refactor: Rename ofCrAnFieldOp to ofCrAnOp
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jstoobysmith
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8cc273fe38
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2d561dd89d
7 changed files with 165 additions and 165 deletions
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@ -27,16 +27,16 @@ variable {𝓕 : FieldSpecification}
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lemma normalOrder_eq_ι_normalOrderF (a : 𝓕.FieldOpFreeAlgebra) :
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𝓝(ι a) = ι 𝓝ᶠ(a) := rfl
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lemma normalOrder_ofCrAnFieldOpList (φs : List 𝓕.CrAnFieldOp) :
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𝓝(ofCrAnFieldOpList φs) = normalOrderSign φs • ofCrAnFieldOpList (normalOrderList φs) := by
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rw [ofCrAnFieldOpList, normalOrder_eq_ι_normalOrderF, normalOrderF_ofCrAnListF]
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lemma normalOrder_ofCrAnOpList (φs : List 𝓕.CrAnFieldOp) :
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𝓝(ofCrAnOpList φs) = normalOrderSign φs • ofCrAnOpList (normalOrderList φs) := by
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rw [ofCrAnOpList, normalOrder_eq_ι_normalOrderF, normalOrderF_ofCrAnListF]
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rfl
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@[simp]
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lemma normalOrder_one_eq_one : normalOrder (𝓕 := 𝓕) 1 = 1 := by
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have h1 : 1 = ofCrAnFieldOpList (𝓕 := 𝓕) [] := by simp [ofCrAnFieldOpList]
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have h1 : 1 = ofCrAnOpList (𝓕 := 𝓕) [] := by simp [ofCrAnOpList]
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rw [h1]
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rw [normalOrder_ofCrAnFieldOpList]
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rw [normalOrder_ofCrAnOpList]
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simp
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@[simp]
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@ -48,14 +48,14 @@ lemma normalOrder_ofFieldOpList_nil : normalOrder (𝓕 := 𝓕) (ofFieldOpList
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simp
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@[simp]
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lemma normalOrder_ofCrAnFieldOpList_nil : normalOrder (𝓕 := 𝓕) (ofCrAnFieldOpList []) = 1 := by
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rw [normalOrder_ofCrAnFieldOpList]
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lemma normalOrder_ofCrAnOpList_nil : normalOrder (𝓕 := 𝓕) (ofCrAnOpList []) = 1 := by
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rw [normalOrder_ofCrAnOpList]
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simp only [normalOrderSign_nil, normalOrderList_nil, one_smul]
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rfl
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lemma ofCrAnFieldOpList_eq_normalOrder (φs : List 𝓕.CrAnFieldOp) :
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ofCrAnFieldOpList (normalOrderList φs) = normalOrderSign φs • 𝓝(ofCrAnFieldOpList φs) := by
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rw [normalOrder_ofCrAnFieldOpList, smul_smul, normalOrderSign, Wick.koszulSign_mul_self,
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lemma ofCrAnOpList_eq_normalOrder (φs : List 𝓕.CrAnFieldOp) :
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ofCrAnOpList (normalOrderList φs) = normalOrderSign φs • 𝓝(ofCrAnOpList φs) := by
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rw [normalOrder_ofCrAnOpList, smul_smul, normalOrderSign, Wick.koszulSign_mul_self,
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one_smul]
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lemma normalOrder_normalOrder_mid (a b c : 𝓕.FieldOpAlgebra) :
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@ -165,16 +165,16 @@ lemma normalOrder_ofFieldOp_ofFieldOp_swap (φ φ' : 𝓕.FieldOp) :
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rw [ofFieldOp_mul_ofFieldOp_eq_superCommute]
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simp
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lemma normalOrder_ofCrAnFieldOp_ofCrAnFieldOpList (φ : 𝓕.CrAnFieldOp)
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(φs : List 𝓕.CrAnFieldOp) : 𝓝(ofCrAnFieldOp φ * ofCrAnFieldOpList φs) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • 𝓝(ofCrAnFieldOpList φs * ofCrAnFieldOp φ) := by
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rw [← ofCrAnFieldOpList_singleton, ofCrAnFieldOpList_mul_ofCrAnFieldOpList_eq_superCommute]
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lemma normalOrder_ofCrAnOp_ofCrAnOpList (φ : 𝓕.CrAnFieldOp)
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(φs : List 𝓕.CrAnFieldOp) : 𝓝(ofCrAnOp φ * ofCrAnOpList φs) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • 𝓝(ofCrAnOpList φs * ofCrAnOp φ) := by
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rw [← ofCrAnOpList_singleton, ofCrAnOpList_mul_ofCrAnOpList_eq_superCommute]
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simp
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lemma normalOrder_ofCrAnFieldOp_ofFieldOpList_swap (φ : 𝓕.CrAnFieldOp) (φ' : List 𝓕.FieldOp) :
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𝓝(ofCrAnFieldOp φ * ofFieldOpList φ') = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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𝓝(ofFieldOpList φ' * ofCrAnFieldOp φ) := by
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rw [← ofCrAnFieldOpList_singleton, ofCrAnFieldOpList_mul_ofFieldOpList_eq_superCommute]
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lemma normalOrder_ofCrAnOp_ofFieldOpList_swap (φ : 𝓕.CrAnFieldOp) (φ' : List 𝓕.FieldOp) :
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𝓝(ofCrAnOp φ * ofFieldOpList φ') = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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𝓝(ofFieldOpList φ' * ofCrAnOp φ) := by
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rw [← ofCrAnOpList_singleton, ofCrAnOpList_mul_ofFieldOpList_eq_superCommute]
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simp
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lemma normalOrder_anPart_ofFieldOpList_swap (φ : 𝓕.FieldOp) (φ' : List 𝓕.FieldOp) :
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@ -184,11 +184,11 @@ lemma normalOrder_anPart_ofFieldOpList_swap (φ : 𝓕.FieldOp) (φ' : List 𝓕
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simp
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| .position φ =>
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simp only [anPart_position, instCommGroup.eq_1]
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rw [normalOrder_ofCrAnFieldOp_ofFieldOpList_swap]
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rw [normalOrder_ofCrAnOp_ofFieldOpList_swap]
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rfl
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| .outAsymp φ =>
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simp only [anPart_posAsymp, instCommGroup.eq_1]
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rw [normalOrder_ofCrAnFieldOp_ofFieldOpList_swap]
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rw [normalOrder_ofCrAnOp_ofFieldOpList_swap]
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rfl
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lemma normalOrder_ofFieldOpList_anPart_swap (φ : 𝓕.FieldOp) (φ' : List 𝓕.FieldOp) :
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@ -224,18 +224,18 @@ The proof of this result ultimetly depends on
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- `superCommuteF_ofCrAnListF_ofFieldOpListF_eq_sum`
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- `normalOrderSign_eraseIdx`
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-/
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lemma ofCrAnFieldOp_superCommute_normalOrder_ofCrAnFieldOpList_sum (φ : 𝓕.CrAnFieldOp)
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(φs : List 𝓕.CrAnFieldOp) : [ofCrAnFieldOp φ, 𝓝(ofCrAnFieldOpList φs)]ₛ = ∑ n : Fin φs.length,
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) • [ofCrAnFieldOp φ, ofCrAnFieldOp φs[n]]ₛ
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* 𝓝(ofCrAnFieldOpList (φs.eraseIdx n)) := by
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rw [normalOrder_ofCrAnFieldOpList, map_smul]
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rw [superCommute_ofCrAnFieldOp_ofCrAnFieldOpList_eq_sum, Finset.smul_sum,
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lemma ofCrAnOp_superCommute_normalOrder_ofCrAnOpList_sum (φ : 𝓕.CrAnFieldOp)
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(φs : List 𝓕.CrAnFieldOp) : [ofCrAnOp φ, 𝓝(ofCrAnOpList φs)]ₛ = ∑ n : Fin φs.length,
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) • [ofCrAnOp φ, ofCrAnOp φs[n]]ₛ
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* 𝓝(ofCrAnOpList (φs.eraseIdx n)) := by
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rw [normalOrder_ofCrAnOpList, map_smul]
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rw [superCommute_ofCrAnOp_ofCrAnOpList_eq_sum, Finset.smul_sum,
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sum_normalOrderList_length]
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congr
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funext n
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simp only [instCommGroup.eq_1, List.get_eq_getElem, normalOrderList_get_normalOrderEquiv,
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normalOrderList_eraseIdx_normalOrderEquiv, Algebra.smul_mul_assoc, Fin.getElem_fin]
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rw [ofCrAnFieldOpList_eq_normalOrder, mul_smul_comm, smul_smul, smul_smul]
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rw [ofCrAnOpList_eq_normalOrder, mul_smul_comm, smul_smul, smul_smul]
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by_cases hs : (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[n])
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· congr
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erw [normalOrderSign_eraseIdx, ← hs]
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@ -250,14 +250,14 @@ lemma ofCrAnFieldOp_superCommute_normalOrder_ofCrAnFieldOpList_sum (φ : 𝓕.Cr
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· erw [superCommute_diff_statistic hs]
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simp
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lemma ofCrAnFieldOp_superCommute_normalOrder_ofFieldOpList_sum (φ : 𝓕.CrAnFieldOp)
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lemma ofCrAnOp_superCommute_normalOrder_ofFieldOpList_sum (φ : 𝓕.CrAnFieldOp)
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(φs : List 𝓕.FieldOp) :
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[ofCrAnFieldOp φ, 𝓝(ofFieldOpList φs)]ₛ = ∑ n : Fin φs.length, 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •
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[ofCrAnFieldOp φ, ofFieldOp φs[n]]ₛ * 𝓝(ofFieldOpList (φs.eraseIdx n)) := by
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[ofCrAnOp φ, 𝓝(ofFieldOpList φs)]ₛ = ∑ n : Fin φs.length, 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •
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[ofCrAnOp φ, ofFieldOp φs[n]]ₛ * 𝓝(ofFieldOpList (φs.eraseIdx n)) := by
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conv_lhs =>
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rw [ofFieldOpList_eq_sum, map_sum, map_sum]
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enter [2, s]
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rw [ofCrAnFieldOp_superCommute_normalOrder_ofCrAnFieldOpList_sum, CrAnSection.sum_over_length]
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rw [ofCrAnOp_superCommute_normalOrder_ofCrAnOpList_sum, CrAnSection.sum_over_length]
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enter [2, n]
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rw [CrAnSection.take_statistics_eq_take_state_statistics, smul_mul_assoc]
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rw [Finset.sum_comm]
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@ -285,13 +285,13 @@ lemma anPart_superCommute_normalOrder_ofFieldOpList_sum (φ : 𝓕.FieldOp) (φs
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simp
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| .position φ =>
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simp only [anPart_position, instCommGroup.eq_1, Fin.getElem_fin, Algebra.smul_mul_assoc]
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rw [ofCrAnFieldOp_superCommute_normalOrder_ofFieldOpList_sum]
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rw [ofCrAnOp_superCommute_normalOrder_ofFieldOpList_sum]
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simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnFieldOpToFieldOp_prod,
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Fin.getElem_fin, Algebra.smul_mul_assoc]
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rfl
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| .outAsymp φ =>
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simp only [anPart_posAsymp, instCommGroup.eq_1, Fin.getElem_fin, Algebra.smul_mul_assoc]
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rw [ofCrAnFieldOp_superCommute_normalOrder_ofFieldOpList_sum]
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rw [ofCrAnOp_superCommute_normalOrder_ofFieldOpList_sum]
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simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnFieldOpToFieldOp_prod,
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Fin.getElem_fin, Algebra.smul_mul_assoc]
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rfl
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@ -339,7 +339,7 @@ For a field specification `𝓕`, the following relation holds in the algebra `
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`φ * 𝓝(φ₀φ₁…φₙ) = 𝓝(φφ₀φ₁…φₙ) + ∑ i, (𝓢(φ,φ₀φ₁…φᵢ₋₁) • [anPartF φ, φᵢ]ₛ) * 𝓝(φ₀φ₁…φᵢ₋₁φᵢ₊₁…φₙ)`.
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The proof of this ultimently depends on :
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- `ofCrAnFieldOp_superCommute_normalOrder_ofCrAnFieldOpList_sum`
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- `ofCrAnOp_superCommute_normalOrder_ofCrAnOpList_sum`
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-/
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lemma ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
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ofFieldOp φ * 𝓝(ofFieldOpList φs) =
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