refactor: rename timeOrder for CrAnAlgebra
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jstoobysmith
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c53299d40a
4 changed files with 95 additions and 95 deletions
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@ -19,7 +19,7 @@ open FieldStatistic
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namespace FieldOpAlgebra
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variable {𝓕 : FieldSpecification}
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lemma ι_timeOrder_superCommuteF_superCommuteF_eq_time_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
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lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
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(φs1 φs2 : List 𝓕.CrAnStates) (h :
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crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
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crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
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@ -38,7 +38,7 @@ lemma ι_timeOrder_superCommuteF_superCommuteF_eq_time_ofCrAnList {φ1 φ2 φ3 :
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ2, φ3]) * ι (ofCrAnList l2)) := by
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ1, φ2, φ3] φs2
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ1 :: φ2 :: φ3 :: φs2 = φs1 ++ [φ1, φ2, φ3] ++ φs2
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rw [timeOrderF_ofCrAnList, show φs1 ++ φ1 :: φ2 :: φ3 :: φs2 = φs1 ++ [φ1, φ2, φ3] ++ φs2
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by simp, crAnTimeOrderList, h1]
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simp only [List.append_assoc, List.singleton_append, decide_not,
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Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
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@ -47,7 +47,7 @@ lemma ι_timeOrder_superCommuteF_superCommuteF_eq_time_ofCrAnList {φ1 φ2 φ3 :
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ3, φ2]) * ι (ofCrAnList l2)) := by
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ1, φ3, φ2] φs2
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ1 :: φ3 :: φ2 :: φs2 = φs1 ++ [φ1, φ3, φ2] ++ φs2
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rw [timeOrderF_ofCrAnList, show φs1 ++ φ1 :: φ3 :: φ2 :: φs2 = φs1 ++ [φ1, φ3, φ2] ++ φs2
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by simp, crAnTimeOrderList, h1]
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simp only [List.singleton_append, decide_not,
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Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
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@ -72,7 +72,7 @@ lemma ι_timeOrder_superCommuteF_superCommuteF_eq_time_ofCrAnList {φ1 φ2 φ3 :
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ2, φ3, φ1]) * ι (ofCrAnList l2)) := by
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ2, φ3, φ1] φs2
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ2 :: φ3 :: φ1 :: φs2 = φs1 ++ [φ2, φ3, φ1] ++ φs2
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rw [timeOrderF_ofCrAnList, show φs1 ++ φ2 :: φ3 :: φ1 :: φs2 = φs1 ++ [φ2, φ3, φ1] ++ φs2
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by simp, crAnTimeOrderList, h1]
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simp only [List.singleton_append, decide_not,
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Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
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@ -90,7 +90,7 @@ lemma ι_timeOrder_superCommuteF_superCommuteF_eq_time_ofCrAnList {φ1 φ2 φ3 :
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ3, φ2, φ1]) * ι (ofCrAnList l2)) := by
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ3, φ2, φ1] φs2
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ3 :: φ2 :: φ1 :: φs2 = φs1 ++ [φ3, φ2, φ1] ++ φs2
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rw [timeOrderF_ofCrAnList, show φs1 ++ φ3 :: φ2 :: φ1 :: φs2 = φs1 ++ [φ3, φ2, φ1] ++ φs2
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by simp, crAnTimeOrderList, h1]
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simp only [List.singleton_append, decide_not,
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Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
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@ -139,7 +139,7 @@ lemma ι_timeOrder_superCommuteF_superCommuteF_eq_time_ofCrAnList {φ1 φ2 φ3 :
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map_smul, smul_sub]
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simp_all
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lemma ι_timeOrder_superCommuteF_superCommuteF_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
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lemma ι_timeOrderF_superCommuteF_superCommuteF_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
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(φs1 φs2 : List 𝓕.CrAnStates) :
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ι 𝓣ᶠ(ofCrAnList φs1 * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs2)
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= 0 := by
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@ -147,13 +147,13 @@ lemma ι_timeOrder_superCommuteF_superCommuteF_ofCrAnList {φ1 φ2 φ3 : 𝓕.Cr
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crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
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crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
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crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2
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· exact ι_timeOrder_superCommuteF_superCommuteF_eq_time_ofCrAnList φs1 φs2 h
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· rw [timeOrder_timeOrder_mid]
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rw [timeOrder_superCommuteF_ofCrAnState_superCommuteF_all_not_crAnTimeOrderRel _ _ _ h]
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· exact ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnList φs1 φs2 h
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· rw [timeOrderF_timeOrderF_mid]
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rw [timeOrderF_superCommuteF_ofCrAnState_superCommuteF_all_not_crAnTimeOrderRel _ _ _ h]
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simp
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@[simp]
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lemma ι_timeOrder_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.CrAnAlgebra) :
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lemma ι_timeOrderF_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.CrAnAlgebra) :
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ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0 := by
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let pb (b : 𝓕.CrAnAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
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Prop := ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0
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@ -169,7 +169,7 @@ lemma ι_timeOrder_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnStates} (
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· intro x hx
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obtain ⟨φs', rfl⟩ := hx
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simp only [ofListBasis_eq_ofList, pa]
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exact ι_timeOrder_superCommuteF_superCommuteF_ofCrAnList φs' φs
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exact ι_timeOrderF_superCommuteF_superCommuteF_ofCrAnList φs' φs
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· simp [pa]
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· intro x y hx hy hpx hpy
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simp_all [pa,mul_add, add_mul]
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@ -181,7 +181,7 @@ lemma ι_timeOrder_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnStates} (
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· intro x hx hpx
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simp_all [pb, hpx]
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lemma ι_timeOrder_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
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lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
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(hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.CrAnAlgebra) :
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ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
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ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b)) := by
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@ -204,7 +204,7 @@ lemma ι_timeOrder_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
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conv_lhs =>
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [mul_sub, sub_mul, ← ofCrAnList_append]
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rw [timeOrder_ofCrAnList, timeOrder_ofCrAnList]
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rw [timeOrderF_ofCrAnList, timeOrderF_ofCrAnList]
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have h1 : crAnTimeOrderSign (φs' ++ φ :: ψ :: φs) =
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crAnTimeOrderSign (φs' ++ ψ :: φ :: φs) := by
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trans crAnTimeOrderSign (φs' ++ [φ, ψ] ++ φs)
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@ -260,7 +260,7 @@ lemma ι_timeOrder_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
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· rw [ι_superCommuteF_of_diff_statistic hq]
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simp
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· rw [crAnTimeOrderSign, Wick.koszulSign_eq_rel_eq_stat _ _, ← crAnTimeOrderSign]
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rw [timeOrder_ofCrAnList]
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rw [timeOrderF_ofCrAnList]
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simp only [map_smul, Algebra.mul_smul_comm]
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simp only [List.nil_append]
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exact hψφ
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@ -277,30 +277,30 @@ lemma ι_timeOrder_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
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· intro x hx hpx
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simp_all [pb, hpx]
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lemma ι_timeOrder_superCommuteF_neq_time {φ ψ : 𝓕.CrAnStates}
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lemma ι_timeOrderF_superCommuteF_neq_time {φ ψ : 𝓕.CrAnStates}
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(hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) (a b : 𝓕.CrAnAlgebra) :
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ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
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rw [timeOrder_timeOrder_mid]
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rw [timeOrderF_timeOrderF_mid]
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have hφψ : ¬ (crAnTimeOrderRel φ ψ) ∨ ¬ (crAnTimeOrderRel ψ φ) := by
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exact Decidable.not_and_iff_or_not.mp hφψ
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rcases hφψ with hφψ | hφψ
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· rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel]
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· rw [timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel]
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simp_all only [false_and, not_false_eq_true, false_or, mul_zero, zero_mul, map_zero]
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simp_all
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· rw [superCommuteF_ofCrAnState_ofCrAnState_symm]
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simp only [instCommGroup.eq_1, neg_smul, map_neg, map_smul, mul_neg, Algebra.mul_smul_comm,
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neg_mul, Algebra.smul_mul_assoc, neg_eq_zero, smul_eq_zero]
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rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel]
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rw [timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel]
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simp only [mul_zero, zero_mul, map_zero, or_true]
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simp_all
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/-!
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## Defining normal order for `FiedOpAlgebra`.
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## Defining time order for `FiedOpAlgebra`.
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-/
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lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
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lemma ι_timeOrderF_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
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(h : a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι 𝓣ᶠ(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 𝓣ᶠ(a) = 0
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@ -314,11 +314,11 @@ lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
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match hc with
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| Or.inl hc =>
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obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
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simp only [ι_timeOrder_superCommuteF_superCommuteF]
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simp only [ι_timeOrderF_superCommuteF_superCommuteF]
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| Or.inr (Or.inl hc) =>
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obtain ⟨φa, hφa, φb, hφb, rfl⟩ := hc
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by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
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· rw [ι_timeOrder_superCommuteF_eq_time]
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· rw [ι_timeOrderF_superCommuteF_eq_time]
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simp only [map_mul]
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rw [ι_superCommuteF_of_create_create]
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simp only [zero_mul]
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@ -326,11 +326,11 @@ lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
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· exact hφb
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· exact heqt.1
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· exact heqt.2
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· rw [ι_timeOrder_superCommuteF_neq_time heqt]
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· rw [ι_timeOrderF_superCommuteF_neq_time heqt]
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| Or.inr (Or.inr (Or.inl hc)) =>
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obtain ⟨φa, hφa, φb, hφb, rfl⟩ := hc
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by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
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· rw [ι_timeOrder_superCommuteF_eq_time]
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· rw [ι_timeOrderF_superCommuteF_eq_time]
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simp only [map_mul]
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rw [ι_superCommuteF_of_annihilate_annihilate]
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simp only [zero_mul]
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@ -338,18 +338,18 @@ lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
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· exact hφb
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· exact heqt.1
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· exact heqt.2
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· rw [ι_timeOrder_superCommuteF_neq_time heqt]
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· rw [ι_timeOrderF_superCommuteF_neq_time heqt]
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| Or.inr (Or.inr (Or.inr hc)) =>
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obtain ⟨φa, φb, hdiff, rfl⟩ := hc
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by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
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· rw [ι_timeOrder_superCommuteF_eq_time]
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· rw [ι_timeOrderF_superCommuteF_eq_time]
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simp only [map_mul]
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rw [ι_superCommuteF_of_diff_statistic]
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simp only [zero_mul]
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· exact hdiff
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· exact heqt.1
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· exact heqt.2
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· rw [ι_timeOrder_superCommuteF_neq_time heqt]
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· rw [ι_timeOrderF_superCommuteF_neq_time heqt]
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· simp [p]
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· intro x y hx hy
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simp only [map_add, p]
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@ -358,16 +358,16 @@ lemma ι_timeOrder_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_timeOrder_eq_of_equiv (a b : 𝓕.CrAnAlgebra) (h : a ≈ b) :
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lemma ι_timeOrderF_eq_of_equiv (a b : 𝓕.CrAnAlgebra) (h : a ≈ b) :
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ι 𝓣ᶠ(a) = ι 𝓣ᶠ(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_timeOrder_zero_of_mem_ideal (a - b) h
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exact ι_timeOrderF_zero_of_mem_ideal (a - b) h
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/-- Time ordering on `FieldOpAlgebra`. -/
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noncomputable def timeOrder : FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
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toFun := Quotient.lift (ι.toLinearMap ∘ₗ CrAnAlgebra.timeOrder) ι_timeOrder_eq_of_equiv
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toFun := Quotient.lift (ι.toLinearMap ∘ₗ timeOrderF) ι_timeOrderF_eq_of_equiv
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map_add' x y := by
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obtain ⟨x, hx⟩ := ι_surjective x
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obtain ⟨y, hy⟩ := ι_surjective y
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