196 lines
7.5 KiB
Text
196 lines
7.5 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.Tensors.ComplexLorentz.PauliMatrices.Basic
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import HepLean.Tensors.Tree.NodeIdentities.ProdContr
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import HepLean.Tensors.Tree.NodeIdentities.PermContr
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import HepLean.Tensors.Tree.NodeIdentities.PermProd
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import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
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import HepLean.Tensors.Tree.NodeIdentities.ContrContr
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import HepLean.Tensors.Tree.NodeIdentities.ProdComm
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import HepLean.Tensors.Tree.NodeIdentities.Congr
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import HepLean.Tensors.Tree.NodeIdentities.ProdAssoc
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/-!
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## Bispinors
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-/
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open IndexNotation
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open CategoryTheory
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open MonoidalCategory
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open Matrix
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open MatrixGroups
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open Complex
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open TensorProduct
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open IndexNotation
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open CategoryTheory
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open TensorTree
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open OverColor.Discrete
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open Fermion
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noncomputable section
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namespace complexLorentzTensor
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open Lorentz
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/-!
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## Definitions
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-/
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/-- A bispinor `pᵃᵃ` created from a lorentz vector `p^μ`. -/
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def contrBispinorUp (p : complexContr) :=
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{pauliCo | μ α β ⊗ p | μ}ᵀ.tensor
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/-- A bispinor `pₐₐ` created from a lorentz vector `p^μ`. -/
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def contrBispinorDown (p : complexContr) :=
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{εL' | α α' ⊗ εR' | β β' ⊗ contrBispinorUp p | α β}ᵀ.tensor
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/-- A bispinor `pᵃᵃ` created from a lorentz vector `p_μ`. -/
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def coBispinorUp (p : complexCo) := {pauliContr | μ α β ⊗ p | μ}ᵀ.tensor
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/-- A bispinor `pₐₐ` created from a lorentz vector `p_μ`. -/
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def coBispinorDown (p : complexCo) :=
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{εL' | α α' ⊗ εR' | β β' ⊗ coBispinorUp p | α β}ᵀ.tensor
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/-!
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## Tensor nodes
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-/
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/-- The definitional tensor node relation for `contrBispinorUp`. -/
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lemma tensorNode_contrBispinorUp (p : complexContr) :
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{contrBispinorUp p | α β}ᵀ.tensor = {pauliCo | μ α β ⊗ p | μ}ᵀ.tensor := by
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rw [contrBispinorUp, tensorNode_tensor]
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/-- The definitional tensor node relation for `contrBispinorDown`. -/
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lemma tensorNode_contrBispinorDown (p : complexContr) :
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{contrBispinorDown p | α β}ᵀ.tensor =
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{εL' | α α' ⊗ εR' | β β' ⊗ contrBispinorUp p | α β}ᵀ.tensor := by
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rw [contrBispinorDown, tensorNode_tensor]
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/-- The definitional tensor node relation for `coBispinorUp`. -/
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lemma tensorNode_coBispinorUp (p : complexCo) :
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{coBispinorUp p | α β}ᵀ.tensor = {pauliContr | μ α β ⊗ p | μ}ᵀ.tensor := by
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rw [coBispinorUp, tensorNode_tensor]
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/-- The definitional tensor node relation for `coBispinorDown`. -/
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lemma tensorNode_coBispinorDown (p : complexCo) :
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{coBispinorDown p | α β}ᵀ.tensor =
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{εL' | α α' ⊗ εR' | β β' ⊗ coBispinorUp p | α β}ᵀ.tensor := by
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rw [coBispinorDown, tensorNode_tensor]
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/-!
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## Basic equalities.
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-/
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informal_lemma contrBispinorUp_eq_metric_contr_contrBispinorDown where
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math :≈ "{contrBispinorUp p | α β = εL | α α' ⊗ εR | β β'⊗ contrBispinorDown p | α' β' }ᵀ"
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proof :≈ "Expand `contrBispinorDown` and use fact that metrics contract to the identity."
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deps :≈ [``contrBispinorUp, ``contrBispinorDown, ``leftMetric, ``rightMetric]
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informal_lemma coBispinorUp_eq_metric_contr_coBispinorDown where
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math :≈ "{coBispinorUp p | α β = εL | α α' ⊗ εR | β β'⊗ coBispinorDown p | α' β' }ᵀ"
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proof :≈ "Expand `coBispinorDown` and use fact that metrics contract to the identity."
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deps :≈ [``coBispinorUp, ``coBispinorDown, ``leftMetric, ``rightMetric]
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lemma contrBispinorDown_expand (p : complexContr) :
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{contrBispinorDown p | α β}ᵀ.tensor =
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{εL' | α α' ⊗ εR' | β β' ⊗
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(pauliCo | μ α β ⊗ p | μ)}ᵀ.tensor := by
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rw [tensorNode_contrBispinorDown p]
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rw [contr_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_snd <| tensorNode_contrBispinorUp p]
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lemma coBispinorDown_expand (p : complexCo) :
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{coBispinorDown p | α β}ᵀ.tensor =
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{εL' | α α' ⊗ εR' | β β' ⊗
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(pauliContr | μ α β ⊗ p | μ)}ᵀ.tensor := by
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rw [tensorNode_coBispinorDown p]
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rw [contr_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_snd <| tensorNode_coBispinorUp p]
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set_option maxRecDepth 5000 in
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lemma contrBispinorDown_eq_pauliCoDown_contr (p : complexContr) :
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{contrBispinorDown p | α β = pauliCoDown | μ α β ⊗ p | μ}ᵀ := by
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conv =>
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rhs
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rw [perm_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_fst <|
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pauliCoDown_eq_metric_mul_pauliCo]
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rw [perm_tensor_eq <| contr_tensor_eq <| prod_perm_left _ _ _ _]
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rw [perm_tensor_eq <| perm_contr_congr 2 2]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_tensor_eq <| contr_prod _ _ _]
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rw [perm_tensor_eq <| perm_contr_congr 2 2]
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rw [perm_perm]
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apply (perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_eq_id _ rfl _).trans
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rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_prod _ _ _]
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rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr_congr 1 3]
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rw [perm_tensor_eq <| perm_contr_congr 2 2]
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rw [perm_perm]
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erw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <|
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perm_eq_id _ rfl _]
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rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <|
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prod_assoc' _ _ _ _ _ _]
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rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_contr_congr 0 4]
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rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr_congr 1 3]
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rw [perm_tensor_eq <| perm_contr_congr 2 2]
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rw [perm_perm]
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conv =>
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lhs
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rw [contrBispinorDown_expand p]
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rw [contr_tensor_eq <| contr_tensor_eq <| prod_contr _ _ _]
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rw [contr_tensor_eq <| perm_contr_congr 0 3]
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rw [perm_contr_congr 1 2]
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apply (perm_tensor_eq <| contr_tensor_eq <| contr_contr _ _ _).trans
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rw [perm_tensor_eq <| perm_contr _ _]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_contr _ _ _]
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rw [perm_perm]
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apply perm_congr _ rfl
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decide
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set_option maxRecDepth 5000 in
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lemma coBispinorDown_eq_pauliContrDown_contr (p : complexCo) :
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{coBispinorDown p | α β = pauliContrDown | μ α β ⊗ p | μ}ᵀ := by
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conv =>
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rhs
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rw [perm_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_fst <|
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pauliContrDown_eq_metric_mul_pauliContr]
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rw [perm_tensor_eq <| contr_tensor_eq <| prod_perm_left _ _ _ _]
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rw [perm_tensor_eq <| perm_contr_congr 2 2]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_tensor_eq <| contr_prod _ _ _]
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rw [perm_tensor_eq <| perm_contr_congr 2 2]
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rw [perm_perm]
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apply (perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_eq_id _ rfl _).trans
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rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_prod _ _ _]
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rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr_congr 1 3]
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rw [perm_tensor_eq <| perm_contr_congr 2 2]
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rw [perm_perm]
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erw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <|
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perm_eq_id _ rfl _]
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rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <|
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prod_assoc' _ _ _ _ _ _]
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rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_contr_congr 0 4]
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rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr_congr 1 3]
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rw [perm_tensor_eq <| perm_contr_congr 2 2]
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rw [perm_perm]
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conv =>
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lhs
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rw [coBispinorDown_expand p]
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rw [contr_tensor_eq <| contr_tensor_eq <| prod_contr _ _ _]
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rw [contr_tensor_eq <| perm_contr_congr 0 3]
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rw [perm_contr_congr 1 2]
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apply (perm_tensor_eq <| contr_tensor_eq <| contr_contr _ _ _).trans
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rw [perm_tensor_eq <| perm_contr _ _]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_contr _ _ _]
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rw [perm_perm]
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apply perm_congr _ rfl
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decide
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end complexLorentzTensor
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end
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