refactor: Lint
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jstoobysmith
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22636db606
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32aefb7eb7
9 changed files with 209 additions and 166 deletions
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@ -20,33 +20,35 @@ namespace FieldOpAlgebra
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variable {𝓕 : FieldSpecification}
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lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
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(φs1 φs2 : List 𝓕.CrAnStates) (h :
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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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crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2):
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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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crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2) :
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ι 𝓣ᶠ(ofCrAnList φs1 * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca *
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ofCrAnList φs2) = 0 := by
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let l1 :=
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(List.takeWhile (fun c => ¬ crAnTimeOrderRel φ1 c) ((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
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(List.takeWhile (fun c => ¬ crAnTimeOrderRel φ1 c)
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((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
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++ (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ crAnTimeOrderRel c φ1) φs1)
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let l2 := (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ crAnTimeOrderRel c φ1) φs2)
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++ (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ ¬ crAnTimeOrderRel c φ1) ((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
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++ (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ ¬ crAnTimeOrderRel c φ1)
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((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
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have h123 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)) =
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crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ2, φ3]) * ι (ofCrAnList l2)):= by
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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 by simp,
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crAnTimeOrderList, h1]
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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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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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have h132 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ1 :: φ3 :: φ2 :: φs2)) =
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crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ3, φ2]) * ι (ofCrAnList l2)):= by
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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 by simp,
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crAnTimeOrderList, h1]
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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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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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congr 1
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@ -67,11 +69,11 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 :
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exact List.Perm.swap φ1 φ2 [φ3]
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have h231 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ2 :: φ3 :: φ1 :: φs2)) =
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crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ2, φ3, φ1]) * ι (ofCrAnList l2)):= by
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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 by simp,
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crAnTimeOrderList, h1]
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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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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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congr 1
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@ -85,11 +87,11 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 :
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simp_all
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have h321 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ3 :: φ2 :: φ1 :: φs2)) =
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crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ3, φ2, φ1]) * ι (ofCrAnList l2)):= by
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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 by simp,
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crAnTimeOrderList, h1]
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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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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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congr 1
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@ -125,7 +127,8 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 :
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repeat rw [mul_assoc]
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rw [← mul_sub, ← mul_sub, ← mul_sub]
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rw [← sub_mul, ← sub_mul, ← sub_mul]
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trans ι (ofCrAnList l1) * ι [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ι (ofCrAnList l2)
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trans ι (ofCrAnList l1) * ι [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca *
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ι (ofCrAnList l2)
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rw [mul_assoc]
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congr
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
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@ -137,7 +140,7 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 :
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simp_all
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lemma ι_timeOrder_superCommute_superCommute_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
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(φs1 φs2 : List 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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by_cases h :
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@ -150,7 +153,7 @@ lemma ι_timeOrder_superCommute_superCommute_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAn
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simp
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@[simp]
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lemma ι_timeOrder_superCommute_superCommute {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.CrAnAlgebra):
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lemma ι_timeOrder_superCommute_superCommute {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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@ -215,20 +218,21 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
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rw [h1]
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simp only [map_smul]
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [φ, ψ] φs
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(by simp_all)
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(by simp_all)
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rw [crAnTimeOrderList, show φs' ++ φ :: ψ :: φs = φs' ++ [φ, ψ] ++ φs by simp, h1]
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have h2 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [ψ, φ] φs
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(by simp_all)
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(by simp_all)
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rw [crAnTimeOrderList, show φs' ++ ψ :: φ :: φs = φs' ++ [ψ, φ] ++ φs by simp, h2]
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repeat rw [ofCrAnList_append]
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rw [smul_smul, mul_comm, ← smul_smul, ← smul_sub]
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rw [map_mul, map_mul, map_mul, map_mul, map_mul, map_mul, map_mul, map_mul]
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rw [← mul_smul_comm]
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rw [mul_assoc, mul_assoc, mul_assoc ,mul_assoc ,mul_assoc ,mul_assoc]
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rw [mul_assoc, mul_assoc, mul_assoc, mul_assoc, mul_assoc, mul_assoc]
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rw [← mul_sub, ← mul_sub, mul_smul_comm, mul_smul_comm, ← smul_mul_assoc,
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← smul_mul_assoc]
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rw [← sub_mul]
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have h1 : (ι (ofCrAnList [φ, ψ]) - (exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics ψ) • ι (ofCrAnList [ψ, φ])) =
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have h1 : (ι (ofCrAnList [φ, ψ]) -
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(exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics ψ) • ι (ofCrAnList [ψ, φ])) =
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ι [ofCrAnState φ, ofCrAnState ψ]ₛca := by
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rw [superCommute_ofCrAnState_ofCrAnState]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_append]
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@ -237,7 +241,8 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
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rw [← ofCrAnList_append]
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simp
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rw [h1]
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have hc : ι ((superCommute (ofCrAnState φ)) (ofCrAnState ψ)) ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
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have hc : ι ((superCommute (ofCrAnState φ)) (ofCrAnState ψ)) ∈
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Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
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apply ι_superCommute_ofCrAnState_ofCrAnState_mem_center
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rw [Subalgebra.mem_center_iff] at hc
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repeat rw [← mul_assoc]
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@ -272,7 +277,6 @@ lemma ι_timeOrder_superCommute_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_superCommute_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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@ -280,18 +284,16 @@ lemma ι_timeOrder_superCommute_neq_time {φ ψ : 𝓕.CrAnStates}
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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_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel ]
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have ht := IsTotal.total (r := crAnTimeOrderRel) φ ψ
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· rw [timeOrder_superCommute_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 [superCommute_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_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel ]
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rw [timeOrder_superCommute_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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