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refactor: Rename ofCrAnState and ofCrAnList

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jstoobysmith 2025-02-03 11:21:11 +00:00
parent 93d06895c6
commit 171e80fc04
11 changed files with 601 additions and 601 deletions
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130
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/TimeOrder.lean
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@ -19,13 +19,13 @@ open FieldStatistic
namespace FieldOpAlgebra
variable {𝓕 : FieldSpecification}
lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnListF {φ1 φ2 φ3 : 𝓕.CrAnStates}
(φs1 φs2 : List 𝓕.CrAnStates) (h :
crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2) :
ι 𝓣ᶠ(ofCrAnList φs1 * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca *
ofCrAnList φs2) = 0 := by
ι 𝓣ᶠ(ofCrAnListF φs1 * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca *
ofCrAnListF φs2) = 0 := by
let l1 :=
(List.takeWhile (fun c => ¬ crAnTimeOrderRel φ1 c)
((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
@ -33,24 +33,24 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnList {φ1 φ2 φ3
let l2 := (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ crAnTimeOrderRel c φ1) φs2)
++ (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ ¬ crAnTimeOrderRel c φ1)
((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
have h123 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)) =
have h123 : ι 𝓣ᶠ(ofCrAnListF (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)) =
crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
• (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ2, φ3]) * ι (ofCrAnList l2)) := by
• (ι (ofCrAnListF l1) * ι (ofCrAnListF [φ1, φ2, φ3]) * ι (ofCrAnListF l2)) := by
have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ1, φ2, φ3] φs2
(by simp_all)
rw [timeOrderF_ofCrAnList, show φs1 ++ φ1 :: φ2 :: φ3 :: φs2 = φs1 ++ [φ1, φ2, φ3] ++ φs2
rw [timeOrderF_ofCrAnListF, show φs1 ++ φ1 :: φ2 :: φ3 :: φs2 = φs1 ++ [φ1, φ2, φ3] ++ φs2
by simp, crAnTimeOrderList, h1]
simp only [List.append_assoc, List.singleton_append, decide_not,
Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
have h132 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ1 :: φ3 :: φ2 :: φs2)) =
Bool.decide_and, ofCrAnListF_append, map_smul, map_mul, l1, l2, mul_assoc]
have h132 : ι 𝓣ᶠ(ofCrAnListF (φs1 ++ φ1 :: φ3 :: φ2 :: φs2)) =
crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
• (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ3, φ2]) * ι (ofCrAnList l2)) := by
• (ι (ofCrAnListF l1) * ι (ofCrAnListF [φ1, φ3, φ2]) * ι (ofCrAnListF l2)) := by
have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ1, φ3, φ2] φs2
(by simp_all)
rw [timeOrderF_ofCrAnList, show φs1 ++ φ1 :: φ3 :: φ2 :: φs2 = φs1 ++ [φ1, φ3, φ2] ++ φs2
rw [timeOrderF_ofCrAnListF, show φs1 ++ φ1 :: φ3 :: φ2 :: φs2 = φs1 ++ [φ1, φ3, φ2] ++ φs2
by simp, crAnTimeOrderList, h1]
simp only [List.singleton_append, decide_not,
Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
Bool.decide_and, ofCrAnListF_append, map_smul, map_mul, l1, l2, mul_assoc]
congr 1
have hp : List.Perm [φ1, φ3, φ2] [φ1, φ2, φ3] := by
refine List.Perm.cons φ1 ?_
@ -67,15 +67,15 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnList {φ1 φ2 φ3
refine List.Perm.trans (l₂ := [φ2, φ1, φ3]) ?_ ?_
refine List.Perm.cons φ2 (List.Perm.swap φ1 φ3 [])
exact List.Perm.swap φ1 φ2 [φ3]
have h231 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ2 :: φ3 :: φ1 :: φs2)) =
have h231 : ι 𝓣ᶠ(ofCrAnListF (φs1 ++ φ2 :: φ3 :: φ1 :: φs2)) =
crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
• (ι (ofCrAnList l1) * ι (ofCrAnList [φ2, φ3, φ1]) * ι (ofCrAnList l2)) := by
• (ι (ofCrAnListF l1) * ι (ofCrAnListF [φ2, φ3, φ1]) * ι (ofCrAnListF l2)) := by
have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ2, φ3, φ1] φs2
(by simp_all)
rw [timeOrderF_ofCrAnList, show φs1 ++ φ2 :: φ3 :: φ1 :: φs2 = φs1 ++ [φ2, φ3, φ1] ++ φs2
rw [timeOrderF_ofCrAnListF, show φs1 ++ φ2 :: φ3 :: φ1 :: φs2 = φs1 ++ [φ2, φ3, φ1] ++ φs2
by simp, crAnTimeOrderList, h1]
simp only [List.singleton_append, decide_not,
Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
Bool.decide_and, ofCrAnListF_append, map_smul, map_mul, l1, l2, mul_assoc]
congr 1
rw [crAnTimeOrderSign, Wick.koszulSign_perm_eq _ _ φ1 _ _ _ _ _ hp231, ← crAnTimeOrderSign]
· simp
@ -85,15 +85,15 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnList {φ1 φ2 φ3
all_goals
subst hφ4
simp_all
have h321 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ3 :: φ2 :: φ1 :: φs2)) =
have h321 : ι 𝓣ᶠ(ofCrAnListF (φs1 ++ φ3 :: φ2 :: φ1 :: φs2)) =
crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
• (ι (ofCrAnList l1) * ι (ofCrAnList [φ3, φ2, φ1]) * ι (ofCrAnList l2)) := by
• (ι (ofCrAnListF l1) * ι (ofCrAnListF [φ3, φ2, φ1]) * ι (ofCrAnListF l2)) := by
have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ3, φ2, φ1] φs2
(by simp_all)
rw [timeOrderF_ofCrAnList, show φs1 ++ φ3 :: φ2 :: φ1 :: φs2 = φs1 ++ [φ3, φ2, φ1] ++ φs2
rw [timeOrderF_ofCrAnListF, show φs1 ++ φ3 :: φ2 :: φ1 :: φs2 = φs1 ++ [φ3, φ2, φ1] ++ φs2
by simp, crAnTimeOrderList, h1]
simp only [List.singleton_append, decide_not,
Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
Bool.decide_and, ofCrAnListF_append, map_smul, map_mul, l1, l2, mul_assoc]
congr 1
have hp : List.Perm [φ3, φ2, φ1] [φ1, φ2, φ3] := by
refine List.Perm.trans ?_ hp231
@ -106,12 +106,12 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnList {φ1 φ2 φ3
all_goals
subst hφ4
simp_all
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
rw [superCommuteF_ofCrAnList_ofCrAnList]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
rw [superCommuteF_ofCrAnListF_ofCrAnListF]
simp only [List.singleton_append, instCommGroup.eq_1, ofList_singleton, map_sub, map_smul]
rw [superCommuteF_ofCrAnList_ofCrAnList, superCommuteF_ofCrAnList_ofCrAnList]
rw [superCommuteF_ofCrAnListF_ofCrAnListF, superCommuteF_ofCrAnListF_ofCrAnListF]
simp only [List.cons_append, List.nil_append, instCommGroup.eq_1, ofList_singleton, mul_sub, ←
ofCrAnList_append, Algebra.mul_smul_comm, sub_mul, List.append_assoc, Algebra.smul_mul_assoc,
ofCrAnListF_append, Algebra.mul_smul_comm, sub_mul, List.append_assoc, Algebra.smul_mul_assoc,
map_sub, map_smul]
rw [h123, h132, h231, h321]
simp only [smul_smul]
@ -127,49 +127,49 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnList {φ1 φ2 φ3
repeat rw [mul_assoc]
rw [← mul_sub, ← mul_sub, ← mul_sub]
rw [← sub_mul, ← sub_mul, ← sub_mul]
trans ι (ofCrAnList l1) * ι [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca *
ι (ofCrAnList l2)
trans ι (ofCrAnListF l1) * ι [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca *
ι (ofCrAnListF l2)
rw [mul_assoc]
congr
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
rw [superCommuteF_ofCrAnList_ofCrAnList]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
rw [superCommuteF_ofCrAnListF_ofCrAnListF]
simp only [List.singleton_append, instCommGroup.eq_1, ofList_singleton, map_sub, map_smul]
rw [superCommuteF_ofCrAnList_ofCrAnList, superCommuteF_ofCrAnList_ofCrAnList]
rw [superCommuteF_ofCrAnListF_ofCrAnListF, superCommuteF_ofCrAnListF_ofCrAnListF]
simp only [List.cons_append, List.nil_append, instCommGroup.eq_1, ofList_singleton, map_sub,
map_smul, smul_sub]
simp_all
lemma ι_timeOrderF_superCommuteF_superCommuteF_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
lemma ι_timeOrderF_superCommuteF_superCommuteF_ofCrAnListF {φ1 φ2 φ3 : 𝓕.CrAnStates}
(φs1 φs2 : List 𝓕.CrAnStates) :
ι 𝓣ᶠ(ofCrAnList φs1 * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs2)
ι 𝓣ᶠ(ofCrAnListF φs1 * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca * ofCrAnListF φs2)
= 0 := by
by_cases h :
crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2
· exact ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnList φs1 φs2 h
· exact ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnListF φs1 φs2 h
· rw [timeOrderF_timeOrderF_mid]
rw [timeOrderF_superCommuteF_ofCrAnState_superCommuteF_all_not_crAnTimeOrderRel _ _ _ h]
rw [timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_all_not_crAnTimeOrderRel _ _ _ h]
simp
@[simp]
lemma ι_timeOrderF_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.FieldOpFreeAlgebra) :
ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0 := by
let pb (b : 𝓕.FieldOpFreeAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0
ι 𝓣ᶠ(a * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca * b) = 0 := by
let pb (b : 𝓕.FieldOpFreeAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca * b) = 0
change pb b (Basis.mem_span _ b)
apply Submodule.span_induction
· intro x hx
obtain ⟨φs, rfl⟩ := hx
simp only [ofListBasis_eq_ofList, pb]
let pa (a : 𝓕.FieldOpFreeAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs) = 0
let pa (a : 𝓕.FieldOpFreeAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca * ofCrAnListF φs) = 0
change pa a (Basis.mem_span _ a)
apply Submodule.span_induction
· intro x hx
obtain ⟨φs', rfl⟩ := hx
simp only [ofListBasis_eq_ofList, pa]
exact ι_timeOrderF_superCommuteF_superCommuteF_ofCrAnList φs' φs
exact ι_timeOrderF_superCommuteF_superCommuteF_ofCrAnListF φs' φs
· simp [pa]
· intro x y hx hy hpx hpy
simp_all [pa,mul_add, add_mul]
@ -183,28 +183,28 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnStates}
lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
(hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.FieldOpFreeAlgebra) :
ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b)) := by
let pb (b : 𝓕.FieldOpFreeAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b))
ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * b) =
ι ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * 𝓣ᶠ(a * b)) := by
let pb (b : 𝓕.FieldOpFreeAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * b) =
ι ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * 𝓣ᶠ(a * b))
change pb b (Basis.mem_span _ b)
apply Submodule.span_induction
· intro x hx
obtain ⟨φs, rfl⟩ := hx
simp only [ofListBasis_eq_ofList, map_mul, pb]
let pa (a : 𝓕.FieldOpFreeAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * ofCrAnList φs) =
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a* ofCrAnList φs))
let pa (a : 𝓕.FieldOpFreeAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * ofCrAnListF φs) =
ι ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * 𝓣ᶠ(a* ofCrAnListF φs))
change pa a (Basis.mem_span _ a)
apply Submodule.span_induction
· intro x hx
obtain ⟨φs', rfl⟩ := hx
simp only [ofListBasis_eq_ofList, map_mul, pa]
conv_lhs =>
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [mul_sub, sub_mul, ← ofCrAnList_append]
rw [timeOrderF_ofCrAnList, timeOrderF_ofCrAnList]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [mul_sub, sub_mul, ← ofCrAnListF_append]
rw [timeOrderF_ofCrAnListF, timeOrderF_ofCrAnListF]
have h1 : crAnTimeOrderSign (φs' ++ φ :: ψ :: φs) =
crAnTimeOrderSign (φs' ++ ψ :: φ :: φs) := by
trans crAnTimeOrderSign (φs' ++ [φ, ψ] ++ φs)
@ -223,7 +223,7 @@ lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
have h2 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [ψ, φ] φs
(by simp_all)
rw [crAnTimeOrderList, show φs' ++ ψ :: φ :: φs = φs' ++ [ψ, φ] ++ φs by simp, h2]
repeat rw [ofCrAnList_append]
repeat rw [ofCrAnListF_append]
rw [smul_smul, mul_comm, ← smul_smul, ← smul_sub]
rw [map_mul, map_mul, map_mul, map_mul, map_mul, map_mul, map_mul, map_mul]
rw [← mul_smul_comm]
@ -231,26 +231,26 @@ lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
rw [← mul_sub, ← mul_sub, mul_smul_comm, mul_smul_comm, ← smul_mul_assoc,
← smul_mul_assoc]
rw [← sub_mul]
have h1 : (ι (ofCrAnList [φ, ψ]) -
(exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics ψ) • ι (ofCrAnList [ψ, φ])) =
ι [ofCrAnState φ, ofCrAnState ψ]ₛca := by
rw [superCommuteF_ofCrAnState_ofCrAnState]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_append]
have h1 : (ι (ofCrAnListF [φ, ψ]) -
(exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics ψ) • ι (ofCrAnListF [ψ, φ])) =
ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca := by
rw [superCommuteF_ofCrAnOpF_ofCrAnOpF]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_append]
simp only [instCommGroup.eq_1, List.singleton_append, Algebra.smul_mul_assoc, map_sub,
map_smul]
rw [← ofCrAnList_append]
rw [← ofCrAnListF_append]
simp
rw [h1]
have hc : ι ((superCommuteF (ofCrAnState φ)) (ofCrAnState ψ)) ∈
have hc : ι ((superCommuteF (ofCrAnOpF φ)) (ofCrAnOpF ψ)) ∈
Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
apply ι_superCommuteF_ofCrAnState_ofCrAnState_mem_center
apply ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_mem_center
rw [Subalgebra.mem_center_iff] at hc
repeat rw [← mul_assoc]
rw [hc]
repeat rw [mul_assoc]
rw [smul_mul_assoc]
rw [← map_mul, ← map_mul, ← map_mul, ← map_mul]
rw [← ofCrAnList_append, ← ofCrAnList_append, ← ofCrAnList_append, ← ofCrAnList_append]
rw [← ofCrAnListF_append, ← ofCrAnListF_append, ← ofCrAnListF_append, ← ofCrAnListF_append]
have h1 := insertionSort_of_takeWhile_filter 𝓕.crAnTimeOrderRel φ φs' φs
simp only [decide_not, Bool.decide_and, List.append_assoc, List.cons_append,
List.singleton_append, Algebra.mul_smul_comm, map_mul] at h1 ⊢
@ -260,7 +260,7 @@ lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
· rw [ι_superCommuteF_of_diff_statistic hq]
simp
· rw [crAnTimeOrderSign, Wick.koszulSign_eq_rel_eq_stat _ _, ← crAnTimeOrderSign]
rw [timeOrderF_ofCrAnList]
rw [timeOrderF_ofCrAnListF]
simp only [map_smul, Algebra.mul_smul_comm]
simp only [List.nil_append]
exact hψφ
@ -279,18 +279,18 @@ lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
lemma ι_timeOrderF_superCommuteF_neq_time {φ ψ : 𝓕.CrAnStates}
(hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) (a b : 𝓕.FieldOpFreeAlgebra) :
ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * b) = 0 := by
rw [timeOrderF_timeOrderF_mid]
have hφψ : ¬ (crAnTimeOrderRel φ ψ) ∨ ¬ (crAnTimeOrderRel ψ φ) := by
exact Decidable.not_and_iff_or_not.mp hφψ
rcases hφψ with hφψ | hφψ
· rw [timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel]
· rw [timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel]
simp_all only [false_and, not_false_eq_true, false_or, mul_zero, zero_mul, map_zero]
simp_all
· rw [superCommuteF_ofCrAnState_ofCrAnState_symm]
· rw [superCommuteF_ofCrAnOpF_ofCrAnOpF_symm]
simp only [instCommGroup.eq_1, neg_smul, map_neg, map_smul, mul_neg, Algebra.mul_smul_comm,
neg_mul, Algebra.smul_mul_assoc, neg_eq_zero, smul_eq_zero]
rw [timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel]
rw [timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel]
simp only [mul_zero, zero_mul, map_zero, or_true]
simp_all
@ -458,7 +458,7 @@ lemma timeOrder_superCommute_neq_time {φ ψ : 𝓕.CrAnStates}
rw [ofCrAnFieldOp, ofCrAnFieldOp]
rw [superCommute_eq_ι_superCommuteF]
rw [timeOrder_eq_ι_timeOrderF]
trans ι (timeOrderF (1 * (superCommuteF (ofCrAnState φ)) (ofCrAnState ψ) * 1))
trans ι (timeOrderF (1 * (superCommuteF (ofCrAnOpF φ)) (ofCrAnOpF ψ) * 1))
simp only [one_mul, mul_one]
rw [ι_timeOrderF_superCommuteF_neq_time]
exact hφψ