refactor: Lint
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jstoobysmith
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05b4d134ec
9 changed files with 51 additions and 40 deletions
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@ -34,7 +34,7 @@ lemma contr_two_two_inner_tprod (c : S.C) (x : S.F.obj (OverColor.mk ![c, c]))
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CoeSort.coe (S.FD.obj { as := (OverColor.mk ![S.τ c, S.τ c]).hom i }))
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(hx : x = PiTensorProduct.tprod S.k fx)
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(hy : y = PiTensorProduct.tprod S.k fy) :
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{x | μ ν ⊗ y| ν ρ}ᵀ.tensor = (S.F.map (OverColor.mkIso (by
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{x | μ ν ⊗ y| ν ρ}ᵀ.tensor = (S.F.map (OverColor.mkIso (by
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funext x
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fin_cases x <;> rfl)).hom).hom ((OverColor.Discrete.pairIsoSep S.FD).hom.hom
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(((S.FD.obj (Discrete.mk c)) ◁ (λ_ (S.FD.obj (Discrete.mk (S.τ c)))).hom).hom
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@ -45,13 +45,14 @@ lemma contr_two_two_inner_tprod (c : S.C) (x : S.F.obj (OverColor.mk ![c, c]))
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((α_ (S.FD.obj (Discrete.mk (c))) (S.FD.obj (Discrete.mk (c)))
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(S.FD.obj (Discrete.mk (S.τ c)) ⊗ S.FD.obj (Discrete.mk (S.τ c)))).hom.hom
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(((OverColor.Discrete.pairIsoSep S.FD).inv.hom x ⊗ₜ
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(OverColor.Discrete.pairIsoSep S.FD).inv.hom y))))))):= by
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(OverColor.Discrete.pairIsoSep S.FD).inv.hom y))))))) := by
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subst hx
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subst hy
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rw [Discrete.pairIsoSep_inv_tprod S.FD fx, Discrete.pairIsoSep_inv_tprod S.FD fy]
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change _ = (S.F.map (OverColor.mkIso _).hom).hom ((OverColor.Discrete.pairIsoSep S.FD).hom.hom
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((fx (0 : Fin 2) ⊗ₜ[S.k] (λ_ (S.FD.obj { as := S.τ c }).V).hom
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((S.contr.app { as := c }).hom (fx (1 : Fin 2) ⊗ₜ[S.k] fy (0 : Fin 2)) ⊗ₜ[S.k] fy (1 : Fin 2)))))
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((fx (0 : Fin 2) ⊗ₜ[S.k] (λ_ (S.FD.obj { as := S.τ c }).V).hom
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((S.contr.app { as := c }).hom (fx (1 : Fin 2)
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⊗ₜ[S.k] fy (0 : Fin 2)) ⊗ₜ[S.k] fy (1 : Fin 2)))))
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simp only [F_def, Functor.id_obj, mk_hom, Action.instMonoidalCategory_tensorObj_V,
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Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
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Action.FunctorCategoryEquivalence.functor_obj_obj, Monoidal.tensorUnit_obj,
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@ -86,7 +87,7 @@ lemma contr_two_two_inner_tprod (c : S.C) (x : S.F.obj (OverColor.mk ![c, c]))
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Functor.id_obj, instMonoidalCategoryStruct_tensorObj_hom, LinearEquiv.ofLinear_apply]
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rfl
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/- The tensor. -/
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· rw (config := { transparency := .instances }) [Discrete.pairIsoSep_tmul,
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· rw (config := { transparency := .instances }) [Discrete.pairIsoSep_tmul,
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OverColor.lift.map_tprod]
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apply congrArg
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funext k
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@ -97,7 +98,7 @@ lemma contr_two_two_inner_tprod (c : S.C) (x : S.F.obj (OverColor.mk ![c, c]))
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/-- Expands the inner contraction of two 2-tensors in terms of basic categorical
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constructions and fields of the tensor species. -/
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lemma contr_two_two_inner (c : S.C) (x : S.F.obj (OverColor.mk ![c, c]))
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(y : S.F.obj (OverColor.mk ![(S.τ c), (S.τ c)])):
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(y : S.F.obj (OverColor.mk ![(S.τ c), (S.τ c)])) :
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{x | μ ν ⊗ y| ν ρ}ᵀ.tensor = (S.F.map (OverColor.mkIso (by
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funext x
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fin_cases x <;> rfl)).hom).hom ((OverColor.Discrete.pairIsoSep S.FD).hom.hom
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@ -109,7 +110,7 @@ lemma contr_two_two_inner (c : S.C) (x : S.F.obj (OverColor.mk ![c, c]))
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((α_ (S.FD.obj (Discrete.mk (c))) (S.FD.obj (Discrete.mk (c)))
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(S.FD.obj (Discrete.mk (S.τ c)) ⊗ S.FD.obj (Discrete.mk (S.τ c)))).hom.hom
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(((OverColor.Discrete.pairIsoSep S.FD).inv.hom x ⊗ₜ
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(OverColor.Discrete.pairIsoSep S.FD).inv.hom y))))))):= by
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(OverColor.Discrete.pairIsoSep S.FD).inv.hom y))))))) := by
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simp only [Nat.reduceAdd, Fin.isValue, contr_tensor, prod_tensor, Functor.id_obj, mk_hom,
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Action.instMonoidalCategory_tensorObj_V, Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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@ -33,22 +33,22 @@ def metricTensor (S : TensorSpecies) (c : S.C) : S.F.obj (OverColor.mk ![c, c])
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variable {S : TensorSpecies}
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lemma metricTensor_congr {c c' : S.C} (h : c = c') : {S.metricTensor c | μ ν}ᵀ.tensor =
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(perm (OverColor.equivToHomEq (Equiv.refl _) (fun x => by subst h; fin_cases x <;> rfl ))
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(perm (OverColor.equivToHomEq (Equiv.refl _) (fun x => by subst h; fin_cases x <;> rfl))
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{S.metricTensor c' | μ ν}ᵀ).tensor := by
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subst h
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change _ = (S.F.map (𝟙 _)).hom (S.metricTensor c)
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simp
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lemma pairIsoSep_inv_metricTensor (c : S.C) :
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(Discrete.pairIsoSep S.FD).inv.hom (S.metricTensor c) =
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(S.metric.app (Discrete.mk c)).hom (1 : S.k) := by
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simp [metricTensor]
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simp only [Action.instMonoidalCategory_tensorObj_V, Nat.succ_eq_add_one, Nat.reduceAdd,
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metricTensor, Monoidal.tensorUnit_obj, Action.instMonoidalCategory_tensorUnit_V]
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erw [Discrete.rep_iso_inv_hom_apply]
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/-- Contraction of a metric tensor with a metric tensor gives the unit.
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Like `S.contr_metric` but with the braiding appearing on the side of the unit. -/
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lemma contr_metric_braid_unit (c : S.C) : (((S.FD.obj (Discrete.mk c)) ◁
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lemma contr_metric_braid_unit (c : S.C) : (((S.FD.obj (Discrete.mk c)) ◁
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(λ_ (S.FD.obj (Discrete.mk (S.τ c)))).hom).hom
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(((S.FD.obj (Discrete.mk c)) ◁ ((S.contr.app (Discrete.mk c)) ▷
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(S.FD.obj (Discrete.mk (S.τ c))))).hom
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@ -63,12 +63,13 @@ lemma contr_metric_braid_unit (c : S.C) : (((S.FD.obj (Discrete.mk c)) ◁
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apply (β_ _ _).toLinearEquiv.toEquiv.injective
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rw [pairIsoSep_inv_metricTensor, pairIsoSep_inv_metricTensor]
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erw [S.contr_metric c]
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change _ = (β_ (S.FD.obj { as := S.τ c }) (S.FD.obj { as := c })).inv.hom
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change _ = (β_ (S.FD.obj { as := S.τ c }) (S.FD.obj { as := c })).inv.hom
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((β_ (S.FD.obj { as := S.τ c }) (S.FD.obj { as := c })).hom.hom _)
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rw [Discrete.rep_iso_inv_hom_apply]
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lemma metricTensor_contr_dual_metricTensor_perm_cond (c : S.C) : ∀ (x : Fin (Nat.succ 0).succ),
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((Sum.elim ![c, c] ![S.τ c, S.τ c] ∘ ⇑finSumFinEquiv.symm) ∘ Fin.succAbove 1 ∘ Fin.succAbove 1) x =
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((Sum.elim ![c, c] ![S.τ c, S.τ c] ∘ ⇑finSumFinEquiv.symm) ∘
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Fin.succAbove 1 ∘ Fin.succAbove 1) x =
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(![S.τ c, c] ∘ ⇑(finMapToEquiv ![1, 0] ![1, 0]).symm) x := by
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intro x
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fin_cases x
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@ -81,12 +82,12 @@ lemma metricTensor_contr_dual_metricTensor_eq_unit (c : S.C) :
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perm (OverColor.equivToHomEq (finMapToEquiv ![1, 0] ![1, 0])
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(metricTensor_contr_dual_metricTensor_perm_cond c))).tensor := by
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rw [contr_two_two_inner, contr_metric_braid_unit, Discrete.pairIsoSep_β]
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change (S.F.map _ ≫ S.F.map _ ).hom _ = _
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change (S.F.map _ ≫ S.F.map _).hom _ = _
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rw [← S.F.map_comp]
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rfl
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/-- The contraction of a metric tensor with its dual via the outer indices gives the unit. -/
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lemma metricTensor_contr_dual_metricTensor_outer_eq_unit (c : S.C) :
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lemma metricTensor_contr_dual_metricTensor_outer_eq_unit (c : S.C) :
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{S.metricTensor c | ν μ ⊗ S.metricTensor (S.τ c) | ρ ν}ᵀ.tensor = ({S.unitTensor c | μ ρ}ᵀ |>
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perm (OverColor.equivToHomEq
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(finMapToEquiv ![1, 0] ![1, 0]) (fun x => by fin_cases x <;> rfl))).tensor := by
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@ -31,21 +31,21 @@ open TensorTree
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/-- The relation between two units of colors which are equal. -/
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lemma unitTensor_congr {c c' : S.C} (h : c = c') : {S.unitTensor c | μ ν}ᵀ.tensor =
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(perm (OverColor.equivToHomEq (Equiv.refl _) (fun x => by subst h; fin_cases x <;> rfl ))
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(perm (OverColor.equivToHomEq (Equiv.refl _) (fun x => by subst h; fin_cases x <;> rfl))
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{S.unitTensor c' | μ ν}ᵀ).tensor := by
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subst h
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change _ = (S.F.map (𝟙 _)).hom (S.unitTensor c)
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simp
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lemma unitTensor_eq_dual_perm_eq (c : S.C) : ∀ (x : Fin (Nat.succ 0).succ),
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![S.τ c, c] x = (![S.τ (S.τ c), S.τ c] ∘ ⇑(finMapToEquiv ![1, 0] ![1, 0]).symm) x := fun x => by
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![S.τ c, c] x = (![S.τ (S.τ c), S.τ c] ∘ ⇑(finMapToEquiv ![1, 0] ![1, 0]).symm) x := fun x => by
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fin_cases x
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· rfl
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· exact (S.τ_involution c).symm
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/-- The unit tensor is equal to a permutation of indices of the dual tensor. -/
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lemma unitTensor_eq_dual_perm (c : S.C) : {S.unitTensor c | μ ν}ᵀ.tensor =
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(perm (OverColor.equivToHomEq (finMapToEquiv ![1,0] ![1, 0]) (unitTensor_eq_dual_perm_eq c))
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(perm (OverColor.equivToHomEq (finMapToEquiv ![1,0] ![1, 0]) (unitTensor_eq_dual_perm_eq c))
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{S.unitTensor (S.τ c) | ν μ }ᵀ).tensor := by
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simp [unitTensor, tensorNode_tensor, perm_tensor]
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have h1 := S.unit_symm c
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@ -85,7 +85,7 @@ lemma unitTensor_eq_dual_perm (c : S.C) : {S.unitTensor c | μ ν}ᵀ.tensor =
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Nat.succ_eq_add_one, Nat.reduceAdd, lift.discreteFunctorMapEqIso, Functor.mapIso_hom,
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eqToIso.hom, Functor.mapIso_inv, eqToIso.inv, LinearEquiv.ofLinear_apply]
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rfl
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exact congrFun (congrArg (fun f => f.toFun) hg) _
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exact congrFun (congrArg (fun f => f.toFun) hg) _
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lemma dual_unitTensor_eq_perm_eq (c : S.C) : ∀ (x : Fin (Nat.succ 0).succ),
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![S.τ (S.τ c), S.τ c] x = (![S.τ c, c] ∘ ⇑(finMapToEquiv ![1, 0] ![1, 0]).symm) x := fun x => by
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