refactor: Lint
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jstoobysmith
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4 changed files with 33 additions and 38 deletions
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@ -209,6 +209,5 @@ lemma basis_contr (c : complexLorentzTensor.C) (i : Fin (complexLorentzTensor.re
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| Color.up => Lorentz.contrCoContraction_basis _ _
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| Color.down => Lorentz.coContrContraction_basis _ _
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end complexLorentzTensor
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end
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@ -177,7 +177,6 @@ lemma basis_contr_pauliMatrix_basis_tree_expand' {n : ℕ} {c : Fin n → comple
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<| contr_tensor_eq <| prod_basisVector_tree _ _]
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rfl
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def pauliMatrixBasisProdMap
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{n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(b : Π k, Fin (complexLorentzTensor.repDim (c k))) (i1 i2 i3 : Fin 4) :
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@ -190,10 +189,11 @@ def pauliMatrixBasisProdMap
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def basisVectorContrPauli {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(i : Fin (n + 3)) (j : Fin (n +2))
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(b : Π k, Fin (complexLorentzTensor.repDim (c k)))
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(i1 i2 i3 : Fin 4) :=
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(i1 i2 i3 : Fin 4) :=
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let c' := (Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ finSumFinEquiv.symm)
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∘ Fin.succAbove i ∘ Fin.succAbove j
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let b' (i1 i2 i3 : Fin 4) := fun k => (pauliMatrixBasisProdMap b i1 i2 i3) (i.succAbove (j.succAbove k))
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let b' (i1 i2 i3 : Fin 4) := fun k => (pauliMatrixBasisProdMap b i1 i2 i3)
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(i.succAbove (j.succAbove k))
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basisVector c' (b' i1 i2 i3)
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lemma basis_contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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@ -203,7 +203,8 @@ lemma basis_contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complex
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(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
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let c' := (Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ finSumFinEquiv.symm)
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∘ Fin.succAbove i ∘ Fin.succAbove j
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let b' (i1 i2 i3 : Fin 4) := fun k => (pauliMatrixBasisProdMap b i1 i2 i3) (i.succAbove (j.succAbove k))
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let b' (i1 i2 i3 : Fin 4) := fun k => (pauliMatrixBasisProdMap b i1 i2 i3)
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(i.succAbove (j.succAbove k))
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(contr i j h (TensorTree.prod (tensorNode (basisVector c b))
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(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
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PauliMatrix.asConsTensor))).tensor = (((
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@ -215,13 +216,16 @@ lemma basis_contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complex
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(tensorNode (basisVector c' (b' 1 0 1))))).add
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(((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 1 1 0))
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(tensorNode (basisVector c' (b' 1 1 0))))).add
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((TensorTree.smul (-I) ((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 0 1))
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((TensorTree.smul (-I) ((TensorTree.smul
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(contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 0 1))
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(tensorNode (basisVector c' (b' 2 0 1)))))).add
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((TensorTree.smul I ((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 1 0))
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((TensorTree.smul I ((TensorTree.smul
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(contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 1 0))
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(tensorNode (basisVector c' (b' 2 1 0)))))).add
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(((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 0 0))
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(tensorNode (basisVector c' (b' 3 0 0))))).add
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(TensorTree.smul (-1) ((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 1 1)) (tensorNode
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(TensorTree.smul (-1) ((TensorTree.smul
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(contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 1 1)) (tensorNode
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(basisVector c' (b' 3 1 1))))))))))))).tensor := by
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rw [basis_contr_pauliMatrix_basis_tree_expand']
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/- Contracting basis vectors. -/
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@ -243,35 +247,30 @@ lemma basis_contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complex
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smul_tensor_eq <| contr_basisVector_tree _]
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rfl
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def pauilMatrixBasisContrMap {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(i : Fin (n + 3)) (j : Fin (n +2))
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(b : Π k, Fin (complexLorentzTensor.repDim (c k))) (i1 i2 i3 : Fin 4) :
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(k : Fin (n + 1)) →
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Fin
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(complexLorentzTensor.repDim
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(Sum.elim c ![Color.up, Color.upL, Color.upR] (finSumFinEquiv.symm (i.succAbove (j.succAbove k))))) :=
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fun k => (pauliMatrixBasisProdMap b i1 i2 i3) (i.succAbove (j.succAbove k))
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lemma basis_contr_pauliMatrix_basis_tree_expand_tensor {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(i : Fin (n + 3)) (j : Fin (n +2))
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(h : (pauliMatrixContrMap c) (i.succAbove j) = complexLorentzTensor.τ
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((pauliMatrixContrMap c) i))
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(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
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let c' := (Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ finSumFinEquiv.symm)
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∘ Fin.succAbove i ∘ Fin.succAbove j
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let b' (i1 i2 i3 : Fin 4) := fun k => (pauliMatrixBasisProdMap b i1 i2 i3) (i.succAbove (j.succAbove k))
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(contr i j h (TensorTree.prod (tensorNode (basisVector c b))
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(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
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PauliMatrix.asConsTensor))).tensor =
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(contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 0 0)) • (basisVectorContrPauli i j b 0 0 0)
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+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 1 1)) • (basisVectorContrPauli i j b 0 1 1)
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+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 1 0 1)) • (basisVectorContrPauli i j b 1 0 1)
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+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 1 1 0)) • (basisVectorContrPauli i j b 1 1 0)
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+ (-I) • (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 0 1)) • (basisVectorContrPauli i j b 2 0 1)
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+ I • (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 1 0)) • (basisVectorContrPauli i j b 2 1 0)
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+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 0 0)) • (basisVectorContrPauli i j b 3 0 0)
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+ (-1 : ℂ) • (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 1 1)) • (basisVectorContrPauli i j b 3 1 1) := by
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(contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 0 0)) •
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(basisVectorContrPauli i j b 0 0 0)
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+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 1 1)) •
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(basisVectorContrPauli i j b 0 1 1)
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+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 1 0 1)) •
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(basisVectorContrPauli i j b 1 0 1)
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+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 1 1 0)) •
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(basisVectorContrPauli i j b 1 1 0)
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+ (-I) • (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 0 1)) •
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(basisVectorContrPauli i j b 2 0 1)
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+ I • (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 1 0)) •
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(basisVectorContrPauli i j b 2 1 0)
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+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 0 0)) •
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(basisVectorContrPauli i j b 3 0 0)
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+ (-1 : ℂ) • (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 1 1)) •
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(basisVectorContrPauli i j b 3 1 1) := by
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rw [basis_contr_pauliMatrix_basis_tree_expand]
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, cons_val_one, head_cons, Fin.val_zero,
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Nat.cast_zero, cons_val_two, Fin.val_one, Nat.cast_one, add_tensor, smul_tensor,
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@ -279,8 +278,6 @@ lemma basis_contr_pauliMatrix_basis_tree_expand_tensor {n : ℕ} {c : Fin n →
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simp_all only [Function.comp_apply, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue]
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rfl
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end complexLorentzTensor
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end
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@ -76,7 +76,7 @@ lemma pauliMatrix_contr_down_0 :
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fin_cases k <;> rfl
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lemma pauliMatrix_contr_down_1 :
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{(basisVector ![Color.down, Color.down] fun x => 1) | ν μ ⊗
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{(basisVector ![Color.down, Color.down] fun x => 1) | ν μ ⊗
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PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
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= basisVector pauliCoMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
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+ basisVector pauliCoMap (fun | 0 => 1 | 1 => 1 | 2 => 0) := by
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@ -268,7 +268,6 @@ lemma pauliCo_basis_expand : pauliCo
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simp only [neg_smul, one_smul]
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abel
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lemma pauliCo_basis_expand_tree : pauliCo
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= (TensorTree.add (tensorNode
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(basisVector pauliCoMap (fun | 0 => 0 | 1 => 0 | 2 => 0))) <|
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