Merge pull request #93 from HEPLean/Tensors
feat: Associativity of multiplication of Lorentz tensors
This commit is contained in:
Joseph Tooby-Smith
GitHub
commit
653feb5efa
3 changed files with 547 additions and 44 deletions
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@ -8,6 +8,7 @@ import Mathlib.Data.Real.Basic
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import Mathlib.Data.Fintype.BigOperators
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import Mathlib.Logic.Equiv.Fin
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import Mathlib.Tactic.FinCases
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import Mathlib.Logic.Equiv.Fintype
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/-!
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# Real Lorentz Tensors
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@ -26,7 +27,7 @@ This will relation should be made explicit in the future.
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/-! TODO: Generalize to maps into Lorentz tensors. -/
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/-- The possible `colors` of an index for a RealLorentzTensor.
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There are two possiblities, `up` and `down`. -/
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There are two possibilities, `up` and `down`. -/
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inductive RealLorentzTensor.Colors where
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| up : RealLorentzTensor.Colors
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| down : RealLorentzTensor.Colors
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@ -87,7 +88,7 @@ lemma color_eq_dual_symm {μ ν : Colors} (h : μ = τ ν) : ν = τ μ :=
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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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/-- An equivalence of `ColorsIndex` types given an equality of 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 (congrArg (ColorsIndex d) h)
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@ -165,7 +166,7 @@ lemma indexValueIso_symm (d : ℕ) (f : X ≃ Y) (h : i = j ∘ f) :
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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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simp only [Function.comp_apply, colorsIndexCast, Equiv.cast_symm, Equiv.cast_apply, cast_cast]
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apply cast_eq_iff_heq.mpr
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rw [Equiv.apply_symm_apply]
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@ -280,7 +281,8 @@ lemma mapIso_refl : mapIso d (Equiv.refl X) = Equiv.refl _ := rfl
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## Sums
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-/
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/-- An equivalence splitting elements of `IndexValue d (Sum.elim TX TY)` into two components. -/
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/-- An equivalence that splits elements of `IndexValue d (Sum.elim TX TY)` into two components. -/
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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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@ -365,7 +367,7 @@ lemma splitIndexValue_sum {T : Marked d X n} [Fintype X] [DecidableEq X]
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∑ i, P (splitIndexValue i) = ∑ j, ∑ k, P (j, k) := by
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rw [Equiv.sum_comp splitIndexValue, Fintype.sum_prod_type]
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/-- Contruction of marked index values for the case of 1 marked index. -/
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/-- Construction of marked index values for the case of 1 marked index. -/
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def oneMarkedIndexValue {T : Marked d X 1} :
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ColorsIndex d (T.color (markedPoint X 0)) ≃ T.MarkedIndexValue where
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toFun x := fun i => match i with
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@ -377,7 +379,7 @@ def oneMarkedIndexValue {T : Marked d X 1} :
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fin_cases x
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rfl
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/-- Contruction of marked index values for the case of 2 marked index. -/
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/-- Construction of marked index values for the case of 2 marked index. -/
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def twoMarkedIndexValue (T : Marked d X 2) (x : ColorsIndex d (T.color (markedPoint X 0)))
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(y : ColorsIndex d <| T.color <| markedPoint X 1) :
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T.MarkedIndexValue := fun i =>
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@ -386,16 +388,236 @@ def twoMarkedIndexValue (T : Marked d X 2) (x : ColorsIndex d (T.color (markedPo
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| 1 => y
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/-- An equivalence of types used to turn the first marked index into an unmarked index. -/
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def unmarkFirstSet (X : Type) (n : ℕ) : (X ⊕ Fin n.succ) ≃
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(X ⊕ Fin 1) ⊕ Fin n :=
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trans (Equiv.sumCongr (Equiv.refl _) $
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def unmarkFirstSet (X : Type) (n : ℕ) : (X ⊕ Fin n.succ) ≃ (X ⊕ Fin 1) ⊕ Fin n :=
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trans (Equiv.sumCongr (Equiv.refl _)
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(((Fin.castOrderIso (Nat.succ_eq_one_add n)).toEquiv.trans finSumFinEquiv.symm)))
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(Equiv.sumAssoc _ _ _).symm
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/-- Unmark the first marked index of a marked thensor. -/
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/-- Unmark the first marked index of a marked tensor. -/
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def unmarkFirst {X : Type} : Marked d X n.succ ≃ Marked d (X ⊕ Fin 1) n :=
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mapIso d (unmarkFirstSet X n)
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/-!
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## Marking elements.
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-/
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section markingElements
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variable [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
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/-- Splits a type based on an embedding from `Fin n` into elements not in the image of the
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embedding, and elements in the image. -/
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def markEmbeddingSet {n : ℕ} (f : Fin n ↪ X) :
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X ≃ {x // x ∈ (Finset.image f Finset.univ)ᶜ} ⊕ Fin n :=
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(Equiv.Set.sumCompl (Set.range ⇑f)).symm.trans <|
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(Equiv.sumComm _ _).trans <|
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Equiv.sumCongr ((Equiv.subtypeEquivRight (by simp))) <|
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(Function.Embedding.toEquivRange f).symm
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lemma markEmbeddingSet_on_mem {n : ℕ} (f : Fin n ↪ X) (x : X)
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(hx : x ∈ Finset.image f Finset.univ) : markEmbeddingSet f x =
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Sum.inr (f.toEquivRange.symm ⟨x, by simpa using hx⟩) := by
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rw [markEmbeddingSet]
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simp only [Equiv.trans_apply, Equiv.sumComm_apply, Equiv.sumCongr_apply]
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rw [Equiv.Set.sumCompl_symm_apply_of_mem]
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rfl
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lemma markEmbeddingSet_on_not_mem {n : ℕ} (f : Fin n ↪ X) (x : X)
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(hx : ¬ x ∈ (Finset.image f Finset.univ)) : markEmbeddingSet f x =
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Sum.inl ⟨x, by simpa using hx⟩ := by
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rw [markEmbeddingSet]
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simp only [Equiv.trans_apply, Equiv.sumComm_apply, Equiv.sumCongr_apply]
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rw [Equiv.Set.sumCompl_symm_apply_of_not_mem]
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rfl
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simpa using hx
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/-- Marks the indices of tensor in the image of an embedding. -/
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@[simps!]
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def markEmbedding {n : ℕ} (f : Fin n ↪ X) :
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RealLorentzTensor d X ≃ Marked d {x // x ∈ (Finset.image f Finset.univ)ᶜ} n :=
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mapIso d (markEmbeddingSet f)
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lemma markEmbeddingSet_on_mem_indexValue_apply {n : ℕ} (f : Fin n ↪ X) (T : RealLorentzTensor d X)
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(i : IndexValue d (markEmbedding f T).color) (x : X) (hx : x ∈ (Finset.image f Finset.univ)) :
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i (markEmbeddingSet f x) = colorsIndexCast (congrArg ((markEmbedding f) T).color
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(markEmbeddingSet_on_mem f x hx).symm)
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(i (Sum.inr (f.toEquivRange.symm ⟨x, by simpa using hx⟩))) := by
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simp [colorsIndexCast]
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symm
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apply cast_eq_iff_heq.mpr
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rw [markEmbeddingSet_on_mem f x hx]
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lemma markEmbeddingSet_on_not_mem_indexValue_apply {n : ℕ}
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(f : Fin n ↪ X) (T : RealLorentzTensor d X) (i : IndexValue d (markEmbedding f T).color)
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(x : X) (hx : ¬ x ∈ (Finset.image f Finset.univ)) :
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i (markEmbeddingSet f x) = colorsIndexCast (congrArg ((markEmbedding f) T).color
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(markEmbeddingSet_on_not_mem f x hx).symm) (i (Sum.inl ⟨x, by simpa using hx⟩)) := by
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simp [colorsIndexCast]
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symm
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apply cast_eq_iff_heq.mpr
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rw [markEmbeddingSet_on_not_mem f x hx]
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/-- An equivalence between the IndexValues for a tensor `T` and the corresponding
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tensor with indices maked by an embedding. -/
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@[simps!]
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def markEmbeddingIndexValue {n : ℕ} (f : Fin n ↪ X) (T : RealLorentzTensor d X) :
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IndexValue d T.color ≃ IndexValue d (markEmbedding f T).color :=
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indexValueIso d (markEmbeddingSet f) (
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(Equiv.comp_symm_eq (markEmbeddingSet f) ((markEmbedding f) T).color T.color).mp rfl)
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lemma markEmbeddingIndexValue_apply_symm_on_mem {n : ℕ}
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(f : Fin n.succ ↪ X) (T : RealLorentzTensor d X) (i : IndexValue d (markEmbedding f T).color)
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(x : X) (hx : x ∈ (Finset.image f Finset.univ)) :
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(markEmbeddingIndexValue f T).symm i x = (colorsIndexCast ((congrFun ((Equiv.comp_symm_eq
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(markEmbeddingSet f) ((markEmbedding f) T).color T.color).mp rfl) x).trans
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(congrArg ((markEmbedding f) T).color (markEmbeddingSet_on_mem f x hx)))).symm
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(i (Sum.inr (f.toEquivRange.symm ⟨x, by simpa using hx⟩))) := by
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rw [markEmbeddingIndexValue, indexValueIso_symm_apply']
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rw [markEmbeddingSet_on_mem_indexValue_apply f T i x hx]
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simp [colorsIndexCast]
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lemma markEmbeddingIndexValue_apply_symm_on_not_mem {n : ℕ} (f : Fin n.succ ↪ X)
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(T : RealLorentzTensor d X) (i : IndexValue d (markEmbedding f T).color) (x : X)
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(hx : ¬ x ∈ (Finset.image f Finset.univ)) : (markEmbeddingIndexValue f T).symm i x =
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(colorsIndexCast ((congrFun ((Equiv.comp_symm_eq
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(markEmbeddingSet f) ((markEmbedding f) T).color T.color).mp rfl) x).trans
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((congrArg ((markEmbedding f) T).color (markEmbeddingSet_on_not_mem f x hx))))).symm
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(i (Sum.inl ⟨x, by simpa using hx⟩)) := by
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rw [markEmbeddingIndexValue, indexValueIso_symm_apply']
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rw [markEmbeddingSet_on_not_mem_indexValue_apply f T i x hx]
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simp only [Nat.succ_eq_add_one, Function.comp_apply, markEmbedding_apply_color, colorsIndexCast,
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Equiv.cast_symm, id_eq, eq_mp_eq_cast, eq_mpr_eq_cast, Equiv.cast_apply, cast_cast, cast_eq,
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Equiv.cast_refl, Equiv.refl_symm]
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rfl
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/-- Given an equivalence of types, an embedding `f` to an embedding `g`, the equivalence
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taking the complement of the image of `f` to the complement of the image of `g`. -/
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@[simps!]
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def equivEmbedCompl (e : X ≃ Y) {f : Fin n ↪ X} {g : Fin n ↪ Y} (he : f.trans e = g) :
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{x // x ∈ (Finset.image f Finset.univ)ᶜ} ≃ {y // y ∈ (Finset.image g Finset.univ)ᶜ} :=
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(Equiv.subtypeEquivOfSubtype' e).trans <|
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(Equiv.subtypeEquivRight (fun x => by simp [← he, Equiv.eq_symm_apply]))
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lemma markEmbedding_mapIso_right (e : X ≃ Y) (f : Fin n ↪ X) (g : Fin n ↪ Y) (he : f.trans e = g)
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(T : RealLorentzTensor d X) : markEmbedding g (mapIso d e T) =
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mapIso d (Equiv.sumCongr (equivEmbedCompl e he) (Equiv.refl (Fin n))) (markEmbedding f T) := by
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rw [markEmbedding, markEmbedding]
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erw [← Equiv.trans_apply, ← Equiv.trans_apply]
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rw [mapIso_trans, mapIso_trans]
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apply congrFun
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repeat apply congrArg
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refine Equiv.ext (fun x => ?_)
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simp only [Equiv.trans_apply, Equiv.sumCongr_apply, Equiv.coe_refl]
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by_cases hx : x ∈ Finset.image f Finset.univ
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· rw [markEmbeddingSet_on_mem f x hx, markEmbeddingSet_on_mem g (e x) (by simpa [← he] using hx)]
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subst he
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simp only [Sum.map_inr, id_eq, Sum.inr.injEq, Equiv.symm_apply_eq,
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Function.Embedding.toEquivRange_apply, Function.Embedding.trans_apply, Equiv.coe_toEmbedding,
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Subtype.mk.injEq, EmbeddingLike.apply_eq_iff_eq]
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change x = f.toEquivRange _
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rw [Equiv.apply_symm_apply]
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· rw [markEmbeddingSet_on_not_mem f x hx,
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markEmbeddingSet_on_not_mem g (e x) (by simpa [← he] using hx)]
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subst he
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rfl
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lemma markEmbedding_mapIso_left {n m : ℕ} (e : Fin n ≃ Fin m) (f : Fin n ↪ X) (g : Fin m ↪ X)
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(he : e.symm.toEmbedding.trans f = g) (T : RealLorentzTensor d X) :
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markEmbedding g T = mapIso d (Equiv.sumCongr (Equiv.subtypeEquivRight (fun x => by
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simpa [← he] using Equiv.forall_congr_left e)) e) (markEmbedding f T) := by
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rw [markEmbedding, markEmbedding]
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erw [← Equiv.trans_apply]
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rw [mapIso_trans]
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apply congrFun
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repeat apply congrArg
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refine Equiv.ext (fun x => ?_)
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simp only [Equiv.trans_apply, Equiv.sumCongr_apply]
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by_cases hx : x ∈ Finset.image f Finset.univ
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· rw [markEmbeddingSet_on_mem f x hx, markEmbeddingSet_on_mem g x (by
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simp [← he, hx]
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obtain ⟨y, _, hy2⟩ := Finset.mem_image.mp hx
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use e y
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simpa using hy2)]
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subst he
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simp [Equiv.symm_apply_eq]
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change x = f.toEquivRange _
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rw [Equiv.apply_symm_apply]
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· rw [markEmbeddingSet_on_not_mem f x hx, markEmbeddingSet_on_not_mem g x (by
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simpa [← he, hx] using fun x => not_exists.mp (Finset.mem_image.mpr.mt hx) (e.symm x))]
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subst he
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rfl
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/-!
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## Marking a single element
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-/
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/-- An embedding from `Fin 1` into `X` given an element `x ∈ X`. -/
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@[simps!]
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def embedSingleton (x : X) : Fin 1 ↪ X :=
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⟨![x], fun x y => by fin_cases x; fin_cases y; simp⟩
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lemma embedSingleton_toEquivRange_symm (x : X) :
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(embedSingleton x).toEquivRange.symm ⟨x, by simp⟩ = 0 := by
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exact Fin.fin_one_eq_zero _
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/-- Equivalence, taking a tensor to a tensor with a single marked index. -/
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@[simps!]
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def markSingle (x : X) : RealLorentzTensor d X ≃ Marked d {x' // x' ≠ x} 1 :=
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(markEmbedding (embedSingleton x)).trans
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(mapIso d (Equiv.sumCongr (Equiv.subtypeEquivRight (fun x => by simp)) (Equiv.refl _)))
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/-- Equivalence between index values of a tensor and the corresponding tensor with a single
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marked index. -/
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@[simps!]
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def markSingleIndexValue (T : RealLorentzTensor d X) (x : X) :
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IndexValue d T.color ≃ IndexValue d (markSingle x T).color :=
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(markEmbeddingIndexValue (embedSingleton x) T).trans <|
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indexValueIso d (Equiv.sumCongr (Equiv.subtypeEquivRight (fun x => by simp)) (Equiv.refl _))
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(by funext x_1; simp)
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/-- Given an equivalence of types, taking `x` to `y` the corresponding equivalence of
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subtypes of elements not equal to `x` and not equal to `y` respectively. -/
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@[simps!]
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def equivSingleCompl (e : X ≃ Y) {x : X} {y : Y} (he : e x = y) :
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{x' // x' ≠ x} ≃ {y' // y' ≠ y} :=
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(Equiv.subtypeEquivOfSubtype' e).trans <|
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Equiv.subtypeEquivRight (fun a => by simp [Equiv.symm_apply_eq, he])
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lemma markSingle_mapIso (e : X ≃ Y) (x : X) (y : Y) (he : e x = y)
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(T : RealLorentzTensor d X) : markSingle y (mapIso d e T) =
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mapIso d (Equiv.sumCongr (equivSingleCompl e he) (Equiv.refl _)) (markSingle x T) := by
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rw [markSingle, Equiv.trans_apply]
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rw [markEmbedding_mapIso_right e (embedSingleton x) (embedSingleton y)
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(Function.Embedding.ext_iff.mp (fun a => by simpa using he)), markSingle, markEmbedding]
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erw [← Equiv.trans_apply, ← Equiv.trans_apply, ← Equiv.trans_apply]
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rw [mapIso_trans, mapIso_trans, mapIso_trans, mapIso_trans]
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apply congrFun
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repeat apply congrArg
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refine Equiv.ext fun x => ?_
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simp only [ne_eq, Fintype.univ_ofSubsingleton, Fin.zero_eta, Fin.isValue, Equiv.sumCongr_trans,
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Equiv.trans_refl, Equiv.trans_apply, Equiv.sumCongr_apply, Equiv.coe_trans, Equiv.coe_refl,
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Sum.map_map, CompTriple.comp_eq]
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subst he
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rfl
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/-!
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## Marking two elements
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-/
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/-- An embedding from `Fin 2` given two inequivalent elements. -/
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@[simps!]
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def embedDoubleton (x y : X) (h : x ≠ y) : Fin 2 ↪ X :=
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⟨![x, y], fun a b => by
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fin_cases a <;> fin_cases b <;> simp [h]
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exact h.symm⟩
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end markingElements
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end Marked
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/-!
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@ -18,6 +18,42 @@ We will derive properties of these constructors.
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-/
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namespace RealLorentzTensor
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/-!
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# Tensors from and to the reals
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An important point that we shall see here is that there is a well defined map
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to the real numbers, i.e. which is invariant under transformations of mapIso.
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-/
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/-- An equivalence from Real tensors on an empty set to `ℝ`. -/
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@[simps!]
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def toReal (d : ℕ) {X : Type} (h : IsEmpty X) : RealLorentzTensor d X ≃ ℝ where
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toFun := fun T => T.coord (IsEmpty.elim h)
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invFun := fun r => {
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color := fun x => IsEmpty.elim h x,
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coord := fun _ => r}
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left_inv T := by
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refine ext ?_ ?_
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funext x
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exact IsEmpty.elim h x
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funext a
|
||||
change T.coord (IsEmpty.elim h) = _
|
||||
apply congrArg
|
||||
funext x
|
||||
exact IsEmpty.elim h x
|
||||
right_inv x := rfl
|
||||
|
||||
/-- The real number obtained from a tensor is invariant under `mapIso`. -/
|
||||
lemma toReal_mapIso (d : ℕ) {X Y : Type} (h : IsEmpty X) (f : X ≃ Y) :
|
||||
(mapIso d f).trans (toReal d (Equiv.isEmpty f.symm)) = toReal d h := by
|
||||
apply Equiv.ext
|
||||
intro x
|
||||
simp only [Equiv.trans_apply, toReal_apply, mapIso_apply_color, mapIso_apply_coord]
|
||||
apply congrArg
|
||||
funext x
|
||||
exact IsEmpty.elim h x
|
||||
|
||||
/-!
|
||||
|
||||
|
|
@ -28,11 +64,6 @@ action of the Lorentz group. They are provided for constructive purposes.
|
|||
|
||||
-/
|
||||
|
||||
/-- A 0-tensor from a real number. -/
|
||||
def ofReal (d : ℕ) (r : ℝ) : RealLorentzTensor d Empty where
|
||||
color := fun _ => Colors.up
|
||||
coord := fun _ => r
|
||||
|
||||
/-- A marked 1-tensor with a single up index constructed from a vector.
|
||||
|
||||
Note: This is not the same as rising indices on `ofVecDown`. -/
|
||||
|
|
@ -186,14 +217,14 @@ open Marked
|
|||
|
||||
/-- Contracting the indices of `ofMatUpDown` returns the trace of the matrix. -/
|
||||
lemma contr_ofMatUpDown_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
|
||||
contr (ofMatUpDown M) (by rfl) = ofReal d M.trace := by
|
||||
contr (ofMatUpDown M) (by rfl) = (toReal d instIsEmptyEmpty).symm M.trace := by
|
||||
refine ext ?_ rfl
|
||||
· funext i
|
||||
exact Empty.elim i
|
||||
|
||||
/-- Contracting the indices of `ofMatDownUp` returns the trace of the matrix. -/
|
||||
lemma contr_ofMatDownUp_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
|
||||
contr (ofMatDownUp M) (by rfl) = ofReal d M.trace := by
|
||||
contr (ofMatDownUp M) (by rfl) = (toReal d instIsEmptyEmpty).symm M.trace := by
|
||||
refine ext ?_ rfl
|
||||
· funext i
|
||||
exact Empty.elim i
|
||||
|
|
@ -207,20 +238,18 @@ lemma contr_ofMatDownUp_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1
|
|||
/-- Multiplying `ofVecUp` with `ofVecDown` gives the dot product. -/
|
||||
@[simp]
|
||||
lemma mul_ofVecUp_ofVecDown_eq_dot_prod {d : ℕ} (v₁ v₂ : Fin 1 ⊕ Fin d → ℝ) :
|
||||
mapIso d (@Equiv.equivEmpty (Empty ⊕ Empty) instIsEmptySum)
|
||||
(mul (ofVecUp v₁) (ofVecDown v₂) (by rfl)) = ofReal d (v₁ ⬝ᵥ v₂) := by
|
||||
mul (ofVecUp v₁) (ofVecDown v₂) rfl = (toReal d instIsEmptySum).symm (v₁ ⬝ᵥ v₂) := by
|
||||
refine ext ?_ rfl
|
||||
· funext i
|
||||
exact Empty.elim i
|
||||
exact IsEmpty.elim instIsEmptySum i
|
||||
|
||||
/-- Multiplying `ofVecDown` with `ofVecUp` gives the dot product. -/
|
||||
@[simp]
|
||||
lemma mul_ofVecDown_ofVecUp_eq_dot_prod {d : ℕ} (v₁ v₂ : Fin 1 ⊕ Fin d → ℝ) :
|
||||
mapIso d (Equiv.equivEmpty (Empty ⊕ Empty))
|
||||
(mul (ofVecDown v₁) (ofVecUp v₂) rfl) = ofReal d (v₁ ⬝ᵥ v₂) := by
|
||||
mul (ofVecDown v₁) (ofVecUp v₂) rfl = (toReal d instIsEmptySum).symm (v₁ ⬝ᵥ v₂) := by
|
||||
refine ext ?_ rfl
|
||||
· funext i
|
||||
exact Empty.elim i
|
||||
exact IsEmpty.elim instIsEmptySum i
|
||||
|
||||
lemma mul_ofMatUpDown_ofVecUp_eq_mulVec {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
|
||||
(v : Fin 1 ⊕ Fin d → ℝ) :
|
||||
|
|
@ -255,8 +284,9 @@ open Matrix
|
|||
|
||||
/-- The action of the Lorentz group on `ofReal v` is trivial. -/
|
||||
@[simp]
|
||||
lemma lorentzAction_ofReal (r : ℝ) : Λ • ofReal d r = ofReal d r :=
|
||||
lorentzAction_on_isEmpty Λ (ofReal d r)
|
||||
lemma lorentzAction_toReal (h : IsEmpty X) (r : ℝ) :
|
||||
Λ • (toReal d h).symm r = (toReal d h).symm r :=
|
||||
lorentzAction_on_isEmpty Λ ((toReal d h).symm r)
|
||||
|
||||
/-- The action of the Lorentz group on `ofVecUp v` is by vector multiplication. -/
|
||||
@[simp]
|
||||
|
|
|
|||
|
|
@ -15,7 +15,6 @@ marked index. This is equivalent to contracting two Lorentz tensors.
|
|||
We prove various results about this multiplication.
|
||||
|
||||
-/
|
||||
/-! TODO: Generalize to contracting any marked index of a marked tensor. -/
|
||||
/-! TODO: Set up a good notation for the multiplication. -/
|
||||
|
||||
namespace RealLorentzTensor
|
||||
|
|
@ -38,6 +37,31 @@ def mul {X Y : Type} (T : Marked d X 1) (S : Marked d Y 1)
|
|||
S.coord (splitIndexValue.symm ((indexValueSumEquiv i).2,
|
||||
oneMarkedIndexValue $ colorsIndexDualCast h x))
|
||||
|
||||
/-- The index value appearing in the multiplication of Marked tensors on the left. -/
|
||||
def mulFstArg {X Y : Type} {T : Marked d X 1} {S : Marked d Y 1}
|
||||
(i : IndexValue d (Sum.elim T.unmarkedColor S.unmarkedColor))
|
||||
(x : ColorsIndex d (T.color (markedPoint X 0))) : IndexValue d T.color :=
|
||||
splitIndexValue.symm ((indexValueSumEquiv i).1, oneMarkedIndexValue x)
|
||||
|
||||
lemma mulFstArg_inr {X Y : Type} {T : Marked d X 1} {S : Marked d Y 1}
|
||||
(i : IndexValue d (Sum.elim T.unmarkedColor S.unmarkedColor))
|
||||
(x : ColorsIndex d (T.color (markedPoint X 0))) :
|
||||
mulFstArg i x (Sum.inr 0) = x := by
|
||||
rfl
|
||||
|
||||
lemma mulFstArg_inl {X Y : Type} {T : Marked d X 1} {S : Marked d Y 1}
|
||||
(i : IndexValue d (Sum.elim T.unmarkedColor S.unmarkedColor))
|
||||
(x : ColorsIndex d (T.color (markedPoint X 0))) (c : X):
|
||||
mulFstArg i x (Sum.inl c) = i (Sum.inl c) := by
|
||||
rfl
|
||||
|
||||
/-- The index value appearing in the multiplication of Marked tensors on the right. -/
|
||||
def mulSndArg {X Y : Type} {T : Marked d X 1} {S : Marked d Y 1}
|
||||
(i : IndexValue d (Sum.elim T.unmarkedColor S.unmarkedColor))
|
||||
(x : ColorsIndex d (T.color (markedPoint X 0))) (h : T.markedColor 0 = τ (S.markedColor 0)) :
|
||||
IndexValue d S.color :=
|
||||
splitIndexValue.symm ((indexValueSumEquiv i).2, oneMarkedIndexValue $ colorsIndexDualCast h x)
|
||||
|
||||
/-!
|
||||
|
||||
## Expressions for multiplication
|
||||
|
|
@ -117,7 +141,6 @@ lemma mul_mapIso {X Y Z : Type} (T : Marked d X 1) (S : Marked d Y 1) (f : X ≃
|
|||
rw [mapIso_apply_coord, mapIso_apply_coord]
|
||||
refine Mathlib.Tactic.Ring.mul_congr ?_ ?_ rfl
|
||||
· apply congrArg
|
||||
simp only [IndexValue]
|
||||
exact (Equiv.symm_apply_eq splitIndexValue).mpr rfl
|
||||
· apply congrArg
|
||||
exact (Equiv.symm_apply_eq splitIndexValue).mpr rfl
|
||||
|
|
@ -188,40 +211,38 @@ lemma mul_unmarkedLorentzAction (T : Marked d X 1) (S : Marked d Y 1)
|
|||
T.coord (splitIndexValue.symm (j, oneMarkedIndexValue x)))*
|
||||
toTensorRepMat Λ (indexValueSumEquiv i).2 k *
|
||||
S.coord (splitIndexValue.symm (k, oneMarkedIndexValue $ colorsIndexDualCast h x))
|
||||
apply Finset.sum_congr rfl (fun x _ => ?_)
|
||||
rw [Finset.sum_mul_sum]
|
||||
apply Finset.sum_congr rfl (fun j _ => ?_)
|
||||
apply Finset.sum_congr rfl (fun k _ => ?_)
|
||||
ring
|
||||
· apply Finset.sum_congr rfl (fun x _ => ?_)
|
||||
rw [Finset.sum_mul_sum ]
|
||||
apply Finset.sum_congr rfl (fun j _ => ?_)
|
||||
apply Finset.sum_congr rfl (fun k _ => ?_)
|
||||
ring
|
||||
rw [Finset.sum_comm]
|
||||
trans ∑ j, ∑ k, ∑ x, (toTensorRepMat Λ (indexValueSumEquiv i).1 j *
|
||||
T.coord (splitIndexValue.symm (j, oneMarkedIndexValue x)))*
|
||||
toTensorRepMat Λ (indexValueSumEquiv i).2 k *
|
||||
S.coord (splitIndexValue.symm (k, oneMarkedIndexValue $ colorsIndexDualCast h x))
|
||||
apply Finset.sum_congr rfl (fun j _ => ?_)
|
||||
rw [Finset.sum_comm]
|
||||
· exact Finset.sum_congr rfl (fun j _ => Finset.sum_comm)
|
||||
trans ∑ j, ∑ k,
|
||||
((toTensorRepMat Λ (indexValueSumEquiv i).1 j) * toTensorRepMat Λ (indexValueSumEquiv i).2 k)
|
||||
* ∑ x, (T.coord (splitIndexValue.symm (j, oneMarkedIndexValue x)))
|
||||
* S.coord (splitIndexValue.symm (k, oneMarkedIndexValue $ colorsIndexDualCast h x))
|
||||
apply Finset.sum_congr rfl (fun j _ => Finset.sum_congr rfl (fun k _ => ?_))
|
||||
rw [Finset.mul_sum]
|
||||
apply Finset.sum_congr rfl (fun x _ => ?_)
|
||||
ring
|
||||
· apply Finset.sum_congr rfl (fun j _ => Finset.sum_congr rfl (fun k _ => ?_))
|
||||
rw [Finset.mul_sum]
|
||||
apply Finset.sum_congr rfl (fun x _ => ?_)
|
||||
ring
|
||||
trans ∑ j, ∑ k, toTensorRepMat Λ i (indexValueSumEquiv.symm (j, k)) *
|
||||
∑ x, (T.coord (splitIndexValue.symm (j, oneMarkedIndexValue x)))
|
||||
* S.coord (splitIndexValue.symm (k, oneMarkedIndexValue $ colorsIndexDualCast h x))
|
||||
apply Finset.sum_congr rfl (fun j _ => Finset.sum_congr rfl (fun k _ => ?_))
|
||||
rw [toTensorRepMat_of_indexValueSumEquiv']
|
||||
congr
|
||||
simp only [IndexValue, Finset.mem_univ, Prod.mk.eta, Equiv.symm_apply_apply, mul_color]
|
||||
· rw [toTensorRepMat_of_indexValueSumEquiv']
|
||||
congr
|
||||
simp only [IndexValue, Finset.mem_univ, Prod.mk.eta, Equiv.symm_apply_apply, mul_color]
|
||||
trans ∑ p, toTensorRepMat Λ i p *
|
||||
∑ x, (T.coord (splitIndexValue.symm ((indexValueSumEquiv p).1, oneMarkedIndexValue x)))
|
||||
* S.coord (splitIndexValue.symm ((indexValueSumEquiv p).2,
|
||||
oneMarkedIndexValue $ colorsIndexDualCast h x))
|
||||
erw [← Equiv.sum_comp indexValueSumEquiv.symm]
|
||||
rw [Fintype.sum_prod_type]
|
||||
rfl
|
||||
· erw [← Equiv.sum_comp indexValueSumEquiv.symm]
|
||||
exact Eq.symm Fintype.sum_prod_type
|
||||
rfl
|
||||
|
||||
/-- The Lorentz action commutes with multiplication. -/
|
||||
|
|
@ -231,4 +252,234 @@ lemma mul_lorentzAction (T : Marked d X 1) (S : Marked d Y 1)
|
|||
simp only [← marked_unmarked_action_eq_action]
|
||||
rw [mul_markedLorentzAction, mul_unmarkedLorentzAction]
|
||||
|
||||
/-!
|
||||
|
||||
## Multiplication on selected indices
|
||||
|
||||
-/
|
||||
|
||||
variable {n m : ℕ} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
|
||||
{X' Y' Z : Type} [Fintype X'] [DecidableEq X'] [Fintype Y'] [DecidableEq Y']
|
||||
[Fintype Z] [DecidableEq Z]
|
||||
|
||||
/-- The multiplication of two real Lorentz Tensors along specified indices. -/
|
||||
@[simps!]
|
||||
def mulS (T : RealLorentzTensor d X) (S : RealLorentzTensor d Y) (x : X) (y : Y)
|
||||
(h : T.color x = τ (S.color y)) : RealLorentzTensor d ({x' // x' ≠ x} ⊕ {y' // y' ≠ y}) :=
|
||||
mul (markSingle x T) (markSingle y S) h
|
||||
|
||||
/-- The first index value appearing in the multiplication of two Lorentz tensors. -/
|
||||
def mulSFstArg {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
|
||||
(i : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor))
|
||||
(a : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0))) :
|
||||
IndexValue d T.color := (markSingleIndexValue T x).symm (mulFstArg i a)
|
||||
|
||||
lemma mulSFstArg_ext {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
|
||||
{i j : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor)}
|
||||
{a b : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0))}
|
||||
(hij : i = j) (hab : a = b) : mulSFstArg i a = mulSFstArg j b := by
|
||||
subst hij; subst hab
|
||||
rfl
|
||||
|
||||
lemma mulSFstArg_on_mem {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
|
||||
(i : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor))
|
||||
(a : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0))) :
|
||||
mulSFstArg i a x = a := by
|
||||
rw [mulSFstArg, markSingleIndexValue]
|
||||
simp only [ne_eq, Fintype.univ_ofSubsingleton, Fin.zero_eta, Fin.isValue, Equiv.symm_trans_apply,
|
||||
Sum.map_inr, id_eq]
|
||||
erw [markEmbeddingIndexValue_apply_symm_on_mem]
|
||||
swap
|
||||
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Finset.univ_unique, Fin.default_eq_zero,
|
||||
Fin.isValue, Finset.image_singleton, embedSingleton_apply, Finset.mem_singleton]
|
||||
rw [indexValueIso_symm_apply']
|
||||
erw [Equiv.symm_apply_eq, Equiv.symm_apply_eq]
|
||||
simp only [Function.comp_apply, colorsIndexCast, Equiv.cast_symm, Equiv.cast_apply, cast_cast]
|
||||
symm
|
||||
apply cast_eq_iff_heq.mpr
|
||||
rw [embedSingleton_toEquivRange_symm]
|
||||
rfl
|
||||
|
||||
lemma mulSFstArg_on_not_mem {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
|
||||
(i : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor))
|
||||
(a : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0)))
|
||||
(c : X) (hc : c ≠ x) : mulSFstArg i a c = i (Sum.inl ⟨c, hc⟩) := by
|
||||
rw [mulSFstArg, markSingleIndexValue]
|
||||
simp only [ne_eq, Fintype.univ_ofSubsingleton, Fin.zero_eta, Fin.isValue, Equiv.symm_trans_apply,
|
||||
Sum.map_inr, id_eq]
|
||||
erw [markEmbeddingIndexValue_apply_symm_on_not_mem]
|
||||
swap
|
||||
simpa using hc
|
||||
rfl
|
||||
|
||||
/-- The second index value appearing in the multiplication of two Lorentz tensors. -/
|
||||
def mulSSndArg {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
|
||||
(i : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor))
|
||||
(a : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0)))
|
||||
(h : T.color x = τ (S.color y)) : IndexValue d S.color :=
|
||||
(markSingleIndexValue S y).symm (mulSndArg i a h)
|
||||
|
||||
lemma mulSSndArg_on_mem {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
|
||||
(i : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor))
|
||||
(a : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0)))
|
||||
(h : T.color x = τ (S.color y)) : mulSSndArg i a h y = colorsIndexDualCast h a := by
|
||||
rw [mulSSndArg, markSingleIndexValue]
|
||||
simp only [ne_eq, Fintype.univ_ofSubsingleton, Fin.zero_eta, Fin.isValue, Equiv.symm_trans_apply,
|
||||
Sum.map_inr, id_eq]
|
||||
erw [markEmbeddingIndexValue_apply_symm_on_mem]
|
||||
swap
|
||||
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Finset.univ_unique, Fin.default_eq_zero,
|
||||
Fin.isValue, Finset.image_singleton, embedSingleton_apply, Finset.mem_singleton]
|
||||
rw [indexValueIso_symm_apply']
|
||||
erw [Equiv.symm_apply_eq, Equiv.symm_apply_eq]
|
||||
simp only [Function.comp_apply, colorsIndexCast, Equiv.cast_symm, Equiv.cast_apply, cast_cast]
|
||||
symm
|
||||
apply cast_eq_iff_heq.mpr
|
||||
rw [embedSingleton_toEquivRange_symm]
|
||||
rfl
|
||||
|
||||
lemma mulSSndArg_on_not_mem {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
|
||||
(i : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor))
|
||||
(a : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0)))
|
||||
(h : T.color x = τ (S.color y)) (c : Y) (hc : c ≠ y) :
|
||||
mulSSndArg i a h c = i (Sum.inr ⟨c, hc⟩) := by
|
||||
rw [mulSSndArg, markSingleIndexValue]
|
||||
simp only [ne_eq, Fintype.univ_ofSubsingleton, Fin.zero_eta, Fin.isValue, Equiv.symm_trans_apply,
|
||||
Sum.map_inr, id_eq]
|
||||
erw [markEmbeddingIndexValue_apply_symm_on_not_mem]
|
||||
swap
|
||||
simpa using hc
|
||||
rfl
|
||||
|
||||
lemma mulSSndArg_ext {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
|
||||
{i j : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor)}
|
||||
{a b : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0))}
|
||||
(h : T.color x = τ (S.color y)) (hij : i = j) (hab : a = b) :
|
||||
mulSSndArg i a h = mulSSndArg j b h := by
|
||||
subst hij
|
||||
subst hab
|
||||
rfl
|
||||
|
||||
lemma mulS_coord_arg (T : RealLorentzTensor d X) (S : RealLorentzTensor d Y) (x : X) (y : Y)
|
||||
(h : T.color x = τ (S.color y))
|
||||
(i : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor)) :
|
||||
(mulS T S x y h).coord i = ∑ a, T.coord (mulSFstArg i a) * S.coord (mulSSndArg i a h) := by
|
||||
rfl
|
||||
|
||||
lemma mulS_mapIso (T : RealLorentzTensor d X) (S : RealLorentzTensor d Y)
|
||||
(eX : X ≃ X') (eY : Y ≃ Y') (x : X) (y : Y) (x' : X') (y' : Y') (hx : eX x = x')
|
||||
(hy : eY y = y') (h : T.color x = τ (S.color y)) :
|
||||
mulS (mapIso d eX T) (mapIso d eY S) x' y' (by subst hx hy; simpa using h) =
|
||||
mapIso d (Equiv.sumCongr (equivSingleCompl eX hx) (equivSingleCompl eY hy))
|
||||
(mulS T S x y h) := by
|
||||
rw [mulS, mulS, mul_mapIso]
|
||||
congr 1 <;> rw [markSingle_mapIso]
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lemma mulS_lorentzAction (T : RealLorentzTensor d X) (S : RealLorentzTensor d Y)
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(x : X) (y : Y) (h : T.color x = τ (S.color y)) (Λ : LorentzGroup d) :
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mulS (Λ • T) (Λ • S) x y h = Λ • mulS T S x y h := by
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rw [mulS, mulS, ← mul_lorentzAction]
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congr 1
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all_goals
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rw [markSingle, markEmbedding, Equiv.trans_apply]
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erw [lorentzAction_mapIso, lorentzAction_mapIso]
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rfl
|
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|
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lemma mulS_symm (T : RealLorentzTensor d X) (S : RealLorentzTensor d Y)
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(x : X) (y : Y) (h : T.color x = τ (S.color y)) :
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mapIso d (Equiv.sumComm _ _) (mulS T S x y h) = mulS S T y x (color_eq_dual_symm h) := by
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rw [mulS, mulS, mul_symm]
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|
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/-- An equivalence of types associated with multiplying two consecutive indices,
|
||||
with the second index appearing on the left. -/
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def mulSSplitLeft {y y' : Y} (hy : y ≠ y') (z : Z) :
|
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{yz // yz ≠ (Sum.inl ⟨y, hy⟩ : {y'' // y'' ≠ y'} ⊕ {z' // z' ≠ z})} ≃
|
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{y'' // y'' ≠ y' ∧ y'' ≠ y} ⊕ {z' // z' ≠ z} :=
|
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Equiv.subtypeSum.trans <|
|
||||
Equiv.sumCongr (
|
||||
(Equiv.subtypeEquivRight (fun a => by
|
||||
obtain ⟨a, p⟩ := a; simp only [ne_eq, Sum.inl.injEq, Subtype.mk.injEq])).trans
|
||||
(Equiv.subtypeSubtypeEquivSubtypeInter _ _)) <|
|
||||
Equiv.subtypeUnivEquiv (fun a => Sum.inr_ne_inl)
|
||||
|
||||
/-- An equivalence of types associated with multiplying two consecutive indices with the
|
||||
second index appearing on the right. -/
|
||||
def mulSSplitRight {y y' : Y} (hy : y ≠ y') (z : Z) :
|
||||
{yz // yz ≠ (Sum.inr ⟨y', hy.symm⟩ : {z' // z' ≠ z} ⊕ {y'' // y'' ≠ y})} ≃
|
||||
{z' // z' ≠ z} ⊕ {y'' // y'' ≠ y' ∧ y'' ≠ y} :=
|
||||
Equiv.subtypeSum.trans <|
|
||||
Equiv.sumCongr (Equiv.subtypeUnivEquiv (fun a => Sum.inl_ne_inr)) <|
|
||||
(Equiv.subtypeEquivRight (fun a => by
|
||||
obtain ⟨a, p⟩ := a; simp only [ne_eq, Sum.inr.injEq, Subtype.mk.injEq])).trans <|
|
||||
((Equiv.subtypeSubtypeEquivSubtypeInter _ _).trans
|
||||
(Equiv.subtypeEquivRight (fun y'' => And.comm)))
|
||||
|
||||
/-- An equivalence of types associated with the associativity property of multiplication. -/
|
||||
def mulSAssocIso (x : X) {y y' : Y} (hy : y ≠ y') (z : Z) :
|
||||
{x' // x' ≠ x} ⊕ {yz // yz ≠ (Sum.inl ⟨y, hy⟩ : {y'' // y'' ≠ y'} ⊕ {z' // z' ≠ z})}
|
||||
≃ {xy // xy ≠ (Sum.inr ⟨y', hy.symm⟩ : {x' // x' ≠ x} ⊕ {y'' // y'' ≠ y})} ⊕ {z' // z' ≠ z} :=
|
||||
(Equiv.sumCongr (Equiv.refl _) (mulSSplitLeft hy z)).trans <|
|
||||
(Equiv.sumAssoc _ _ _).symm.trans <|
|
||||
(Equiv.sumCongr (mulSSplitRight hy x).symm (Equiv.refl _))
|
||||
|
||||
lemma mulS_assoc_color {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y}
|
||||
{U : RealLorentzTensor d Z} {x : X} {y y' : Y} (hy : y ≠ y') {z : Z}
|
||||
(h : T.color x = τ (S.color y))
|
||||
(h' : S.color y' = τ (U.color z)) : (mulS (mulS T S x y h) U (Sum.inr ⟨y', hy.symm⟩) z h').color
|
||||
= (mapIso d (mulSAssocIso x hy z) (mulS T (mulS S U y' z h') x (Sum.inl ⟨y, hy⟩) h)).color := by
|
||||
funext a
|
||||
match a with
|
||||
| .inl ⟨.inl _, _⟩ => rfl
|
||||
| .inl ⟨.inr _, _⟩ => rfl
|
||||
| .inr _ => rfl
|
||||
|
||||
/-- An equivalence of index values associated with the associativity property of multiplication. -/
|
||||
def mulSAssocIndexValue {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y}
|
||||
{U : RealLorentzTensor d Z} {x : X} {y y' : Y} (hy : y ≠ y') {z : Z}
|
||||
(h : T.color x = τ (S.color y)) (h' : S.color y' = τ (U.color z)) :
|
||||
IndexValue d ((T.mulS S x y h).mulS U (Sum.inr ⟨y', hy.symm⟩) z h').color ≃
|
||||
IndexValue d (T.mulS (S.mulS U y' z h') x (Sum.inl ⟨y, hy⟩) h).color :=
|
||||
indexValueIso d (mulSAssocIso x hy z).symm (mulS_assoc_color hy h h')
|
||||
|
||||
/-- Multiplication of indices is associative, up to a `mapIso` equivalence. -/
|
||||
lemma mulS_assoc (T : RealLorentzTensor d X) (S : RealLorentzTensor d Y) (U : RealLorentzTensor d Z)
|
||||
(x : X) (y y' : Y) (hy : y ≠ y') (z : Z) (h : T.color x = τ (S.color y))
|
||||
(h' : S.color y' = τ (U.color z)) : mulS (mulS T S x y h) U (Sum.inr ⟨y', hy.symm⟩) z h' =
|
||||
mapIso d (mulSAssocIso x hy z) (mulS T (mulS S U y' z h') x (Sum.inl ⟨y, hy⟩) h) := by
|
||||
apply ext (mulS_assoc_color _ _ _) ?_
|
||||
funext i
|
||||
trans ∑ a, (∑ b, T.coord (mulSFstArg (mulSFstArg i a) b) *
|
||||
S.coord (mulSSndArg (mulSFstArg i a) b h)) * U.coord (mulSSndArg i a h')
|
||||
rfl
|
||||
trans ∑ a, T.coord (mulSFstArg (mulSAssocIndexValue hy h h' i) a) *
|
||||
(∑ b, S.coord (mulSFstArg (mulSSndArg (mulSAssocIndexValue hy h h' i) a h) b) *
|
||||
U.coord (mulSSndArg (mulSSndArg (mulSAssocIndexValue hy h h' i) a h) b h'))
|
||||
swap
|
||||
rw [mapIso_apply_coord, mulS_coord_arg, indexValueIso_symm]
|
||||
rfl
|
||||
rw [Finset.sum_congr rfl (fun x _ => Finset.sum_mul _ _ _)]
|
||||
rw [Finset.sum_congr rfl (fun x _ => Finset.mul_sum _ _ _)]
|
||||
rw [Finset.sum_comm]
|
||||
refine Finset.sum_congr rfl (fun a _ => Finset.sum_congr rfl (fun b _ => ?_))
|
||||
rw [mul_assoc]
|
||||
refine Mathlib.Tactic.Ring.mul_congr rfl (Mathlib.Tactic.Ring.mul_congr ?_ rfl rfl) rfl
|
||||
apply congrArg
|
||||
funext c
|
||||
by_cases hcy : c = y
|
||||
· subst hcy
|
||||
rw [mulSSndArg_on_mem, mulSFstArg_on_not_mem, mulSSndArg_on_mem]
|
||||
rfl
|
||||
· by_cases hcy' : c = y'
|
||||
· subst hcy'
|
||||
rw [mulSFstArg_on_mem, mulSSndArg_on_not_mem, mulSFstArg_on_mem]
|
||||
· rw [mulSFstArg_on_not_mem, mulSSndArg_on_not_mem, mulSSndArg_on_not_mem,
|
||||
mulSFstArg_on_not_mem]
|
||||
rw [mulSAssocIndexValue, indexValueIso_eq_symm, indexValueIso_symm_apply']
|
||||
simp only [ne_eq, Function.comp_apply, Equiv.symm_symm_apply, mulS_color, Sum.elim_inr,
|
||||
colorsIndexCast, Equiv.cast_refl, Equiv.refl_symm]
|
||||
erw [Equiv.refl_apply]
|
||||
rfl
|
||||
exact hcy'
|
||||
simpa using hcy
|
||||
|
||||
end RealLorentzTensor
|
||||
|
|
|
|||
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Add table
Add a link
Reference in a new issue