
             cat_pr1 (cat_binprod a1 unit))) $@
           (cat_idl (cat_pr1 (cat_binprod a1 unit)) $@
            (cat_idr (cat_pr1 (cat_binprod a1 unit)))^$))
          ((cat_assoc_opp (cat_pr1 (cat_binprod a1 unit))
              (cat_binprod_corec (cat_binprod a1 unit) 
                 (Id a1) (mor_terminal a1 unit))
              (cat_pr2 (cat_binprod a1 unit)) $@
            (cat_binprod_beta_pr2 (cat_binprod a1 unit) 
               (Id a1) (mor_terminal a1 unit) $@R
             cat_pr1 (cat_binprod a1 unit))) $@
           ((mor_terminal_unique (cat_binprod a1 unit) unit
               (mor_terminal a1 unit $o cat_pr1 (cat_binprod a1 unit)))^$ $@
            mor_terminal_unique (cat_binprod a1 unit) unit
              (cat_pr2 (cat_binprod a1 unit) $o Id (cat_binprod a1 unit)))))
     :
     flip cat_binprod unit a1 $<~> idmap a1) a0))
  b $o
fmap (flip cat_binprod unit) f $==
fmap idmap f $o
(fun a0 : A =>
 cate_fun
   ((fun a1 : A =>
     cate_adjointify (cat_pr1 (cat_binprod a1 unit))
       (cat_binprod_corec (cat_binprod a1 unit) (Id a1)
          (mor_terminal a1 unit))
       (cat_binprod_beta_pr1 (cat_binprod a1 unit) 
          (Id a1) (mor_terminal a1 unit))
       (cat_binprod_eta_pr (cat_binprod a1 unit)
          (cat_binprod_corec (cat_binprod a1 unit) 
             (Id a1) (mor_terminal a1 unit) $o
           cat_pr1 (cat_binprod a1 unit))
          (Id (cat_binprod a1 unit))
          ((cat_assoc_opp (cat_pr1 (cat_binprod a1 unit))
              (cat_binprod_corec (cat_binprod a1 unit) 
                 (Id a1) (mor_terminal a1 unit))
              (cat_pr1 (cat_binprod a1 unit)) $@
            (cat_binprod_beta_pr1 (cat_binprod a1 unit) 
               (Id a1) (mor_terminal a1 unit) $@R
             cat_pr1 (cat_binprod a1 unit))) $@
           (cat_idl (cat_pr1 (cat_binprod a1 unit)) $@
            (cat_idr (cat_pr1 (cat_binprod a1 unit)))^$))
          ((cat_assoc_opp (cat_pr1 (cat_binprod a1 unit))
              (cat_binprod_corec (cat_binprod a1 unit) 
                 (Id a1) (mor_terminal a1 unit))
              (cat_pr2 (cat_binprod a1 unit)) $@
            (cat_binprod_beta_pr2 (cat_binprod a1 unit) 
               (Id a1) (mor_terminal a1 unit) $@R
             cat_pr1 (cat_binprod a1 unit))) $@
           ((mor_terminal_unique (cat_binprod a1 unit) unit
               (mor_terminal a1 unit $o cat_pr1 (cat_binprod a1 unit)))^$ $@
            mor_terminal_unique (cat_binprod a1 unit) unit
              (cat_pr2 (cat_binprod a1 unit) $o Id (cat_binprod a1 unit)))))
     :
     flip cat_binprod unit a1 $<~> idmap a1) a0))
  a
      refine ((_ $@R _) $@ _ $@ (_ $@L _^$)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
f: a $-> b
cate_adjointify (cat_pr1 (cat_binprod b unit))
  (cat_binprod_corec (cat_binprod b unit) (Id b) (mor_terminal b unit))
  (cat_binprod_beta_pr1 (cat_binprod b unit) (Id b) (mor_terminal b unit))
  (cat_binprod_eta_pr (cat_binprod b unit)
     (cat_binprod_corec (cat_binprod b unit) (Id b) (mor_terminal b unit) $o
      cat_pr1 (cat_binprod b unit))
     (Id (cat_binprod b unit))
     ((cat_assoc_opp (cat_pr1 (cat_binprod b unit))
         (cat_binprod_corec (cat_binprod b unit) (Id b) (mor_terminal b unit))
         (cat_pr1 (cat_binprod b unit)) $@
       (cat_binprod_beta_pr1 (cat_binprod b unit) 
          (Id b) (mor_terminal b unit) $@R
        cat_pr1 (cat_binprod b unit))) $@
      (cat_idl (cat_pr1 (cat_binprod b unit)) $@
       (cat_idr (cat_pr1 (cat_binprod b unit)))^$))
     ((cat_assoc_opp (cat_pr1 (cat_binprod b unit))
         (cat_binprod_corec (cat_binprod b unit) (Id b) (mor_terminal b unit))
         (cat_pr2 (cat_binprod b unit)) $@
       (cat_binprod_beta_pr2 (cat_binprod b unit) 
          (Id b) (mor_terminal b unit) $@R
        cat_pr1 (cat_binprod b unit))) $@
      ((mor_terminal_unique (cat_binprod b unit) unit
          (mor_terminal b unit $o cat_pr1 (cat_binprod b unit)))^$ $@
       mor_terminal_unique (cat_binprod b unit) unit
         (cat_pr2 (cat_binprod b unit) $o Id (cat_binprod b unit))))) $==
?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
f: a $-> b
?Goal $o fmap (flip cat_binprod unit) f $== fmap idmap f $o ?Goal2
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
f: a $-> b
cate_adjointify (cat_pr1 (cat_binprod a unit))
  (cat_binprod_corec (cat_binprod a unit) (Id a) (mor_terminal a unit))
  (cat_binprod_beta_pr1 (cat_binprod a unit) (Id a) (mor_terminal a unit))
  (cat_binprod_eta_pr (cat_binprod a unit)
     (cat_binprod_corec (cat_binprod a unit) (Id a) (mor_terminal a unit) $o
      cat_pr1 (cat_binprod a unit))
     (Id (cat_binprod a unit))
     ((cat_assoc_opp (cat_pr1 (cat_binprod a unit))
         (cat_binprod_corec (cat_binprod a unit) (Id a) (mor_terminal a unit))
         (cat_pr1 (cat_binprod a unit)) $@
       (cat_binprod_beta_pr1 (cat_binprod a unit) 
          (Id a) (mor_terminal a unit) $@R
        cat_pr1 (cat_binprod a unit))) $@
      (cat_idl (cat_pr1 (cat_binprod a unit)) $@
       (cat_idr (cat_pr1 (cat_binprod a unit)))^$))
     ((cat_assoc_opp (cat_pr1 (cat_binprod a unit))
         (cat_binprod_corec (cat_binprod a unit) (Id a) (mor_terminal a unit))
         (cat_pr2 (cat_binprod a unit)) $@
       (cat_binprod_beta_pr2 (cat_binprod a unit) 
          (Id a) (mor_terminal a unit) $@R
        cat_pr1 (cat_binprod a unit))) $@
      ((mor_terminal_unique (cat_binprod a unit) unit
          (mor_terminal a unit $o cat_pr1 (cat_binprod a unit)))^$ $@
       mor_terminal_unique (cat_binprod a unit) unit
         (cat_pr2 (cat_binprod a unit) $o Id (cat_binprod a unit))))) $->
?Goal2
      1,3: napply cate_buildequiv_fun.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
f: a $-> b
cat_pr1 (cat_binprod b unit) $o fmap (flip cat_binprod unit) f $==
fmap idmap f $o cat_pr1 (cat_binprod a unit)
      napply cat_binprod_beta_pr1.
  Defined.

  Local Existing Instance left_unitor_twist.

  Local Instance triangle_binprod
    : TriangleIdentity cat_binprod unit.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
TriangleIdentity cat_binprod unit
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
TriangleIdentity cat_binprod unit
    snapply triangle_twist.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
forall a b : A,
fmap01 cat_binprod a (right_unitor_binprod b) $==
symmetricbraiding_binprod b a $o
fmap01 cat_binprod b (right_unitor_binprod a) $o cat_binprod_twist a b unit
    intros a b.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
fmap01 cat_binprod a (right_unitor_binprod b) $==
symmetricbraiding_binprod b a $o
fmap01 cat_binprod b (right_unitor_binprod a) $o cat_binprod_twist a b unit
    refine (fmap02 _ _ _ $@ _ $@ ((_ $@L fmap02 _ _ _^$) $@R _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
right_unitor_binprod b $== ?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
fmap01 cat_binprod a ?Goal $==
symmetricbraiding_binprod b a $o fmap01 cat_binprod b ?Goal2 $o
cat_binprod_twist a b unit
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
right_unitor_binprod a $-> ?Goal2
    1,3: napply cate_buildequiv_fun.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
fmap01 cat_binprod a (cat_pr1 (cat_binprod b unit)) $==
symmetricbraiding_binprod b a $o
fmap01 cat_binprod b (cat_pr1 (cat_binprod a unit)) $o
cat_binprod_twist a b unit
    napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
cat_pr1 (cat_binprod a b) $o
fmap01 cat_binprod a (cat_pr1 (cat_binprod b unit)) $==
cat_pr1 (cat_binprod a b) $o
(symmetricbraiding_binprod b a $o
 fmap01 cat_binprod b (cat_pr1 (cat_binprod a unit)) $o
 cat_binprod_twist a b unit)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
cat_pr2 (cat_binprod a b) $o
fmap01 cat_binprod a (cat_pr1 (cat_binprod b unit)) $==
cat_pr2 (cat_binprod a b) $o
(symmetricbraiding_binprod b a $o
 fmap01 cat_binprod b (cat_pr1 (cat_binprod a unit)) $o
 cat_binprod_twist a b unit)
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
cat_pr1 (cat_binprod a b) $o
fmap01 cat_binprod a (cat_pr1 (cat_binprod b unit)) $==
cat_pr1 (cat_binprod a b) $o
(symmetricbraiding_binprod b a $o
 fmap01 cat_binprod b (cat_pr1 (cat_binprod a unit)) $o
 cat_binprod_twist a b unit) nrefine (cat_pr1_fmap01_binprod _ _ $@ _ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
cat_pr1 (cat_binprod a (flip cat_binprod unit b)) $==
cat_pr1 (cat_binprod a b) $o
(symmetricbraiding_binprod b a $o
 fmap01 cat_binprod b (cat_pr1 (cat_binprod a unit))) $o
cat_binprod_twist a b unit
      refine (_ $@ (((_^$ $@R _) $@ cat_assoc _ _ _) $@R _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
cat_pr1 (cat_binprod a (flip cat_binprod unit b)) $==
?Goal1 $o fmap01 cat_binprod b (cat_pr1 (cat_binprod a unit)) $o
cat_binprod_twist a b unit
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
cat_pr1 (cat_binprod a b) $o symmetricbraiding_binprod b a $-> ?Goal1
      2: napply cat_binprod_beta_pr1.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
cat_pr1 (cat_binprod a (flip cat_binprod unit b)) $==
cat_pr2 (cat_binprod (fst (b, a)) (snd (b, a))) $o
fmap01 cat_binprod b (cat_pr1 (cat_binprod a unit)) $o
cat_binprod_twist a b unit
      refine ((_ $@R _) $@ _)^$.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
cat_pr2 (cat_binprod b a) $o
fmap01 cat_binprod b (cat_pr1 (cat_binprod a unit)) $== 
?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
?Goal0 $o cat_binprod_twist a b unit $==
cat_pr1 (cat_binprod a (flip cat_binprod unit b))
      1: napply cat_pr2_fmap01_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
cat_pr1 (cat_binprod a unit) $o
cat_pr2 (cat_binprod b (flip cat_binprod unit a)) $o
cat_binprod_twist a b unit $==
cat_pr1 (cat_binprod a (flip cat_binprod unit b))
      napply cat_binprod_pr1_pr2_twist.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
cat_pr2 (cat_binprod a b) $o
fmap01 cat_binprod a (cat_pr1 (cat_binprod b unit)) $==
cat_pr2 (cat_binprod a b) $o
(symmetricbraiding_binprod b a $o
 fmap01 cat_binprod b (cat_pr1 (cat_binprod a unit)) $o
 cat_binprod_twist a b unit) nrefine (cat_pr2_fmap01_binprod _ _ $@ _ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
cat_pr1 (cat_binprod b unit) $o
cat_pr2 (cat_binprod a (flip cat_binprod unit b)) $==
cat_pr2 (cat_binprod a b) $o
(symmetricbraiding_binprod b a $o
 fmap01 cat_binprod b (cat_pr1 (cat_binprod a unit))) $o
cat_binprod_twist a b unit
      refine (_ $@ (((cat_binprod_beta_pr2 _ _ _)^$ $@R _) $@ cat_assoc _ _ _ $@R _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
cat_pr1 (cat_binprod b unit) $o
cat_pr2 (cat_binprod a (flip cat_binprod unit b)) $==
cat_pr1 (cat_binprod b a) $o
fmap01 cat_binprod b (cat_pr1 (cat_binprod a unit)) $o
cat_binprod_twist a b unit
      refine ((_ $@R _) $@ _)^$.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
cat_pr1 (cat_binprod b a) $o
fmap01 cat_binprod b (cat_pr1 (cat_binprod a unit)) $== 
?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
?Goal $o cat_binprod_twist a b unit $==
cat_pr1 (cat_binprod b unit) $o
cat_pr2 (cat_binprod a (flip cat_binprod unit b))
      1: napply cat_pr1_fmap01_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b: A
cat_pr1 (cat_binprod b (flip cat_binprod unit a)) $o
cat_binprod_twist a b unit $==
cat_pr1 (cat_binprod b unit) $o
cat_pr2 (cat_binprod a (flip cat_binprod unit b))
      napply cat_binprod_beta_pr1.
  Defined.

  #[export] Instance pentagon_binprod
    : PentagonIdentity cat_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
PentagonIdentity cat_binprod
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
PentagonIdentity cat_binprod
    intros a b c d.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
associator_cat_binprod (cat_binprod a b) c d $o
associator_cat_binprod a b (cat_binprod c d) $==
fmap10 cat_binprod (associator_cat_binprod a b c) d $o
associator_cat_binprod a (cat_binprod b c) d $o
fmap01 cat_binprod a (associator_cat_binprod b c d)
    napply cat_binprod_eta_pr_xx_x.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(associator_cat_binprod (cat_binprod a b) c d $o
 associator_cat_binprod a b (cat_binprod c d)) $==
cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(fmap10 cat_binprod (associator_cat_binprod a b c) d $o
 associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(associator_cat_binprod (cat_binprod a b) c d $o
 associator_cat_binprod a b (cat_binprod c d)) $==
cat_pr2 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(fmap10 cat_binprod (associator_cat_binprod a b c) d $o
 associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(associator_cat_binprod (cat_binprod a b) c d $o
 associator_cat_binprod a b (cat_binprod c d)) $==
cat_pr2 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(fmap10 cat_binprod (associator_cat_binprod a b c) d $o
 associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(associator_cat_binprod (cat_binprod a b) c d $o
 associator_cat_binprod a b (cat_binprod c d)) $==
cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(fmap10 cat_binprod (associator_cat_binprod a b c) d $o
 associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d)) nrefine (cat_assoc_opp _ _ _ $@ (_ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
associator_cat_binprod (cat_binprod a b) c d $== ?Goal1
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
?Goal1 $o associator_cat_binprod a b (cat_binprod c d) $==
cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(fmap10 cat_binprod (associator_cat_binprod a b c) d $o
 associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
      1: napply cat_pr1_pr1_associator_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_binprod (cat_binprod a b) (cat_binprod c d)) $o
associator_cat_binprod a b (cat_binprod c d) $==
cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(fmap10 cat_binprod (associator_cat_binprod a b c) d $o
 associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
      refine (_ $@ (_ $@L ((((_^$ $@R _) $@ cat_assoc _ _ _) $@R _)
        $@ cat_assoc _ _ _)) $@ cat_assoc_opp _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_binprod (cat_binprod a b) (cat_binprod c d)) $o
associator_cat_binprod a b (cat_binprod c d) $==
cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
(?Goal2 $o associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
fmap10 cat_binprod (associator_cat_binprod a b c) d $-> 
?Goal2
      2: napply cat_pr1_fmap10_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_binprod (cat_binprod a b) (cat_binprod c d)) $o
associator_cat_binprod a b (cat_binprod c d) $==
cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
(associator_cat_binprod a b c $o
 cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
 associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
      do 2 nrefine (_ $@ (_ $@L cat_assoc_opp _ _ _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_binprod (cat_binprod a b) (cat_binprod c d)) $o
associator_cat_binprod a b (cat_binprod c d) $==
cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
(associator_cat_binprod a b c $o
 (cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
  (associator_cat_binprod a (cat_binprod b c) d $o
   fmap01 cat_binprod a (associator_cat_binprod b c d))))
      napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_binprod a b) $o
(cat_pr1 (cat_binprod (cat_binprod a b) (cat_binprod c d)) $o
 associator_cat_binprod a b (cat_binprod c d)) $==
cat_pr1 (cat_binprod a b) $o
(cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
 (associator_cat_binprod a b c $o
  (cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
   (associator_cat_binprod a (cat_binprod b c) d $o
    fmap01 cat_binprod a (associator_cat_binprod b c d)))))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_binprod a b) $o
(cat_pr1 (cat_binprod (cat_binprod a b) (cat_binprod c d)) $o
 associator_cat_binprod a b (cat_binprod c d)) $==
cat_pr2 (cat_binprod a b) $o
(cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
 (associator_cat_binprod a b c $o
  (cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
   (associator_cat_binprod a (cat_binprod b c) d $o
    fmap01 cat_binprod a (associator_cat_binprod b c d)))))
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_binprod a b) $o
(cat_pr1 (cat_binprod (cat_binprod a b) (cat_binprod c d)) $o
 associator_cat_binprod a b (cat_binprod c d)) $==
cat_pr1 (cat_binprod a b) $o
(cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
 (associator_cat_binprod a b c $o
  (cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
   (associator_cat_binprod a (cat_binprod b c) d $o
    fmap01 cat_binprod a (associator_cat_binprod b c d))))) nrefine (cat_assoc_opp _ _ _ $@ _ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_binprod a b) $o
cat_pr1 (cat_binprod (cat_binprod a b) (cat_binprod c d)) $o
associator_cat_binprod a b (cat_binprod c d) $==
cat_pr1 (cat_binprod a b) $o
cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
(associator_cat_binprod a b c $o
 (cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
  (associator_cat_binprod a (cat_binprod b c) d $o
   fmap01 cat_binprod a (associator_cat_binprod b c d))))
        refine (_ $@ _ $@ (_^$ $@R _) $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_binprod a b) $o
cat_pr1 (cat_binprod (cat_binprod a b) (cat_binprod c d)) $o
associator_cat_binprod a b (cat_binprod c d) $== ?Goal4
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
?Goal4 $==
?Goal7 $o
(cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
 (associator_cat_binprod a (cat_binprod b c) d $o
  fmap01 cat_binprod a (associator_cat_binprod b c d)))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_binprod a b) $o
cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
associator_cat_binprod a b c $-> ?Goal7
        1,3: napply cat_pr1_pr1_associator_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_binprod a (cat_binprod b (cat_binprod c d))) $==
cat_pr1 (cat_binprod a (cat_binprod b c)) $o
(cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
 (associator_cat_binprod a (cat_binprod b c) d $o
  fmap01 cat_binprod a (associator_cat_binprod b c d)))
        do 2 nrefine (_ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_binprod a (cat_binprod b (cat_binprod c d))) $==
cat_pr1 (cat_binprod a (cat_binprod b c)) $o
cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
associator_cat_binprod a (cat_binprod b c) d $o
fmap01 cat_binprod a (associator_cat_binprod b c d)
        refine (_^$ $@ (_^$ $@R _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
?Goal3 $o fmap01 cat_binprod a (associator_cat_binprod b c d) $->
cat_pr1 (cat_binprod a (cat_binprod b (cat_binprod c d)))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_binprod a (cat_binprod b c)) $o
cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
associator_cat_binprod a (cat_binprod b c) d $-> ?Goal3
        2: napply cat_pr1_pr1_associator_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_binprod a (cat_binprod (cat_binprod b c) d)) $o
fmap01 cat_binprod a (associator_cat_binprod b c d) $->
cat_pr1 (cat_binprod a (cat_binprod b (cat_binprod c d)))
        napply cat_pr1_fmap01_binprod.
      +A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_binprod a b) $o
(cat_pr1 (cat_binprod (cat_binprod a b) (cat_binprod c d)) $o
 associator_cat_binprod a b (cat_binprod c d)) $==
cat_pr2 (cat_binprod a b) $o
(cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
 (associator_cat_binprod a b c $o
  (cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
   (associator_cat_binprod a (cat_binprod b c) d $o
    fmap01 cat_binprod a (associator_cat_binprod b c d))))) nrefine (cat_assoc_opp _ _ _ $@ _ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_binprod a b) $o
cat_pr1 (cat_binprod (cat_binprod a b) (cat_binprod c d)) $o
associator_cat_binprod a b (cat_binprod c d) $==
cat_pr2 (cat_binprod a b) $o
cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
(associator_cat_binprod a b c $o
 (cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
  (associator_cat_binprod a (cat_binprod b c) d $o
   fmap01 cat_binprod a (associator_cat_binprod b c d))))
        refine (_ $@ _ $@ (_^$ $@R _) $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_binprod a b) $o
cat_pr1 (cat_binprod (cat_binprod a b) (cat_binprod c d)) $o
associator_cat_binprod a b (cat_binprod c d) $== ?Goal1
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
?Goal1 $==
?Goal4 $o
(cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
 (associator_cat_binprod a (cat_binprod b c) d $o
  fmap01 cat_binprod a (associator_cat_binprod b c d)))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_binprod a b) $o
cat_pr1 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
associator_cat_binprod a b c $-> ?Goal4
        1,3: napply cat_pr2_pr1_associator_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d))) $==
cat_pr1 (cat_prod (Bool_rec A b c)) $o
cat_pr2 (cat_binprod a (cat_binprod b c)) $o
(cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
 (associator_cat_binprod a (cat_binprod b c) d $o
  fmap01 cat_binprod a (associator_cat_binprod b c d)))
        do 2 nrefine (_ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d))) $==
cat_pr1 (cat_prod (Bool_rec A b c)) $o
cat_pr2 (cat_binprod a (cat_binprod b c)) $o
cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
associator_cat_binprod a (cat_binprod b c) d $o
fmap01 cat_binprod a (associator_cat_binprod b c d)
        refine (_ $@ ((cat_assoc _ _ _ $@ (_ $@L (_^$ $@ cat_assoc _ _ _))
          $@ cat_assoc_opp _ _ _ $@ cat_assoc_opp _ _ _) $@R _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d))) $==
cat_pr1 (cat_prod (Bool_rec A b c)) $o ?Goal4 $o ?Goal3 $o
fmap01 cat_binprod a (associator_cat_binprod b c d)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_binprod a (cat_binprod b c)) $o
cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
associator_cat_binprod a (cat_binprod b c) d $-> ?Goal4 $o ?Goal3
        2: napply cat_pr2_pr1_associator_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d))) $==
cat_pr1 (cat_prod (Bool_rec A b c)) $o
cat_pr1 (cat_prod (Bool_rec A (cat_binprod b c) d)) $o
cat_pr2 (cat_binprod a (cat_binprod (cat_binprod b c) d)) $o
fmap01 cat_binprod a (associator_cat_binprod b c d)
        refine (_^$ $@ (_ $@L _^$) $@ cat_assoc_opp _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A b c)) $o
cat_pr1 (cat_prod (Bool_rec A (cat_binprod b c) d)) $o 
?Goal2 $->
cat_pr1 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_binprod a (cat_binprod (cat_binprod b c) d)) $o
fmap01 cat_binprod a (associator_cat_binprod b c d) $-> 
?Goal2
        2: napply cat_pr2_fmap01_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A b c)) $o
cat_pr1 (cat_prod (Bool_rec A (cat_binprod b c) d)) $o
(associator_cat_binprod b c d $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $->
cat_pr1 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))
        nrefine (cat_assoc_opp _ _ _ $@ (_ $@R _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A b c)) $o
cat_pr1 (cat_prod (Bool_rec A (cat_binprod b c) d)) $o
associator_cat_binprod b c d $==
cat_pr1 (cat_prod (Bool_rec A b (cat_binprod c d)))
        napply cat_pr1_pr1_associator_binprod.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(associator_cat_binprod (cat_binprod a b) c d $o
 associator_cat_binprod a b (cat_binprod c d)) $==
cat_pr2 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(fmap10 cat_binprod (associator_cat_binprod a b c) d $o
 associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d)) nrefine (cat_assoc_opp _ _ _ $@ (_ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
associator_cat_binprod (cat_binprod a b) c d $== ?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
?Goal0 $o associator_cat_binprod a b (cat_binprod c d) $==
cat_pr2 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(fmap10 cat_binprod (associator_cat_binprod a b c) d $o
 associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
      1: napply cat_pr2_pr1_associator_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A c d)) $o
cat_pr2 (cat_binprod (cat_binprod a b) (cat_binprod c d)) $o
associator_cat_binprod a b (cat_binprod c d) $==
cat_pr2 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(fmap10 cat_binprod (associator_cat_binprod a b c) d $o
 associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
      nrefine (cat_assoc _ _ _ $@ _ $@ cat_assoc_opp _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_binprod (cat_binprod a b) (cat_binprod c d)) $o
 associator_cat_binprod a b (cat_binprod c d)) $==
cat_pr2 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
(cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
 (fmap10 cat_binprod (associator_cat_binprod a b c) d $o
  associator_cat_binprod a (cat_binprod b c) d $o
  fmap01 cat_binprod a (associator_cat_binprod b c d)))
      nrefine ((_ $@L cat_pr2_associator_binprod _ _ _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $==
cat_pr2 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
(cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
 (fmap10 cat_binprod (associator_cat_binprod a b c) d $o
  associator_cat_binprod a (cat_binprod b c) d $o
  fmap01 cat_binprod a (associator_cat_binprod b c d)))
      refine (_ $@ (_ $@L ((((_^$ $@R _) $@ cat_assoc _ _ _) $@R _) $@ cat_assoc _ _ _))).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $==
cat_pr2 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
(?Goal1 $o associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
fmap10 cat_binprod (associator_cat_binprod a b c) d $-> 
?Goal1
      2: napply cat_pr1_fmap10_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $==
cat_pr2 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
(associator_cat_binprod a b c $o
 cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
 associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
      nrefine (_ $@ (_ $@L (cat_assoc_opp _ _ _ $@ cat_assoc_opp _ _ _))).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $==
cat_pr2 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
(associator_cat_binprod a b c $o
 (cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
  (associator_cat_binprod a (cat_binprod b c) d $o
   fmap01 cat_binprod a (associator_cat_binprod b c d))))
      refine (_ $@ (_^$ $@R _) $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $==
?Goal1 $o
(cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
 (associator_cat_binprod a (cat_binprod b c) d $o
  fmap01 cat_binprod a (associator_cat_binprod b c d)))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_prod (Bool_rec A (cat_binprod a b) c)) $o
associator_cat_binprod a b c $-> ?Goal1
      2: napply cat_pr2_associator_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $==
cat_pr2 (cat_prod (Bool_rec A b c)) $o
cat_pr2 (cat_binprod a (cat_binprod b c)) $o
(cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
 (associator_cat_binprod a (cat_binprod b c) d $o
  fmap01 cat_binprod a (associator_cat_binprod b c d)))
      refine (_ $@ (_ $@L ((_^$ $@R _) $@ cat_assoc _ _ _ $@ cat_assoc _ _ _)) $@ cat_assoc_opp _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $==
cat_pr2 (cat_prod (Bool_rec A b c)) $o
(?Goal1 $o fmap01 cat_binprod a (associator_cat_binprod b c d))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_binprod a (cat_binprod b c)) $o
cat_pr1 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
associator_cat_binprod a (cat_binprod b c) d $-> ?Goal1
      2: napply cat_pr2_pr1_associator_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $==
cat_pr2 (cat_prod (Bool_rec A b c)) $o
(cat_pr1 (cat_prod (Bool_rec A (cat_binprod b c) d)) $o
 cat_pr2 (cat_binprod a (cat_binprod (cat_binprod b c) d)) $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
      refine (_ $@ (_ $@L ((_ $@L _^$) $@ cat_assoc_opp _ _ _))).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $==
cat_pr2 (cat_prod (Bool_rec A b c)) $o
(cat_pr1 (cat_prod (Bool_rec A (cat_binprod b c) d)) $o ?Goal1)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_binprod a (cat_binprod (cat_binprod b c) d)) $o
fmap01 cat_binprod a (associator_cat_binprod b c d) $-> 
?Goal1
      2: napply cat_pr2_fmap01_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr1 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $==
cat_pr2 (cat_prod (Bool_rec A b c)) $o
(cat_pr1 (cat_prod (Bool_rec A (cat_binprod b c) d)) $o
 (associator_cat_binprod b c d $o
  cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))))
      refine (cat_assoc_opp _ _ _ $@ (_^$ $@R _) $@ cat_assoc _ _ _ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_prod (Bool_rec A b c)) $o
cat_pr1 (cat_prod (Bool_rec A (cat_binprod b c) d)) $o
associator_cat_binprod b c d $->
cat_pr1 (cat_prod (Bool_rec A c d)) $o
cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d)))
      napply cat_pr2_pr1_associator_binprod.
    -A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(associator_cat_binprod (cat_binprod a b) c d $o
 associator_cat_binprod a b (cat_binprod c d)) $==
cat_pr2 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(fmap10 cat_binprod (associator_cat_binprod a b c) d $o
 associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d)) nrefine (cat_assoc_opp _ _ _ $@ (cat_pr2_associator_binprod _ _ _ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_prod (Bool_rec A c d)) $o
cat_pr2 (cat_binprod (cat_binprod a b) (cat_binprod c d)) $o
associator_cat_binprod a b (cat_binprod c d) $==
cat_pr2 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(fmap10 cat_binprod (associator_cat_binprod a b c) d $o
 associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
      nrefine (cat_assoc _ _ _ $@ (_ $@L (cat_pr2_associator_binprod _ _ _)) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $==
cat_pr2 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
(fmap10 cat_binprod (associator_cat_binprod a b c) d $o
 associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
      refine (_ $@ (_^$ $@R _) $@ cat_assoc _ _ _ $@ (_ $@L (cat_assoc_opp _ _ _))).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $==
?Goal0 $o
(associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_binprod (cat_binprod (cat_binprod a b) c) d) $o
fmap10 cat_binprod (associator_cat_binprod a b c) d $-> 
?Goal0
      2: napply cat_pr2_fmap10_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $==
cat_pr2 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
(associator_cat_binprod a (cat_binprod b c) d $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
      refine (_ $@ cat_assoc_opp _ _ _ $@ (_^$ $@R _) $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $==
?Goal2 $o (?Goal1 $o fmap01 cat_binprod a (associator_cat_binprod b c d))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_binprod (cat_binprod a (cat_binprod b c)) d) $o
associator_cat_binprod a (cat_binprod b c) d $-> ?Goal2 $o ?Goal1
      2: napply cat_pr2_associator_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_prod (Bool_rec A c d)) $o
(cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d))) $o
 cat_pr2 (cat_binprod a (cat_binprod b (cat_binprod c d)))) $==
cat_pr2 (cat_prod (Bool_rec A (cat_binprod b c) d)) $o
(cat_pr2 (cat_binprod a (cat_binprod (cat_binprod b c) d)) $o
 fmap01 cat_binprod a (associator_cat_binprod b c d))
      refine (cat_assoc_opp _ _ _ $@ (_^$ $@R _) $@ cat_assoc _ _ _
        $@ (_ $@L (cat_pr2_fmap01_binprod _ _)^$)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c, d: A
cat_pr2 (cat_prod (Bool_rec A (cat_binprod b c) d)) $o
associator_cat_binprod b c d $->
cat_pr2 (cat_prod (Bool_rec A c d)) $o
cat_pr2 (cat_prod (Bool_rec A b (cat_binprod c d)))
      napply cat_pr2_associator_binprod.
  Defined.

  #[export] Instance hexagon_identity
    : HexagonIdentity cat_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
HexagonIdentity cat_binprod
  Proof.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
HexagonIdentity cat_binprod
    intros a b c.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
fmap10 cat_binprod (symmetricbraiding_binprod b a) c $o
associator_cat_binprod b a c $o
fmap01 cat_binprod b (symmetricbraiding_binprod c a) $==
associator_cat_binprod a b c $o symmetricbraiding_binprod (cat_binprod b c) a $o
associator_cat_binprod b c a
    nrefine (cat_assoc _ _ _ $@ _ $@ cat_assoc_opp _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
fmap10 cat_binprod (symmetricbraiding_binprod b a) c $o
(associator_cat_binprod b a c $o
 fmap01 cat_binprod b (symmetricbraiding_binprod c a)) $==
associator_cat_binprod a b c $o
(symmetricbraiding_binprod (cat_binprod b c) a $o
 associator_cat_binprod b c a)
    napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod (cat_binprod a b) c) $o
(fmap10 cat_binprod (symmetricbraiding_binprod b a) c $o
 (associator_cat_binprod b a c $o
  fmap01 cat_binprod b (symmetricbraiding_binprod c a))) $==
cat_pr1 (cat_binprod (cat_binprod a b) c) $o
(associator_cat_binprod a b c $o
 (symmetricbraiding_binprod (cat_binprod b c) a $o
  associator_cat_binprod b c a))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod (cat_binprod a b) c) $o
(fmap10 cat_binprod (symmetricbraiding_binprod b a) c $o
 (associator_cat_binprod b a c $o
  fmap01 cat_binprod b (symmetricbraiding_binprod c a))) $==
cat_pr2 (cat_binprod (cat_binprod a b) c) $o
(associator_cat_binprod a b c $o
 (symmetricbraiding_binprod (cat_binprod b c) a $o
  associator_cat_binprod b c a))
    {A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod (cat_binprod a b) c) $o
(fmap10 cat_binprod (symmetricbraiding_binprod b a) c $o
 (associator_cat_binprod b a c $o
  fmap01 cat_binprod b (symmetricbraiding_binprod c a))) $==
cat_pr1 (cat_binprod (cat_binprod a b) c) $o
(associator_cat_binprod a b c $o
 (symmetricbraiding_binprod (cat_binprod b c) a $o
  associator_cat_binprod b c a)) nrefine (cat_assoc_opp _ _ _ $@ (cat_pr1_fmap10_binprod _ _ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
symmetricbraiding_binprod b a $o cat_pr1 (cat_binprod (cat_binprod b a) c) $o
(associator_cat_binprod b a c $o
 fmap01 cat_binprod b (symmetricbraiding_binprod c a)) $==
cat_pr1 (cat_binprod (cat_binprod a b) c) $o
(associator_cat_binprod a b c $o
 (symmetricbraiding_binprod (cat_binprod b c) a $o
  associator_cat_binprod b c a))
      nrefine (cat_assoc _ _ _ $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
symmetricbraiding_binprod b a $o
(cat_pr1 (cat_binprod (cat_binprod b a) c) $o
 (associator_cat_binprod b a c $o
  fmap01 cat_binprod b (symmetricbraiding_binprod c a))) $==
cat_pr1 (cat_binprod (cat_binprod a b) c) $o
(associator_cat_binprod a b c $o
 (symmetricbraiding_binprod (cat_binprod b c) a $o
  associator_cat_binprod b c a))
      napply cat_binprod_eta_pr.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod a b) $o
(symmetricbraiding_binprod b a $o
 (cat_pr1 (cat_binprod (cat_binprod b a) c) $o
  (associator_cat_binprod b a c $o
   fmap01 cat_binprod b (symmetricbraiding_binprod c a)))) $==
cat_pr1 (cat_binprod a b) $o
(cat_pr1 (cat_binprod (cat_binprod a b) c) $o
 (associator_cat_binprod a b c $o
  (symmetricbraiding_binprod (cat_binprod b c) a $o
   associator_cat_binprod b c a)))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod a b) $o
(symmetricbraiding_binprod b a $o
 (cat_pr1 (cat_binprod (cat_binprod b a) c) $o
  (associator_cat_binprod b a c $o
   fmap01 cat_binprod b (symmetricbraiding_binprod c a)))) $==
cat_pr2 (cat_binprod a b) $o
(cat_pr1 (cat_binprod (cat_binprod a b) c) $o
 (associator_cat_binprod a b c $o
  (symmetricbraiding_binprod (cat_binprod b c) a $o
   associator_cat_binprod b c a)))
      {A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod a b) $o
(symmetricbraiding_binprod b a $o
 (cat_pr1 (cat_binprod (cat_binprod b a) c) $o
  (associator_cat_binprod b a c $o
   fmap01 cat_binprod b (symmetricbraiding_binprod c a)))) $==
cat_pr1 (cat_binprod a b) $o
(cat_pr1 (cat_binprod (cat_binprod a b) c) $o
 (associator_cat_binprod a b c $o
  (symmetricbraiding_binprod (cat_binprod b c) a $o
   associator_cat_binprod b c a))) nrefine (cat_assoc_opp _ _ _ $@ _ $@ cat_assoc _ _ _ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod a b) $o symmetricbraiding_binprod b a $o
(cat_pr1 (cat_binprod (cat_binprod b a) c) $o
 (associator_cat_binprod b a c $o
  fmap01 cat_binprod b (symmetricbraiding_binprod c a))) $==
cat_pr1 (cat_binprod a b) $o cat_pr1 (cat_binprod (cat_binprod a b) c) $o
associator_cat_binprod a b c $o
(symmetricbraiding_binprod (cat_binprod b c) a $o
 associator_cat_binprod b c a)
        refine ((_ $@R _) $@ _ $@ (_^$ $@R _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod a b) $o symmetricbraiding_binprod b a $== ?Goal3
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
?Goal3 $o
(cat_pr1 (cat_binprod (cat_binprod b a) c) $o
 (associator_cat_binprod b a c $o
  fmap01 cat_binprod b (symmetricbraiding_binprod c a))) $==
?Goal6 $o
(symmetricbraiding_binprod (cat_binprod b c) a $o
 associator_cat_binprod b c a)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod a b) $o cat_pr1 (cat_binprod (cat_binprod a b) c) $o
associator_cat_binprod a b c $-> ?Goal6
        1: napply cat_binprod_beta_pr1.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod (fst (b, a)) (snd (b, a))) $o
(cat_pr1 (cat_binprod (cat_binprod b a) c) $o
 (associator_cat_binprod b a c $o
  fmap01 cat_binprod b (symmetricbraiding_binprod c a))) $==
?Goal2 $o
(symmetricbraiding_binprod (cat_binprod b c) a $o
 associator_cat_binprod b c a)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod a b) $o cat_pr1 (cat_binprod (cat_binprod a b) c) $o
associator_cat_binprod a b c $-> ?Goal2
        2: napply cat_pr1_pr1_associator_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod (fst (b, a)) (snd (b, a))) $o
(cat_pr1 (cat_binprod (cat_binprod b a) c) $o
 (associator_cat_binprod b a c $o
  fmap01 cat_binprod b (symmetricbraiding_binprod c a))) $==
cat_pr1 (cat_binprod a (cat_binprod b c)) $o
(symmetricbraiding_binprod (cat_binprod b c) a $o
 associator_cat_binprod b c a)
        nrefine (cat_assoc_opp _ _ _ $@ cat_assoc_opp _ _ _ $@ _ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod b a) $o cat_pr1 (cat_binprod (cat_binprod b a) c) $o
associator_cat_binprod b a c $o
fmap01 cat_binprod b (symmetricbraiding_binprod c a) $==
cat_pr1 (cat_binprod a (cat_binprod b c)) $o
symmetricbraiding_binprod (cat_binprod b c) a $o associator_cat_binprod b c a
        refine ((_ $@R _) $@ _ $@ (_^$ $@R _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod b a) $o cat_pr1 (cat_binprod (cat_binprod b a) c) $o
associator_cat_binprod b a c $== ?Goal1
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
?Goal1 $o fmap01 cat_binprod b (symmetricbraiding_binprod c a) $==
?Goal4 $o associator_cat_binprod b c a
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod a (cat_binprod b c)) $o
symmetricbraiding_binprod (cat_binprod b c) a $-> 
?Goal4
        1: napply cat_pr2_pr1_associator_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_prod (Bool_rec A a c)) $o
cat_pr2 (cat_binprod b (cat_binprod a c)) $o
fmap01 cat_binprod b (symmetricbraiding_binprod c a) $==
?Goal2 $o associator_cat_binprod b c a
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod a (cat_binprod b c)) $o
symmetricbraiding_binprod (cat_binprod b c) a $-> 
?Goal2
        2: napply cat_binprod_beta_pr1.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_prod (Bool_rec A a c)) $o
cat_pr2 (cat_binprod b (cat_binprod a c)) $o
fmap01 cat_binprod b (symmetricbraiding_binprod c a) $==
cat_pr2 (cat_binprod (fst (cat_binprod b c, a)) (snd (cat_binprod b c, a))) $o
associator_cat_binprod b c a
        refine (cat_assoc _ _ _ $@ (_ $@L _) $@ cat_assoc_opp _ _ _ $@ (_ $@R _) $@ _^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod b (cat_binprod a c)) $o
fmap01 cat_binprod b (symmetricbraiding_binprod c a) $== 
?Goal4 $o ?Goal3
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_prod (Bool_rec A a c)) $o ?Goal4 $== ?Goal5
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod (cat_binprod b c) a) $o associator_cat_binprod b c a $->
?Goal5 $o ?Goal3
        1: napply cat_pr2_fmap01_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_prod (Bool_rec A a c)) $o symmetricbraiding_binprod c a $==
?Goal1
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod (cat_binprod b c) a) $o associator_cat_binprod b c a $->
?Goal1 $o cat_pr2 (cat_binprod b (cat_binprod c a))
        2: napply cat_pr2_associator_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_prod (Bool_rec A a c)) $o symmetricbraiding_binprod c a $==
cat_pr2 (cat_prod (Bool_rec A c a))
        napply cat_binprod_beta_pr1. }A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod a b) $o
(symmetricbraiding_binprod b a $o
 (cat_pr1 (cat_binprod (cat_binprod b a) c) $o
  (associator_cat_binprod b a c $o
   fmap01 cat_binprod b (symmetricbraiding_binprod c a)))) $==
cat_pr2 (cat_binprod a b) $o
(cat_pr1 (cat_binprod (cat_binprod a b) c) $o
 (associator_cat_binprod a b c $o
  (symmetricbraiding_binprod (cat_binprod b c) a $o
   associator_cat_binprod b c a)))
      nrefine (cat_assoc_opp _ _ _ $@ _ $@ cat_assoc _ _ _ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod a b) $o symmetricbraiding_binprod b a $o
(cat_pr1 (cat_binprod (cat_binprod b a) c) $o
 (associator_cat_binprod b a c $o
  fmap01 cat_binprod b (symmetricbraiding_binprod c a))) $==
cat_pr2 (cat_binprod a b) $o cat_pr1 (cat_binprod (cat_binprod a b) c) $o
associator_cat_binprod a b c $o
(symmetricbraiding_binprod (cat_binprod b c) a $o
 associator_cat_binprod b c a)
      refine ((_ $@R _) $@ _ $@ (_^$ $@R _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod a b) $o symmetricbraiding_binprod b a $== ?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
?Goal0 $o
(cat_pr1 (cat_binprod (cat_binprod b a) c) $o
 (associator_cat_binprod b a c $o
  fmap01 cat_binprod b (symmetricbraiding_binprod c a))) $==
?Goal3 $o
(symmetricbraiding_binprod (cat_binprod b c) a $o
 associator_cat_binprod b c a)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod a b) $o cat_pr1 (cat_binprod (cat_binprod a b) c) $o
associator_cat_binprod a b c $-> ?Goal3
      1: napply cat_binprod_beta_pr2.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod (fst (b, a)) (snd (b, a))) $o
(cat_pr1 (cat_binprod (cat_binprod b a) c) $o
 (associator_cat_binprod b a c $o
  fmap01 cat_binprod b (symmetricbraiding_binprod c a))) $==
?Goal1 $o
(symmetricbraiding_binprod (cat_binprod b c) a $o
 associator_cat_binprod b c a)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod a b) $o cat_pr1 (cat_binprod (cat_binprod a b) c) $o
associator_cat_binprod a b c $-> ?Goal1
      2: napply cat_pr2_pr1_associator_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod (fst (b, a)) (snd (b, a))) $o
(cat_pr1 (cat_binprod (cat_binprod b a) c) $o
 (associator_cat_binprod b a c $o
  fmap01 cat_binprod b (symmetricbraiding_binprod c a))) $==
cat_pr1 (cat_prod (Bool_rec A b c)) $o
cat_pr2 (cat_binprod a (cat_binprod b c)) $o
(symmetricbraiding_binprod (cat_binprod b c) a $o
 associator_cat_binprod b c a)
      nrefine (cat_assoc_opp _ _ _ $@ cat_assoc_opp _ _ _ $@ _ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod b a) $o cat_pr1 (cat_binprod (cat_binprod b a) c) $o
associator_cat_binprod b a c $o
fmap01 cat_binprod b (symmetricbraiding_binprod c a) $==
cat_pr1 (cat_prod (Bool_rec A b c)) $o
cat_pr2 (cat_binprod a (cat_binprod b c)) $o
symmetricbraiding_binprod (cat_binprod b c) a $o associator_cat_binprod b c a
      refine ((_ $@R _) $@ _ $@ (((_ $@L _^$) $@ cat_assoc_opp _ _ _) $@R _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod b a) $o cat_pr1 (cat_binprod (cat_binprod b a) c) $o
associator_cat_binprod b a c $== ?Goal0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
?Goal0 $o fmap01 cat_binprod b (symmetricbraiding_binprod c a) $==
cat_pr1 (cat_prod (Bool_rec A b c)) $o ?Goal3 $o associator_cat_binprod b c a
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod a (cat_binprod b c)) $o
symmetricbraiding_binprod (cat_binprod b c) a $-> 
?Goal3
      1: napply cat_pr1_pr1_associator_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod b (cat_binprod a c)) $o
fmap01 cat_binprod b (symmetricbraiding_binprod c a) $==
cat_pr1 (cat_prod (Bool_rec A b c)) $o ?Goal1 $o associator_cat_binprod b c a
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod a (cat_binprod b c)) $o
symmetricbraiding_binprod (cat_binprod b c) a $-> 
?Goal1
      2: napply cat_binprod_beta_pr2.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_binprod b (cat_binprod a c)) $o
fmap01 cat_binprod b (symmetricbraiding_binprod c a) $==
cat_pr1 (cat_prod (Bool_rec A b c)) $o
cat_pr1 (cat_binprod (fst (cat_binprod b c, a)) (snd (cat_binprod b c, a))) $o
associator_cat_binprod b c a
      refine (cat_pr1_fmap01_binprod _ _ $@ _^$).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr1 (cat_prod (Bool_rec A b c)) $o
cat_pr1 (cat_binprod (cat_binprod b c) a) $o associator_cat_binprod b c a $->
cat_pr1 (cat_binprod b (cat_binprod c a))
      napply cat_pr1_pr1_associator_binprod. }A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod (cat_binprod a b) c) $o
(fmap10 cat_binprod (symmetricbraiding_binprod b a) c $o
 (associator_cat_binprod b a c $o
  fmap01 cat_binprod b (symmetricbraiding_binprod c a))) $==
cat_pr2 (cat_binprod (cat_binprod a b) c) $o
(associator_cat_binprod a b c $o
 (symmetricbraiding_binprod (cat_binprod b c) a $o
  associator_cat_binprod b c a))
    nrefine (cat_assoc_opp _ _ _ $@ _ $@ cat_assoc _ _ _ $@ cat_assoc _ _ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod (cat_binprod a b) c) $o
fmap10 cat_binprod (symmetricbraiding_binprod b a) c $o
(associator_cat_binprod b a c $o
 fmap01 cat_binprod b (symmetricbraiding_binprod c a)) $==
cat_pr2 (cat_binprod (cat_binprod a b) c) $o associator_cat_binprod a b c $o
symmetricbraiding_binprod (cat_binprod b c) a $o associator_cat_binprod b c a
    refine ((_ $@R _) $@ _ $@ ((_^$ $@R _) $@R _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod (cat_binprod a b) c) $o
fmap10 cat_binprod (symmetricbraiding_binprod b a) c $== 
?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
?Goal $o
(associator_cat_binprod b a c $o
 fmap01 cat_binprod b (symmetricbraiding_binprod c a)) $==
?Goal2 $o symmetricbraiding_binprod (cat_binprod b c) a $o
associator_cat_binprod b c a
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod (cat_binprod a b) c) $o associator_cat_binprod a b c $->
?Goal2
    1: napply cat_pr2_fmap10_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod (cat_binprod b a) c) $o
(associator_cat_binprod b a c $o
 fmap01 cat_binprod b (symmetricbraiding_binprod c a)) $==
?Goal0 $o symmetricbraiding_binprod (cat_binprod b c) a $o
associator_cat_binprod b c a
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod (cat_binprod a b) c) $o associator_cat_binprod a b c $->
?Goal0
    2: napply cat_pr2_associator_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod (cat_binprod b a) c) $o
(associator_cat_binprod b a c $o
 fmap01 cat_binprod b (symmetricbraiding_binprod c a)) $==
cat_pr2 (cat_prod (Bool_rec A b c)) $o
cat_pr2 (cat_binprod a (cat_binprod b c)) $o
symmetricbraiding_binprod (cat_binprod b c) a $o associator_cat_binprod b c a
    nrefine (cat_assoc_opp _ _ _ $@ (cat_pr2_associator_binprod _ _ _ $@R _) $@ _).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_prod (Bool_rec A a c)) $o
cat_pr2 (cat_binprod b (cat_binprod a c)) $o
fmap01 cat_binprod b (symmetricbraiding_binprod c a) $==
cat_pr2 (cat_prod (Bool_rec A b c)) $o
cat_pr2 (cat_binprod a (cat_binprod b c)) $o
symmetricbraiding_binprod (cat_binprod b c) a $o associator_cat_binprod b c a
    nrefine (cat_assoc _ _ _ $@ (_ $@L _) $@ _ $@ (cat_assoc_opp _ _ _ $@R _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod b (cat_binprod a c)) $o
fmap01 cat_binprod b (symmetricbraiding_binprod c a) $== 
?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_prod (Bool_rec A a c)) $o ?Goal $==
cat_pr2 (cat_prod (Bool_rec A b c)) $o
(cat_pr2 (cat_binprod a (cat_binprod b c)) $o
 symmetricbraiding_binprod (cat_binprod b c) a) $o
associator_cat_binprod b c a
    1: napply cat_pr2_fmap01_binprod.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_prod (Bool_rec A a c)) $o
(symmetricbraiding_binprod c a $o cat_pr2 (cat_binprod b (cat_binprod c a))) $==
cat_pr2 (cat_prod (Bool_rec A b c)) $o
(cat_pr2 (cat_binprod a (cat_binprod b c)) $o
 symmetricbraiding_binprod (cat_binprod b c) a) $o
associator_cat_binprod b c a
    refine (cat_assoc_opp _ _ _ $@ (_ $@R _) $@ _^$ $@ ((_ $@L _^$) $@R _)).A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_prod (Bool_rec A a c)) $o symmetricbraiding_binprod c a $==
?Goal
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_prod (Bool_rec A b c)) $o ?Goal2 $o associator_cat_binprod b c a $->
?Goal $o cat_pr2 (cat_binprod b (cat_binprod c a))
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_binprod a (cat_binprod b c)) $o
symmetricbraiding_binprod (cat_binprod b c) a $-> 
?Goal2
    1,3: napply cat_binprod_beta_pr2.A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: HasEquivs A
hbp: HasBinaryProducts A
unit: A
IsTerminal0: IsTerminal unit
a, b, c: A
cat_pr2 (cat_prod (Bool_rec A b c)) $o
cat_pr1 (cat_binprod (fst (cat_binprod b c, a)) (snd (cat_binprod b c, a))) $o
associator_cat_binprod b c a $->
cat_pr1 (cat_binprod (fst (c, a)) (snd (c, a))) $o
cat_pr2 (cat_binprod b (cat_binprod c a))
    napply cat_pr2_pr1_associator_binprod.
  Defined.

  Local Instance ismonoidal_cat_binprod
    : IsMonoidal A cat_binprod unit
    := {}.

  (** Many of the above instances are declared to be local because they follow from this one. *)
  #[export] Instance issymmetricmonoidal_cat_binprod
    : IsSymmetricMonoidal A cat_binprod unit
    := {}.

End Associativity.

(** ** Examples *)

(** *** Products in Type *)

(** Since we use the Yoneda lemma in this file, we therefore depend on WildCat.Universe which means these instances have to live here. *)

(** Assuming [Funext], [Type] has all products. *)
Instance hasallproducts_type `{Funext} : HasAllProducts Type.H: Funext
HasAllProducts Type
Proof.H: Funext
HasAllProducts Type
  intros I x.H: Funext
I: Type
x: I -> Type
Product x
  snapply Build_Product.H: Funext
I: Type
x: I -> Type
Type
H: Funext
I: Type
x: I -> Type
forall i : I, ?cat_prod $-> x i
H: Funext
I: Type
x: I -> Type
forall z : Type, (forall i : I, z $-> x i) -> z $-> ?cat_prod
H: Funext
I: Type
x: I -> Type
forall (z : Type) (f : forall i : I, z $-> x i) (i : I),
?cat_pr i $o ?cat_prod_corec z f $== f i
H: Funext
I: Type
x: I -> Type
forall (z : Type) (f g : z $-> ?cat_prod),
(forall i : I, ?cat_pr i $o f $== ?cat_pr i $o g) -> f $== g
  -H: Funext
I: Type
x: I -> Type
Type exact (forall (i : I), x i).
  -H: Funext
I: Type
x: I -> Type
forall i : I, (forall i0 : I, x i0) $-> x i intros i f.H: Funext
I: Type
x: I -> Type
i: I
f: forall i0 : I, x i0
x i exact (f i).
  -H: Funext
I: Type
x: I -> Type
forall z : Type, (forall i : I, z $-> x i) -> z $-> (forall i : I, x i) intros Z f a i.H: Funext
I: Type
x: I -> Type
Z: Type
f: forall i0 : I, Z $-> x i0
a: Z
i: I
x i exact (f i a).
  -H: Funext
I: Type
x: I -> Type
forall (z : Type) (f : forall i : I, z $-> x i) (i : I),
(fun i0 : I =>
 (fun f0 : forall i1 : I, x i1 => f0 i0) : (forall i1 : I, x i1) $-> x i0) i $o
(fun (Z : Type) (f0 : forall i0 : I, Z $-> x i0) =>
 (fun (a : Z) (i0 : I) => f0 i0 a) : Z $-> (forall i0 : I, x i0)) z f $==
f i reflexivity.
  -H: Funext
I: Type
x: I -> Type
forall (z : Type) (f g : z $-> (forall i : I, x i)),
(forall i : I,
 (fun i0 : I =>
  (fun f0 : forall i1 : I, x i1 => f0 i0) : (forall i1 : I, x i1) $-> x i0) i $o
 f $==
 (fun i0 : I =>
  (fun f0 : forall i1 : I, x i1 => f0 i0) : (forall i1 : I, x i1) $-> x i0) i $o
 g) ->
f $== g intros Z f g p a.H: Funext
I: Type
x: I -> Type
Z: Type
f, g: Z $-> (forall i : I, x i)
p: forall i : I,
(fun i0 : I =>
 (fun f0 : forall i1 : I, x i1 => f0 i0) : (forall i1 : I, x i1) $-> x i0) i $o
f $==
(fun i0 : I =>
 (fun f0 : forall i1 : I, x i1 => f0 i0) : (forall i1 : I, x i1) $-> x i0) i $o
g
a: Z
f a = g a
    exact (path_forall _ _ (fun i => p i a)).
Defined.

(** It follows that [Type] has binary products, but we prove this separately to avoid [Funext]. *)
Instance hasbinaryproducts_type : HasBinaryProducts Type.HasBinaryProducts Type
Proof.HasBinaryProducts Type
  intros X Y.X, Y: Type
BinaryProduct X Y
  snapply Build_BinaryProduct.X, Y: Type
Type
X, Y: Type
?cat_binprod $-> X
X, Y: Type
?cat_binprod $-> Y
X, Y: Type
forall z : Type, (z $-> X) -> (z $-> Y) -> z $-> ?cat_binprod
X, Y: Type
forall (z : Type) (f : z $-> X) (g : z $-> Y),
?cat_pr1 $o ?cat_binprod_corec z f g $== f
X, Y: Type
forall (z : Type) (f : z $-> X) (g : z $-> Y),
?cat_pr2 $o ?cat_binprod_corec z f g $== g
X, Y: Type
forall (z : Type) (f g : z $-> ?cat_binprod),
?cat_pr1 $o f $== ?cat_pr1 $o g -> ?cat_pr2 $o f $== ?cat_pr2 $o g -> f $== g
  -X, Y: Type
Type exact (X * Y).
  -X, Y: Type
X * Y $-> X exact fst.
  -X, Y: Type
X * Y $-> Y exact snd.
  -X, Y: Type
forall z : Type, (z $-> X) -> (z $-> Y) -> z $-> X * Y intros Z f g z.X, Y, Z: Type
f: Z $-> X
g: Z $-> Y
z: Z
X * Y exact (f z, g z).
  -X, Y: Type
forall (z : Type) (f : z $-> X) (g : z $-> Y),
fst $o
(fun (Z : Type) (f0 : Z $-> X) (g0 : Z $-> Y) =>
 (fun z0 : Z => (f0 z0, g0 z0)) : Z $-> X * Y) z f g $==
f reflexivity.
  -X, Y: Type
forall (z : Type) (f : z $-> X) (g : z $-> Y),
snd $o
(fun (Z : Type) (f0 : Z $-> X) (g0 : Z $-> Y) =>
 (fun z0 : Z => (f0 z0, g0 z0)) : Z $-> X * Y) z f g $==
g reflexivity.
  -X, Y: Type
forall (z : Type) (f g : z $-> X * Y),
fst $o f $== fst $o g -> snd $o f $== snd $o g -> f $== g intros Z f g p q x.X, Y, Z: Type
f, g: Z $-> X * Y
p: fst $o f $== fst $o g
q: snd $o f $== snd $o g
x: Z
f x = g x
    napply path_prod.X, Y, Z: Type
f, g: Z $-> X * Y
p: fst $o f $== fst $o g
q: snd $o f $== snd $o g
x: Z
fst (f x) = fst (g x)
X, Y, Z: Type
f, g: Z $-> X * Y
p: fst $o f $== fst $o g
q: snd $o f $== snd $o g
x: Z
snd (f x) = snd (g x)
    +X, Y, Z: Type
f, g: Z $-> X * Y
p: fst $o f $== fst $o g
q: snd $o f $== snd $o g
x: Z
fst (f x) = fst (g x) exact (p x).
    +X, Y, Z: Type
f, g: Z $-> X * Y
p: fst $o f $== fst $o g
q: snd $o f $== snd $o g
x: Z
snd (f x) = snd (g x) exact (q x).
Defined.

(** *** Products in ZeroGpd *)

(** Since we use products in ZeroGpd to define general products, we must depend on ZeroGroupoid, which means that these instances have to live here. *)

(** Note that this does not rely on [Funext], since the 1-cells in the product 0-groupoid are *defined* to be homotopies. *)
Instance hasallproducts_0gpd : HasAllProducts ZeroGpd.HasAllProducts ZeroGpd
Proof.HasAllProducts ZeroGpd
  intros I x.I: Type
x: I -> ZeroGpd
Product x
  snapply Build_Product.I: Type
x: I -> ZeroGpd
ZeroGpd
I: Type
x: I -> ZeroGpd
forall i : I, ?cat_prod $-> x i
I: Type
x: I -> ZeroGpd
forall z : ZeroGpd, (forall i : I, z $-> x i) -> z $-> ?cat_prod
I: Type
x: I -> ZeroGpd
forall (z : ZeroGpd) (f : forall i : I, z $-> x i) 
(i : I), ?cat_pr i $o ?cat_prod_corec z f $== f i
I: Type
x: I -> ZeroGpd
forall (z : ZeroGpd) (f g : z $-> ?cat_prod),
(forall i : I, ?cat_pr i $o f $== ?cat_pr i $o g) -> f $== g
  -I: Type
x: I -> ZeroGpd
ZeroGpd exact (prod_0gpd I x).
  -I: Type
x: I -> ZeroGpd
forall i : I, prod_0gpd I x $-> x i exact prod_0gpd_pr.
  -I: Type
x: I -> ZeroGpd
forall z : ZeroGpd, (forall i : I, z $-> x i) -> z $-> prod_0gpd I x intro G.I: Type
x: I -> ZeroGpd
G: ZeroGpd
(forall i : I, G $-> x i) -> G $-> prod_0gpd I x apply equiv_prod_0gpd_corec.
  -I: Type
x: I -> ZeroGpd
forall (z : ZeroGpd) (f : forall i : I, z $-> x i) 
(i : I),
prod_0gpd_pr i $o
(fun G : ZeroGpd => let X := equiv_fun equiv_prod_0gpd_corec in X) z f $==
f i reflexivity.
  -I: Type
x: I -> ZeroGpd
forall (z : ZeroGpd) (f g : z $-> prod_0gpd I x),
(forall i : I, prod_0gpd_pr i $o f $== prod_0gpd_pr i $o g) -> f $== g intros G f g p.I: Type
x: I -> ZeroGpd
G: ZeroGpd
f, g: G $-> prod_0gpd I x
p: forall i : I, prod_0gpd_pr i $o f $== prod_0gpd_pr i $o g
f $== g intro a.I: Type
x: I -> ZeroGpd
G: ZeroGpd
f, g: G $-> prod_0gpd I x
p: forall i : I, prod_0gpd_pr i $o f $== prod_0gpd_pr i $o g
a: zerogpd_graph G
f a $-> g a intro i.I: Type
x: I -> ZeroGpd
G: ZeroGpd
f, g: G $-> prod_0gpd I x
p: forall i0 : I, prod_0gpd_pr i0 $o f $== prod_0gpd_pr i0 $o g
a: zerogpd_graph G
i: I
f a i $-> g a i
    exact (p i a).
Defined.

(** This follows from the previous result, but we prove it separately because using these custom binary products can make certain things easier, and can sometimes avoid the need to use [Funext]. *)
Instance hasbinaryproducts_0gpd : HasBinaryProducts ZeroGpd.HasBinaryProducts ZeroGpd
Proof.HasBinaryProducts ZeroGpd
  intros G H.G, H: ZeroGpd
BinaryProduct G H
  snapply Build_BinaryProduct.G, H: ZeroGpd
ZeroGpd
G, H: ZeroGpd
?cat_binprod $-> G
G, H: ZeroGpd
?cat_binprod $-> H
G, H: ZeroGpd
forall z : ZeroGpd, (z $-> G) -> (z $-> H) -> z $-> ?cat_binprod
G, H: ZeroGpd
forall (z : ZeroGpd) (f : z $-> G) (g : z $-> H),
?cat_pr1 $o ?cat_binprod_corec z f g $== f
G, H: ZeroGpd
forall (z : ZeroGpd) (f : z $-> G) (g : z $-> H),
?cat_pr2 $o ?cat_binprod_corec z f g $== g
G, H: ZeroGpd
forall (z : ZeroGpd) (f g : z $-> ?cat_binprod),
?cat_pr1 $o f $== ?cat_pr1 $o g -> ?cat_pr2 $o f $== ?cat_pr2 $o g -> f $== g
  -G, H: ZeroGpd
ZeroGpd exact (binprod_0gpd G H).
  -G, H: ZeroGpd
binprod_0gpd G H $-> G apply binprod_0gpd_pr1.
  -G, H: ZeroGpd
binprod_0gpd G H $-> H apply binprod_0gpd_pr2.
  -G, H: ZeroGpd
forall z : ZeroGpd, (z $-> G) -> (z $-> H) -> z $-> binprod_0gpd G H intros K f g.G, H, K: ZeroGpd
f: K $-> G
g: K $-> H
K $-> binprod_0gpd G H exact (equiv_binprod_0gpd_corec G H K (f, g)).
  -G, H: ZeroGpd
forall (z : ZeroGpd) (f : z $-> G) (g : z $-> H),
binprod_0gpd_pr1 G H $o
(fun (K : ZeroGpd) (f0 : K $-> G) (g0 : K $-> H) =>
 equiv_binprod_0gpd_corec G H K (f0, g0)) z f g $==
f reflexivity.
  -G, H: ZeroGpd
forall (z : ZeroGpd) (f : z $-> G) (g : z $-> H),
binprod_0gpd_pr2 G H $o
(fun (K : ZeroGpd) (f0 : K $-> G) (g0 : K $-> H) =>
 equiv_binprod_0gpd_corec G H K (f0, g0)) z f g $==
g reflexivity.
  -G, H: ZeroGpd
forall (z : ZeroGpd) (f g : z $-> binprod_0gpd G H),
binprod_0gpd_pr1 G H $o f $== binprod_0gpd_pr1 G H $o g ->
binprod_0gpd_pr2 G H $o f $== binprod_0gpd_pr2 G H $o g -> f $== g intros K f g p q k.G, H, K: ZeroGpd
f, g: K $-> binprod_0gpd G H
p: binprod_0gpd_pr1 G H $o f $== binprod_0gpd_pr1 G H $o g
q: binprod_0gpd_pr2 G H $o f $== binprod_0gpd_pr2 G H $o g
k: zerogpd_graph K
f k $-> g k exact (p k, q k).
Defined.