refactor: Rename States to FieldOps
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jstoobysmith
parent
171e80fc04
commit
8f41de5785
36 changed files with 946 additions and 946 deletions
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@ -129,24 +129,24 @@ lemma superCommute_eq_ι_superCommuteF (a b : 𝓕.FieldOpFreeAlgebra) :
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-/
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lemma superCommute_create_create {φ φ' : 𝓕.CrAnStates}
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lemma superCommute_create_create {φ φ' : 𝓕.CrAnFieldOp}
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(h : 𝓕 |>ᶜ φ = .create) (h' : 𝓕 |>ᶜ φ' = .create) :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ = 0 := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp]
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rw [superCommute_eq_ι_superCommuteF, ι_superCommuteF_of_create_create _ _ h h']
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lemma superCommute_annihilate_annihilate {φ φ' : 𝓕.CrAnStates}
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lemma superCommute_annihilate_annihilate {φ φ' : 𝓕.CrAnFieldOp}
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(h : 𝓕 |>ᶜ φ = .annihilate) (h' : 𝓕 |>ᶜ φ' = .annihilate) :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ = 0 := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp]
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rw [superCommute_eq_ι_superCommuteF, ι_superCommuteF_of_annihilate_annihilate _ _ h h']
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lemma superCommute_diff_statistic {φ φ' : 𝓕.CrAnStates} (h : (𝓕 |>ₛ φ) ≠ 𝓕 |>ₛ φ') :
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lemma superCommute_diff_statistic {φ φ' : 𝓕.CrAnFieldOp} (h : (𝓕 |>ₛ φ) ≠ 𝓕 |>ₛ φ') :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ = 0 := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp]
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rw [superCommute_eq_ι_superCommuteF, ι_superCommuteF_of_diff_statistic h]
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lemma superCommute_ofCrAnFieldOp_ofFieldOp_diff_stat_zero (φ : 𝓕.CrAnStates) (ψ : 𝓕.States)
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lemma superCommute_ofCrAnFieldOp_ofFieldOp_diff_stat_zero (φ : 𝓕.CrAnFieldOp) (ψ : 𝓕.FieldOp)
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(h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) : [ofCrAnFieldOp φ, ofFieldOp ψ]ₛ = 0 := by
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rw [ofFieldOp_eq_sum, map_sum]
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rw [Finset.sum_eq_zero]
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@ -154,33 +154,33 @@ lemma superCommute_ofCrAnFieldOp_ofFieldOp_diff_stat_zero (φ : 𝓕.CrAnStates)
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apply superCommute_diff_statistic
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simpa [crAnStatistics] using h
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lemma superCommute_anPart_ofFieldOpF_diff_grade_zero (φ ψ : 𝓕.States)
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lemma superCommute_anPart_ofFieldOpF_diff_grade_zero (φ ψ : 𝓕.FieldOp)
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(h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) : [anPart φ, ofFieldOp ψ]ₛ = 0 := by
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match φ with
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| States.inAsymp _ =>
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| FieldOp.inAsymp _ =>
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simp
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| States.position φ =>
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| FieldOp.position φ =>
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simp only [anPartF_position]
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apply superCommute_ofCrAnFieldOp_ofFieldOp_diff_stat_zero _ _ _
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simpa [crAnStatistics] using h
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| States.outAsymp _ =>
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| FieldOp.outAsymp _ =>
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simp only [anPartF_posAsymp]
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apply superCommute_ofCrAnFieldOp_ofFieldOp_diff_stat_zero _ _
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simpa [crAnStatistics] using h
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lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp_mem_center (φ φ' : 𝓕.CrAnStates) :
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lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp_mem_center (φ φ' : 𝓕.CrAnFieldOp) :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ ∈ Subalgebra.center ℂ (FieldOpAlgebra 𝓕) := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp, superCommute_eq_ι_superCommuteF]
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exact ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_mem_center φ φ'
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lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp_commute (φ φ' : 𝓕.CrAnStates)
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lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp_commute (φ φ' : 𝓕.CrAnFieldOp)
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(a : FieldOpAlgebra 𝓕) :
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a * [ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ = [ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ * a := by
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have h1 := superCommute_ofCrAnFieldOp_ofCrAnFieldOp_mem_center φ φ'
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rw [@Subalgebra.mem_center_iff] at h1
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exact h1 a
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lemma superCommute_ofCrAnFieldOp_ofFieldOp_mem_center (φ : 𝓕.CrAnStates) (φ' : 𝓕.States) :
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lemma superCommute_ofCrAnFieldOp_ofFieldOp_mem_center (φ : 𝓕.CrAnFieldOp) (φ' : 𝓕.FieldOp) :
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[ofCrAnFieldOp φ, ofFieldOp φ']ₛ ∈ Subalgebra.center ℂ (FieldOpAlgebra 𝓕) := by
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rw [ofFieldOp_eq_sum]
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simp only [map_sum]
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@ -188,22 +188,22 @@ lemma superCommute_ofCrAnFieldOp_ofFieldOp_mem_center (φ : 𝓕.CrAnStates) (φ
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intro x hx
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exact superCommute_ofCrAnFieldOp_ofCrAnFieldOp_mem_center φ ⟨φ', x⟩
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lemma superCommute_ofCrAnFieldOp_ofFieldOp_commute (φ : 𝓕.CrAnStates) (φ' : 𝓕.States)
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lemma superCommute_ofCrAnFieldOp_ofFieldOp_commute (φ : 𝓕.CrAnFieldOp) (φ' : 𝓕.FieldOp)
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(a : FieldOpAlgebra 𝓕) : a * [ofCrAnFieldOp φ, ofFieldOp φ']ₛ =
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[ofCrAnFieldOp φ, ofFieldOp φ']ₛ * a := by
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have h1 := superCommute_ofCrAnFieldOp_ofFieldOp_mem_center φ φ'
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rw [@Subalgebra.mem_center_iff] at h1
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exact h1 a
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lemma superCommute_anPart_ofFieldOp_mem_center (φ φ' : 𝓕.States) :
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lemma superCommute_anPart_ofFieldOp_mem_center (φ φ' : 𝓕.FieldOp) :
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[anPart φ, ofFieldOp φ']ₛ ∈ Subalgebra.center ℂ (FieldOpAlgebra 𝓕) := by
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match φ with
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| States.inAsymp _ =>
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| FieldOp.inAsymp _ =>
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simp only [anPart_negAsymp, map_zero, LinearMap.zero_apply]
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exact Subalgebra.zero_mem (Subalgebra.center ℂ _)
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| States.position φ =>
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| FieldOp.position φ =>
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exact superCommute_ofCrAnFieldOp_ofFieldOp_mem_center _ _
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| States.outAsymp _ =>
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| FieldOp.outAsymp _ =>
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exact superCommute_ofCrAnFieldOp_ofFieldOp_mem_center _ _
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/-!
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@ -212,57 +212,57 @@ lemma superCommute_anPart_ofFieldOp_mem_center (φ φ' : 𝓕.States) :
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-/
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lemma superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList (φs φs' : List 𝓕.CrAnStates) :
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lemma superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList (φs φs' : List 𝓕.CrAnFieldOp) :
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[ofCrAnFieldOpList φs, ofCrAnFieldOpList φs']ₛ =
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ofCrAnFieldOpList (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnFieldOpList (φs' ++ φs) := by
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rw [ofCrAnFieldOpList_eq_ι_ofCrAnListF, ofCrAnFieldOpList_eq_ι_ofCrAnListF]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofCrAnListF_ofCrAnListF]
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rfl
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lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp (φ φ' : 𝓕.CrAnStates) :
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lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp (φ φ' : 𝓕.CrAnFieldOp) :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ = ofCrAnFieldOp φ * ofCrAnFieldOp φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofCrAnFieldOp φ' * ofCrAnFieldOp φ := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofCrAnOpF_ofCrAnOpF]
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rfl
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lemma superCommute_ofCrAnFieldOpList_ofFieldOpList (φcas : List 𝓕.CrAnStates)
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(φs : List 𝓕.States) :
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lemma superCommute_ofCrAnFieldOpList_ofFieldOpList (φcas : List 𝓕.CrAnFieldOp)
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(φs : List 𝓕.FieldOp) :
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[ofCrAnFieldOpList φcas, ofFieldOpList φs]ₛ = ofCrAnFieldOpList φcas * ofFieldOpList φs -
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𝓢(𝓕 |>ₛ φcas, 𝓕 |>ₛ φs) • ofFieldOpList φs * ofCrAnFieldOpList φcas := by
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rw [ofCrAnFieldOpList, ofFieldOpList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofCrAnListF_ofFieldOpFsList]
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rfl
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lemma superCommute_ofFieldOpList_ofFieldOpList (φs φs' : List 𝓕.States) :
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lemma superCommute_ofFieldOpList_ofFieldOpList (φs φs' : List 𝓕.FieldOp) :
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[ofFieldOpList φs, ofFieldOpList φs']ₛ = ofFieldOpList φs * ofFieldOpList φs' -
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofFieldOpList φs := by
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rw [ofFieldOpList, ofFieldOpList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofFieldOpListF_ofFieldOpFsList]
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rfl
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lemma superCommute_ofFieldOp_ofFieldOpList (φ : 𝓕.States) (φs : List 𝓕.States) :
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lemma superCommute_ofFieldOp_ofFieldOpList (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
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[ofFieldOp φ, ofFieldOpList φs]ₛ = ofFieldOp φ * ofFieldOpList φs -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpList φs * ofFieldOp φ := by
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rw [ofFieldOp, ofFieldOpList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofFieldOpF_ofFieldOpFsList]
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rfl
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lemma superCommute_ofFieldOpList_ofFieldOp (φs : List 𝓕.States) (φ : 𝓕.States) :
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lemma superCommute_ofFieldOpList_ofFieldOp (φs : List 𝓕.FieldOp) (φ : 𝓕.FieldOp) :
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[ofFieldOpList φs, ofFieldOp φ]ₛ = ofFieldOpList φs * ofFieldOp φ -
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOp φ * ofFieldOpList φs := by
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rw [ofFieldOpList, ofFieldOp]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofFieldOpListF_ofFieldOpF]
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rfl
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lemma superCommute_anPart_crPart (φ φ' : 𝓕.States) :
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lemma superCommute_anPart_crPart (φ φ' : 𝓕.FieldOp) :
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[anPart φ, crPart φ']ₛ = anPart φ * crPart φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * anPart φ := by
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rw [anPart, crPart]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_crPartF]
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rfl
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lemma superCommute_crPart_anPart (φ φ' : 𝓕.States) :
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lemma superCommute_crPart_anPart (φ φ' : 𝓕.FieldOp) :
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[crPart φ, anPart φ']ₛ = crPart φ * anPart φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * crPart φ := by
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rw [anPart, crPart]
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@ -270,83 +270,83 @@ lemma superCommute_crPart_anPart (φ φ' : 𝓕.States) :
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rfl
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@[simp]
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lemma superCommute_crPart_crPart (φ φ' : 𝓕.States) : [crPart φ, crPart φ']ₛ = 0 := by
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lemma superCommute_crPart_crPart (φ φ' : 𝓕.FieldOp) : [crPart φ, crPart φ']ₛ = 0 := by
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match φ, φ' with
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| States.outAsymp φ, _ =>
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| FieldOp.outAsymp φ, _ =>
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simp
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| _, States.outAsymp φ =>
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| _, FieldOp.outAsymp φ =>
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simp
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| States.position φ, States.position φ' =>
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| FieldOp.position φ, FieldOp.position φ' =>
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simp only [crPart_position]
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apply superCommute_create_create
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· rfl
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· rfl
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| States.position φ, States.inAsymp φ' =>
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| FieldOp.position φ, FieldOp.inAsymp φ' =>
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simp only [crPart_position, crPart_negAsymp]
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apply superCommute_create_create
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· rfl
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· rfl
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| States.inAsymp φ, States.inAsymp φ' =>
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| FieldOp.inAsymp φ, FieldOp.inAsymp φ' =>
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simp only [crPart_negAsymp]
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apply superCommute_create_create
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· rfl
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· rfl
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| States.inAsymp φ, States.position φ' =>
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| FieldOp.inAsymp φ, FieldOp.position φ' =>
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simp only [crPart_negAsymp, crPart_position]
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apply superCommute_create_create
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· rfl
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· rfl
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@[simp]
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lemma superCommute_anPart_anPart (φ φ' : 𝓕.States) : [anPart φ, anPart φ']ₛ = 0 := by
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lemma superCommute_anPart_anPart (φ φ' : 𝓕.FieldOp) : [anPart φ, anPart φ']ₛ = 0 := by
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match φ, φ' with
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| States.inAsymp φ, _ =>
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| FieldOp.inAsymp φ, _ =>
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simp
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| _, States.inAsymp φ =>
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| _, FieldOp.inAsymp φ =>
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simp
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| States.position φ, States.position φ' =>
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| FieldOp.position φ, FieldOp.position φ' =>
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simp only [anPart_position]
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apply superCommute_annihilate_annihilate
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· rfl
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· rfl
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| States.position φ, States.outAsymp φ' =>
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| FieldOp.position φ, FieldOp.outAsymp φ' =>
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simp only [anPart_position, anPart_posAsymp]
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apply superCommute_annihilate_annihilate
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· rfl
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· rfl
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| States.outAsymp φ, States.outAsymp φ' =>
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| FieldOp.outAsymp φ, FieldOp.outAsymp φ' =>
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simp only [anPart_posAsymp]
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apply superCommute_annihilate_annihilate
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· rfl
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· rfl
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| States.outAsymp φ, States.position φ' =>
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| FieldOp.outAsymp φ, FieldOp.position φ' =>
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simp only [anPart_posAsymp, anPart_position]
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apply superCommute_annihilate_annihilate
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· rfl
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· rfl
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lemma superCommute_crPart_ofFieldOpList (φ : 𝓕.States) (φs : List 𝓕.States) :
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lemma superCommute_crPart_ofFieldOpList (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
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[crPart φ, ofFieldOpList φs]ₛ = crPart φ * ofFieldOpList φs -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpList φs * crPart φ := by
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rw [crPart, ofFieldOpList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_ofFieldOpListF]
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rfl
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lemma superCommute_anPart_ofFieldOpList (φ : 𝓕.States) (φs : List 𝓕.States) :
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lemma superCommute_anPart_ofFieldOpList (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
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[anPart φ, ofFieldOpList φs]ₛ = anPart φ * ofFieldOpList φs -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpList φs * anPart φ := by
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rw [anPart, ofFieldOpList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_ofFieldOpListF]
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rfl
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lemma superCommute_crPart_ofFieldOp (φ φ' : 𝓕.States) :
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lemma superCommute_crPart_ofFieldOp (φ φ' : 𝓕.FieldOp) :
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[crPart φ, ofFieldOp φ']ₛ = crPart φ * ofFieldOp φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOp φ' * crPart φ := by
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rw [crPart, ofFieldOp]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_ofFieldOpF]
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rfl
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lemma superCommute_anPart_ofFieldOp (φ φ' : 𝓕.States) :
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lemma superCommute_anPart_ofFieldOp (φ φ' : 𝓕.FieldOp) :
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[anPart φ, ofFieldOp φ']ₛ = anPart φ * ofFieldOp φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOp φ' * anPart φ := by
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rw [anPart, ofFieldOp]
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@ -362,73 +362,73 @@ multiplication with a sign plus the super commutor.
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-/
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lemma ofCrAnFieldOpList_mul_ofCrAnFieldOpList_eq_superCommute (φs φs' : List 𝓕.CrAnStates) :
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lemma ofCrAnFieldOpList_mul_ofCrAnFieldOpList_eq_superCommute (φs φs' : List 𝓕.CrAnFieldOp) :
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ofCrAnFieldOpList φs * ofCrAnFieldOpList φs' =
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnFieldOpList φs' * ofCrAnFieldOpList φs
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+ [ofCrAnFieldOpList φs, ofCrAnFieldOpList φs']ₛ := by
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rw [superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList]
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simp [ofCrAnFieldOpList_append]
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lemma ofCrAnFieldOp_mul_ofCrAnFieldOpList_eq_superCommute (φ : 𝓕.CrAnStates)
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(φs' : List 𝓕.CrAnStates) : ofCrAnFieldOp φ * ofCrAnFieldOpList φs' =
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lemma ofCrAnFieldOp_mul_ofCrAnFieldOpList_eq_superCommute (φ : 𝓕.CrAnFieldOp)
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(φs' : List 𝓕.CrAnFieldOp) : ofCrAnFieldOp φ * ofCrAnFieldOpList φs' =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofCrAnFieldOpList φs' * ofCrAnFieldOp φ
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+ [ofCrAnFieldOp φ, ofCrAnFieldOpList φs']ₛ := by
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rw [← ofCrAnFieldOpList_singleton, ofCrAnFieldOpList_mul_ofCrAnFieldOpList_eq_superCommute]
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simp
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lemma ofFieldOpList_mul_ofFieldOpList_eq_superCommute (φs φs' : List 𝓕.States) :
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lemma ofFieldOpList_mul_ofFieldOpList_eq_superCommute (φs φs' : List 𝓕.FieldOp) :
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ofFieldOpList φs * ofFieldOpList φs' =
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofFieldOpList φs
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+ [ofFieldOpList φs, ofFieldOpList φs']ₛ := by
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rw [superCommute_ofFieldOpList_ofFieldOpList]
|
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simp
|
||||
|
||||
lemma ofFieldOp_mul_ofFieldOpList_eq_superCommute (φ : 𝓕.States) (φs' : List 𝓕.States) :
|
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lemma ofFieldOp_mul_ofFieldOpList_eq_superCommute (φ : 𝓕.FieldOp) (φs' : List 𝓕.FieldOp) :
|
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ofFieldOp φ * ofFieldOpList φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofFieldOp φ
|
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+ [ofFieldOp φ, ofFieldOpList φs']ₛ := by
|
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rw [superCommute_ofFieldOp_ofFieldOpList]
|
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simp
|
||||
|
||||
lemma ofFieldOp_mul_ofFieldOp_eq_superCommute (φ φ' : 𝓕.States) :
|
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lemma ofFieldOp_mul_ofFieldOp_eq_superCommute (φ φ' : 𝓕.FieldOp) :
|
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ofFieldOp φ * ofFieldOp φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOp φ' * ofFieldOp φ
|
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+ [ofFieldOp φ, ofFieldOp φ']ₛ := by
|
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rw [← ofFieldOpList_singleton, ← ofFieldOpList_singleton]
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rw [ofFieldOpList_mul_ofFieldOpList_eq_superCommute, ofFieldOpList_singleton]
|
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simp
|
||||
|
||||
lemma ofFieldOpList_mul_ofFieldOp_eq_superCommute (φs : List 𝓕.States) (φ : 𝓕.States) :
|
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lemma ofFieldOpList_mul_ofFieldOp_eq_superCommute (φs : List 𝓕.FieldOp) (φ : 𝓕.FieldOp) :
|
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ofFieldOpList φs * ofFieldOp φ = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOp φ * ofFieldOpList φs
|
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+ [ofFieldOpList φs, ofFieldOp φ]ₛ := by
|
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rw [superCommute_ofFieldOpList_ofFieldOp]
|
||||
simp
|
||||
|
||||
lemma ofCrAnFieldOpList_mul_ofFieldOpList_eq_superCommute (φs : List 𝓕.CrAnStates)
|
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(φs' : List 𝓕.States) : ofCrAnFieldOpList φs * ofFieldOpList φs' =
|
||||
lemma ofCrAnFieldOpList_mul_ofFieldOpList_eq_superCommute (φs : List 𝓕.CrAnFieldOp)
|
||||
(φs' : List 𝓕.FieldOp) : ofCrAnFieldOpList φs * ofFieldOpList φs' =
|
||||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofCrAnFieldOpList φs
|
||||
+ [ofCrAnFieldOpList φs, ofFieldOpList φs']ₛ := by
|
||||
rw [superCommute_ofCrAnFieldOpList_ofFieldOpList]
|
||||
simp
|
||||
|
||||
lemma crPart_mul_anPart_eq_superCommute (φ φ' : 𝓕.States) :
|
||||
lemma crPart_mul_anPart_eq_superCommute (φ φ' : 𝓕.FieldOp) :
|
||||
crPart φ * anPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * crPart φ
|
||||
+ [crPart φ, anPart φ']ₛ := by
|
||||
rw [superCommute_crPart_anPart]
|
||||
simp
|
||||
|
||||
lemma anPart_mul_crPart_eq_superCommute (φ φ' : 𝓕.States) :
|
||||
lemma anPart_mul_crPart_eq_superCommute (φ φ' : 𝓕.FieldOp) :
|
||||
anPart φ * crPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * anPart φ
|
||||
+ [anPart φ, crPart φ']ₛ := by
|
||||
rw [superCommute_anPart_crPart]
|
||||
simp
|
||||
|
||||
lemma crPart_mul_crPart_swap (φ φ' : 𝓕.States) :
|
||||
lemma crPart_mul_crPart_swap (φ φ' : 𝓕.FieldOp) :
|
||||
crPart φ * crPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * crPart φ := by
|
||||
trans 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * crPart φ + [crPart φ, crPart φ']ₛ
|
||||
· rw [crPart, crPart, superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_crPartF]
|
||||
simp
|
||||
· simp
|
||||
|
||||
lemma anPart_mul_anPart_swap (φ φ' : 𝓕.States) :
|
||||
lemma anPart_mul_anPart_swap (φ φ' : 𝓕.FieldOp) :
|
||||
anPart φ * anPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * anPart φ := by
|
||||
trans 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * anPart φ + [anPart φ, anPart φ']ₛ
|
||||
· rw [anPart, anPart, superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_anPartF]
|
||||
|
|
@ -441,14 +441,14 @@ lemma anPart_mul_anPart_swap (φ φ' : 𝓕.States) :
|
|||
|
||||
-/
|
||||
|
||||
lemma superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList_symm (φs φs' : List 𝓕.CrAnStates) :
|
||||
lemma superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList_symm (φs φs' : List 𝓕.CrAnFieldOp) :
|
||||
[ofCrAnFieldOpList φs, ofCrAnFieldOpList φs']ₛ =
|
||||
(- 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs')) • [ofCrAnFieldOpList φs', ofCrAnFieldOpList φs]ₛ := by
|
||||
rw [ofCrAnFieldOpList, ofCrAnFieldOpList, superCommute_eq_ι_superCommuteF,
|
||||
superCommuteF_ofCrAnListF_ofCrAnListF_symm]
|
||||
rfl
|
||||
|
||||
lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp_symm (φ φ' : 𝓕.CrAnStates) :
|
||||
lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp_symm (φ φ' : 𝓕.CrAnFieldOp) :
|
||||
[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ =
|
||||
(- 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ')) • [ofCrAnFieldOp φ', ofCrAnFieldOp φ]ₛ := by
|
||||
rw [ofCrAnFieldOp, ofCrAnFieldOp, superCommute_eq_ι_superCommuteF,
|
||||
|
|
@ -461,7 +461,7 @@ lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp_symm (φ φ' : 𝓕.CrAnStates) :
|
|||
|
||||
-/
|
||||
|
||||
lemma superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList_eq_sum (φs φs' : List 𝓕.CrAnStates) :
|
||||
lemma superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList_eq_sum (φs φs' : List 𝓕.CrAnFieldOp) :
|
||||
[ofCrAnFieldOpList φs, ofCrAnFieldOpList φs']ₛ =
|
||||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
|
||||
ofCrAnFieldOpList (φs'.take n) * [ofCrAnFieldOpList φs, ofCrAnFieldOp (φs'.get n)]ₛ *
|
||||
|
|
@ -472,8 +472,8 @@ lemma superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList_eq_sum (φs φs' : List
|
|||
rw [map_sum]
|
||||
rfl
|
||||
|
||||
lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOpList_eq_sum (φ : 𝓕.CrAnStates)
|
||||
(φs' : List 𝓕.CrAnStates) : [ofCrAnFieldOp φ, ofCrAnFieldOpList φs']ₛ =
|
||||
lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOpList_eq_sum (φ : 𝓕.CrAnFieldOp)
|
||||
(φs' : List 𝓕.CrAnFieldOp) : [ofCrAnFieldOp φ, ofCrAnFieldOpList φs']ₛ =
|
||||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs'.take n) •
|
||||
[ofCrAnFieldOp φ, ofCrAnFieldOp (φs'.get n)]ₛ * ofCrAnFieldOpList (φs'.eraseIdx n) := by
|
||||
conv_lhs =>
|
||||
|
|
@ -487,8 +487,8 @@ lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOpList_eq_sum (φ : 𝓕.CrAnStates)
|
|||
congr
|
||||
exact Eq.symm (List.eraseIdx_eq_take_drop_succ φs' ↑n)
|
||||
|
||||
lemma superCommute_ofCrAnFieldOpList_ofFieldOpList_eq_sum (φs : List 𝓕.CrAnStates)
|
||||
(φs' : List 𝓕.States) : [ofCrAnFieldOpList φs, ofFieldOpList φs']ₛ =
|
||||
lemma superCommute_ofCrAnFieldOpList_ofFieldOpList_eq_sum (φs : List 𝓕.CrAnFieldOp)
|
||||
(φs' : List 𝓕.FieldOp) : [ofCrAnFieldOpList φs, ofFieldOpList φs']ₛ =
|
||||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
|
||||
ofFieldOpList (φs'.take n) * [ofCrAnFieldOpList φs, ofFieldOp (φs'.get n)]ₛ *
|
||||
ofFieldOpList (φs'.drop (n + 1)) := by
|
||||
|
|
@ -498,7 +498,7 @@ lemma superCommute_ofCrAnFieldOpList_ofFieldOpList_eq_sum (φs : List 𝓕.CrAnS
|
|||
rw [map_sum]
|
||||
rfl
|
||||
|
||||
lemma superCommute_ofCrAnFieldOp_ofFieldOpList_eq_sum (φ : 𝓕.CrAnStates) (φs' : List 𝓕.States) :
|
||||
lemma superCommute_ofCrAnFieldOp_ofFieldOpList_eq_sum (φ : 𝓕.CrAnFieldOp) (φs' : List 𝓕.FieldOp) :
|
||||
[ofCrAnFieldOp φ, ofFieldOpList φs']ₛ =
|
||||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs'.take n) •
|
||||
[ofCrAnFieldOp φ, ofFieldOp (φs'.get n)]ₛ * ofFieldOpList (φs'.eraseIdx n) := by
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue