refactor: Lint
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jstoobysmith
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11 changed files with 179 additions and 182 deletions
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@ -441,38 +441,38 @@ lemma superCommute_ofCrAnList_ofStateList_eq_sum (φs : List 𝓕.CrAnStates) :
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FieldStatistic.ofList_cons_eq_mul, mul_comm]
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lemma summerCommute_jacobi_ofCrAnList (φs1 φs2 φs3 : List 𝓕.CrAnStates) :
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[ofCrAnList φs1, [ofCrAnList φs2, ofCrAnList φs3]ₛca]ₛca =
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𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs3) •
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(- 𝓢(𝓕 |>ₛ φs2, 𝓕 |>ₛ φs3 ) • [ofCrAnList φs3, [ofCrAnList φs1, ofCrAnList φs2]ₛca]ₛca -
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𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs2) • [ofCrAnList φs2, [ofCrAnList φs3, ofCrAnList φs1]ₛca]ₛca) := by
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repeat rw [superCommute_ofCrAnList_ofCrAnList]
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simp
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repeat rw [superCommute_ofCrAnList_ofCrAnList]
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simp only [instCommGroup.eq_1, ofList_append_eq_mul, List.append_assoc]
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by_cases h1 : (𝓕 |>ₛ φs1) = bosonic <;>
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by_cases h2 : (𝓕 |>ₛ φs2) = bosonic <;>
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by_cases h3 : (𝓕 |>ₛ φs3) = bosonic
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· simp [h1, h2, exchangeSign_bosonic, h3, mul_one, one_smul]
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abel
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· simp [h1, h2, exchangeSign_bosonic, bosonic_exchangeSign, mul_one, one_smul]
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abel
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· simp [h1, bosonic_exchangeSign, h3, exchangeSign_bosonic, mul_one, one_smul]
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abel
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· simp at h1 h2 h3
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simp [h1, h2, h3]
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abel
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· simp at h1 h2 h3
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simp [h1, h2, h3]
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abel
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· simp at h1 h2 h3
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simp [h1, h2, h3]
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abel
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· simp at h1 h2 h3
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simp [h1, h2, h3]
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abel
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· simp at h1 h2 h3
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simp [h1, h2, h3]
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abel
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[ofCrAnList φs1, [ofCrAnList φs2, ofCrAnList φs3]ₛca]ₛca =
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𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs3) •
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(- 𝓢(𝓕 |>ₛ φs2, 𝓕 |>ₛ φs3) • [ofCrAnList φs3, [ofCrAnList φs1, ofCrAnList φs2]ₛca]ₛca -
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𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs2) • [ofCrAnList φs2, [ofCrAnList φs3, ofCrAnList φs1]ₛca]ₛca) := by
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repeat rw [superCommute_ofCrAnList_ofCrAnList]
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simp
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repeat rw [superCommute_ofCrAnList_ofCrAnList]
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simp only [instCommGroup.eq_1, ofList_append_eq_mul, List.append_assoc]
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by_cases h1 : (𝓕 |>ₛ φs1) = bosonic <;>
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by_cases h2 : (𝓕 |>ₛ φs2) = bosonic <;>
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by_cases h3 : (𝓕 |>ₛ φs3) = bosonic
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· simp [h1, h2, exchangeSign_bosonic, h3, mul_one, one_smul]
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abel
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· simp [h1, h2, exchangeSign_bosonic, bosonic_exchangeSign, mul_one, one_smul]
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abel
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· simp [h1, bosonic_exchangeSign, h3, exchangeSign_bosonic, mul_one, one_smul]
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abel
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· simp at h1 h2 h3
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simp [h1, h2, h3]
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abel
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· simp at h1 h2 h3
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simp [h1, h2, h3]
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abel
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· simp at h1 h2 h3
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simp [h1, h2, h3]
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abel
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· simp at h1 h2 h3
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simp [h1, h2, h3]
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abel
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· simp at h1 h2 h3
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simp [h1, h2, h3]
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abel
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/-!
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## Interaction with grading.
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@ -483,14 +483,14 @@ lemma superCommute_grade {a b : 𝓕.CrAnAlgebra} {f1 f2 : FieldStatistic}
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(ha : a ∈ statisticSubmodule f1) (hb : b ∈ statisticSubmodule f2) :
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[a, b]ₛca ∈ statisticSubmodule (f1 + f2) := by
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule f2) : Prop :=
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[a, a2]ₛca ∈ statisticSubmodule (f1 + f2)
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[a, a2]ₛca ∈ statisticSubmodule (f1 + f2)
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change p b hb
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apply Submodule.span_induction (p := p)
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· intro x hx
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obtain ⟨φs, rfl, hφs⟩ := hx
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simp [p]
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule f1) : Prop :=
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[a2 , ofCrAnList φs]ₛca ∈ statisticSubmodule (f1 + f2)
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[a2, ofCrAnList φs]ₛca ∈ statisticSubmodule (f1 + f2)
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change p a ha
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apply Submodule.span_induction (p := p)
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· intro x hx
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@ -525,13 +525,13 @@ lemma superCommute_bosonic_bosonic {a b : 𝓕.CrAnAlgebra}
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(ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule bosonic) :
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[a, b]ₛca = a * b - b * a := by
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
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[a, a2]ₛca = a * a2 - a2 * a
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[a, a2]ₛca = a * a2 - a2 * a
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change p b hb
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apply Submodule.span_induction (p := p)
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· intro x hx
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obtain ⟨φs, rfl, hφs⟩ := hx
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
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[a2 , ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
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[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
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change p a ha
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apply Submodule.span_induction (p := p)
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· intro x hx
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@ -554,18 +554,17 @@ lemma superCommute_bosonic_bosonic {a b : 𝓕.CrAnAlgebra}
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simp_all [p, smul_sub]
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· exact hb
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lemma superCommute_bosonic_fermionic {a b : 𝓕.CrAnAlgebra}
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(ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule fermionic) :
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[a, b]ₛca = a * b - b * a := by
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
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[a, a2]ₛca = a * a2 - a2 * a
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[a, a2]ₛca = a * a2 - a2 * a
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change p b hb
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apply Submodule.span_induction (p := p)
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· intro x hx
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obtain ⟨φs, rfl, hφs⟩ := hx
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
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[a2 , ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
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[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
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change p a ha
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apply Submodule.span_induction (p := p)
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· intro x hx
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@ -588,18 +587,17 @@ lemma superCommute_bosonic_fermionic {a b : 𝓕.CrAnAlgebra}
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simp_all [p, smul_sub]
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· exact hb
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lemma superCommute_fermionic_bonsonic {a b : 𝓕.CrAnAlgebra}
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(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule bosonic) :
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[a, b]ₛca = a * b - b * a := by
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
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[a, a2]ₛca = a * a2 - a2 * a
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[a, a2]ₛca = a * a2 - a2 * a
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change p b hb
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apply Submodule.span_induction (p := p)
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· intro x hx
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obtain ⟨φs, rfl, hφs⟩ := hx
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
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[a2 , ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
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[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
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change p a ha
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apply Submodule.span_induction (p := p)
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· intro x hx
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@ -622,7 +620,7 @@ lemma superCommute_fermionic_bonsonic {a b : 𝓕.CrAnAlgebra}
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simp_all [p, smul_sub]
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· exact hb
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lemma superCommute_bonsonic {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
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lemma superCommute_bonsonic {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
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[a, b]ₛca = a * b - b * a := by
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rw [← bosonicProj_add_fermionicProj a]
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simp only [map_add, LinearMap.add_apply]
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@ -630,7 +628,7 @@ lemma superCommute_bonsonic {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmo
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simp only [add_mul, mul_add]
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abel
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lemma bosonic_superCommute {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
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lemma bosonic_superCommute {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
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[a, b]ₛca = a * b - b * a := by
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rw [← bosonicProj_add_fermionicProj b]
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simp only [map_add, LinearMap.add_apply]
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@ -638,12 +636,12 @@ lemma bosonic_superCommute {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmod
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simp only [add_mul, mul_add]
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abel
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lemma superCommute_bonsonic_symm {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
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lemma superCommute_bonsonic_symm {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
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[a, b]ₛca = - [b, a]ₛca := by
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rw [bosonic_superCommute hb, superCommute_bonsonic hb]
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simp
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lemma bonsonic_superCommute_symm {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
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lemma bonsonic_superCommute_symm {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
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[a, b]ₛca = - [b, a]ₛca := by
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rw [bosonic_superCommute ha, superCommute_bonsonic ha]
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simp
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@ -652,13 +650,13 @@ lemma superCommute_fermionic_fermionic {a b : 𝓕.CrAnAlgebra}
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(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule fermionic) :
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[a, b]ₛca = a * b + b * a := by
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
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[a, a2]ₛca = a * a2 + a2 * a
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[a, a2]ₛca = a * a2 + a2 * a
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change p b hb
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apply Submodule.span_induction (p := p)
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· intro x hx
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obtain ⟨φs, rfl, hφs⟩ := hx
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
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[a2 , ofCrAnList φs]ₛca = a2 * ofCrAnList φs + ofCrAnList φs * a2
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[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs + ofCrAnList φs * a2
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change p a ha
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apply Submodule.span_induction (p := p)
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· intro x hx
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@ -701,7 +699,7 @@ lemma superCommute_expand_bosonicProj_fermionicProj (a b : 𝓕.CrAnAlgebra) :
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abel
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lemma superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List 𝓕.CrAnStates) :
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[ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule bosonic ∨
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[ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule bosonic ∨
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[ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule fermionic := by
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by_cases h1 : (𝓕 |>ₛ φs) = bosonic <;> by_cases h2 : (𝓕 |>ₛ φs') = bosonic
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· left
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@ -738,7 +736,7 @@ lemma superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List
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apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h2)
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lemma superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic (φ φ' : 𝓕.CrAnStates) :
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[ofCrAnState φ, ofCrAnState φ']ₛca ∈ statisticSubmodule bosonic ∨
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[ofCrAnState φ, ofCrAnState φ']ₛca ∈ statisticSubmodule bosonic ∨
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[ofCrAnState φ, ofCrAnState φ']ₛca ∈ statisticSubmodule fermionic := by
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
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exact superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ] [φ']
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@ -779,7 +777,7 @@ lemma superCommute_bosonic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List
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ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
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ofCrAnList (φs.drop (n + 1)) := by
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let p (a : 𝓕.CrAnAlgebra) (ha : a ∈ statisticSubmodule bosonic) : Prop :=
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[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length),
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[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length),
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ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
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ofCrAnList (φs.drop (n + 1))
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change p a ha
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@ -802,14 +800,13 @@ lemma superCommute_bosonic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List
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simp_all [p, Finset.smul_sum]
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· exact ha
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lemma superCommute_fermionic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List 𝓕.CrAnStates)
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(ha : a ∈ statisticSubmodule fermionic) :
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[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
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[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
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ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
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ofCrAnList (φs.drop (n + 1)) := by
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let p (a : 𝓕.CrAnAlgebra) (ha : a ∈ statisticSubmodule fermionic) : Prop :=
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[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
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[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
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ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
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ofCrAnList (φs.drop (n + 1))
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change p a ha
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@ -835,7 +832,6 @@ lemma superCommute_fermionic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : Lis
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simp [smul_smul, mul_comm]
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· exact ha
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lemma statistic_neq_of_superCommute_fermionic {φs φs' : List 𝓕.CrAnStates}
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(h : [ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule fermionic) :
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(𝓕 |>ₛ φs) ≠ (𝓕 |>ₛ φs') ∨ [ofCrAnList φs, ofCrAnList φs']ₛca = 0 := by
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