chore: Bump to 4.14.0
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jstoobysmith
parent
c91ca06272
commit
5dfd29ab8d
32 changed files with 376 additions and 334 deletions
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@ -105,11 +105,19 @@ instance : Group S.G := S.G_group
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/-- The field `repDim` of a TensorSpecies is non-zero for all colors. -/
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instance (c : S.C) : NeZero (S.repDim c) := S.repDim_neZero c
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/-- The lift of the functor `S.F` to a monoidal functor. -/
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def F : BraidedFunctor (OverColor S.C) (Rep S.k S.G) := (OverColor.lift).obj S.FD
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/-- The lift of the functor `S.F` to functor. -/
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def F : Functor (OverColor S.C) (Rep S.k S.G) := ((OverColor.lift).obj S.FD).toFunctor
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/- The definition of `F` as a lemma. -/
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lemma F_def : F S = (OverColor.lift).obj S.FD := rfl
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lemma F_def : F S = ((OverColor.lift).obj S.FD).toFunctor := rfl
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instance F_monoidal : Functor.Monoidal S.F := lift.instMonoidalRepObjFunctorDiscreteLaxBraidedFunctor S.FD
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instance F_laxBraided : Functor.LaxBraided S.F := lift.instLaxBraidedRepObjFunctorDiscreteLaxBraidedFunctor S.FD
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instance F_braided : Functor.Braided S.F := Functor.Braided.mk
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(fun X Y => Functor.LaxBraided.braided X Y)
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lemma perm_contr_cond {n : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n.succ.succ → S.C}
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{i : Fin n.succ.succ} {j : Fin n.succ}
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@ -158,7 +166,7 @@ def evalIso {n : ℕ} (c : Fin n.succ → S.C)
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(OverColor.lift.obj S.FD).obj (OverColor.mk (c ∘ i.succAbove)) :=
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(S.F.mapIso (OverColor.equivToIso (HepLean.Fin.finExtractOne i))).trans <|
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(S.F.mapIso (OverColor.mkSum (c ∘ (HepLean.Fin.finExtractOne i).symm))).trans <|
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(S.F.μIso _ _).symm.trans <|
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(Functor.Monoidal.μIso S.F _ _).symm.trans <|
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tensorIso
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((S.F.mapIso (OverColor.mkIso (by ext x; fin_cases x; rfl))).trans
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(OverColor.forgetLiftApp S.FD (c i))) (S.F.mapIso (OverColor.mkIso (by ext x; simp)))
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@ -174,7 +182,7 @@ lemma evalIso_tprod {n : ℕ} {c : Fin n.succ → S.C} (i : Fin n.succ)
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change (((lift.obj S.FD).map (mkIso _).hom).hom ≫
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(forgetLiftApp S.FD (c i)).hom.hom ⊗
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((lift.obj S.FD).map (mkIso _).hom).hom)
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(((lift.obj S.FD).μIso
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((Functor.Monoidal.μIso (lift.obj S.FD).toFunctor
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(OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractOne i).symm) ∘ Sum.inl))
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(OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractOne i).symm) ∘ Sum.inr))).inv.hom
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(((lift.obj S.FD).map (mkSum (c ∘ ⇑(HepLean.Fin.finExtractOne i).symm)).hom).hom
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@ -184,7 +192,7 @@ lemma evalIso_tprod {n : ℕ} {c : Fin n.succ → S.C} (i : Fin n.succ)
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change (((lift.obj S.FD).map (mkIso _).hom).hom ≫
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(forgetLiftApp S.FD (c i)).hom.hom ⊗
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((lift.obj S.FD).map (mkIso _).hom).hom)
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(((lift.obj S.FD).μIso
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((Functor.Monoidal.μIso (lift.obj S.FD).toFunctor
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(OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractOne i).symm) ∘ Sum.inl))
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(OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractOne i).symm) ∘ Sum.inr))).inv.hom
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(((lift.obj S.FD).map (mkSum (c ∘ ⇑(HepLean.Fin.finExtractOne i).symm)).hom).hom
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@ -193,7 +201,7 @@ lemma evalIso_tprod {n : ℕ} {c : Fin n.succ → S.C} (i : Fin n.succ)
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change ((TensorProduct.map (((lift.obj S.FD).map (mkIso _).hom).hom ≫
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(forgetLiftApp S.FD (c i)).hom.hom)
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((lift.obj S.FD).map (mkIso _).hom).hom))
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(((lift.obj S.FD).μIso
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((Functor.Monoidal.μIso (lift.obj S.FD).toFunctor
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(OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractOne i).symm) ∘ Sum.inl))
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(OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractOne i).symm) ∘ Sum.inr))).inv.hom
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((((PiTensorProduct.tprod S.k) _)))) =_
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@ -247,7 +255,7 @@ def evalLinearMap {n : ℕ} {c : Fin n.succ → S.C} (i : Fin n.succ) (e : Fin (
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of representations. -/
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def evalMap {n : ℕ} {c : Fin n.succ → S.C} (i : Fin n.succ) (e : Fin (S.repDim (c i))) :
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(S.F.obj (OverColor.mk c)).V ⟶ (S.F.obj (OverColor.mk (c ∘ i.succAbove))).V :=
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(S.evalIso c i).hom.hom ≫ ((Action.forgetMonoidal _ _).μIso _ _).inv
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(S.evalIso c i).hom.hom ≫ (Functor.Monoidal.μIso (Action.forget _ _) _ _).inv
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≫ ModuleCat.asHom (TensorProduct.map (S.evalLinearMap i e) LinearMap.id) ≫
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ModuleCat.asHom (TensorProduct.lid S.k _).toLinearMap
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@ -258,21 +266,21 @@ lemma evalMap_tprod {n : ℕ} {c : Fin n.succ → S.C} (i : Fin n.succ) (e : Fin
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(PiTensorProduct.tprod S.k
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(fun k => x (i.succAbove k)) : S.F.obj (OverColor.mk (c ∘ i.succAbove))) := by
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rw [evalMap]
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simp only [Nat.succ_eq_add_one, Action.instMonoidalCategory_tensorObj_V,
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Action.forgetMonoidal_toLaxMonoidalFunctor_toFunctor, Action.forget_obj, Functor.id_obj, mk_hom,
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Function.comp_apply, ModuleCat.coe_comp]
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simp only [Nat.succ_eq_add_one, Action.instMonoidalCategory_tensorObj_V, Action.forget_obj,
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Functor.Monoidal.μIso_inv, Functor.CoreMonoidal.toMonoidal_toOplaxMonoidal, Action.forget_δ,
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mk_left, Functor.id_obj, mk_hom, Function.comp_apply, Category.id_comp, ModuleCat.coe_comp]
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erw [evalIso_tprod]
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change ((TensorProduct.lid S.k ↑((lift.obj S.FD).obj (OverColor.mk (c ∘ i.succAbove))).V))
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(((TensorProduct.map (S.evalLinearMap i e) LinearMap.id))
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(((Action.forgetMonoidal (ModuleCat S.k) (MonCat.of S.G)).μIso (S.FD.obj { as := c i })
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((Functor.Monoidal.μIso (Action.forget (ModuleCat S.k) (MonCat.of S.G)) (S.FD.obj { as := c i })
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((lift.obj S.FD).obj (OverColor.mk (c ∘ i.succAbove)))).inv
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(x i ⊗ₜ[S.k] (PiTensorProduct.tprod S.k) fun k => x (i.succAbove k)))) = _
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simp only [Nat.succ_eq_add_one, Action.forgetMonoidal_toLaxMonoidalFunctor_toFunctor,
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Action.forget_obj, Action.instMonoidalCategory_tensorObj_V, MonoidalFunctor.μIso,
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Action.forgetMonoidal_toLaxMonoidalFunctor_μ, asIso_inv, IsIso.inv_id, Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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Functor.id_obj, mk_hom, Function.comp_apply, ModuleCat.id_apply, TensorProduct.map_tmul,
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LinearMap.id_coe, id_eq, TensorProduct.lid_tmul]
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simp only [Nat.succ_eq_add_one, Action.forget_obj, Action.instMonoidalCategory_tensorObj_V,
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Functor.Monoidal.μIso_inv, Functor.CoreMonoidal.toMonoidal_toOplaxMonoidal, Action.forget_δ,
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Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
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Action.FunctorCategoryEquivalence.functor_obj_obj, mk_left, Functor.id_obj, mk_hom,
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Function.comp_apply, ModuleCat.id_apply, TensorProduct.map_tmul, LinearMap.id_coe, id_eq,
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TensorProduct.lid_tmul]
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rfl
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/-!
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@ -30,7 +30,7 @@ def contrFin1Fin1 {n : ℕ} (c : Fin n.succ.succ → S.C)
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(OverColor.Discrete.pairτ S.FD S.τ).obj { as := c i } := by
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apply (S.F.mapIso
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(OverColor.mkSum (((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl)))).trans
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apply (S.F.μIso _ _).symm.trans
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apply (Functor.Monoidal.μIso S.F _ _).symm.trans
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apply tensorIso ?_ ?_
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· symm
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apply (OverColor.forgetLiftApp S.FD (c i)).symm.trans
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@ -57,15 +57,15 @@ lemma contrFin1Fin1_inv_tmul {n : ℕ} (c : Fin n.succ.succ → S.C)
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(eqToHom (by simp [h]))).hom y) := by
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simp only [Nat.succ_eq_add_one, contrFin1Fin1, Functor.comp_obj, Discrete.functor_obj_eq_as,
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Function.comp_apply, Iso.trans_symm, Iso.symm_symm_eq, Iso.trans_inv, tensorIso_inv,
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Iso.symm_inv, Functor.mapIso_hom, tensor_comp, MonoidalFunctor.μIso_hom, Category.assoc,
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LaxMonoidalFunctor.μ_natural, Functor.mapIso_inv, Action.comp_hom,
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Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorHom_hom,
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Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
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Action.FunctorCategoryEquivalence.functor_obj_obj, ModuleCat.coe_comp, Functor.id_obj, mk_hom,
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Fin.isValue]
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Iso.symm_inv, Functor.mapIso_hom, tensor_comp, Functor.Monoidal.μIso_hom,
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Functor.CoreMonoidal.toMonoidal_toLaxMonoidal, Category.assoc, Functor.LaxMonoidal.μ_natural,
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Functor.mapIso_inv, Action.comp_hom, Action.instMonoidalCategory_tensorObj_V,
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Action.instMonoidalCategory_tensorHom_hom, Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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ModuleCat.coe_comp, mk_left, Functor.id_obj, mk_hom, Fin.isValue]
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change (S.F.map (OverColor.mkSum ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl)).inv).hom
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((S.F.map ((OverColor.mkIso _).hom ⊗ (OverColor.mkIso _).hom)).hom
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((S.F.μ (OverColor.mk fun _ => c i) (OverColor.mk fun _ => S.τ (c i))).hom
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((Functor.LaxMonoidal.μ S.F (OverColor.mk fun _ => c i) (OverColor.mk fun _ => S.τ (c i))).hom
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((((OverColor.forgetLiftApp S.FD (c i)).inv.hom x) ⊗ₜ[S.k]
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((OverColor.forgetLiftApp S.FD (S.τ (c i))).inv.hom y))))) = _
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simp only [Nat.succ_eq_add_one, Action.instMonoidalCategory_tensorObj_V, Equivalence.symm_inverse,
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@ -76,7 +76,7 @@ lemma contrFin1Fin1_inv_tmul {n : ℕ} (c : Fin n.succ.succ → S.C)
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change ((OverColor.lift.obj S.FD).map (OverColor.mkSum
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((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl)).inv).hom
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(((OverColor.lift.obj S.FD).map ((OverColor.mkIso _).hom ⊗ (OverColor.mkIso _).hom)).hom
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(((OverColor.lift.obj S.FD).μ (OverColor.mk fun _ => c i)
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((Functor.LaxMonoidal.μ (OverColor.lift.obj S.FD).toFunctor (OverColor.mk fun _ => c i)
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(OverColor.mk fun _ => S.τ (c i))).hom
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(((PiTensorProduct.tprod S.k) fun _ => x) ⊗ₜ[S.k] (PiTensorProduct.tprod S.k) fun _ => y))) = _
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rw [OverColor.lift.obj_μ_tprod_tmul S.FD]
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@ -144,7 +144,7 @@ def contrIso {n : ℕ} (c : Fin n.succ.succ → S.C)
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(OverColor.lift.obj S.FD).obj (OverColor.mk (c ∘ i.succAbove ∘ j.succAbove)) :=
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(S.F.mapIso (OverColor.equivToIso (HepLean.Fin.finExtractTwo i j))).trans <|
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(S.F.mapIso (OverColor.mkSum (c ∘ (HepLean.Fin.finExtractTwo i j).symm))).trans <|
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(S.F.μIso _ _).symm.trans <| by
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(Functor.Monoidal.μIso S.F _ _).symm.trans <| by
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refine tensorIso (S.contrFin1Fin1 c i j h) (S.F.mapIso (OverColor.mkIso (by ext x; simp)))
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lemma contrIso_hom_hom {n : ℕ} {c1 : Fin n.succ.succ → S.C}
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@ -152,7 +152,7 @@ lemma contrIso_hom_hom {n : ℕ} {c1 : Fin n.succ.succ → S.C}
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(S.contrIso c1 i j h).hom.hom =
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(S.F.map (equivToIso (HepLean.Fin.finExtractTwo i j)).hom).hom ≫
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(S.F.map (mkSum (c1 ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)).hom).hom ≫
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(S.F.μIso (OverColor.mk ((c1 ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl))
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(Functor.Monoidal.μIso S.F (OverColor.mk ((c1 ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl))
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(OverColor.mk ((c1 ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inr))).inv.hom ≫
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((S.contrFin1Fin1 c1 i j h).hom.hom ⊗
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(S.F.map (mkIso (contrIso.proof_1 S c1 i j)).hom).hom) := by
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@ -192,38 +192,15 @@ lemma contrMap_tprod {n : ℕ} (c : Fin n.succ.succ → S.C)
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(((S.contr.app { as := c i }).hom ▷ ((lift.obj S.FD).obj
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(OverColor.mk (c ∘ i.succAbove ∘ j.succAbove))).V)
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(((S.contrFin1Fin1 c i j h).hom.hom ⊗ ((lift.obj S.FD).map (mkIso _).hom).hom)
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(((lift.obj S.FD).μIso (OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)
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((Functor.Monoidal.μIso (lift.obj S.FD).toFunctor (OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)
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∘ Sum.inl))
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(OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inr))).inv.hom
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(((lift.obj S.FD).map (mkSum (c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)).hom).hom
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(((lift.obj S.FD).map (equivToIso (HepLean.Fin.finExtractTwo i j)).hom).hom
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((PiTensorProduct.tprod S.k) x)))))) = _
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rw [lift.map_tprod]
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change (λ_ ((lift.obj S.FD).obj (OverColor.mk (c ∘ i.succAbove ∘ j.succAbove)))).hom.hom
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(((S.contr.app { as := c i }).hom ▷
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((lift.obj S.FD).obj (OverColor.mk (c ∘ i.succAbove ∘ j.succAbove))).V)
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(((S.contrFin1Fin1 c i j h).hom.hom ⊗ ((lift.obj S.FD).map (mkIso _).hom).hom)
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(((lift.obj S.FD).μIso (OverColor.mk
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((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl))
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(OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inr))).inv.hom
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(((lift.obj S.FD).map (mkSum (c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)).hom).hom
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((PiTensorProduct.tprod S.k) fun i_1 =>
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(lift.discreteFunctorMapEqIso S.FD _)
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(x ((Hom.toEquiv (equivToIso (HepLean.Fin.finExtractTwo i j)).hom).symm i_1))))))) = _
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rw [lift.map_tprod]
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change (λ_ ((lift.obj S.FD).obj (OverColor.mk (c ∘ i.succAbove ∘ j.succAbove)))).hom.hom
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(((S.contr.app { as := c i }).hom ▷ ((lift.obj S.FD).obj
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(OverColor.mk (c ∘ i.succAbove ∘ j.succAbove))).V)
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(((S.contrFin1Fin1 c i j h).hom.hom ⊗ ((lift.obj S.FD).map (mkIso _).hom).hom)
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(((lift.obj S.FD).μIso
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(OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl))
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(OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inr))).inv.hom
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((PiTensorProduct.tprod S.k) fun i_1 =>
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(lift.discreteFunctorMapEqIso S.FD _)
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((lift.discreteFunctorMapEqIso S.FD _)
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(x ((Hom.toEquiv (equivToIso (HepLean.Fin.finExtractTwo i j)).hom).symm
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((Hom.toEquiv (mkSum (c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)).hom).symm i_1)))))))) = _
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rw [lift.μIso_inv_tprod]
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erw [lift.map_tprod]
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erw [lift.μIso_inv_tprod]
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change (λ_ ((lift.obj S.FD).obj (OverColor.mk (c ∘ i.succAbove ∘ j.succAbove)))).hom.hom
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(((S.contr.app { as := c i }).hom ▷ ((lift.obj S.FD).obj
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(OverColor.mk (c ∘ i.succAbove ∘ j.succAbove))).V)
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@ -38,12 +38,9 @@ def tprod (p : Pure S c) : S.F.obj c := PiTensorProduct.tprod S.k p
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/-- The map `tprod` is equivariant with respect to the group action. -/
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lemma tprod_equivariant (g : S.G) (p : Pure S c) : (ρ g p).tprod = (S.F.obj c).ρ g p.tprod := by
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simp only [F_def, OverColor.lift, OverColor.lift.obj', OverColor.lift.objObj',
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OverColor.instMonoidalCategoryStruct_tensorUnit_left, Functor.id_obj,
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OverColor.instMonoidalCategoryStruct_tensorUnit_hom,
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OverColor.instMonoidalCategoryStruct_tensorObj_left,
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OverColor.instMonoidalCategoryStruct_tensorObj_hom, Rep.coe_of, tprod, Rep.of_ρ,
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MonoidHom.coe_mk, OneHom.coe_mk, PiTensorProduct.map_tprod]
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simp only [F_def, OverColor.lift, OverColor.lift.obj', LaxBraidedFunctor.of_toFunctor,
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OverColor.lift.objObj', Functor.id_obj, Rep.coe_of, tprod, Rep.of_ρ, MonoidHom.coe_mk,
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OneHom.coe_mk, PiTensorProduct.map_tprod]
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rfl
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end Pure
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