feat: unit contract vec
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jstoobysmith
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3 changed files with 55 additions and 23 deletions
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@ -6,6 +6,10 @@ Authors: Joseph Tooby-Smith
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import HepLean.Tensors.Tree.Elab
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import HepLean.Tensors.Tree.NodeIdentities.Basic
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import HepLean.Tensors.Tree.NodeIdentities.Congr
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import HepLean.Tensors.Tree.NodeIdentities.ProdComm
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import HepLean.Tensors.Tree.NodeIdentities.PermProd
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import HepLean.Tensors.Tree.NodeIdentities.PermContr
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import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
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import HepLean.Tensors.TensorSpecies.ContractLemmas
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/-!
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@ -38,6 +42,8 @@ lemma unitTensor_congr {c c' : S.C} (h : c = c') : {S.unitTensor c | μ ν}ᵀ.t
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change _ = (S.F.map (𝟙 _)).hom (S.unitTensor c)
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simp
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/-- Applying `Discrete.pairIsoSep` inv to `unitTensor` returns the unit natural transformation
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evaluated at `1`. -/
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lemma pairIsoSep_inv_unitTensor (c : S.C) :
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(Discrete.pairIsoSep S.FD).inv.hom (S.unitTensor c) =
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(S.unit.app (Discrete.mk c)).hom (1 : S.k) := by
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@ -45,23 +51,19 @@ lemma pairIsoSep_inv_unitTensor (c : S.C) :
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unitTensor, Monoidal.tensorUnit_obj, Action.instMonoidalCategory_tensorUnit_V]
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erw [Discrete.rep_iso_inv_hom_apply]
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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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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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{S.unitTensor (S.τ c) | ν μ }ᵀ).tensor := by
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({S.unitTensor (S.τ c) | ν μ }ᵀ |>
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perm (OverColor.equivToHomEq (finMapToEquiv ![1,0] ![1, 0])
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(fun x => match x with | 0 => by rfl | 1 => (S.τ_involution c).symm))).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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erw [h1]
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have hg : (Discrete.pairIsoSep S.FD).hom.hom ∘ₗ (S.FD.obj { as := S.τ c } ◁
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S.FD.map (Discrete.eqToHom (S.τ_involution c))).hom ∘ₗ
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(β_ (S.FD.obj { as := S.τ (S.τ c) }) (S.FD.obj { as := S.τ c })).hom.hom =
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(S.F.map (equivToHomEq (finMapToEquiv ![1, 0] ![1, 0]) (unitTensor_eq_dual_perm_eq c))).hom
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(S.F.map (equivToHomEq (finMapToEquiv ![1, 0] ![1, 0])
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(fun x => match x with | 0 => by rfl | 1 => (S.τ_involution c).symm))).hom
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∘ₗ (Discrete.pairIsoSep S.FD).hom.hom := by
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apply TensorProduct.ext'
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intro x y
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@ -95,15 +97,12 @@ lemma unitTensor_eq_dual_perm (c : S.C) : {S.unitTensor c | μ ν}ᵀ.tensor =
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rfl
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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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fin_cases x
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· exact (S.τ_involution c)
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· rfl
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lemma dual_unitTensor_eq_perm (c : S.C) : {S.unitTensor (S.τ c) | ν μ}ᵀ.tensor =
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(perm (OverColor.equivToHomEq (finMapToEquiv ![1, 0] ![1, 0]) (dual_unitTensor_eq_perm_eq c))
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{S.unitTensor c | μ ν}ᵀ).tensor := by
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/-- The unit tensor of the dual of a color `c` is equal to the unit tensor of `c`
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with indicees permuted. -/
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lemma dual_unitTensor_eq_perm (c : S.C) :
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{S.unitTensor (S.τ c) | ν μ}ᵀ.tensor = ({S.unitTensor c | μ ν}ᵀ |>
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perm (OverColor.equivToHomEq (finMapToEquiv ![1, 0] ![1, 0])
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(fun x => match x with | 0 => (S.τ_involution c) | 1 => by rfl))).tensor := by
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rw [unitTensor_eq_dual_perm]
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conv =>
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lhs
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@ -113,8 +112,8 @@ lemma dual_unitTensor_eq_perm (c : S.C) : {S.unitTensor (S.τ c) | ν μ}ᵀ.ten
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue]
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rfl
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lemma contrOneTwoLeft_unitTensor {c1 : S.C}
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(x : S.F.obj (OverColor.mk ![c1])) :
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/-- Applying `contrOneTwoLeft` with the unit tensor is the identity map. -/
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lemma contrOneTwoLeft_unitTensor {c1 : S.C} (x : S.F.obj (OverColor.mk ![c1])) :
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contrOneTwoLeft x (S.unitTensor c1) = x := by
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, contrOneTwoLeft,
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Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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@ -125,15 +124,39 @@ lemma contrOneTwoLeft_unitTensor {c1 : S.C}
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erw [pairIsoSep_inv_unitTensor (S := S) (c := c1)]
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change (S.F.mapIso (mkIso _)).hom.hom _ = _
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rw [Discrete.rep_iso_apply_iff, Discrete.rep_iso_inv_apply_iff]
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simpa using S.contr_unit c1 ((OverColor.forgetLiftApp S.FD c1).hom.hom ((S.F.map (OverColor.mkIso (by
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funext x; fin_cases x; rfl)).hom).hom x))
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simpa using S.contr_unit c1 ((OverColor.forgetLiftApp S.FD c1).hom.hom ((S.F.map (OverColor.mkIso
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(by funext x; fin_cases x; rfl)).hom).hom x))
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lemma one_contr_unitTensor_eq_self {c1 : S.C} (x : S.F.obj (OverColor.mk ![c1])) :
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/-- Contracting a vector with a unit tensor returns the vector. -/
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lemma vec_contr_unitTensor_eq_self {c1 : S.C} (x : S.F.obj (OverColor.mk ![c1])) :
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{x | μ ⊗ S.unitTensor c1 | μ ν}ᵀ.tensor = ({x | ν}ᵀ |>
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perm (OverColor.equivToHomEq (Equiv.refl _) (fun x => by fin_cases x; rfl ))).tensor := by
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rw [contr_one_two_left_eq_contrOneTwoLeft, contrOneTwoLeft_unitTensor]
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rfl
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/-- Contracting a unit tensor with a vector returns the unit vector. -/
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lemma unitTensor_contr_vec_eq_self {c1 : S.C} (x : S.F.obj (OverColor.mk ![S.τ c1])) :
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{S.unitTensor c1 | ν μ ⊗ x | μ}ᵀ.tensor = ({x | ν}ᵀ |>
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perm (OverColor.equivToHomEq (Equiv.refl _) (fun x => by fin_cases x; rfl ))).tensor := by
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conv_lhs =>
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rw [contr_tensor_eq <| prod_comm _ _ _ _]
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rw [perm_contr_congr 2 0 (by simp; decide) (by simp; decide)]
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rw [perm_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_snd <| unitTensor_eq_dual_perm _]
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rw [perm_tensor_eq <| contr_tensor_eq <| prod_perm_right _ _ _ _]
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rw [perm_tensor_eq <| perm_contr_congr 1 0 (by simp; decide) (by simp; decide)]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_swap _ _]
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rw [perm_perm]
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erw [perm_tensor_eq <| vec_contr_unitTensor_eq_self _]
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rw [perm_perm]
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refine perm_congr (OverColor.Hom.fin_ext _ _ fun i => ?_) rfl
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simp only [mk_left, mk_hom, Function.comp_apply, equivToHomEq_toEquiv, Hom.hom_comp,
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Over.comp_left, equivToHomEq_hom_left, types_comp_apply, ContrPair.contrSwapHom_hom_left_apply,
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mkIso_hom_hom_left_apply, extractTwo_hom_left_apply, permProdRight_toEquiv,
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finExtractOnePerm_apply, finExtractOne_symm_inr_apply, braidPerm_toEquiv]
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fin_cases i
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· decide
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end TensorSpecies
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end
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