diff --git a/HepLean/Tensors/ComplexLorentz/Basic.lean b/HepLean/Tensors/ComplexLorentz/Basic.lean
index b1dd57f..0cc1723 100644
--- a/HepLean/Tensors/ComplexLorentz/Basic.lean
+++ b/HepLean/Tensors/ComplexLorentz/Basic.lean
@@ -18,9 +18,6 @@ import HepLean.SpaceTime.PauliMatrices.AsTensor
 ## Complex Lorentz tensors
 
 -/
-
-namespace Fermion
-
 open Matrix
 open MatrixGroups
 open Complex
@@ -29,6 +26,8 @@ open IndexNotation
 open CategoryTheory
 open MonoidalCategory
 
+namespace complexLorentzTensor
+
 /-- The colors associated with complex representations of SL(2, ℂ) of intrest to physics. -/
 inductive Color
   | upL : Color
@@ -78,11 +77,13 @@ instance : DecidableEq Color := fun x y =>
   | Color.down, Color.downR => isFalse fun h => Color.noConfusion h
   | Color.down, Color.up => isFalse fun h => Color.noConfusion h
 
-noncomputable section
+end complexLorentzTensor
 
+noncomputable section
+open complexLorentzTensor in
 /-- The tensor structure for complex Lorentz tensors. -/
 def complexLorentzTensor : TensorSpecies where
-  C := Fermion.Color
+  C := complexLorentzTensor.Color
   G := SL(2, ℂ)
   G_group := inferInstance
   k := ℂ
@@ -191,8 +192,8 @@ def complexLorentzTensor : TensorSpecies where
     | Color.downR => by simpa using Fermion.altRightContraction_apply_metric
     | Color.up => by simpa using Lorentz.contrCoContraction_apply_metric
     | Color.down => by simpa using Lorentz.coContrContraction_apply_metric
-
-instance : DecidableEq complexLorentzTensor.C := Fermion.instDecidableEqColor
+namespace complexLorentzTensor
+instance : DecidableEq complexLorentzTensor.C := complexLorentzTensor.instDecidableEqColor
 
 lemma basis_contr (c : complexLorentzTensor.C) (i : Fin (complexLorentzTensor.repDim c))
     (j : Fin (complexLorentzTensor.repDim (complexLorentzTensor.τ c))) :
@@ -208,5 +209,6 @@ lemma basis_contr (c : complexLorentzTensor.C) (i : Fin (complexLorentzTensor.re
   | Color.up => Lorentz.contrCoContraction_basis _ _
   | Color.down => Lorentz.coContrContraction_basis _ _
 
+
+end complexLorentzTensor
 end
-end Fermion
diff --git a/HepLean/Tensors/ComplexLorentz/Basis.lean b/HepLean/Tensors/ComplexLorentz/Basis.lean
index 3acf860..d3859b3 100644
--- a/HepLean/Tensors/ComplexLorentz/Basis.lean
+++ b/HepLean/Tensors/ComplexLorentz/Basis.lean
@@ -32,8 +32,8 @@ open IndexNotation
 open CategoryTheory
 open TensorTree
 open OverColor.Discrete
+open Fermion
 noncomputable section
-namespace Fermion
 namespace complexLorentzTensor
 
 /-- Basis vectors for complex Lorentz tensors. -/
@@ -95,7 +95,7 @@ def contrBasisVectorMul {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.
 
 lemma contrBasisVectorMul_neg {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
     {i : Fin n.succ.succ} {j : Fin n.succ} {b : Π k, Fin (complexLorentzTensor.repDim (c k))}
-    (h : ¬ (b i).val = (b (i.succAbove j)).val := by decide) :
+    (h : ¬ (b i).val = (b (i.succAbove j)).val) :
     contrBasisVectorMul i j b = 0 := by
   rw [contrBasisVectorMul]
   simp only [ite_eq_else, one_ne_zero, imp_false]
@@ -103,7 +103,7 @@ lemma contrBasisVectorMul_neg {n : ℕ} {c : Fin n.succ.succ → complexLorentzT
 
 lemma contrBasisVectorMul_pos {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
     {i : Fin n.succ.succ} {j : Fin n.succ} {b : Π k, Fin (complexLorentzTensor.repDim (c k))}
-    (h : (b i).val = (b (i.succAbove j)).val := by decide) :
+    (h : (b i).val = (b (i.succAbove j)).val) :
     contrBasisVectorMul i j b = 1 := by
   rw [contrBasisVectorMul]
   simp only [ite_eq_then, zero_ne_one, imp_false, Decidable.not_not]
@@ -156,7 +156,7 @@ lemma contr_basisVector_tree_neg {n : ℕ} {c : Fin n.succ.succ → complexLoren
     (tensorNode 0).tensor := by
   rw [contr_basisVector_tree, contrBasisVectorMul]
   rw [if_neg hn]
-  simp only [Nat.succ_eq_add_one, smul_tensor, tensorNode_tensor, zero_smul]
+  simp only [Nat.succ_eq_add_one, smul_tensor, tensorNode_tensor, _root_.zero_smul]
 
 /-- Equivalence of Fin types appearing in the product of two basis vectors. -/
 def prodBasisVecEquiv {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
@@ -214,7 +214,7 @@ lemma eval_basisVector {n : ℕ} {c : Fin n.succ → complexLorentzTensor.C}
     basisVector (c ∘ Fin.succAbove i) (fun k => b (i.succAbove k)) := by
   rw [eval_tensor, basisVector, basisVector]
   simp only [Nat.succ_eq_add_one, Functor.id_obj, OverColor.mk_hom, tensorNode_tensor,
-    Function.comp_apply, one_smul, zero_smul]
+    Function.comp_apply, one_smul, _root_.zero_smul]
   erw [TensorSpecies.evalMap_tprod]
   congr 1
   have h1 : Fin.ofNat' ↑j (@Fin.size_pos' (complexLorentzTensor.repDim (c i)) _) = j := by
@@ -448,5 +448,4 @@ lemma pauliMatrix_basis_expand_tree : {PauliMatrix.asConsTensor | μ α β}ᵀ.t
   rfl
 
 end complexLorentzTensor
-end Fermion
 end
diff --git a/HepLean/Tensors/ComplexLorentz/BasisTrees.lean b/HepLean/Tensors/ComplexLorentz/BasisTrees.lean
index 23ed781..9dd69eb 100644
--- a/HepLean/Tensors/ComplexLorentz/BasisTrees.lean
+++ b/HepLean/Tensors/ComplexLorentz/BasisTrees.lean
@@ -36,8 +36,8 @@ open OverColor.Discrete
 
 noncomputable section
 
-namespace Fermion
-open complexLorentzTensor
+namespace complexLorentzTensor
+open Fermion
 
 /-!
 
@@ -177,6 +177,25 @@ lemma basis_contr_pauliMatrix_basis_tree_expand' {n : ℕ} {c : Fin n → comple
     <| contr_tensor_eq <| prod_basisVector_tree _ _]
   rfl
 
+
+def pauliMatrixBasisProdMap
+    {n : ℕ} {c : Fin n → complexLorentzTensor.C}
+    (b : Π k, Fin (complexLorentzTensor.repDim (c k))) (i1 i2 i3 : Fin 4) :
+    (i : Fin (n + (Nat.succ 0).succ.succ)) →
+      Fin (complexLorentzTensor.repDim (Sum.elim c ![Color.up, Color.upL, Color.upR]
+          (finSumFinEquiv.symm i))) := fun i => prodBasisVecEquiv (finSumFinEquiv.symm i)
+      ((HepLean.PiTensorProduct.elimPureTensor b (fun | (0 : Fin 3) => i1 | 1 => i2 | 2 => i3))
+      (finSumFinEquiv.symm i))
+
+def basisVectorContrPauli {n : ℕ} {c : Fin n → complexLorentzTensor.C}
+    (i : Fin (n + 3)) (j : Fin (n +2))
+    (b : Π k, Fin (complexLorentzTensor.repDim (c k)))
+    (i1 i2 i3 : Fin 4)  :=
+  let c' := (Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ finSumFinEquiv.symm)
+      ∘ Fin.succAbove i ∘ Fin.succAbove j
+  let b' (i1 i2 i3 : Fin 4) := fun k => (pauliMatrixBasisProdMap b i1 i2 i3) (i.succAbove (j.succAbove k))
+  basisVector c' (b' i1 i2 i3)
+
 lemma basis_contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
     (i : Fin (n + 3)) (j : Fin (n +2))
     (h : (pauliMatrixContrMap c) (i.succAbove j) = complexLorentzTensor.τ
@@ -184,30 +203,25 @@ lemma basis_contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complex
     (b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
     let c' := (Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ finSumFinEquiv.symm)
       ∘ Fin.succAbove i ∘ Fin.succAbove j
-    let b'' (i1 i2 i3 : Fin 4) : (i : Fin (n + (Nat.succ 0).succ.succ)) →
-        Fin (complexLorentzTensor.repDim (Sum.elim c ![Color.up, Color.upL, Color.upR]
-          (finSumFinEquiv.symm i))) := fun i => prodBasisVecEquiv (finSumFinEquiv.symm i)
-      ((HepLean.PiTensorProduct.elimPureTensor b (fun | (0 : Fin 3) => i1 | 1 => i2 | 2 => i3))
-      (finSumFinEquiv.symm i))
-    let b' (i1 i2 i3 : Fin 4) := fun k => (b'' i1 i2 i3) (i.succAbove (j.succAbove k))
+    let b' (i1 i2 i3 : Fin 4) := fun k => (pauliMatrixBasisProdMap b i1 i2 i3) (i.succAbove (j.succAbove k))
     (contr i j h (TensorTree.prod (tensorNode (basisVector c b))
     (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
     PauliMatrix.asConsTensor))).tensor = (((
-    TensorTree.smul (contrBasisVectorMul i j (b'' 0 0 0))
+    TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 0 0))
       (tensorNode (basisVector c' (b' 0 0 0))))).add
-    (((TensorTree.smul (contrBasisVectorMul i j (b'' 0 1 1))
+    (((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 1 1))
       (tensorNode (basisVector c' (b' 0 1 1))))).add
-    (((TensorTree.smul (contrBasisVectorMul i j (b'' 1 0 1))
+    (((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 1 0 1))
       (tensorNode (basisVector c' (b' 1 0 1))))).add
-    (((TensorTree.smul (contrBasisVectorMul i j (b'' 1 1 0))
+    (((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 1 1 0))
       (tensorNode (basisVector c' (b' 1 1 0))))).add
-    ((TensorTree.smul (-I) ((TensorTree.smul (contrBasisVectorMul i j (b'' 2 0 1))
+    ((TensorTree.smul (-I) ((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 0 1))
       (tensorNode (basisVector c' (b' 2 0 1)))))).add
-    ((TensorTree.smul I ((TensorTree.smul (contrBasisVectorMul i j (b'' 2 1 0))
+    ((TensorTree.smul I ((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 1 0))
       (tensorNode (basisVector c' (b' 2 1 0)))))).add
-    (((TensorTree.smul (contrBasisVectorMul i j (b'' 3 0 0))
+    (((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 0 0))
       (tensorNode (basisVector c' (b' 3 0 0))))).add
-    (TensorTree.smul (-1) ((TensorTree.smul (contrBasisVectorMul i j (b'' 3 1 1)) (tensorNode
+    (TensorTree.smul (-1) ((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 1 1)) (tensorNode
       (basisVector c' (b' 3 1 1))))))))))))).tensor := by
   rw [basis_contr_pauliMatrix_basis_tree_expand']
   /- Contracting basis vectors. -/
@@ -227,7 +241,46 @@ lemma basis_contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complex
   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 <|
     smul_tensor_eq <| contr_basisVector_tree _]
+  rfl
 
-end Fermion
+def pauilMatrixBasisContrMap {n : ℕ} {c : Fin n → complexLorentzTensor.C}
+    (i : Fin (n + 3)) (j : Fin (n +2))
+    (b : Π k, Fin (complexLorentzTensor.repDim (c k))) (i1 i2 i3 : Fin 4)  :
+     (k : Fin (n + 1)) →
+        Fin
+          (complexLorentzTensor.repDim
+            (Sum.elim c ![Color.up, Color.upL, Color.upR] (finSumFinEquiv.symm (i.succAbove (j.succAbove k))))) :=
+    fun k => (pauliMatrixBasisProdMap b i1 i2 i3) (i.succAbove (j.succAbove k))
+
+
+lemma basis_contr_pauliMatrix_basis_tree_expand_tensor {n : ℕ} {c : Fin n → complexLorentzTensor.C}
+    (i : Fin (n + 3)) (j : Fin (n +2))
+    (h : (pauliMatrixContrMap c) (i.succAbove j) = complexLorentzTensor.τ
+      ((pauliMatrixContrMap c) i))
+    (b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
+    let c' := (Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ finSumFinEquiv.symm)
+      ∘ Fin.succAbove i ∘ Fin.succAbove j
+    let b' (i1 i2 i3 : Fin 4) := fun k => (pauliMatrixBasisProdMap b i1 i2 i3) (i.succAbove (j.succAbove k))
+    (contr i j h (TensorTree.prod (tensorNode (basisVector c b))
+    (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
+    PauliMatrix.asConsTensor))).tensor =
+    (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 0 0)) • (basisVectorContrPauli i j b 0 0 0)
+    + (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 1 1)) • (basisVectorContrPauli i j b 0 1 1)
+    + (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 1 0 1)) • (basisVectorContrPauli i j b 1 0 1)
+    + (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 1 1 0)) • (basisVectorContrPauli i j b 1 1 0)
+    + (-I) • (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 0 1)) • (basisVectorContrPauli i j b 2 0 1)
+    + I • (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 1 0)) • (basisVectorContrPauli i j b 2 1 0)
+    + (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 0 0)) • (basisVectorContrPauli i j b 3 0 0)
+    + (-1 : ℂ) • (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 1 1)) • (basisVectorContrPauli i j b 3 1 1) := by
+  rw [basis_contr_pauliMatrix_basis_tree_expand]
+  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, cons_val_one, head_cons, Fin.val_zero,
+    Nat.cast_zero, cons_val_two, Fin.val_one, Nat.cast_one, add_tensor, smul_tensor,
+    tensorNode_tensor, neg_smul, one_smul, Int.reduceNeg]
+  simp_all only [Function.comp_apply, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue]
+  rfl
+
+
+
+end complexLorentzTensor
 
 end
diff --git a/HepLean/Tensors/ComplexLorentz/Bispinors/Basic.lean b/HepLean/Tensors/ComplexLorentz/Bispinors/Basic.lean
index 6e2c614..950f01f 100644
--- a/HepLean/Tensors/ComplexLorentz/Bispinors/Basic.lean
+++ b/HepLean/Tensors/ComplexLorentz/Bispinors/Basic.lean
@@ -3,15 +3,7 @@ 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.Tree.Elab
-import HepLean.Tensors.ComplexLorentz.Basic
-import Mathlib.LinearAlgebra.TensorProduct.Basis
-import HepLean.Tensors.Tree.NodeIdentities.Basic
-import HepLean.Tensors.Tree.NodeIdentities.PermProd
-import HepLean.Tensors.Tree.NodeIdentities.PermContr
-import HepLean.Tensors.Tree.NodeIdentities.ProdComm
-import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
-import HepLean.Tensors.Tree.NodeIdentities.ContrContr
+import HepLean.Tensors.ComplexLorentz.PauliLower
 /-!
 
 ## Bispinors
@@ -30,18 +22,17 @@ open TensorTree
 open OverColor.Discrete
 open Fermion
 noncomputable section
-namespace Lorentz
-namespace complexContr
+namespace complexLorentzTensor
+open Lorentz
 
 /-- A bispinor `pᵃᵃ` created from a lorentz vector  `p^μ`. -/
-def bispinorUp (p : complexContr) :=
-  {p | μ ⊗ (Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | ν α β)}ᵀ.tensor
+def contrBispinorUp (p : complexContr) :=
+  {p | μ ⊗ pauliCo | μ α β}ᵀ.tensor
 
 /-- A bispinor `pₐₐ` created from a lorentz vector  `p^μ`. -/
 def bispinorDown (p : complexContr) :=
   {Fermion.altRightMetric | β β' ⊗ Fermion.altLeftMetric | α α' ⊗
     (complexContr.bispinorUp p) | α β}ᵀ.tensor
 
-end complexContr
-end Lorentz
+end complexLorentzTensor
 end
diff --git a/HepLean/Tensors/ComplexLorentz/Lemmas.lean b/HepLean/Tensors/ComplexLorentz/Lemmas.lean
index 0f432d5..ebdbe65 100644
--- a/HepLean/Tensors/ComplexLorentz/Lemmas.lean
+++ b/HepLean/Tensors/ComplexLorentz/Lemmas.lean
@@ -22,8 +22,8 @@ open TensorTree
 open OverColor.Discrete
 noncomputable section
 
-namespace Fermion
-open complexLorentzTensor
+namespace complexLorentzTensor
+open Fermion
 set_option maxRecDepth 20000 in
 lemma contr_rank_2_symm {T1 : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
     {T2 : (Lorentz.complexCo ⊗ Lorentz.complexCo).V} :
@@ -159,6 +159,6 @@ lemma leftMetric_mul_rightMetric_tree :
     smul_tensor, neg_smul, one_smul]
   rfl
 
-end Fermion
+end complexLorentzTensor
 
 end
diff --git a/HepLean/Tensors/ComplexLorentz/PauliContr.lean b/HepLean/Tensors/ComplexLorentz/PauliContr.lean
index d65380d..fd091ee 100644
--- a/HepLean/Tensors/ComplexLorentz/PauliContr.lean
+++ b/HepLean/Tensors/ComplexLorentz/PauliContr.lean
@@ -29,28 +29,52 @@ open TensorTree
 open OverColor.Discrete
 noncomputable section
 
-namespace Fermion
-open complexLorentzTensor
+namespace complexLorentzTensor
+open Fermion
 
 /-- 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 1 ∘ 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 ⊗
+    {(basisVector pauliCoMap (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]
+  conv =>
+    lhs
+    rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; rhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs
+    simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
+    rw [basisVectorContrPauli, basisVectorContrPauli]
   /- 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
@@ -60,18 +84,42 @@ lemma pauliMatrix_contr_lower_0_0_0 :
     fin_cases k <;> rfl
 
 lemma pauliMatrix_contr_lower_0_1_1 :
-    {(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)) | μ α β ⊗
+    {(basisVector pauliCoMap (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]
+  conv =>
+    lhs
+    rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; rhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs
+    simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
+    rw [basisVectorContrPauli, basisVectorContrPauli]
   /- 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
@@ -81,18 +129,42 @@ lemma pauliMatrix_contr_lower_0_1_1 :
     fin_cases k <;> rfl
 
 lemma pauliMatrix_contr_lower_1_0_1 :
-    {(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)) | μ α β ⊗
+    {(basisVector pauliCoMap (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]
+  conv =>
+    lhs
+    rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; rhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs
+    simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
+    rw [basisVectorContrPauli, basisVectorContrPauli]
   /- 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
@@ -102,18 +174,42 @@ lemma pauliMatrix_contr_lower_1_0_1 :
     fin_cases k <;> rfl
 
 lemma pauliMatrix_contr_lower_1_1_0 :
-    {(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)) | μ α β ⊗
+    {(basisVector pauliCoMap (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]
+  conv =>
+    lhs
+    rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; rhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs
+    simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
+    rw [basisVectorContrPauli, basisVectorContrPauli]
   /- 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
@@ -123,19 +219,43 @@ lemma pauliMatrix_contr_lower_1_1_0 :
     fin_cases k <;> rfl
 
 lemma pauliMatrix_contr_lower_2_0_1 :
-    {(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)) | μ α β ⊗
+    {(basisVector pauliCoMap (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]
+  conv =>
+    lhs
+    rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; rhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs
+    simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
+    rw [basisVectorContrPauli, basisVectorContrPauli]
   /- 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
@@ -145,19 +265,43 @@ lemma pauliMatrix_contr_lower_2_0_1 :
     fin_cases k <;> rfl
 
 lemma pauliMatrix_contr_lower_2_1_0 :
-    {(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)) | μ α β ⊗
+    {(basisVector pauliCoMap (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]
+  conv =>
+    lhs
+    rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; rhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs
+    simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
+    rw [basisVectorContrPauli, basisVectorContrPauli]
   /- 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
@@ -167,19 +311,43 @@ lemma pauliMatrix_contr_lower_2_1_0 :
     fin_cases k <;> rfl
 
 lemma pauliMatrix_contr_lower_3_0_0 :
-    {(basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)) | μ α β ⊗
+    {(basisVector pauliCoMap (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]
+  conv =>
+    lhs
+    rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; rhs; lhs;
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; rhs; rhs; lhs;
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs
+    simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
+    rw [basisVectorContrPauli, basisVectorContrPauli]
   /- 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
@@ -189,19 +357,43 @@ lemma pauliMatrix_contr_lower_3_0_0 :
     fin_cases k <;> rfl
 
 lemma pauliMatrix_contr_lower_3_1_1 :
-    {(basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1)) | μ α β ⊗
+    {(basisVector pauliCoMap (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]
+  conv =>
+    lhs
+    rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; rhs; lhs;
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; rhs; rhs; lhs;
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs
+    simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
+    rw [basisVectorContrPauli, basisVectorContrPauli]
   /- 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
@@ -210,35 +402,34 @@ lemma pauliMatrix_contr_lower_3_1_1 :
     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'`.
+/-! TODO: Work out why `pauliCo_prod_basis_expand'` is needed. -/
+/-- This lemma is exactly the same as `pauliCo_prod_basis_expand`.
   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}
+private lemma pauliCo_prod_basis_expand' {n : ℕ} {c : Fin n → complexLorentzTensor.C}
     (t : TensorTree complexLorentzTensor c) :
-    (prod {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ t).tensor =
+    (TensorTree.prod (tensorNode pauliCo) t).tensor =
     ((((tensorNode
-      (basisVector pauliMatrixLowerMap fun | 0 => 0 | 1 => 0 | 2 => 0)).prod t).add
+      (basisVector pauliCoMap fun | 0 => 0 | 1 => 0 | 2 => 0)).prod t).add
     (((tensorNode
-      (basisVector pauliMatrixLowerMap fun | 0 => 0 | 1 => 1 | 2 => 1)).prod t).add
+      (basisVector pauliCoMap 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
+      (basisVector pauliCoMap 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
+      (basisVector pauliCoMap 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
+      (basisVector pauliCoMap 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
+      (basisVector pauliCoMap 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
+      (basisVector pauliCoMap fun | 0 => 3 | 1 => 0 | 2 => 0)).prod t)).add
     ((tensorNode
-      (basisVector pauliMatrixLowerMap fun | 0 => 3 | 1 => 1 | 2 => 1)).prod
+      (basisVector pauliCoMap fun | 0 => 3 | 1 => 1 | 2 => 1)).prod
       t))))))))).tensor := by
-    exact pauliMatrix_lower_basis_expand_prod _
+    exact pauliCo_prod_basis_expand _
 
-lemma pauliMatrix_contract_pauliMatrix_aux :
-    {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗
-    PauliMatrix.asConsTensor | ν α' β'}ᵀ.tensor
+lemma pauliCo_contr_pauliContr_expand_aux :
+    {pauliCo | μ α β ⊗ PauliMatrix.asConsTensor | μ 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
@@ -266,7 +457,7 @@ lemma pauliMatrix_contract_pauliMatrix_aux :
         ((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' _]
+  rw [contr_tensor_eq <| pauliCo_prod_basis_expand' _]
   /- Moving contraction through addition. -/
   rw [contr_add]
   rw [add_tensor_eq_snd <| contr_add _ _]
@@ -309,9 +500,8 @@ lemma pauliMatrix_contract_pauliMatrix_aux :
     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 =
+lemma pauliCo_contr_pauliContr_expand :
+    {pauliCo | ν 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)
@@ -326,9 +516,8 @@ lemma pauliMatrix_contract_pauliMatrix_expand :
   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' =
+theorem pauliCo_contr_pauliContr :
+    {pauliCo | ν 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]
@@ -370,4 +559,4 @@ theorem pauliMatrix_contract_pauliMatrix :
   · funext i
     fin_cases i <;> rfl
 
-end Fermion
+end complexLorentzTensor
diff --git a/HepLean/Tensors/ComplexLorentz/PauliLower.lean b/HepLean/Tensors/ComplexLorentz/PauliLower.lean
index d84422e..a283de0 100644
--- a/HepLean/Tensors/ComplexLorentz/PauliLower.lean
+++ b/HepLean/Tensors/ComplexLorentz/PauliLower.lean
@@ -26,209 +26,292 @@ namespace Fermion
 open complexLorentzTensor
 
 /-- The map to color one gets when lowering the indices of pauli matrices. -/
-def pauliMatrixLowerMap := ((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘
-  ⇑finSumFinEquiv.symm) ∘ Fin.succAbove 0 ∘ Fin.succAbove 1)
+def pauliCoMap := ((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘
+  ⇑finSumFinEquiv.symm) ∘ Fin.succAbove 1 ∘ Fin.succAbove 1)
 
 lemma pauliMatrix_contr_down_0 :
-    (contr 0 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
+    (contr 1 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
     (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
       PauliMatrix.asConsTensor)))).tensor
-    = basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
-    + basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 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]
-  simp only [smul_tensor, add_tensor, tensorNode_tensor]
-  simp only [one_smul, zero_smul, smul_zero, add_zero]
+    = basisVector pauliCoMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
+    + basisVector pauliCoMap (fun | 0 => 0 | 1 => 1 | 2 => 1) := by
+  conv =>
+    lhs
+    rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; rhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs
+    simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
   congr 1
-  · congr 1
+  · rw [basisVectorContrPauli]
+    congr 1
     funext k
     fin_cases k <;> rfl
-  · congr 1
+  · rw [basisVectorContrPauli]
+    congr 1
     funext k
     fin_cases k <;> rfl
 
-lemma pauliMatrix_contr_down_0_tree :
-    (contr 0 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
-    (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
-      PauliMatrix.asConsTensor)))).tensor
-    = (TensorTree.add (tensorNode
-      (basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)))
-    (tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)))).tensor := by
-  exact pauliMatrix_contr_down_0
-
 lemma pauliMatrix_contr_down_1 :
-    {(basisVector ![Color.down, Color.down] fun x => 1) | μ ν ⊗
+    {(basisVector ![Color.down, Color.down] fun x => 1) | ν μ  ⊗
       PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
-    = basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
-    + basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 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]
-  simp only [smul_tensor, add_tensor, tensorNode_tensor]
-  simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
+    = basisVector pauliCoMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
+    + basisVector pauliCoMap (fun | 0 => 1 | 1 => 1 | 2 => 0) := by
+  conv =>
+    lhs
+    rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; rhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs
+    simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
   congr 1
-  · congr 1
+  · rw [basisVectorContrPauli]
+    congr 1
     funext k
     fin_cases k <;> rfl
-  · congr 1
+  · rw [basisVectorContrPauli]
+    congr 1
     funext k
     fin_cases k <;> rfl
 
-lemma pauliMatrix_contr_down_1_tree :
-    {(basisVector ![Color.down, Color.down] fun x => 1) | μ ν ⊗
-      PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
-    = (TensorTree.add (tensorNode
-      (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)))
-    (tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)))).tensor := by
-  exact pauliMatrix_contr_down_1
-
 lemma pauliMatrix_contr_down_2 :
     {(basisVector ![Color.down, Color.down] fun x => 2) | μ ν ⊗
-      PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
-    = (- I) • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
-    + (I) • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 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]
+      PauliMatrix.asConsTensor | ν α β}ᵀ.tensor
+    = (- I) • basisVector pauliCoMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
+    + (I) • basisVector pauliCoMap (fun | 0 => 2 | 1 => 1 | 2 => 0) := by
+  conv =>
+    lhs
+    rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; lhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; rhs; rhs; lhs;
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs
+    simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
   congr 1
-  · congr 2
+  · rw [basisVectorContrPauli]
+    congr 2
     funext k
     fin_cases k <;> rfl
-  · congr 2
+  · rw [basisVectorContrPauli]
+    congr 2
     funext k
     fin_cases k <;> rfl
 
-lemma pauliMatrix_contr_down_2_tree :
-    {(basisVector ![Color.down, Color.down] fun x => 2) | μ ν ⊗
-      PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
-    (TensorTree.add
-    (smul (- I) (tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1))))
-    (smul I (tensorNode (basisVector
-      pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0))))).tensor := by
-  exact pauliMatrix_contr_down_2
-
 lemma pauliMatrix_contr_down_3 :
     {(basisVector ![Color.down, Color.down] fun x => 3) | μ ν ⊗
-    PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
-    = basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
-    + (- 1 : ℂ) • basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 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]
+    PauliMatrix.asConsTensor | ν α β}ᵀ.tensor
+    = basisVector pauliCoMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
+    + (- 1 : ℂ) • basisVector pauliCoMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
+  conv =>
+    lhs
+    rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; lhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; lhs; rhs; rhs; lhs
+    rw [contrBasisVectorMul_neg (by decide)]
+  conv =>
+    lhs; lhs; rhs; lhs;
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs; rhs; rhs; lhs;
+    rw [contrBasisVectorMul_pos (by decide)]
+  conv =>
+    lhs
+    simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
   congr 1
-  · congr 2
+  · rw [basisVectorContrPauli]
+    congr 1
     funext k
     fin_cases k <;> rfl
-  · congr 2
+  · rw [basisVectorContrPauli]
+    congr 1
+    congr 1
     funext k
     fin_cases k <;> rfl
 
-lemma pauliMatrix_contr_down_3_tree : {(basisVector ![Color.down, Color.down] fun x => 3) | μ ν ⊗
-    PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
-    (TensorTree.add
-      ((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0))))
-      (smul (-1) (tensorNode (basisVector pauliMatrixLowerMap
-      (fun | 0 => 3 | 1 => 1 | 2 => 1))))).tensor := by
-  exact pauliMatrix_contr_down_3
+def pauliCo := {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | ν α β}ᵀ.tensor
 
-lemma pauliMatrix_lower : {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
-    = basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
-    + basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)
-    - basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
-    - basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)
-    + I • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
-    - I • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)
-    - basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
-    + basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
-  rw [contr_tensor_eq <| prod_tensor_eq_fst <| coMetric_basis_expand_tree]
-  /- Moving the prod through additions. -/
-  rw [contr_tensor_eq <| add_prod _ _ _]
-  rw [contr_tensor_eq <| add_tensor_eq_snd <| add_prod _ _ _]
-  rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
-  /- Moving the prod through smuls. -/
-  rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_prod _ _ _]
-  rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
-    <| smul_prod _ _ _]
-  rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
-    <| smul_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 _ _]
-  /- Moving contraction through smul. -/
-  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_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 <| contr_smul _ _]
-  /- Replacing the contractions. -/
-  rw [add_tensor_eq_fst <| pauliMatrix_contr_down_0_tree]
-  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| pauliMatrix_contr_down_1_tree]
-  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
-    pauliMatrix_contr_down_2_tree]
-  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq <|
-    pauliMatrix_contr_down_3_tree]
-  /- Simplifying -/
-  simp only [add_tensor, smul_tensor, tensorNode_tensor, smul_add,_root_.smul_smul]
-  simp only [Nat.reduceAdd, Fin.isValue, neg_smul, one_smul, mul_neg, neg_mul, one_mul,
-    _root_.neg_neg, mul_one]
+lemma tensoreNode_pauliCo : (tensorNode pauliCo).tensor =
+    {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | ν α β}ᵀ.tensor := by
+  rw [pauliCo]
   rfl
 
-lemma pauliMatrix_lower_tree : {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
+set_option profiler true
+set_option profiler.threshold 10
+lemma pauliCo_basis_expand : pauliCo
+    = basisVector pauliCoMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
+    + basisVector pauliCoMap (fun | 0 => 0 | 1 => 1 | 2 => 1)
+    - basisVector pauliCoMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
+    - basisVector pauliCoMap (fun | 0 => 1 | 1 => 1 | 2 => 0)
+    + I • basisVector pauliCoMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
+    - I • basisVector pauliCoMap (fun | 0 => 2 | 1 => 1 | 2 => 0)
+    - basisVector pauliCoMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
+    + basisVector pauliCoMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
+  conv =>
+    lhs
+    rw [pauliCo]
+    rw [contr_tensor_eq <| prod_tensor_eq_fst <| coMetric_basis_expand_tree]
+    /- Moving the prod through additions. -/
+    rw [contr_tensor_eq <| add_prod _ _ _]
+    rw [contr_tensor_eq <| add_tensor_eq_snd <| add_prod _ _ _]
+    rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
+    /- Moving the prod through smuls. -/
+    rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_prod _ _ _]
+    rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+        <| smul_prod _ _ _]
+    rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+      <| smul_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 _ _]
+    /- Moving contraction through smul. -/
+    rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_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 <| contr_smul _ _]
+    simp only [tensorNode_tensor, add_tensor, smul_tensor]
+    simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, neg_smul, one_smul]
+  conv =>
+    lhs; lhs;
+    rw [pauliMatrix_contr_down_0]
+  conv =>
+    lhs; rhs; lhs; rhs;
+    rw [pauliMatrix_contr_down_1]
+  conv =>
+    lhs; rhs; rhs; lhs; rhs;
+    rw [pauliMatrix_contr_down_2]
+  conv =>
+    lhs; rhs; rhs; rhs; rhs;
+    rw [pauliMatrix_contr_down_3]
+  simp only [neg_smul, one_smul]
+  abel
+
+
+lemma pauliCo_basis_expand_tree : pauliCo
     = (TensorTree.add (tensorNode
-      (basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0))) <|
+      (basisVector pauliCoMap (fun | 0 => 0 | 1 => 0 | 2 => 0))) <|
     TensorTree.add (tensorNode
-      (basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1))) <|
+      (basisVector pauliCoMap (fun | 0 => 0 | 1 => 1 | 2 => 1))) <|
     TensorTree.add (TensorTree.smul (-1) (tensorNode
-      (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)))) <|
+      (basisVector pauliCoMap (fun | 0 => 1 | 1 => 0 | 2 => 1)))) <|
     TensorTree.add (TensorTree.smul (-1) (tensorNode
-      (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)))) <|
+      (basisVector pauliCoMap (fun | 0 => 1 | 1 => 1 | 2 => 0)))) <|
     TensorTree.add (TensorTree.smul I (tensorNode
-      (basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)))) <|
+      (basisVector pauliCoMap (fun | 0 => 2 | 1 => 0 | 2 => 1)))) <|
     TensorTree.add (TensorTree.smul (-I) (tensorNode
-      (basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)))) <|
+      (basisVector pauliCoMap (fun | 0 => 2 | 1 => 1 | 2 => 0)))) <|
     TensorTree.add (TensorTree.smul (-1) (tensorNode
-      (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)))) <|
-      (tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1)))).tensor := by
-  rw [pauliMatrix_lower]
-  simp only [Nat.reduceAdd, Fin.isValue, add_tensor,
-    tensorNode_tensor, smul_tensor, neg_smul, one_smul]
+      (basisVector pauliCoMap (fun | 0 => 3 | 1 => 0 | 2 => 0)))) <|
+      (tensorNode (basisVector pauliCoMap (fun | 0 => 3 | 1 => 1 | 2 => 1)))).tensor := by
+  rw [pauliCo_basis_expand]
+  simp only [Nat.reduceAdd, Fin.isValue, add_tensor, tensorNode_tensor, smul_tensor, neg_smul,
+    one_smul]
   rfl
 
-lemma pauliMatrix_lower_basis_expand_prod {n : ℕ} {c : Fin n → complexLorentzTensor.C}
+lemma pauliCo_prod_basis_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
     (t : TensorTree complexLorentzTensor c) :
-    (prod {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ t).tensor =
+    (prod (tensorNode pauliCo) t).tensor =
     (((tensorNode
-      (basisVector pauliMatrixLowerMap fun | 0 => 0 | 1 => 0 | 2 => 0)).prod t).add
+      (basisVector pauliCoMap fun | 0 => 0 | 1 => 0 | 2 => 0)).prod t).add
     (((tensorNode
-      (basisVector pauliMatrixLowerMap fun | 0 => 0 | 1 => 1 | 2 => 1)).prod t).add
+      (basisVector pauliCoMap 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
+      (basisVector pauliCoMap 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
+      (basisVector pauliCoMap 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
+      (basisVector pauliCoMap 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
+      (basisVector pauliCoMap 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
+      (basisVector pauliCoMap fun | 0 => 3 | 1 => 0 | 2 => 0)).prod t)).add
     ((tensorNode
-      (basisVector pauliMatrixLowerMap fun | 0 => 3 | 1 => 1 | 2 => 1)).prod
+      (basisVector pauliCoMap fun | 0 => 3 | 1 => 1 | 2 => 1)).prod
       t)))))))).tensor := by
-  rw [prod_tensor_eq_fst <| pauliMatrix_lower_tree]
+  rw [prod_tensor_eq_fst <| tensoreNode_pauliCo]
+  rw [prod_tensor_eq_fst <| pauliCo_basis_expand_tree]
   /- Moving the prod through additions. -/
   rw [add_prod _ _ _]
   rw [add_tensor_eq_snd <| add_prod _ _ _]
diff --git a/HepLean/Tensors/Tree/Basic.lean b/HepLean/Tensors/Tree/Basic.lean
index 8941602..013c1be 100644
--- a/HepLean/Tensors/Tree/Basic.lean
+++ b/HepLean/Tensors/Tree/Basic.lean
@@ -770,10 +770,63 @@ lemma smul_tensor_eq {T1 T2 : TensorTree S c} {a : S.k} (h : T1.tensor = T2.tens
   simp only [smul_tensor]
   rw [h]
 
+lemma smul_mul_eq {T1 : TensorTree S c} {a b : S.k} (h : a = b) :
+    (smul a T1).tensor = (smul b T1).tensor := by
+  rw [h]
+
 lemma eq_tensorNode_of_eq_tensor {T1 : TensorTree S c} {t : S.F.obj (OverColor.mk c)}
     (h : T1.tensor = t) : T1.tensor = (tensorNode t).tensor := by
   simpa using h
 
+/-!
+
+## The zero tensor tree
+
+-/
+
+/-- The zero tensor tree. -/
+def zeroTree {n : ℕ} {c : Fin n → S.C} : TensorTree S c := tensorNode 0
+
+@[simp]
+lemma zeroTree_tensor {n : ℕ} {c : Fin n → S.C} : (zeroTree (c := c)).tensor = 0 := by
+  rfl
+
+lemma zero_smul {T1  : TensorTree S c} :
+    (smul 0 T1).tensor = zeroTree.tensor := by
+  simp only [smul_tensor, _root_.zero_smul, zeroTree_tensor]
+
+lemma smul_zero {a : S.k} : (smul a (zeroTree (c :=c ))).tensor = zeroTree.tensor := by
+  simp only [smul_tensor, zeroTree_tensor, _root_.smul_zero]
+
+lemma zero_add {T1 : TensorTree S c} : (add zeroTree T1).tensor = T1.tensor := by
+  simp only [add_tensor, zeroTree_tensor, _root_.zero_add]
+
+lemma add_zero {T1 : TensorTree S c} : (add T1 zeroTree).tensor = T1.tensor := by
+  simp only [add_tensor, zeroTree_tensor, _root_.add_zero]
+
+lemma perm_zero {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C} (σ : (OverColor.mk c) ⟶
+    (OverColor.mk c1)) : (perm σ zeroTree).tensor = zeroTree.tensor := by
+  simp only [perm_tensor, zeroTree_tensor, map_zero]
+
+lemma neg_zero  : (neg (zeroTree (c := c))).tensor = zeroTree.tensor := by
+  simp only [neg_tensor, zeroTree_tensor, _root_.neg_zero]
+
+lemma contr_zero  {n : ℕ} {c : Fin n.succ.succ → S.C} {i : Fin n.succ.succ} {j : Fin n.succ}
+    {h : c (i.succAbove j) = S.τ (c i)}  : (contr i j h zeroTree).tensor = zeroTree.tensor := by
+  simp only [contr_tensor, zeroTree_tensor, map_zero]
+
+lemma zero_prod {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C} (t : TensorTree S c1) :
+    (prod (zeroTree (c := c)) t).tensor = zeroTree.tensor := by
+  simp only [prod_tensor, Functor.id_obj, OverColor.mk_hom, Action.instMonoidalCategory_tensorObj_V,
+    Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
+    Action.FunctorCategoryEquivalence.functor_obj_obj, zeroTree_tensor, zero_tmul, map_zero]
+
+lemma prod_zero {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C} (t : TensorTree S c) :
+    (prod t (zeroTree (c := c1))).tensor = zeroTree.tensor := by
+  simp only [prod_tensor, Functor.id_obj, OverColor.mk_hom, Action.instMonoidalCategory_tensorObj_V,
+    Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
+    Action.FunctorCategoryEquivalence.functor_obj_obj, zeroTree_tensor, tmul_zero, map_zero]
+
 /-- A structure containing a pair of indices (i, j) to be contracted in a tensor.
   This is used in some proofs of node identities for tensor trees. -/
 structure ContrPair {n : ℕ} (c : Fin n.succ.succ → S.C) where
diff --git a/HepLean/Tensors/Tree/Elab.lean b/HepLean/Tensors/Tree/Elab.lean
index 07d217e..ac1a863 100644
--- a/HepLean/Tensors/Tree/Elab.lean
+++ b/HepLean/Tensors/Tree/Elab.lean
@@ -23,7 +23,7 @@ open Lean.Elab
 open Lean.Elab.Term
 open Lean Meta Elab Tactic
 open IndexNotation
-
+open complexLorentzTensor
 namespace TensorTree
 
 /-!
@@ -182,60 +182,60 @@ def stringToTerm (str : String) : TermElabM Term := do
 /-- Specific types of tensors which appear which we want to elaborate in specific ways. -/
 def specialTypes : List (String × (Term → Term)) := [
   ("CoeSort.coe Lorentz.complexCo", fun T =>
-    Syntax.mkApp (mkIdent ``TensorTree.vecNodeE) #[mkIdent ``Fermion.complexLorentzTensor,
-      mkIdent ``Fermion.Color.down, T]),
+    Syntax.mkApp (mkIdent ``TensorTree.vecNodeE) #[mkIdent ``complexLorentzTensor,
+      mkIdent ``complexLorentzTensor.Color.down, T]),
   ("CoeSort.coe Lorentz.complexContr", fun T =>
-    Syntax.mkApp (mkIdent ``TensorTree.vecNodeE) #[mkIdent ``Fermion.complexLorentzTensor,
-      mkIdent ``Fermion.Color.up, T]),
+    Syntax.mkApp (mkIdent ``TensorTree.vecNodeE) #[mkIdent ``complexLorentzTensor,
+      mkIdent ``complexLorentzTensor.Color.up, T]),
   ("ModuleCat.carrier (Lorentz.complexContr ⊗ Lorentz.complexCo).V", fun T =>
-    Syntax.mkApp (mkIdent ``TensorTree.twoNodeE) #[mkIdent ``Fermion.complexLorentzTensor,
-        mkIdent ``Fermion.Color.up, mkIdent ``Fermion.Color.down, T]),
+    Syntax.mkApp (mkIdent ``TensorTree.twoNodeE) #[mkIdent ``complexLorentzTensor,
+        mkIdent ``complexLorentzTensor.Color.up, mkIdent ``complexLorentzTensor.Color.down, T]),
   ("ModuleCat.carrier (Lorentz.complexContr ⊗ Lorentz.complexContr).V", fun T =>
-    Syntax.mkApp (mkIdent ``TensorTree.twoNodeE) #[mkIdent ``Fermion.complexLorentzTensor,
-        mkIdent ``Fermion.Color.up, mkIdent ``Fermion.Color.up, T]),
+    Syntax.mkApp (mkIdent ``TensorTree.twoNodeE) #[mkIdent ``complexLorentzTensor,
+        mkIdent ``complexLorentzTensor.Color.up, mkIdent ``complexLorentzTensor.Color.up, T]),
   ("ModuleCat.carrier (Lorentz.complexCo ⊗ Lorentz.complexCo).V", fun T =>
-    Syntax.mkApp (mkIdent ``TensorTree.twoNodeE) #[mkIdent ``Fermion.complexLorentzTensor,
-        mkIdent ``Fermion.Color.down, mkIdent ``Fermion.Color.down, T]),
+    Syntax.mkApp (mkIdent ``TensorTree.twoNodeE) #[mkIdent ``complexLorentzTensor,
+        mkIdent ``complexLorentzTensor.Color.down, mkIdent ``complexLorentzTensor.Color.down, T]),
   ("ModuleCat.carrier (Lorentz.complexCo ⊗ Lorentz.complexContr).V", fun T =>
     Syntax.mkApp (mkIdent ``TensorTree.twoNodeE) #[
-        mkIdent ``Fermion.complexLorentzTensor,
-        mkIdent ``Fermion.Color.down,
-        mkIdent ``Fermion.Color.up, T]),
+        mkIdent ``complexLorentzTensor,
+        mkIdent ``complexLorentzTensor.Color.down,
+        mkIdent ``complexLorentzTensor.Color.up, T]),
   ("𝟙_ (Rep ℂ SL(2, ℂ)) ⟶ Lorentz.complexCo ⊗ Lorentz.complexCo", fun T =>
     Syntax.mkApp (mkIdent ``TensorTree.constTwoNodeE) #[
-      mkIdent ``Fermion.complexLorentzTensor,
-      mkIdent ``Fermion.Color.down,
-      mkIdent ``Fermion.Color.down, T]),
+      mkIdent ``complexLorentzTensor,
+      mkIdent ``complexLorentzTensor.Color.down,
+      mkIdent ``complexLorentzTensor.Color.down, T]),
   ("𝟙_ (Rep ℂ SL(2, ℂ)) ⟶ Lorentz.complexContr ⊗ Lorentz.complexContr", fun T =>
     Syntax.mkApp (mkIdent ``TensorTree.constTwoNodeE) #[
-      mkIdent ``Fermion.complexLorentzTensor,
-      mkIdent ``Fermion.Color.up,
-      mkIdent ``Fermion.Color.up, T]),
+      mkIdent ``complexLorentzTensor,
+      mkIdent ``complexLorentzTensor.Color.up,
+      mkIdent ``complexLorentzTensor.Color.up, T]),
   ("𝟙_ (Rep ℂ SL(2, ℂ)) ⟶ Lorentz.complexContr ⊗ Fermion.leftHanded ⊗ Fermion.rightHanded", fun T =>
     Syntax.mkApp (mkIdent ``TensorTree.constThreeNodeE) #[
-      mkIdent ``Fermion.complexLorentzTensor, mkIdent ``Fermion.Color.up,
-      mkIdent ``Fermion.Color.upL,
-      mkIdent ``Fermion.Color.upR, T]),
+      mkIdent ``complexLorentzTensor, mkIdent ``complexLorentzTensor.Color.up,
+      mkIdent ``complexLorentzTensor.Color.upL,
+      mkIdent ``complexLorentzTensor.Color.upR, T]),
   ("𝟙_ (Rep ℂ SL(2, ℂ)) ⟶ Fermion.leftHanded ⊗ Fermion.leftHanded", fun T =>
     Syntax.mkApp (mkIdent ``TensorTree.constTwoNodeE) #[
-      mkIdent ``Fermion.complexLorentzTensor,
-      mkIdent ``Fermion.Color.upL,
-      mkIdent ``Fermion.Color.upL, T]),
+      mkIdent ``complexLorentzTensor,
+      mkIdent ``complexLorentzTensor.Color.upL,
+      mkIdent ``complexLorentzTensor.Color.upL, T]),
   ("𝟙_ (Rep ℂ SL(2, ℂ)) ⟶ Fermion.altLeftHanded ⊗ Fermion.altLeftHanded", fun T =>
     Syntax.mkApp (mkIdent ``TensorTree.constTwoNodeE) #[
-      mkIdent ``Fermion.complexLorentzTensor,
-      mkIdent ``Fermion.Color.downL,
-      mkIdent ``Fermion.Color.downL, T]),
+      mkIdent ``complexLorentzTensor,
+      mkIdent ``complexLorentzTensor.Color.downL,
+      mkIdent ``complexLorentzTensor.Color.downL, T]),
   ("𝟙_ (Rep ℂ SL(2, ℂ)) ⟶ Fermion.altRightHanded ⊗ Fermion.altRightHanded", fun T =>
     Syntax.mkApp (mkIdent ``TensorTree.constTwoNodeE) #[
-      mkIdent ``Fermion.complexLorentzTensor,
-      mkIdent ``Fermion.Color.downR,
-      mkIdent ``Fermion.Color.downR, T]),
+      mkIdent ``complexLorentzTensor,
+      mkIdent ``complexLorentzTensor.Color.downR,
+      mkIdent ``complexLorentzTensor.Color.downR, T]),
   ("𝟙_ (Rep ℂ SL(2, ℂ)) ⟶ Fermion.rightHanded ⊗ Fermion.rightHanded", fun T =>
     Syntax.mkApp (mkIdent ``TensorTree.constTwoNodeE) #[
-      mkIdent ``Fermion.complexLorentzTensor,
-      mkIdent ``Fermion.Color.upR,
-      mkIdent ``Fermion.Color.upR, T])]
+      mkIdent ``complexLorentzTensor,
+      mkIdent ``complexLorentzTensor.Color.upR,
+      mkIdent ``complexLorentzTensor.Color.upR, T])]
 
 /-- The syntax associated with a terminal node of a tensor tree. -/
 def termNodeSyntax (T : Term) : TermElabM Term := do