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refactor: Fix problem with elab and do lint

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jstoobysmith 2024-10-24 07:36:54 +00:00
parent 95857993b5
commit 1e8efdb16a
6 changed files with 217 additions and 152 deletions
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325
HepLean/Tensors/ComplexLorentz/Lemmas.lean
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@ -119,7 +119,8 @@ And related results.
-/
open complexLorentzTensor
def leftMetricMulRightMap := (Sum.elim ![Color.upL, Color.upL] ![Color.upR, Color.upR]) ∘ finSumFinEquiv.symm
def leftMetricMulRightMap := (Sum.elim ![Color.upL, Color.upL] ![Color.upR, Color.upR]) ∘
finSumFinEquiv.symm
lemma leftMetric_mul_rightMetric : {Fermion.leftMetric | α α' ⊗ Fermion.rightMetric | β β'}ᵀ.tensor
= basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
@ -141,7 +142,7 @@ lemma leftMetric_mul_rightMetric : {Fermion.leftMetric | α α' ⊗ Fermion.righ
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| prod_basisVector_tree _ _]
rw [← add_assoc]
simp only [add_tensor, smul_tensor, tensorNode_tensor]
change _ = basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
change _ = basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
+- basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
+- basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
+ basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)
@ -153,10 +154,11 @@ lemma leftMetric_mul_rightMetric : {Fermion.leftMetric | α α' ⊗ Fermion.righ
funext x
fin_cases x <;> rfl
def pauliMatrixLowerMap := ((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
Fin.succAbove 0 ∘ Fin.succAbove 1)
def pauliMatrixLowerMap := ((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘
⇑finSumFinEquiv.symm) ∘ Fin.succAbove 0 ∘ Fin.succAbove 1)
abbrev pauliMatrixContrMap {n : ℕ} (c : Fin n → complexLorentzTensor.C) := (Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm)
abbrev pauliMatrixContrMap {n : ℕ} (c : Fin n → complexLorentzTensor.C) :=
(Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm)
lemma prod_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
(t : TensorTree complexLorentzTensor c) :
@ -164,12 +166,12 @@ lemma prod_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentz
PauliMatrix.asConsTensor)).tensor = (((t.prod (tensorNode
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 0 | 2 => 0)))).add
(((t.prod (tensorNode
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 1 | 2 => 1)))).add
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 1 | 2 => 1)))).add
(((t.prod (tensorNode
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 0 | 2 => 1)))).add
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 0 | 2 => 1)))).add
(((t.prod (tensorNode
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 1 | 2 => 0)))).add
((TensorTree.smul (-I) ((t.prod (tensorNode
((TensorTree.smul (-I) ((t.prod (tensorNode
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 0 | 2 => 1))))).add
((TensorTree.smul I ((t.prod (tensorNode
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 1 | 2 => 0))))).add
@ -180,7 +182,7 @@ lemma prod_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentz
fun | 0 => 3 | 1 => 1 | 2 => 1))))))))))).tensor := by
rw [prod_tensor_eq_snd <| pauliMatrix_basis_expand_tree]
rw [prod_add _ _ _]
rw [add_tensor_eq_snd <| prod_add _ _ _]
rw [add_tensor_eq_snd <| prod_add _ _ _]
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
@ -188,39 +190,41 @@ lemma prod_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentz
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
<| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_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 <| prod_add _ _ _]
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
/- Moving smuls. -/
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
<| add_tensor_eq_snd <| add_tensor_eq_fst <| prod_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 <| prod_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_snd <| add_tensor_eq_snd<| add_tensor_eq_snd
<| add_tensor_eq_snd <| prod_smul _ _ _]
rfl
lemma contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
(t : TensorTree complexLorentzTensor c) (i : Fin (n + 3)) (j : Fin (n +2))
(h : (pauliMatrixContrMap c) (i.succAbove j) = complexLorentzTensor.τ ((pauliMatrixContrMap c) i)) :
(contr i j h (TensorTree.prod t (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
(h : (pauliMatrixContrMap c) (i.succAbove j) =
complexLorentzTensor.τ ((pauliMatrixContrMap c) i)) :
(contr i j h (TensorTree.prod t
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor))).tensor =
((contr i j h (t.prod (tensorNode
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 0 | 2 => 0)))).add
((contr i j h (t.prod (tensorNode
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 1 | 2 => 1)))).add
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 1 | 2 => 1)))).add
((contr i j h (t.prod (tensorNode
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 0 | 2 => 1)))).add
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 0 | 2 => 1)))).add
((contr i j h (t.prod (tensorNode
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 1 | 2 => 0)))).add
((TensorTree.smul (-I) (contr i j h (t.prod (tensorNode
((TensorTree.smul (-I) (contr i j h (t.prod (tensorNode
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 0 | 2 => 1))))).add
((TensorTree.smul I (contr i j h (t.prod (tensorNode
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 1 | 2 => 0))))).add
((contr i j h (t.prod (tensorNode
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 3 | 1 => 0 | 2 => 0)))).add
(TensorTree.smul (-1) (contr i j h (t.prod (tensorNode
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 3 | 1 => 1 | 2 => 1)))))))))))).tensor := by
(basisVector ![Color.up, Color.upL, Color.upR]
fun | 0 => 3 | 1 => 1 | 2 => 1)))))))))))).tensor := by
rw [contr_tensor_eq <| prod_pauliMatrix_basis_tree_expand _]
/- Moving contr over add. -/
rw [contr_add]
@ -244,12 +248,15 @@ lemma contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorent
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 ((pauliMatrixContrMap c) i))
(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
(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
let b' (i1 i2 i3 : Fin 4) := fun i => prodBasisVecEquiv (finSumFinEquiv.symm i)
((HepLean.PiTensorProduct.elimPureTensor b (fun | 0 => i1 | 1 => i2 | 2 => i3)) (finSumFinEquiv.symm i))
(contr i j h (TensorTree.prod (tensorNode (basisVector c b)) (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
((HepLean.PiTensorProduct.elimPureTensor b (fun | 0 => i1 | 1 => i2 | 2 => i3))
(finSumFinEquiv.symm i))
(contr i j h (TensorTree.prod (tensorNode (basisVector c b))
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor))).tensor = ((contr i j h ((tensorNode
(basisVector c' (b' 0 0 0))))).add
((contr i j h ((tensorNode (basisVector c' (b' 0 1 1))))).add
@ -276,31 +283,41 @@ lemma basis_contr_pauliMatrix_basis_tree_expand' {n : ℕ} {c : Fin n → comple
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_tensor_eq
<| prod_basisVector_tree _ _]
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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_tensor_eq <| prod_basisVector_tree _ _]
rfl
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 ((pauliMatrixContrMap c) i))
(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
(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) : (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))
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))
(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)) (tensorNode (basisVector c' (b' 0 0 0))))).add
(((TensorTree.smul (contrBasisVectorMul i j (b'' 0 1 1)) (tensorNode (basisVector c' (b' 0 1 1))))).add
(((TensorTree.smul (contrBasisVectorMul i j (b'' 1 0 1)) (tensorNode (basisVector c' (b' 1 0 1))))).add
(((TensorTree.smul (contrBasisVectorMul i j (b'' 1 1 0)) (tensorNode (basisVector c' (b' 1 1 0))))).add
((TensorTree.smul (-I) ((TensorTree.smul (contrBasisVectorMul i j (b'' 2 0 1)) (tensorNode (basisVector c' (b' 2 0 1)))))).add
((TensorTree.smul I ((TensorTree.smul (contrBasisVectorMul i j (b'' 2 1 0)) (tensorNode (basisVector c' (b' 2 1 0)))))).add
(((TensorTree.smul (contrBasisVectorMul i j (b'' 3 0 0)) (tensorNode (basisVector c' (b' 3 0 0))))).add
TensorTree.smul (contrBasisVectorMul i j (b'' 0 0 0))
(tensorNode (basisVector c' (b' 0 0 0))))).add
(((TensorTree.smul (contrBasisVectorMul i j (b'' 0 1 1))
(tensorNode (basisVector c' (b' 0 1 1))))).add
(((TensorTree.smul (contrBasisVectorMul i j (b'' 1 0 1))
(tensorNode (basisVector c' (b' 1 0 1))))).add
(((TensorTree.smul (contrBasisVectorMul i j (b'' 1 1 0))
(tensorNode (basisVector c' (b' 1 1 0))))).add
((TensorTree.smul (-I) ((TensorTree.smul (contrBasisVectorMul i j (b'' 2 0 1))
(tensorNode (basisVector c' (b' 2 0 1)))))).add
((TensorTree.smul I ((TensorTree.smul (contrBasisVectorMul i j (b'' 2 1 0))
(tensorNode (basisVector c' (b' 2 1 0)))))).add
(((TensorTree.smul (contrBasisVectorMul i j (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
(basisVector c' (b' 3 1 1))))))))))))).tensor := by
rw [basis_contr_pauliMatrix_basis_tree_expand']
@ -319,8 +336,8 @@ 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_fst <| contr_basisVector_tree _]
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 _]
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
smul_tensor_eq <| contr_basisVector_tree _]
lemma pauliMatrix_contr_down_0 :
(contr 0 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
@ -344,7 +361,7 @@ lemma pauliMatrix_contr_down_0 :
fin_cases k <;> rfl
lemma pauliMatrix_contr_down_0_tree :
(contr 0 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
(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
@ -352,10 +369,9 @@ lemma pauliMatrix_contr_down_0_tree :
(tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)))).tensor := by
exact pauliMatrix_contr_down_0
lemma pauliMatrix_contr_down_1 : (contr 0 1 rfl
(((tensorNode (basisVector ![Color.down, Color.down] fun x => 1)).prod
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor)))).tensor
lemma pauliMatrix_contr_down_1 :
{(basisVector ![Color.down, Color.down] fun x => 1) | μ ν ⊗
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]
@ -373,20 +389,19 @@ lemma pauliMatrix_contr_down_1 : (contr 0 1 rfl
funext k
fin_cases k <;> rfl
lemma pauliMatrix_contr_down_1_tree : (contr 0 1 rfl
(((tensorNode (basisVector ![Color.down, Color.down] fun x => 1)).prod
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor)))).tensor
= (TensorTree.add (tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)))
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 : (contr 0 1 rfl
(((tensorNode (basisVector ![Color.down, Color.down] fun x => 2)).prod
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor)))).tensor
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
+ (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,
@ -403,21 +418,19 @@ lemma pauliMatrix_contr_down_2 : (contr 0 1 rfl
funext k
fin_cases k <;> rfl
lemma pauliMatrix_contr_down_2_tree : (contr 0 1 rfl
(((tensorNode (basisVector ![Color.down, Color.down] fun x => 2)).prod
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor)))).tensor =
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 : (contr 0 1 rfl
(((tensorNode (basisVector ![Color.down, Color.down] fun x => 3)).prod
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor)))).tensor
= basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
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,
@ -435,27 +448,23 @@ lemma pauliMatrix_contr_down_3 : (contr 0 1 rfl
funext k
fin_cases k <;> rfl
lemma pauliMatrix_contr_down_3_tree : (contr 0 1 rfl
(((tensorNode (basisVector ![Color.down, Color.down] fun x => 3)).prod
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor)))).tensor =
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
(smul (-1) (tensorNode (basisVector pauliMatrixLowerMap
(fun | 0 => 3 | 1 => 1 | 2 => 1))))).tensor := by
exact pauliMatrix_contr_down_3
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)
((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 : (contr 0 2 rfl
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0))).prod
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor)))).tensor = basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
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,
@ -473,10 +482,10 @@ lemma pauliMatrix_contr_lower_0_0_0 : (contr 0 2 rfl
funext k
fin_cases k <;> rfl
lemma pauliMatrix_contr_lower_0_1_1 : (contr 0 2 rfl
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1))).prod
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor)))).tensor = basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
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,
@ -494,10 +503,10 @@ lemma pauliMatrix_contr_lower_0_1_1 : (contr 0 2 rfl
funext k
fin_cases k <;> rfl
lemma pauliMatrix_contr_lower_1_0_1 : (contr 0 2 rfl
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1))).prod
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor)))).tensor = basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
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,
@ -515,10 +524,10 @@ lemma pauliMatrix_contr_lower_1_0_1 : (contr 0 2 rfl
funext k
fin_cases k <;> rfl
lemma pauliMatrix_contr_lower_1_1_0 : (contr 0 2 rfl
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0))).prod
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor)))).tensor = basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
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,
@ -536,12 +545,12 @@ lemma pauliMatrix_contr_lower_1_1_0 : (contr 0 2 rfl
funext k
fin_cases k <;> rfl
lemma pauliMatrix_contr_lower_2_0_1 : (contr 0 2 rfl
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1))).prod
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor)))).tensor =
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
+ (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,
@ -558,12 +567,12 @@ lemma pauliMatrix_contr_lower_2_0_1 : (contr 0 2 rfl
funext k
fin_cases k <;> rfl
lemma pauliMatrix_contr_lower_2_1_0 : (contr 0 2 rfl
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0))).prod
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor)))).tensor =
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
+ (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,
@ -580,13 +589,12 @@ lemma pauliMatrix_contr_lower_2_1_0 : (contr 0 2 rfl
funext k
fin_cases k <;> rfl
lemma pauliMatrix_contr_lower_3_0_0 : (contr 0 2 rfl
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0))).prod
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor)))).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
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,
@ -603,12 +611,12 @@ lemma pauliMatrix_contr_lower_3_0_0 : (contr 0 2 rfl
funext k
fin_cases k <;> rfl
lemma pauliMatrix_contr_lower_3_1_1 : (contr 0 2 rfl
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1))).prod
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
PauliMatrix.asConsTensor)))).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
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,
@ -656,8 +664,10 @@ lemma pauliMatrix_lower : {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor
/- 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]
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,
@ -686,7 +696,8 @@ lemma pauliMatrix_lower_tree : {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsT
rfl
lemma pauliMatrix_contract_pauliMatrix_aux :
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗ PauliMatrix.asConsTensor | ν α' β'}ᵀ.tensor
{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
@ -701,57 +712,89 @@ lemma pauliMatrix_contract_pauliMatrix_aux :
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
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
(-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
(-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap
fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1))))))))).tensor := by
rw [contr_tensor_eq <| prod_tensor_eq_fst <| pauliMatrix_lower_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 _ _ _]
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
rw [contr_tensor_eq <| 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_prod _ _ _]
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
add_prod _ _ _]
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
add_tensor_eq_snd <| add_prod _ _ _]
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
rw [contr_tensor_eq <| 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_prod _ _ _]
/- Moving the prod through smuls. -/
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 <| add_tensor_eq_fst <| smul_prod _ _ _]
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| 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 <| 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 <| add_tensor_eq_snd <| 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_fst <|
smul_prod _ _ _]
rw [contr_tensor_eq <| add_tensor_eq_snd <| 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 <|
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 <|
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 <|
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
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 _ _]
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 _ _]
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 _ _]
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]
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 :
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗ PauliMatrix.asConsTensor | ν α' β'}ᵀ.tensor =
{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)
@ -762,7 +805,7 @@ lemma pauliMatrix_contract_pauliMatrix :
neg_mul, _root_.neg_neg]
ring_nf
rw [Complex.I_sq]
simp only [ neg_smul, one_smul, _root_.neg_neg]
simp only [neg_smul, one_smul, _root_.neg_neg]
abel
end Fermion