feat: Prove multiplication commute Lorentz action
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5 changed files with 649 additions and 348 deletions
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@ -6,6 +6,8 @@ Authors: Joseph Tooby-Smith
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import Mathlib.Logic.Function.CompTypeclasses
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import Mathlib.Data.Real.Basic
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import Mathlib.Analysis.Normed.Field.Basic
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import Mathlib.CategoryTheory.Core
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import Mathlib.CategoryTheory.Types
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/-!
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# Real Lorentz Tensors
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@ -59,35 +61,11 @@ structure RealLorentzTensor (d : ℕ) (X : Type) where
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coord : RealLorentzTensor.IndexValue d color → ℝ
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namespace RealLorentzTensor
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open Matrix
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open Matrix CategoryTheory
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universe u1
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variable {d : ℕ} {X Y Z : Type}
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variable (c : X → Colors)
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/-!
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## Some equivalences of types
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These come in use casting Lorentz tensors.
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There is likely a better way to deal with these castings.
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-/
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/-- An equivalence from `Empty ⊕ PUnit.{1}` to `Empty ⊕ Σ _ : Fin 1, PUnit`. -/
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def equivPUnitToSigma :
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(Empty ⊕ PUnit.{1}) ≃ (Empty ⊕ Σ _ : Fin 1, PUnit) where
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toFun x := match x with
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| Sum.inr x => Sum.inr ⟨0, x⟩
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invFun x := match x with
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| Sum.inr ⟨0, x⟩ => Sum.inr x
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left_inv x := match x with
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| Sum.inr _ => rfl
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right_inv x := match x with
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| Sum.inr ⟨0, _⟩ => rfl
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/-!
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## Colors
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@ -105,11 +83,19 @@ lemma τ_involutive : Function.Involutive τ := by
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intro x
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cases x <;> rfl
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lemma color_eq_dual_symm {μ ν : Colors} (h : μ = τ ν) : ν = τ μ :=
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(Function.Involutive.eq_iff τ_involutive).mp h.symm
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/-- The color associated with an element of `x ∈ X` for a tensor `T`. -/
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def ch {X : Type} (x : X) (T : RealLorentzTensor d X) : Colors := T.color x
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/-- An equivalence of `ColorsIndex` types given an equality of a colors. -/
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def colorsIndexCast {d : ℕ} {μ₁ μ₂ : RealLorentzTensor.Colors} (h : μ₁ = μ₂) :
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ColorsIndex d μ₁ ≃ ColorsIndex d μ₂ :=
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Equiv.cast (by rw [h])
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/-- An equivalence of `ColorsIndex` between that of a color and that of its dual. -/
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def dualColorsIndex {d : ℕ} {μ : RealLorentzTensor.Colors}:
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def colorsIndexDualCastSelf {d : ℕ} {μ : RealLorentzTensor.Colors}:
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ColorsIndex d μ ≃ ColorsIndex d (τ μ) where
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toFun x :=
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match μ with
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@ -122,26 +108,18 @@ def dualColorsIndex {d : ℕ} {μ : RealLorentzTensor.Colors}:
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left_inv x := by cases μ <;> rfl
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right_inv x := by cases μ <;> rfl
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/-- An equivalence of `ColorsIndex` types given an equality of a colors. -/
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def castColorsIndex {d : ℕ} {μ₁ μ₂ : RealLorentzTensor.Colors} (h : μ₁ = μ₂) :
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ColorsIndex d μ₁ ≃ ColorsIndex d μ₂ :=
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Equiv.cast (by rw [h])
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/-- An equivalence of `ColorsIndex` types given an equality of a color and the dual of a color. -/
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def congrColorsDual {μ ν : Colors} (h : μ = τ ν) :
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def colorsIndexDualCast {μ ν : Colors} (h : μ = τ ν) :
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ColorsIndex d μ ≃ ColorsIndex d ν :=
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(castColorsIndex h).trans dualColorsIndex.symm
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(colorsIndexCast h).trans colorsIndexDualCastSelf.symm
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lemma congrColorsDual_symm {μ ν : Colors} (h : μ = τ ν) :
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(congrColorsDual h).symm =
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@congrColorsDual d _ _ ((Function.Involutive.eq_iff τ_involutive).mp h.symm) := by
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lemma colorsIndexDualCast_symm {μ ν : Colors} (h : μ = τ ν) :
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(colorsIndexDualCast h).symm =
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@colorsIndexDualCast d _ _ ((Function.Involutive.eq_iff τ_involutive).mp h.symm) := by
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match μ, ν with
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| Colors.up, Colors.down => rfl
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| Colors.down, Colors.up => rfl
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lemma color_eq_dual_symm {μ ν : Colors} (h : μ = τ ν) : ν = τ μ :=
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(Function.Involutive.eq_iff τ_involutive).mp h.symm
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/-!
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## Index values
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@ -153,19 +131,54 @@ instance [Fintype X] [DecidableEq X] : Fintype (IndexValue d c) := Pi.fintype
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instance [Fintype X] [DecidableEq X] : DecidableEq (IndexValue d c) :=
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Fintype.decidablePiFintype
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/-- An equivalence of Index values from an equality of color maps. -/
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def castIndexValue {X : Type} {T S : X → Colors} (h : T = S) :
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IndexValue d T ≃ IndexValue d S where
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toFun i := (fun μ => castColorsIndex (congrFun h μ) (i μ))
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invFun i := (fun μ => (castColorsIndex (congrFun h μ)).symm (i μ))
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left_inv i := by
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simp
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right_inv i := by
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simp
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/-!
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lemma indexValue_eq {T₁ T₂ : X → RealLorentzTensor.Colors} (d : ℕ) (h : T₁ = T₂) :
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IndexValue d T₁ = IndexValue d T₂ :=
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pi_congr fun a => congrArg (ColorsIndex d) (congrFun h a)
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## Induced isomorphisms between IndexValue sets
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-/
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@[simps!]
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def indexValueIso (d : ℕ) (f : X ≃ Y) {i : X → Colors} {j : Y → Colors} (h : i = j ∘ f) :
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IndexValue d i ≃ IndexValue d j :=
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(Equiv.piCongrRight (fun μ => colorsIndexCast (congrFun h μ))).trans $
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Equiv.piCongrLeft (fun y => RealLorentzTensor.ColorsIndex d (j y)) f
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lemma indexValueIso_symm_apply' (d : ℕ) (f : X ≃ Y) {i : X → Colors} {j : Y → Colors}
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(h : i = j ∘ f) (y : IndexValue d j) (x : X) :
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(indexValueIso d f h).symm y x = (colorsIndexCast (congrFun h x)).symm (y (f x)) := by
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rfl
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@[simp]
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lemma indexValueIso_trans (d : ℕ) (f : X ≃ Y) (g : Y ≃ Z) {i : X → Colors}
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{j : Y → Colors} {k : Z → Colors} (h : i = j ∘ f) (h' : j = k ∘ g) :
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(indexValueIso d f h).trans (indexValueIso d g h') =
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indexValueIso d (f.trans g) (by rw [h, h', Function.comp.assoc]; rfl) := by
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have h1 : ((indexValueIso d f h).trans (indexValueIso d g h')).symm =
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(indexValueIso d (f.trans g) (by rw [h, h', Function.comp.assoc]; rfl)).symm := by
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subst h' h
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ext x : 2
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rfl
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simpa only [Equiv.symm_symm] using congrArg (fun e => e.symm) h1
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lemma indexValueIso_symm (d : ℕ) (f : X ≃ Y) (h : i = j ∘ f) :
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(indexValueIso d f h).symm = indexValueIso d f.symm (by rw [h, Function.comp.assoc]; simp) := by
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ext i : 1
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rw [← Equiv.symm_apply_eq]
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funext y
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rw [indexValueIso_symm_apply', indexValueIso_symm_apply']
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simp [colorsIndexCast]
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apply cast_eq_iff_heq.mpr
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rw [Equiv.apply_symm_apply]
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lemma indexValueIso_eq_symm (d : ℕ) (f : X ≃ Y) (h : i = j ∘ f) :
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indexValueIso d f h = (indexValueIso d f.symm (by rw [h, Function.comp.assoc]; simp)).symm := by
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rw [indexValueIso_symm]
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congr
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@[simp]
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lemma indexValueIso_refl (d : ℕ) (i : X → Colors) :
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indexValueIso d (Equiv.refl X) (rfl : i = i) = Equiv.refl _ := by
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rfl
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/-!
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@ -174,19 +187,7 @@ lemma indexValue_eq {T₁ T₂ : X → RealLorentzTensor.Colors} (d : ℕ) (h :
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-/
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lemma ext {T₁ T₂ : RealLorentzTensor d X} (h : T₁.color = T₂.color)
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(h' : T₁.coord = T₂.coord ∘ Equiv.cast (indexValue_eq d h)) : T₁ = T₂ := by
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cases T₁
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cases T₂
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simp_all only [IndexValue, mk.injEq]
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apply And.intro h
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simp only at h
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subst h
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simp only [Equiv.cast_refl, Equiv.coe_refl, CompTriple.comp_eq] at h'
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subst h'
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rfl
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lemma ext' {T₁ T₂ : RealLorentzTensor d X} (h : T₁.color = T₂.color)
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(h' : T₁.coord = fun i => T₂.coord (castIndexValue h i)) :
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(h' : T₁.coord = fun i => T₂.coord (indexValueIso d (Equiv.refl X) h i)) :
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T₁ = T₂ := by
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cases T₁
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cases T₂
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@ -199,60 +200,69 @@ lemma ext' {T₁ T₂ : RealLorentzTensor d X} (h : T₁.color = T₂.color)
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/-!
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## Congruence
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## Mapping isomorphisms.
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-/
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/-- An equivalence between `X → Fin 1 ⊕ Fin d` and `Y → Fin 1 ⊕ Fin d` given an isomorphism
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between `X` and `Y`. -/
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@[simps!]
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def congrSetIndexValue (d : ℕ) (f : X ≃ Y) (i : X → Colors) :
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IndexValue d i ≃ IndexValue d (i ∘ f.symm) :=
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Equiv.piCongrLeft' _ f
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@[simp]
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lemma castColorsIndex_comp_congrSetIndexValue (c : X → Colors) (j : IndexValue d c) (f : X ≃ Y)
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(h1 : (c <| f.symm <| f <| x) = c x) : (castColorsIndex h1 <| congrSetIndexValue d f c j <| f x)
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= j x := by
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rw [congrSetIndexValue_apply]
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refine cast_eq_iff_heq.mpr ?_
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rw [Equiv.symm_apply_apply]
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/-- Given an equivalence of indexing sets, a map on Lorentz tensors. -/
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@[simps!]
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def congrSetMap (f : X ≃ Y) (T : RealLorentzTensor d X) : RealLorentzTensor d Y where
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color := T.color ∘ f.symm
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coord := T.coord ∘ (congrSetIndexValue d f T.color).symm
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lemma congrSetMap_trans (f : X ≃ Y) (g : Y ≃ Z) (T : RealLorentzTensor d X) :
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congrSetMap g (congrSetMap f T) = congrSetMap (f.trans g) T := by
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apply ext (by rfl)
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have h1 : congrSetIndexValue d (f.trans g) T.color = (congrSetIndexValue d f T.color).trans
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(congrSetIndexValue d g $ Equiv.piCongrLeft' (fun _ => Colors) f T.color) := by
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exact Equiv.coe_inj.mp rfl
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simp only [congrSetMap, Equiv.piCongrLeft'_apply, IndexValue, Equiv.symm_trans_apply, h1,
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Equiv.cast_refl, Equiv.coe_refl, CompTriple.comp_eq]
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rfl
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/-- An equivalence of Tensors given an equivalence of underlying sets. -/
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@[simps!]
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def congrSet (f : X ≃ Y) : RealLorentzTensor d X ≃ RealLorentzTensor d Y where
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toFun := congrSetMap f
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invFun := congrSetMap f.symm
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def mapIso (d : ℕ) (f : X ≃ Y) : RealLorentzTensor d X ≃ RealLorentzTensor d Y where
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toFun T := {
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color := T.color ∘ f.symm,
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coord := T.coord ∘ (indexValueIso d f (by simp : T.color = T.color ∘ f.symm ∘ f)).symm}
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invFun T := {
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color := T.color ∘ f,
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coord := T.coord ∘ (indexValueIso d f.symm (by simp : T.color = T.color ∘ f ∘ f.symm)).symm}
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left_inv T := by
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rw [congrSetMap_trans, Equiv.self_trans_symm]
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rfl
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refine ext ?_ ?_
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· simp [Function.comp.assoc]
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· funext i
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simp only [IndexValue, Function.comp_apply, Function.comp_id]
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apply congrArg
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funext x
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erw [indexValueIso_symm_apply', indexValueIso_symm_apply', indexValueIso_eq_symm,
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indexValueIso_symm_apply']
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rw [← Equiv.apply_eq_iff_eq_symm_apply]
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simp only [Equiv.refl_symm, Equiv.coe_refl, Function.comp_apply, id_eq, colorsIndexCast,
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Equiv.cast_symm, Equiv.cast_apply, cast_cast, Equiv.refl_apply]
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apply cast_eq_iff_heq.mpr
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congr
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exact Equiv.symm_apply_apply f x
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right_inv T := by
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rw [congrSetMap_trans, Equiv.symm_trans_self]
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rfl
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refine ext ?_ ?_
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· simp [Function.comp.assoc]
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· funext i
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simp only [IndexValue, Function.comp_apply, Function.comp_id]
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apply congrArg
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funext x
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erw [indexValueIso_symm_apply', indexValueIso_symm_apply', indexValueIso_eq_symm,
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indexValueIso_symm_apply']
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rw [← Equiv.apply_eq_iff_eq_symm_apply]
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simp only [Equiv.refl_symm, Equiv.coe_refl, Function.comp_apply, id_eq, colorsIndexCast,
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Equiv.cast_symm, Equiv.cast_apply, cast_cast, Equiv.refl_apply]
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apply cast_eq_iff_heq.mpr
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congr
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exact Equiv.apply_symm_apply f x
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lemma congrSet_trans (f : X ≃ Y) (g : Y ≃ Z) :
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(@congrSet d _ _ f).trans (congrSet g) = congrSet (f.trans g) := by
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@[simp]
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lemma mapIso_trans (f : X ≃ Y) (g : Y ≃ Z) :
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(mapIso d f).trans (mapIso d g) = mapIso d (f.trans g) := by
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refine Equiv.coe_inj.mp ?_
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funext T
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exact congrSetMap_trans f g T
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refine ext rfl ?_
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simp only [Equiv.trans_apply, IndexValue, mapIso_apply_color, Equiv.symm_trans_apply,
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indexValueIso_refl, Equiv.refl_apply, mapIso_apply_coord]
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funext i
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rw [mapIso_apply_coord, mapIso_apply_coord]
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apply congrArg
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rw [← indexValueIso_trans]
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rfl
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simp only [Function.comp.assoc, Equiv.symm_comp_self, CompTriple.comp_eq]
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lemma congrSet_refl : @congrSet d _ _ (Equiv.refl X) = Equiv.refl _ := rfl
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lemma mapIso_symm (f : X ≃ Y) : (mapIso d f).symm = mapIso d f.symm := by
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rfl
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lemma mapIso_refl : mapIso d (Equiv.refl X) = Equiv.refl _ := rfl
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/-!
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@ -260,54 +270,24 @@ lemma congrSet_refl : @congrSet d _ _ (Equiv.refl X) = Equiv.refl _ := rfl
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-/
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/-- The sum of two color maps. -/
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def sumElimIndexColor (Tc : X → Colors) (Sc : Y → Colors) :
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(X ⊕ Y) → Colors :=
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Sum.elim Tc Sc
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/-- The symmetry property on `sumElimIndexColor`. -/
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lemma sumElimIndexColor_symm (Tc : X → Colors) (Sc : Y → Colors) : sumElimIndexColor Tc Sc =
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Equiv.piCongrLeft' _ (Equiv.sumComm X Y).symm (sumElimIndexColor Sc Tc) := by
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ext1 x
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simp_all only [Equiv.piCongrLeft'_apply, Equiv.sumComm_symm, Equiv.sumComm_apply]
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cases x <;> rfl
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/-- The sum of two index values for different color maps. -/
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@[simp]
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def sumElimIndexValue {X Y : Type} {TX : X → Colors} {TY : Y → Colors}
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(i : IndexValue d TX) (j : IndexValue d TY) :
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IndexValue d (sumElimIndexColor TX TY) :=
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fun c => match c with
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| Sum.inl x => i x
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| Sum.inr x => j x
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/-- The projection of an index value on a sum of color maps to its left component. -/
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def inlIndexValue {Tc : X → Colors} {Sc : Y → Colors} (i : IndexValue d (sumElimIndexColor Tc Sc)) :
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IndexValue d Tc := fun x => i (Sum.inl x)
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/-- The projection of an index value on a sum of color maps to its right component. -/
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def inrIndexValue {Tc : X → Colors} {Sc : Y → Colors}
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(i : IndexValue d (sumElimIndexColor Tc Sc)) :
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IndexValue d Sc := fun y => i (Sum.inr y)
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def indexValueSumEquiv {X Y : Type} {TX : X → Colors} {TY : Y → Colors} :
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IndexValue d (Sum.elim TX TY) ≃ IndexValue d TX × IndexValue d TY where
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toFun i := (fun x => i (Sum.inl x), fun x => i (Sum.inr x))
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invFun p := fun c => match c with
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| Sum.inl x => (p.1 x)
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| Sum.inr x => (p.2 x)
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left_inv i := by
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simp only [IndexValue]
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ext1 x
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cases x with
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| inl val => rfl
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| inr val_1 => rfl
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right_inv p := rfl
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/-- An equivalence between index values formed by commuting sums. -/
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def sumCommIndexValue {X Y : Type} (Tc : X → Colors) (Sc : Y → Colors) :
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IndexValue d (sumElimIndexColor Tc Sc) ≃ IndexValue d (sumElimIndexColor Sc Tc) :=
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(congrSetIndexValue d (Equiv.sumComm X Y) (sumElimIndexColor Tc Sc)).trans
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(castIndexValue (sumElimIndexColor_symm Sc Tc).symm)
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lemma sumCommIndexValue_inlIndexValue {X Y : Type} {Tc : X → Colors} {Sc : Y → Colors}
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(i : IndexValue d <| sumElimIndexColor Tc Sc) :
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inlIndexValue (sumCommIndexValue Tc Sc i) = inrIndexValue i := rfl
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lemma sumCommIndexValue_inrIndexValue {X Y : Type} {Tc : X → Colors} {Sc : Y → Colors}
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(i : IndexValue d <| sumElimIndexColor Tc Sc) :
|
||||
inrIndexValue (sumCommIndexValue Tc Sc i) = inlIndexValue i := rfl
|
||||
|
||||
/-- Equivalence between sets of `RealLorentzTensor` formed by commuting sums. -/
|
||||
@[simps!]
|
||||
def sumComm : RealLorentzTensor d (X ⊕ Y) ≃ RealLorentzTensor d (Y ⊕ X) :=
|
||||
congrSet (Equiv.sumComm X Y)
|
||||
def indexValueSumComm {X Y : Type} (Tc : X → Colors) (Sc : Y → Colors) :
|
||||
IndexValue d (Sum.elim Tc Sc) ≃ IndexValue d (Sum.elim Sc Tc) :=
|
||||
indexValueIso d (Equiv.sumComm X Y) (by aesop)
|
||||
|
||||
/-!
|
||||
|
||||
|
|
@ -325,7 +305,6 @@ namespace Marked
|
|||
|
||||
variable {n m : ℕ}
|
||||
|
||||
|
||||
/-- The marked point. -/
|
||||
def markedPoint (X : Type) (i : Fin n) : (X ⊕ Fin n) :=
|
||||
Sum.inr i
|
||||
|
|
@ -342,43 +321,31 @@ def markedColor (T : Marked d X n) : Fin n → Colors :=
|
|||
def UnmarkedIndexValue (T : Marked d X n) : Type :=
|
||||
IndexValue d T.unmarkedColor
|
||||
|
||||
instance [Fintype X] [DecidableEq X] (T : Marked d X n) : Fintype T.UnmarkedIndexValue :=
|
||||
instance [Fintype X] [DecidableEq X] (T : Marked d X n) : Fintype T.UnmarkedIndexValue :=
|
||||
Pi.fintype
|
||||
|
||||
instance [Fintype X] [DecidableEq X] (T : Marked d X n) : DecidableEq T.UnmarkedIndexValue :=
|
||||
Fintype.decidablePiFintype
|
||||
|
||||
/-- The index values restricted to marked indices. -/
|
||||
def MarkedIndexValue (T : Marked d X n) : Type :=
|
||||
IndexValue d T.markedColor
|
||||
|
||||
instance [Fintype X] [DecidableEq X] (T : Marked d X n) : Fintype T.MarkedIndexValue :=
|
||||
instance [Fintype X] [DecidableEq X] (T : Marked d X n) : Fintype T.MarkedIndexValue :=
|
||||
Pi.fintype
|
||||
|
||||
lemma sumElimIndexColor_of_marked (T : Marked d X n) :
|
||||
sumElimIndexColor T.unmarkedColor T.markedColor = T.color := by
|
||||
instance [Fintype X] [DecidableEq X] (T : Marked d X n) : DecidableEq T.MarkedIndexValue :=
|
||||
Fintype.decidablePiFintype
|
||||
|
||||
lemma color_eq_elim (T : Marked d X n) :
|
||||
T.color = Sum.elim T.unmarkedColor T.markedColor := by
|
||||
ext1 x
|
||||
cases' x <;> rfl
|
||||
|
||||
def toUnmarkedIndexValue {T : Marked d X n} (i : IndexValue d T.color) : UnmarkedIndexValue T :=
|
||||
inlIndexValue <| castIndexValue T.sumElimIndexColor_of_marked.symm <| i
|
||||
|
||||
def toMarkedIndexValue {T : Marked d X n} (i : IndexValue d T.color) : MarkedIndexValue T :=
|
||||
inrIndexValue <| castIndexValue T.sumElimIndexColor_of_marked.symm <| i
|
||||
|
||||
def splitIndexValue {T : Marked d X n} :
|
||||
IndexValue d T.color ≃ UnmarkedIndexValue T × MarkedIndexValue T where
|
||||
toFun i := ⟨toUnmarkedIndexValue i, toMarkedIndexValue i⟩
|
||||
invFun p := castIndexValue T.sumElimIndexColor_of_marked $
|
||||
sumElimIndexValue p.1 p.2
|
||||
left_inv i := by
|
||||
simp_all only [IndexValue]
|
||||
ext1 x
|
||||
cases x with
|
||||
| inl _ => rfl
|
||||
| inr _ => rfl
|
||||
right_inv p := by
|
||||
simp_all only [IndexValue]
|
||||
obtain ⟨fst, snd⟩ := p
|
||||
simp_all only [Prod.mk.injEq]
|
||||
apply And.intro rfl rfl
|
||||
IndexValue d T.color ≃ T.UnmarkedIndexValue × T.MarkedIndexValue :=
|
||||
(indexValueIso d (Equiv.refl _) T.color_eq_elim).trans
|
||||
indexValueSumEquiv
|
||||
|
||||
@[simp]
|
||||
lemma splitIndexValue_sum {T : Marked d X n} [Fintype X] [DecidableEq X]
|
||||
|
|
@ -406,7 +373,6 @@ def twoMarkedIndexValue (T : Marked d X 2) (x : ColorsIndex d (T.color (markedPo
|
|||
| 0 => x
|
||||
| 1 => y
|
||||
|
||||
|
||||
/-- An equivalence of types used to turn the first marked index into an unmarked index. -/
|
||||
def unmarkFirstSet (X : Type) (n : ℕ) : (X ⊕ Fin n.succ) ≃
|
||||
(X ⊕ Fin 1) ⊕ Fin n :=
|
||||
|
|
@ -416,62 +382,24 @@ def unmarkFirstSet (X : Type) (n : ℕ) : (X ⊕ Fin n.succ) ≃
|
|||
|
||||
/-- Unmark the first marked index of a marked thensor. -/
|
||||
def unmarkFirst {X : Type} : Marked d X n.succ ≃ Marked d (X ⊕ Fin 1) n :=
|
||||
congrSet (unmarkFirstSet X n)
|
||||
mapIso d (unmarkFirstSet X n)
|
||||
|
||||
end Marked
|
||||
|
||||
/-!
|
||||
|
||||
## Multiplication
|
||||
|
||||
-/
|
||||
open Marked
|
||||
|
||||
/-- The contraction of the marked indices of two tensors each with one marked index, which
|
||||
is dual to the others. The contraction is done via
|
||||
`φ^μ ψ_μ = φ^0 ψ_0 + φ^1 ψ_1 + ...`. -/
|
||||
@[simps!]
|
||||
def mul {X Y : Type} (T : Marked d X 1) (S : Marked d Y 1)
|
||||
(h : T.markedColor 0 = τ (S.markedColor 0)) :
|
||||
RealLorentzTensor d (X ⊕ Y) where
|
||||
color := sumElimIndexColor T.unmarkedColor S.unmarkedColor
|
||||
coord := fun i => ∑ x,
|
||||
T.coord (splitIndexValue.symm (inlIndexValue i, oneMarkedIndexValue x)) *
|
||||
S.coord (splitIndexValue.symm (inrIndexValue i, oneMarkedIndexValue $ congrColorsDual h x))
|
||||
|
||||
/-- Multiplication is well behaved with regard to swapping tensors. -/
|
||||
lemma sumComm_mul {X Y : Type} (T : Marked d X 1) (S : Marked d Y 1)
|
||||
(h : T.markedColor 0 = τ (S.markedColor 0)) :
|
||||
sumComm (mul T S h) = mul S T (color_eq_dual_symm h) := by
|
||||
refine ext' (sumElimIndexColor_symm S.unmarkedColor T.unmarkedColor).symm ?_
|
||||
change (mul T S h).coord ∘
|
||||
(congrSetIndexValue d (Equiv.sumComm X Y) (mul T S h).color).symm = _
|
||||
rw [Equiv.comp_symm_eq]
|
||||
funext i
|
||||
simp only [mul_coord, IndexValue, mul_color, Function.comp_apply, sumComm_apply_color]
|
||||
erw [sumCommIndexValue_inlIndexValue, sumCommIndexValue_inrIndexValue,
|
||||
← Equiv.sum_comp (congrColorsDual h)]
|
||||
refine Fintype.sum_congr _ _ (fun a => ?_)
|
||||
rw [mul_comm]
|
||||
repeat apply congrArg
|
||||
rw [← congrColorsDual_symm h]
|
||||
exact (Equiv.apply_eq_iff_eq_symm_apply <| congrColorsDual h).mp rfl
|
||||
|
||||
/-! TODO: Following the ethos of modular operads, prove properties of multiplication. -/
|
||||
|
||||
/-! TODO: Use `mul` to generalize to any pair of marked index. -/
|
||||
/-!
|
||||
|
||||
## Contraction of indices
|
||||
|
||||
-/
|
||||
|
||||
open Marked
|
||||
|
||||
/-- The contraction of the marked indices in a tensor with two marked indices. -/
|
||||
def contr {X : Type} (T : Marked d X 2) (h : T.markedColor 0 = τ (T.markedColor 1)) :
|
||||
RealLorentzTensor d X where
|
||||
color := T.unmarkedColor
|
||||
coord := fun i =>
|
||||
∑ x, T.coord (splitIndexValue.symm (i, T.twoMarkedIndexValue x $ congrColorsDual h x))
|
||||
∑ x, T.coord (splitIndexValue.symm (i, T.twoMarkedIndexValue x $ colorsIndexDualCast h x))
|
||||
|
||||
/-! TODO: Following the ethos of modular operads, prove properties of contraction. -/
|
||||
|
||||
|
|
|
|||
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Add table
Add a link
Reference in a new issue