feat: Double contraction of indices lemma
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jstoobysmith
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9123431424
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d02e94886d
10 changed files with 543 additions and 362 deletions
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@ -19,6 +19,105 @@ noncomputable section
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open TensorProduct
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namespace TensorColor
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variable {𝓒 : TensorColor} [DecidableEq 𝓒.Color] [Fintype 𝓒.Color]
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variable {d : ℕ} {X Y Y' Z W C P : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
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[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z] [Fintype W] [DecidableEq W]
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[Fintype C] [DecidableEq C] [Fintype P] [DecidableEq P]
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/-!
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## Dual maps
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-/
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namespace ColorMap
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variable (cX : 𝓒.ColorMap X)
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def partDual (e : C ⊕ P ≃ X) : 𝓒.ColorMap X :=
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(Sum.elim (𝓒.τ ∘ cX ∘ e ∘ Sum.inl) (cX ∘ e ∘ Sum.inr) ∘ e.symm)
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/-- Two color maps are said to be dual if their quotents are dual. -/
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def DualMap (c₁ c₂ : 𝓒.ColorMap X) : Prop :=
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𝓒.colorQuot ∘ c₁ = 𝓒.colorQuot ∘ c₂
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namespace DualMap
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variable {c₁ c₂ c₃ : 𝓒.ColorMap X}
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variable {n : ℕ}
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/-- The bool which if `𝓒.colorQuot (c₁ i) = 𝓒.colorQuot (c₂ i)` is true for all `i`. -/
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def boolFin (c₁ c₂ : 𝓒.ColorMap (Fin n)) : Bool :=
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(Fin.list n).all fun i => if 𝓒.colorQuot (c₁ i) = 𝓒.colorQuot (c₂ i) then true else false
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lemma boolFin_DualMap {c₁ c₂ : 𝓒.ColorMap (Fin n)} (h : boolFin c₁ c₂ = true) :
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DualMap c₁ c₂ := by
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simp [boolFin] at h
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simp [DualMap]
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funext x
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have h2 {n : ℕ} (m : Fin n) : m ∈ Fin.list n := by
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have h1' : (Fin.list n)[m] = m := by
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erw [Fin.getElem_list]
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rfl
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rw [← h1']
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apply List.getElem_mem
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exact h x (h2 _)
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lemma refl : DualMap c₁ c₁ := by
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simp [DualMap]
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lemma symm (h : DualMap c₁ c₂) : DualMap c₂ c₁ := by
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rw [DualMap] at h ⊢
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exact h.symm
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lemma trans (h : DualMap c₁ c₂) (h' : DualMap c₂ c₃) : DualMap c₁ c₃ := by
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rw [DualMap] at h h' ⊢
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exact h.trans h'
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/-- The splitting of `X` given two color maps based on the equality of the color. -/
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def split (c₁ c₂ : 𝓒.ColorMap X) : { x // c₁ x ≠ c₂ x} ⊕ { x // c₁ x = c₂ x} ≃ X :=
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((Equiv.Set.sumCompl {x | c₁ x = c₂ x}).symm.trans (Equiv.sumComm _ _)).symm
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lemma dual_eq_of_neq (h : DualMap c₁ c₂) {x : X} (h' : c₁ x ≠ c₂ x) :
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𝓒.τ (c₁ x) = c₂ x := by
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rw [DualMap] at h
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have h1 := congrFun h x
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simp [colorQuot, HasEquiv.Equiv, Setoid.r, colorRel] at h1
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simp_all only [ne_eq, false_or]
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exact 𝓒.τ_involutive (c₂ x)
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@[simp]
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lemma split_dual (h : DualMap c₁ c₂) : c₁.partDual (split c₁ c₂) = c₂ := by
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rw [partDual, Equiv.comp_symm_eq]
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funext x
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match x with
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| Sum.inl x =>
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exact h.dual_eq_of_neq x.2
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| Sum.inr x =>
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exact x.2
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@[simp]
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lemma split_dual' (h : DualMap c₁ c₂) : c₂.partDual (split c₁ c₂) = c₁ := by
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rw [partDual, Equiv.comp_symm_eq]
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funext x
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match x with
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| Sum.inl x =>
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change 𝓒.τ (c₂ x) = c₁ x
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rw [← h.dual_eq_of_neq x.2]
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exact (𝓒.τ_involutive (c₁ x))
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| Sum.inr x =>
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exact x.2.symm
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end DualMap
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end ColorMap
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end TensorColor
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variable {R : Type} [CommSemiring R]
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namespace TensorStructure
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@ -28,8 +127,8 @@ variable (𝓣 : TensorStructure R)
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variable {d : ℕ} {X Y Y' Z W C P : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
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[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z] [Fintype W] [DecidableEq W]
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[Fintype C] [DecidableEq C] [Fintype P] [DecidableEq P]
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{cX cX2 : X → 𝓣.Color} {cY : Y → 𝓣.Color} {cZ : Z → 𝓣.Color}
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{cW : W → 𝓣.Color} {cY' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
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{cX cX2 : 𝓣.ColorMap X} {cY : 𝓣.ColorMap Y} {cZ : 𝓣.ColorMap Z}
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{cW : 𝓣.ColorMap W} {cY' : 𝓣.ColorMap Y'} {μ ν: 𝓣.Color}
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variable {G H : Type} [Group G] [Group H] [MulActionTensor G 𝓣]
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local infixl:101 " • " => 𝓣.rep
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@ -199,12 +298,9 @@ lemma dualize_cond' (e : C ⊕ P ≃ X) :
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| Sum.inl x => simp
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| Sum.inr x => simp
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/-! TODO: Show that `dualize` is equivariant with respect to the group action. -/
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/-- Given an equivalence `C ⊕ P ≃ X` dualizes those indices of a tensor which correspond to
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`C` whilst leaving the indices `P` invariant. -/
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def dualize (e : C ⊕ P ≃ X) : 𝓣.Tensor cX ≃ₗ[R]
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𝓣.Tensor (Sum.elim (𝓣.τ ∘ cX ∘ e ∘ Sum.inl) (cX ∘ e ∘ Sum.inr) ∘ e.symm) :=
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def dualize (e : C ⊕ P ≃ X) : 𝓣.Tensor cX ≃ₗ[R] 𝓣.Tensor (cX.partDual e) :=
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𝓣.mapIso e.symm (𝓣.dualize_cond e) ≪≫ₗ
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(𝓣.tensoratorEquiv _ _).symm ≪≫ₗ
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TensorProduct.congr 𝓣.dualizeAll (LinearEquiv.refl _ _) ≪≫ₗ
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@ -232,95 +328,3 @@ lemma dualize_equivariant_apply (e : C ⊕ P ≃ X) (g : G) (x : 𝓣.Tensor cX)
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end TensorStructure
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end
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namespace TensorColor
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variable {𝓒 : TensorColor} [DecidableEq 𝓒.Color] [Fintype 𝓒.Color]
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variable {d : ℕ} {X Y Y' Z W C P : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
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[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z] [Fintype W] [DecidableEq W]
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[Fintype C] [DecidableEq C] [Fintype P] [DecidableEq P]
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/-!
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## Dual maps
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-/
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/-- Two color maps are said to be dual if their quotents are dual. -/
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def DualMap (c₁ : X → 𝓒.Color) (c₂ : X → 𝓒.Color) : Prop :=
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𝓒.colorQuot ∘ c₁ = 𝓒.colorQuot ∘ c₂
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namespace DualMap
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variable {c₁ c₂ c₃ : X → 𝓒.Color}
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variable {n : ℕ}
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/-- The bool which if `𝓒.colorQuot (c₁ i) = 𝓒.colorQuot (c₂ i)` is true for all `i`. -/
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def boolFin (c₁ c₂ : Fin n → 𝓒.Color) : Bool :=
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(Fin.list n).all fun i => if 𝓒.colorQuot (c₁ i) = 𝓒.colorQuot (c₂ i) then true else false
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lemma boolFin_DualMap {c₁ c₂ : Fin n → 𝓒.Color} (h : boolFin c₁ c₂ = true) :
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DualMap c₁ c₂ := by
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simp [boolFin] at h
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simp [DualMap]
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funext x
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have h2 {n : ℕ} (m : Fin n) : m ∈ Fin.list n := by
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have h1' : (Fin.list n)[m] = m := by
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erw [Fin.getElem_list]
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rfl
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rw [← h1']
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apply List.getElem_mem
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exact h x (h2 _)
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lemma refl : DualMap c₁ c₁ := by
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simp [DualMap]
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lemma symm (h : DualMap c₁ c₂) : DualMap c₂ c₁ := by
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rw [DualMap] at h ⊢
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exact h.symm
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lemma trans (h : DualMap c₁ c₂) (h' : DualMap c₂ c₃) : DualMap c₁ c₃ := by
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rw [DualMap] at h h' ⊢
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exact h.trans h'
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/-- The splitting of `X` given two color maps based on the equality of the color. -/
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def split (c₁ c₂ : X → 𝓒.Color) : { x // c₁ x ≠ c₂ x} ⊕ { x // c₁ x = c₂ x} ≃ X :=
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((Equiv.Set.sumCompl {x | c₁ x = c₂ x}).symm.trans (Equiv.sumComm _ _)).symm
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lemma dual_eq_of_neq (h : DualMap c₁ c₂) {x : X} (h' : c₁ x ≠ c₂ x) :
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𝓒.τ (c₁ x) = c₂ x := by
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rw [DualMap] at h
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have h1 := congrFun h x
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simp [colorQuot, HasEquiv.Equiv, Setoid.r, colorRel] at h1
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simp_all only [ne_eq, false_or]
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exact 𝓒.τ_involutive (c₂ x)
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@[simp]
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lemma split_dual (h : DualMap c₁ c₂) :
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Sum.elim (𝓒.τ ∘ c₁ ∘ (split c₁ c₂) ∘ Sum.inl) (c₁ ∘ (split c₁ c₂) ∘ Sum.inr)
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∘ (split c₁ c₂).symm = c₂ := by
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rw [Equiv.comp_symm_eq]
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funext x
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match x with
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| Sum.inl x =>
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exact h.dual_eq_of_neq x.2
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| Sum.inr x =>
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exact x.2
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@[simp]
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lemma split_dual' (h : DualMap c₁ c₂) :
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Sum.elim (𝓒.τ ∘ c₂ ∘ (split c₁ c₂) ∘ Sum.inl) (c₂ ∘ (split c₁ c₂) ∘ Sum.inr) ∘
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(split c₁ c₂).symm = c₁ := by
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rw [Equiv.comp_symm_eq]
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funext x
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match x with
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| Sum.inl x =>
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change 𝓒.τ (c₂ x) = c₁ x
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rw [← h.dual_eq_of_neq x.2]
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exact (𝓒.τ_involutive (c₁ x))
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| Sum.inr x =>
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exact x.2.symm
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end DualMap
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end TensorColor
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