refactor: Rename CrAnAlgebra
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jstoobysmith
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16 changed files with 214 additions and 214 deletions
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@ -3,7 +3,7 @@ Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.NormalOrder
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import HepLean.PerturbationTheory.Algebras.FieldOpFreeAlgebra.NormalOrder
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import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.SuperCommute
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/-!
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@ -12,7 +12,7 @@ import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.SuperCommute
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-/
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namespace FieldSpecification
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open CrAnAlgebra
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open FieldOpFreeAlgebra
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open HepLean.List
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open FieldStatistic
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@ -52,12 +52,12 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_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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(a : 𝓕.FieldOpFreeAlgebra) : ι 𝓝ᶠ(ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca * a) = 0 := by
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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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simp only [FieldOpFreeAlgebra.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_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero φa φa' φs l
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change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
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@ -66,7 +66,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero
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simp
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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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(a b : 𝓕.FieldOpFreeAlgebra) :
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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 normalOrderF ∘ₗ mulLinearMap.flip
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@ -75,7 +75,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrA
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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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simp only [mulLinearMap, CrAnAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp,
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simp only [mulLinearMap, FieldOpFreeAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp,
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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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@ -85,7 +85,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrA
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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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(a b : 𝓕.FieldOpFreeAlgebra) :
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ι 𝓝ᶠ(a * [ofCrAnState φa, ofCrAnList φs]ₛca * b) = 0 := by
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rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList_eq_sum]
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rw [Finset.mul_sum, Finset.sum_mul]
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@ -97,7 +97,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul (φa :
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rw [ι_normalOrderF_superCommuteF_ofCrAnState_eq_zero_mul]
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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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(φs : List 𝓕.CrAnStates) (a b : 𝓕.FieldOpFreeAlgebra) :
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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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@ -106,7 +106,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnState_eq_zero_mul (φa :
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simp
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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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(φs φs' : List 𝓕.CrAnStates) (a b : 𝓕.FieldOpFreeAlgebra) :
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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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rw [map_sum, map_sum]
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@ -118,7 +118,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_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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(a b c : 𝓕.FieldOpFreeAlgebra) :
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ι 𝓝ᶠ(a * [ofCrAnList φs, c]ₛca * b) = 0 := by
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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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@ -126,7 +126,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero_mul
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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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simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, FieldOpFreeAlgebra.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_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul]
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@ -135,14 +135,14 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero_mul
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@[simp]
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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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(a b c d : 𝓕.FieldOpFreeAlgebra) : ι 𝓝ᶠ(a * [d, c]ₛca * b) = 0 := by
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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 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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simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, FieldOpFreeAlgebra.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_normalOrderF_superCommuteF_ofCrAnList_eq_zero_mul]
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@ -150,26 +150,26 @@ lemma ι_normalOrderF_superCommuteF_eq_zero_mul
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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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lemma ι_normalOrder_superCommuteF_eq_zero_mul_right (b c d : 𝓕.FieldOpFreeAlgebra) :
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ι 𝓝ᶠ([d, c]ₛca * b) = 0 := by
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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_normalOrderF_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.FieldOpFreeAlgebra) :
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ι 𝓝ᶠ(a * [d, c]ₛca) = 0 := by
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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_normalOrderF_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.FieldOpFreeAlgebra) :
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ι 𝓝ᶠ(a * [d, c]ₛca * b1 * b2) = 0 := by
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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_normalOrderF_superCommuteF_eq_zero (c d : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ([d, c]ₛca) = 0 := by
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lemma ι_normalOrderF_superCommuteF_eq_zero (c d : 𝓕.FieldOpFreeAlgebra) : 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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@ -179,10 +179,10 @@ lemma ι_normalOrderF_superCommuteF_eq_zero (c d : 𝓕.CrAnAlgebra) : ι 𝓝
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-/
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lemma ι_normalOrderF_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
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lemma ι_normalOrderF_zero_of_mem_ideal (a : 𝓕.FieldOpFreeAlgebra)
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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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let p {k : Set 𝓕.FieldOpFreeAlgebra} (a : FieldOpFreeAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) := ι 𝓝ᶠ(a) = 0
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change p a h
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apply AddSubgroup.closure_induction
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· intro x hx
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@ -211,7 +211,7 @@ lemma ι_normalOrderF_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_normalOrderF_eq_of_equiv (a b : 𝓕.CrAnAlgebra) (h : a ≈ b) :
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lemma ι_normalOrderF_eq_of_equiv (a b : 𝓕.FieldOpFreeAlgebra) (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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@ -241,7 +241,7 @@ scoped[FieldSpecification.FieldOpAlgebra] notation "𝓝(" a ")" => normalOrder
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-/
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lemma normalOrder_eq_ι_normalOrderF (a : 𝓕.CrAnAlgebra) :
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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.CrAnStates) :
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