proposition: Proposition lassociator_hexagon_two (VC : category) (TV : Terminal VC) (VP : BinProducts VC) (expV : Exponentials VP) (V : sym_mon_closed_cat := sym_mon_closed_cartesian_cat VC VP TV expV) (x y z : V) : mon_lassociator x y z · sym_mon_braiding V x _ · mon_lassociator y z x · sym_mon_braiding V y _ · mon_lassociator z x y = sym_mon_braiding V _ z.
proof: Proof. use BinProductArrowsEq ; cbn. - rewrite !assoc'. rewrite !BinProductPr1Commutes. etrans. { do 3 apply maponpaths. rewrite !assoc. rewrite BinProductPr1Commutes. apply idpath. } rewrite !cartesian_semi_cart_tensor_pr2. etrans. { do 2 apply maponpaths. rewrite !assoc. apply maponpaths_2. apply BinProductPr2Commutes. } rewrite BinProductPr1Commutes. etrans. { apply maponpaths. rewrite !assoc. apply maponpaths_2. apply BinProductPr1Commutes. } rewrite !assoc. rewrite !BinProductPr2Commutes. apply idpath. - rewrite !assoc'. rewrite !BinProductPr2Commutes. rewrite !cartesian_semi_cart_tensor_pr1, !cartesian_semi_cart_tensor_pr2. use BinProductArrowsEq. + rewrite !assoc'. rewrite !BinProductPr1Commutes. etrans. { do 3 apply maponpaths. rewrite !assoc. rewrite BinProductPr1Commutes. apply idpath. } etrans. { do 2 apply maponpaths. rewrite !assoc. rewrite !BinProductPr2Commutes. apply idpath. } rewrite !BinProductPr2Commutes. rewrite !BinProductPr1Commutes. apply idpath. + rewrite !assoc'. rewrite !BinProductPr2Commutes. etrans. { do 2 apply maponpaths. rewrite BinProductPr1Commutes. apply idpath. } etrans. { apply maponpaths. rewrite !assoc. rewrite !BinProductPr1Commutes. apply idpath. } rewrite !assoc. rewrite !BinProductPr2Commutes. rewrite !BinProductPr1Commutes. apply idpath. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.exponentials UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Structure.Cartesian UniMath.CategoryTheory.Monoidal.Structure.Symmetric UniMath.CategoryTheory.Monoidal.Structure.Closed UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Categories.HSET.Limits UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Adjunctions.Core