refactor: Lint

2024-10-24 12:08:35 +00:00 · 2024-10-24 12:08:35 +00:00 · c9c7b25ea8
commit c9c7b25ea8
parent 1e8efdb16a
9 changed files with 946 additions and 696 deletions
--- a/HepLean/Tensors/ComplexLorentz/PauliContr.lean
+++ b/HepLean/Tensors/ComplexLorentz/PauliContr.lean
@ -0,0 +1,373 @@
+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.Tensors.ComplexLorentz.PauliLower
+import HepLean.Tensors.ComplexLorentz.Lemmas
+/-!
+
+## Contractiong of indices of Pauli matrix.
+
+The main result of this file is `pauliMatrix_contract_pauliMatrix` which states that
+`η_{μν} σ^{μ α dot β} σ^{ν α' dot β'} = 2 ε^{αα'} ε^{dot β dot β'}`.
+
+The current way this result is proved is by using tensor tree manipulations.
+There is likely a more direct path to this result.
+
+-/
+open IndexNotation
+open CategoryTheory
+open MonoidalCategory
+open Matrix
+open MatrixGroups
+open Complex
+open TensorProduct
+open IndexNotation
+open CategoryTheory
+open TensorTree
+open OverColor.Discrete
+noncomputable section
+
+namespace Fermion
+open complexLorentzTensor
+
+/-- The map to colors one gets when contracting the 4-vector indices pauli matrices. -/
+def pauliMatrixContrPauliMatrixMap := ((Sum.elim
+  ((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
+  Fin.succAbove 0 ∘ Fin.succAbove 1) ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
+  Fin.succAbove 0 ∘ Fin.succAbove 2)
+
+lemma pauliMatrix_contr_lower_0_0_0 :
+    {(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)) | μ α β ⊗
+    PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
+    basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
+    + basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
+  rw [basis_contr_pauliMatrix_basis_tree_expand]
+  rw [contrBasisVectorMul_pos, contrBasisVectorMul_pos,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+
+lemma pauliMatrix_contr_lower_0_1_1 :
+    {(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)) | μ α β ⊗
+    PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
+    basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
+    + basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
+  rw [basis_contr_pauliMatrix_basis_tree_expand]
+  rw [contrBasisVectorMul_pos, contrBasisVectorMul_pos,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+
+lemma pauliMatrix_contr_lower_1_0_1 :
+    {(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)) | μ α β ⊗
+    PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
+    basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
+    + basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
+  rw [basis_contr_pauliMatrix_basis_tree_expand]
+  rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_pos, contrBasisVectorMul_pos,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+
+lemma pauliMatrix_contr_lower_1_1_0 :
+    {(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)) | μ α β ⊗
+    PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
+    basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
+    + basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
+  rw [basis_contr_pauliMatrix_basis_tree_expand]
+  rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_pos, contrBasisVectorMul_pos,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+
+lemma pauliMatrix_contr_lower_2_0_1 :
+    {(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)) | μ α β ⊗
+    PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
+      (-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
+    + (I) •
+      basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
+  rw [basis_contr_pauliMatrix_basis_tree_expand]
+  rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_pos, contrBasisVectorMul_pos,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+
+lemma pauliMatrix_contr_lower_2_1_0 :
+    {(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)) | μ α β ⊗
+    PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
+      (-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
+    + (I) •
+      basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
+  rw [basis_contr_pauliMatrix_basis_tree_expand]
+  rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_pos, contrBasisVectorMul_pos,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+
+lemma pauliMatrix_contr_lower_3_0_0 :
+    {(basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)) | μ α β ⊗
+    PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
+    basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
+    + (-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap
+    (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
+  rw [basis_contr_pauliMatrix_basis_tree_expand]
+  rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_pos, contrBasisVectorMul_pos]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+
+lemma pauliMatrix_contr_lower_3_1_1 :
+    {(basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1)) | μ α β ⊗
+    PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
+    basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
+    + (-1 : ℂ) •
+      basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
+  rw [basis_contr_pauliMatrix_basis_tree_expand]
+  rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_neg, contrBasisVectorMul_neg,
+      contrBasisVectorMul_pos, contrBasisVectorMul_pos]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+
+/-! TODO: Work out why `pauliMatrix_lower_basis_expand_prod'` is needed. -/
+/-- This lemma is exactly the same as `pauliMatrix_lower_basis_expand_prod'`.
+  It is needed here for `pauliMatrix_contract_pauliMatrix_aux`. It is unclear why
+  `pauliMatrix_lower_basis_expand_prod` does not work. -/
+private lemma pauliMatrix_lower_basis_expand_prod' {n : ℕ} {c : Fin n → complexLorentzTensor.C}
+    (t : TensorTree complexLorentzTensor c) :
+    (prod {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ t).tensor =
+    ((((tensorNode
+      (basisVector pauliMatrixLowerMap fun | 0 => 0 | 1 => 0 | 2 => 0)).prod t).add
+    (((tensorNode
+      (basisVector pauliMatrixLowerMap fun | 0 => 0 | 1 => 1 | 2 => 1)).prod t).add
+    ((TensorTree.smul (-1) ((tensorNode
+      (basisVector pauliMatrixLowerMap fun | 0 => 1 | 1 => 0 | 2 => 1)).prod t)).add
+    ((TensorTree.smul (-1) ((tensorNode
+      (basisVector pauliMatrixLowerMap fun | 0 => 1 | 1 => 1 | 2 => 0)).prod t)).add
+    ((TensorTree.smul I ((tensorNode
+      (basisVector pauliMatrixLowerMap fun | 0 => 2 | 1 => 0 | 2 => 1)).prod t)).add
+    ((TensorTree.smul (-I) ((tensorNode
+      (basisVector pauliMatrixLowerMap fun | 0 => 2 | 1 => 1 | 2 => 0)).prod t)).add
+    ((TensorTree.smul (-1) ((tensorNode
+      (basisVector pauliMatrixLowerMap fun | 0 => 3 | 1 => 0 | 2 => 0)).prod t)).add
+    ((tensorNode
+      (basisVector pauliMatrixLowerMap fun | 0 => 3 | 1 => 1 | 2 => 1)).prod
+      t))))))))).tensor := by
+    exact pauliMatrix_lower_basis_expand_prod _
+
+lemma pauliMatrix_contract_pauliMatrix_aux :
+  {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗
+    PauliMatrix.asConsTensor | ν α' β'}ᵀ.tensor
+  = ((tensorNode
+      ((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0) +
+        basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1)).add
+    ((tensorNode
+      ((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0) +
+      basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1)).add
+    ((TensorTree.smul (-1) (tensorNode
+      ((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1) +
+      basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0))).add
+    ((TensorTree.smul (-1) (tensorNode
+      ((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) +
+      basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0))).add
+    ((TensorTree.smul I (tensorNode
+      ((-I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1) +
+        I •
+        basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0))).add
+    ((TensorTree.smul (-I) (tensorNode
+      ((-I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) +
+      I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0))).add
+      ((TensorTree.smul (-1) (tensorNode
+        ((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0) +
+        (-1 : ℂ) •
+        basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1))).add
+      (tensorNode
+        ((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0) +
+        (-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap
+          fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1))))))))).tensor := by
+  rw [contr_tensor_eq <| pauliMatrix_lower_basis_expand_prod' _]
+  /- Moving contraction through addition. -/
+  rw [contr_add]
+  rw [add_tensor_eq_snd <| contr_add _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+    contr_add _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+    add_tensor_eq_snd <| contr_add _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+    add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
+  /- Moving contraction through smul. -/
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
+    contr_smul _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+    add_tensor_eq_fst <| contr_smul _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+    add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+    add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
+  /- Replacing the contractions. -/
+  rw [add_tensor_eq_fst <| eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_0_0_0]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| eq_tensorNode_of_eq_tensor <|
+    pauliMatrix_contr_lower_0_1_1]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
+    eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_1_0_1]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
+    smul_tensor_eq <| eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_1_1_0]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+    add_tensor_eq_fst <| smul_tensor_eq <| eq_tensorNode_of_eq_tensor <|
+    pauliMatrix_contr_lower_2_0_1]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+    add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| eq_tensorNode_of_eq_tensor
+    <| pauliMatrix_contr_lower_2_1_0]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+    add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
+    eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_3_0_0]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+    add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| eq_tensorNode_of_eq_tensor <|
+    pauliMatrix_contr_lower_3_1_1]
+
+lemma pauliMatrix_contract_pauliMatrix_expand :
+  {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗
+  PauliMatrix.asConsTensor | ν α' β'}ᵀ.tensor =
+    2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1)
+    + 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
+    - 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
+    - 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) := by
+  rw [pauliMatrix_contract_pauliMatrix_aux]
+  simp only [Nat.reduceAdd, Fin.isValue, Fin.succAbove_zero, neg_smul,
+    one_smul, add_tensor, tensorNode_tensor, smul_tensor, smul_add, smul_neg, _root_.smul_smul,
+    neg_mul, _root_.neg_neg]
+  ring_nf
+  rw [Complex.I_sq]
+  simp only [neg_smul, one_smul, _root_.neg_neg]
+  abel
+
+/-- The statement that `η_{μν} σ^{μ α dot β} σ^{ν α' dot β'} = 2 ε^{αα'} ε^{dot β dot β'}`. -/
+theorem pauliMatrix_contract_pauliMatrix :
+    {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗
+    PauliMatrix.asConsTensor | ν α' β' =
+    2 •ₜ Fermion.leftMetric | α α' ⊗ Fermion.rightMetric | β β'}ᵀ := by
+  rw [pauliMatrix_contract_pauliMatrix_expand]
+  rw [perm_tensor_eq <| smul_tensor_eq <| leftMetric_mul_rightMetric_tree]
+  rw [perm_smul]
+  /- Moving perm through adds. -/
+  rw [smul_tensor_eq <| perm_add _ _ _]
+  rw [smul_tensor_eq <| add_tensor_eq_snd <| perm_add _ _ _]
+  rw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| perm_add _ _ _]
+  /- Moving perm through smul. -/
+  rw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_fst <| perm_smul _ _ _]
+  rw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| perm_smul _ _ _]
+  /- Perm acting on basis. -/
+  erw [smul_tensor_eq <| add_tensor_eq_fst <| perm_basisVector_tree _ _]
+  erw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
+    perm_basisVector_tree _ _]
+  erw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
+    smul_tensor_eq <| perm_basisVector_tree _ _]
+  erw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+    perm_basisVector_tree _ _]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  have h1 (b0011 b1100 b0110 b1001 : CoeSort.coe (complexLorentzTensor.F.obj
+      (OverColor.mk pauliMatrixContrPauliMatrixMap))) :
+      ((2 • b0011 + 2 • b1100) - 2 • b0110 - 2 • b1001) = (2 : ℂ) • ((b0011) +
+      (((-1 : ℂ)• b0110) + (((-1 : ℂ) •b1001) + b1100))) := by
+    trans (2 : ℂ) • b0011 + (2 : ℂ) • b1100 - ((2 : ℂ) • b0110) - ((2 : ℂ) • b1001)
+    · repeat rw [two_smul]
+    · simp only [neg_smul, one_smul, smul_add, smul_neg]
+      abel
+  rw [h1]
+  congr
+  · funext i
+    fin_cases i <;> rfl
+  · funext i
+    fin_cases i <;> rfl
+  · funext i
+    fin_cases i <;> rfl
+  · funext i
+    fin_cases i <;> rfl
+
+end Fermion