refactor: rename normalOrder for CrAnAlgebra
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jstoobysmith
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5 changed files with 249 additions and 249 deletions
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@ -24,53 +24,53 @@ variable {𝓕 : FieldSpecification}
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## Normal order on super-commutators.
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The main result of this is
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`ι_normalOrder_superCommuteF_eq_zero_mul`
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`ι_normalOrderF_superCommuteF_eq_zero_mul`
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which states that applying `ι` to the normal order of something containing a super-commutator
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is zero.
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-/
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lemma ι_normalOrder_superCommuteF_ofCrAnList_ofCrAnList_eq_zero
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lemma ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero
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(φa φa' : 𝓕.CrAnStates) (φs φs' : List 𝓕.CrAnStates) :
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ι 𝓝ᶠ(ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca * ofCrAnList φs') = 0 := by
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rcases CreateAnnihilate.eq_create_or_annihilate (𝓕 |>ᶜ φa) with hφa | hφa
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<;> rcases CreateAnnihilate.eq_create_or_annihilate (𝓕 |>ᶜ φa') with hφa' | hφa'
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· rw [normalOrder_superCommuteF_ofCrAnList_create_create_ofCrAnList φa φa' hφa hφa' φs φs']
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· rw [normalOrderF_superCommuteF_ofCrAnList_create_create_ofCrAnList φa φa' hφa hφa' φs φs']
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rw [map_smul, map_mul, map_mul, map_mul, ι_superCommuteF_of_create_create φa φa' hφa hφa']
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simp
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· rw [normalOrder_superCommuteF_create_annihilate φa φa' hφa hφa' (ofCrAnList φs)
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· rw [normalOrderF_superCommuteF_create_annihilate φa φa' hφa hφa' (ofCrAnList φs)
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(ofCrAnList φs')]
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simp
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· rw [normalOrder_superCommuteF_annihilate_create φa' φa hφa' hφa (ofCrAnList φs)
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· rw [normalOrderF_superCommuteF_annihilate_create φa' φa hφa' hφa (ofCrAnList φs)
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(ofCrAnList φs')]
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simp
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· rw [normalOrder_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList φa φa' hφa hφa' φs φs']
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· rw [normalOrderF_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList φa φa' hφa hφa' φs φs']
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rw [map_smul, map_mul, map_mul, map_mul,
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ι_superCommuteF_of_annihilate_annihilate φa φa' hφa hφa']
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simp
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lemma ι_normalOrder_superCommuteF_ofCrAnList_eq_zero
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lemma ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero
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(φa φa' : 𝓕.CrAnStates) (φs : List 𝓕.CrAnStates)
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(a : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ(ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca * a) = 0 := by
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have hf : ι.toLinearMap ∘ₗ normalOrder ∘ₗ
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have hf : ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap (ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca) = 0 := by
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apply ofCrAnListBasis.ext
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intro l
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simp only [CrAnAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp, Function.comp_apply,
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AlgHom.toLinearMap_apply, LinearMap.zero_apply]
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exact ι_normalOrder_superCommuteF_ofCrAnList_ofCrAnList_eq_zero φa φa' φs l
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change (ι.toLinearMap ∘ₗ normalOrder ∘ₗ
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exact ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero φa φa' φs l
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change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap ((ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca))) a = 0
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rw [hf]
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simp
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lemma ι_normalOrder_superCommuteF_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrAnStates)
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lemma ι_normalOrderF_superCommuteF_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrAnStates)
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(a b : 𝓕.CrAnAlgebra) :
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ι 𝓝ᶠ(a * [ofCrAnState φa, ofCrAnState φa']ₛca * b) = 0 := by
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rw [mul_assoc]
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change (ι.toLinearMap ∘ₗ normalOrder ∘ₗ mulLinearMap.flip
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change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ mulLinearMap.flip
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([ofCrAnState φa, ofCrAnState φa']ₛca * b)) a = 0
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have hf : ι.toLinearMap ∘ₗ normalOrder ∘ₗ mulLinearMap.flip
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have hf : ι.toLinearMap ∘ₗ normalOrderF ∘ₗ mulLinearMap.flip
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([ofCrAnState φa, ofCrAnState φa']ₛca * b) = 0 := by
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apply ofCrAnListBasis.ext
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intro l
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@ -78,11 +78,11 @@ lemma ι_normalOrder_superCommuteF_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrAn
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Function.comp_apply, LinearMap.flip_apply, LinearMap.coe_mk, AddHom.coe_mk,
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AlgHom.toLinearMap_apply, LinearMap.zero_apply]
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rw [← mul_assoc]
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exact ι_normalOrder_superCommuteF_ofCrAnList_eq_zero φa φa' _ _
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exact ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero φa φa' _ _
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rw [hf]
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simp
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lemma ι_normalOrder_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul (φa : 𝓕.CrAnStates)
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lemma ι_normalOrderF_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul (φa : 𝓕.CrAnStates)
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(φs : List 𝓕.CrAnStates)
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(a b : 𝓕.CrAnAlgebra) :
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ι 𝓝ᶠ(a * [ofCrAnState φa, ofCrAnList φs]ₛca * b) = 0 := by
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@ -93,18 +93,18 @@ lemma ι_normalOrder_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul (φa :
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intro n
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rw [← mul_assoc, ← mul_assoc]
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rw [mul_assoc _ _ b, ofCrAnList_singleton]
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rw [ι_normalOrder_superCommuteF_ofCrAnState_eq_zero_mul]
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rw [ι_normalOrderF_superCommuteF_ofCrAnState_eq_zero_mul]
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lemma ι_normalOrder_superCommuteF_ofCrAnList_ofCrAnState_eq_zero_mul (φa : 𝓕.CrAnStates)
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lemma ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnState_eq_zero_mul (φa : 𝓕.CrAnStates)
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(φs : List 𝓕.CrAnStates) (a b : 𝓕.CrAnAlgebra) :
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ι 𝓝ᶠ(a * [ofCrAnList φs, ofCrAnState φa]ₛca * b) = 0 := by
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rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList_symm, ofCrAnList_singleton]
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simp only [FieldStatistic.instCommGroup.eq_1, FieldStatistic.ofList_singleton, mul_neg,
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Algebra.mul_smul_comm, neg_mul, Algebra.smul_mul_assoc, map_neg, map_smul]
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rw [ι_normalOrder_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul]
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rw [ι_normalOrderF_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul]
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simp
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lemma ι_normalOrder_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul
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lemma ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul
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(φs φs' : List 𝓕.CrAnStates) (a b : 𝓕.CrAnAlgebra) :
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ι 𝓝ᶠ(a * [ofCrAnList φs, ofCrAnList φs']ₛca * b) = 0 := by
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rw [superCommuteF_ofCrAnList_ofCrAnList_eq_sum, Finset.mul_sum, Finset.sum_mul]
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@ -113,63 +113,63 @@ lemma ι_normalOrder_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul
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intro n
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rw [← mul_assoc, ← mul_assoc]
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rw [mul_assoc _ _ b]
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rw [ι_normalOrder_superCommuteF_ofCrAnList_ofCrAnState_eq_zero_mul]
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rw [ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnState_eq_zero_mul]
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lemma ι_normalOrder_superCommuteF_ofCrAnList_eq_zero_mul
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lemma ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero_mul
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(φs : List 𝓕.CrAnStates)
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(a b c : 𝓕.CrAnAlgebra) :
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ι 𝓝ᶠ(a * [ofCrAnList φs, c]ₛca * b) = 0 := by
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change (ι.toLinearMap ∘ₗ normalOrder ∘ₗ
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change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnList φs)) c = 0
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have hf : (ι.toLinearMap ∘ₗ normalOrder ∘ₗ
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have hf : (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnList φs)) = 0 := by
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apply ofCrAnListBasis.ext
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intro φs'
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simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, CrAnAlgebra.ofListBasis_eq_ofList,
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LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
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LinearMap.zero_apply]
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rw [ι_normalOrder_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul]
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rw [ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul]
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rw [hf]
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simp
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@[simp]
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lemma ι_normalOrder_superCommuteF_eq_zero_mul
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lemma ι_normalOrderF_superCommuteF_eq_zero_mul
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(a b c d : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ(a * [d, c]ₛca * b) = 0 := by
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change (ι.toLinearMap ∘ₗ normalOrder ∘ₗ
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change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF.flip c) d = 0
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have hf : (ι.toLinearMap ∘ₗ normalOrder ∘ₗ
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have hf : (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF.flip c) = 0 := by
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apply ofCrAnListBasis.ext
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intro φs
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simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, CrAnAlgebra.ofListBasis_eq_ofList,
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LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
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LinearMap.zero_apply]
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rw [ι_normalOrder_superCommuteF_ofCrAnList_eq_zero_mul]
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rw [ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero_mul]
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rw [hf]
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simp
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@[simp]
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lemma ι_normalOrder_superCommuteF_eq_zero_mul_right (b c d : 𝓕.CrAnAlgebra) :
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ι 𝓝ᶠ([d, c]ₛca * b) = 0 := by
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rw [← ι_normalOrder_superCommuteF_eq_zero_mul 1 b c d]
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rw [← ι_normalOrderF_superCommuteF_eq_zero_mul 1 b c d]
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simp
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@[simp]
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lemma ι_normalOrder_superCommuteF_eq_zero_mul_left (a c d : 𝓕.CrAnAlgebra) :
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lemma ι_normalOrderF_superCommuteF_eq_zero_mul_left (a c d : 𝓕.CrAnAlgebra) :
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ι 𝓝ᶠ(a * [d, c]ₛca) = 0 := by
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rw [← ι_normalOrder_superCommuteF_eq_zero_mul a 1 c d]
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rw [← ι_normalOrderF_superCommuteF_eq_zero_mul a 1 c d]
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simp
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@[simp]
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lemma ι_normalOrder_superCommuteF_eq_zero_mul_mul_right (a b1 b2 c d: 𝓕.CrAnAlgebra) :
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lemma ι_normalOrderF_superCommuteF_eq_zero_mul_mul_right (a b1 b2 c d: 𝓕.CrAnAlgebra) :
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ι 𝓝ᶠ(a * [d, c]ₛca * b1 * b2) = 0 := by
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rw [← ι_normalOrder_superCommuteF_eq_zero_mul a (b1 * b2) c d]
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rw [← ι_normalOrderF_superCommuteF_eq_zero_mul a (b1 * b2) c d]
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congr 2
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noncomm_ring
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@[simp]
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lemma ι_normalOrder_superCommuteF_eq_zero (c d : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ([d, c]ₛca) = 0 := by
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rw [← ι_normalOrder_superCommuteF_eq_zero_mul 1 1 c d]
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lemma ι_normalOrderF_superCommuteF_eq_zero (c d : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ([d, c]ₛca) = 0 := by
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rw [← ι_normalOrderF_superCommuteF_eq_zero_mul 1 1 c d]
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simp
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/-!
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@ -178,7 +178,7 @@ lemma ι_normalOrder_superCommuteF_eq_zero (c d : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ
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-/
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lemma ι_normalOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
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lemma ι_normalOrderF_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
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(h : a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι 𝓝ᶠ(a) = 0 := by
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at h
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let p {k : Set 𝓕.CrAnAlgebra} (a : CrAnAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) := ι 𝓝ᶠ(a) = 0
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@ -210,16 +210,16 @@ lemma ι_normalOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
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· intro x hx
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simp [p]
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lemma ι_normalOrder_eq_of_equiv (a b : 𝓕.CrAnAlgebra) (h : a ≈ b) :
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lemma ι_normalOrderF_eq_of_equiv (a b : 𝓕.CrAnAlgebra) (h : a ≈ b) :
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ι 𝓝ᶠ(a) = ι 𝓝ᶠ(b) := by
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rw [equiv_iff_sub_mem_ideal] at h
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rw [LinearMap.sub_mem_ker_iff.mp]
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simp only [LinearMap.mem_ker, ← map_sub]
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exact ι_normalOrder_zero_of_mem_ideal (a - b) h
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exact ι_normalOrderF_zero_of_mem_ideal (a - b) h
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/-- Normal ordering on `FieldOpAlgebra`. -/
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noncomputable def normalOrder : FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
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toFun := Quotient.lift (ι.toLinearMap ∘ₗ CrAnAlgebra.normalOrder) ι_normalOrder_eq_of_equiv
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toFun := Quotient.lift (ι.toLinearMap ∘ₗ normalOrderF) ι_normalOrderF_eq_of_equiv
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map_add' x y := by
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obtain ⟨x, rfl⟩ := ι_surjective x
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obtain ⟨y, rfl⟩ := ι_surjective y
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