refactor: rename timeOrder for CrAnAlgebra
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jstoobysmith
parent
f5e7bbad59
commit
c53299d40a
4 changed files with 95 additions and 95 deletions
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@ -27,19 +27,19 @@ open HepLean.List
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-/
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/-- Time ordering for the `CrAnAlgebra`. -/
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def timeOrder : CrAnAlgebra 𝓕 →ₗ[ℂ] CrAnAlgebra 𝓕 :=
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def timeOrderF : CrAnAlgebra 𝓕 →ₗ[ℂ] CrAnAlgebra 𝓕 :=
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Basis.constr ofCrAnListBasis ℂ fun φs =>
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crAnTimeOrderSign φs • ofCrAnList (crAnTimeOrderList φs)
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@[inherit_doc timeOrder]
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scoped[FieldSpecification.CrAnAlgebra] notation "𝓣ᶠ(" a ")" => timeOrder a
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@[inherit_doc timeOrderF]
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scoped[FieldSpecification.CrAnAlgebra] notation "𝓣ᶠ(" a ")" => timeOrderF a
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lemma timeOrder_ofCrAnList (φs : List 𝓕.CrAnStates) :
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lemma timeOrderF_ofCrAnList (φs : List 𝓕.CrAnStates) :
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𝓣ᶠ(ofCrAnList φs) = crAnTimeOrderSign φs • ofCrAnList (crAnTimeOrderList φs) := by
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rw [← ofListBasis_eq_ofList]
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simp only [timeOrder, Basis.constr_basis]
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simp only [timeOrderF, Basis.constr_basis]
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lemma timeOrder_timeOrder_mid (a b c : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b * c) = 𝓣ᶠ(a * 𝓣ᶠ(b) * c) := by
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lemma timeOrderF_timeOrderF_mid (a b c : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b * c) = 𝓣ᶠ(a * 𝓣ᶠ(b) * c) := by
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let pc (c : 𝓕.CrAnAlgebra) (hc : c ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
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Prop := 𝓣ᶠ(a * b * c) = 𝓣ᶠ(a * 𝓣ᶠ(b) * c)
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change pc c (Basis.mem_span _ c)
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@ -61,10 +61,10 @@ lemma timeOrder_timeOrder_mid (a b c : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b * c) =
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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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rw [timeOrder_ofCrAnList]
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rw [timeOrderF_ofCrAnList]
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simp only [← ofCrAnList_append, Algebra.mul_smul_comm,
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Algebra.smul_mul_assoc, map_smul]
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rw [timeOrder_ofCrAnList, timeOrder_ofCrAnList, smul_smul]
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rw [timeOrderF_ofCrAnList, timeOrderF_ofCrAnList, smul_smul]
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congr 1
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· simp only [crAnTimeOrderSign, crAnTimeOrderList]
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rw [Wick.koszulSign_of_append_eq_insertionSort, mul_comm]
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@ -87,68 +87,68 @@ lemma timeOrder_timeOrder_mid (a b c : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b * c) =
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· intro x hx h hp
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simp_all [pc]
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lemma timeOrder_timeOrder_right (a b : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b) = 𝓣ᶠ(a * 𝓣ᶠ(b)) := by
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lemma timeOrderF_timeOrderF_right (a b : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b) = 𝓣ᶠ(a * 𝓣ᶠ(b)) := by
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trans 𝓣ᶠ(a * b * 1)
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· simp
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· rw [timeOrder_timeOrder_mid]
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· rw [timeOrderF_timeOrderF_mid]
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simp
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lemma timeOrder_timeOrder_left (a b : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b) = 𝓣ᶠ(𝓣ᶠ(a) * b) := by
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lemma timeOrderF_timeOrderF_left (a b : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b) = 𝓣ᶠ(𝓣ᶠ(a) * b) := by
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trans 𝓣ᶠ(1 * a * b)
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· simp
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· rw [timeOrder_timeOrder_mid]
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· rw [timeOrderF_timeOrderF_mid]
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simp
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lemma timeOrder_ofStateList (φs : List 𝓕.States) :
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lemma timeOrderF_ofStateList (φs : List 𝓕.States) :
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𝓣ᶠ(ofStateList φs) = timeOrderSign φs • ofStateList (timeOrderList φs) := by
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conv_lhs =>
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rw [ofStateList_sum, map_sum]
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enter [2, x]
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rw [timeOrder_ofCrAnList]
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rw [timeOrderF_ofCrAnList]
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simp only [crAnTimeOrderSign_crAnSection]
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rw [← Finset.smul_sum]
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congr
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rw [ofStateList_sum, sum_crAnSections_timeOrder]
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rfl
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lemma timeOrder_ofStateList_nil : timeOrder (𝓕 := 𝓕) (ofStateList []) = 1 := by
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rw [timeOrder_ofStateList]
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lemma timeOrderF_ofStateList_nil : timeOrderF (𝓕 := 𝓕) (ofStateList []) = 1 := by
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rw [timeOrderF_ofStateList]
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simp [timeOrderSign, Wick.koszulSign, timeOrderList]
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@[simp]
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lemma timeOrder_ofStateList_singleton (φ : 𝓕.States) : 𝓣ᶠ(ofStateList [φ]) = ofStateList [φ] := by
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simp [timeOrder_ofStateList, timeOrderSign, timeOrderList]
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lemma timeOrderF_ofStateList_singleton (φ : 𝓕.States) : 𝓣ᶠ(ofStateList [φ]) = ofStateList [φ] := by
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simp [timeOrderF_ofStateList, timeOrderSign, timeOrderList]
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lemma timeOrder_ofState_ofState_ordered {φ ψ : 𝓕.States} (h : timeOrderRel φ ψ) :
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lemma timeOrderF_ofState_ofState_ordered {φ ψ : 𝓕.States} (h : timeOrderRel φ ψ) :
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𝓣ᶠ(ofState φ * ofState ψ) = ofState φ * ofState ψ := by
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rw [← ofStateList_singleton, ← ofStateList_singleton, ← ofStateList_append, timeOrder_ofStateList]
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rw [← ofStateList_singleton, ← ofStateList_singleton, ← ofStateList_append, timeOrderF_ofStateList]
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simp only [List.singleton_append]
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rw [timeOrderSign_pair_ordered h, timeOrderList_pair_ordered h]
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simp
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lemma timeOrder_ofState_ofState_not_ordered {φ ψ : 𝓕.States} (h : ¬ timeOrderRel φ ψ) :
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lemma timeOrderF_ofState_ofState_not_ordered {φ ψ : 𝓕.States} (h : ¬ timeOrderRel φ ψ) :
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𝓣ᶠ(ofState φ * ofState ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • ofState ψ * ofState φ := by
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rw [← ofStateList_singleton, ← ofStateList_singleton,
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← ofStateList_append, timeOrder_ofStateList]
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← ofStateList_append, timeOrderF_ofStateList]
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simp only [List.singleton_append, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [timeOrderSign_pair_not_ordered h, timeOrderList_pair_not_ordered h]
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simp [← ofStateList_append]
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lemma timeOrder_ofState_ofState_not_ordered_eq_timeOrder {φ ψ : 𝓕.States} (h : ¬ timeOrderRel φ ψ) :
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lemma timeOrderF_ofState_ofState_not_ordered_eq_timeOrderF {φ ψ : 𝓕.States} (h : ¬ timeOrderRel φ ψ) :
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𝓣ᶠ(ofState φ * ofState ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • 𝓣ᶠ(ofState ψ * ofState φ) := by
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rw [timeOrder_ofState_ofState_not_ordered h]
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rw [timeOrder_ofState_ofState_ordered]
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rw [timeOrderF_ofState_ofState_not_ordered h]
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rw [timeOrderF_ofState_ofState_ordered]
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simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc]
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have hx := IsTotal.total (r := timeOrderRel) ψ φ
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simp_all
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lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel
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lemma timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) :
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𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca) = 0 := by
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rw [superCommuteF_ofCrAnState_ofCrAnState]
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simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, map_sub, map_smul]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton,
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← ofCrAnList_append, ← ofCrAnList_append, timeOrder_ofCrAnList, timeOrder_ofCrAnList]
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← ofCrAnList_append, ← ofCrAnList_append, timeOrderF_ofCrAnList, timeOrderF_ofCrAnList]
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simp only [List.singleton_append]
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rw [crAnTimeOrderSign_pair_not_ordered h, crAnTimeOrderList_pair_not_ordered h]
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rw [sub_eq_zero, smul_smul]
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@ -160,51 +160,51 @@ lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel
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· rw [crAnTimeOrderList_pair_ordered]
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simp_all
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lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right
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lemma timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.CrAnAlgebra) :
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𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca) = 0 := by
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rw [timeOrder_timeOrder_right,
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timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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rw [timeOrderF_timeOrderF_right,
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timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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simp
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lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left
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lemma timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.CrAnAlgebra) :
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𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca * a) = 0 := by
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rw [timeOrder_timeOrder_left,
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timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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rw [timeOrderF_timeOrderF_left,
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timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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simp
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lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_mid
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lemma timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_mid
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a b : 𝓕.CrAnAlgebra) :
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𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
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rw [timeOrder_timeOrder_mid,
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timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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rw [timeOrderF_timeOrderF_mid,
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timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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simp
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lemma timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel
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lemma timeOrderF_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel
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{φ1 φ2 : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ1 φ2) (a : 𝓕.CrAnAlgebra) :
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𝓣ᶠ([a, [ofCrAnState φ1, ofCrAnState φ2]ₛca]ₛca) = 0 := by
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rw [← bosonicProj_add_fermionicProj a]
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simp only [map_add, LinearMap.add_apply]
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rw [bosonic_superCommuteF (Submodule.coe_mem (bosonicProj a))]
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simp only [map_sub]
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rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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rw [timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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simp only [sub_self, zero_add]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
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rcases superCommuteF_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ1] [φ2] with h' | h'
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· rw [superCommuteF_bonsonic h']
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simp only [ofCrAnList_singleton, map_sub]
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rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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rw [timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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simp
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· rw [superCommuteF_fermionic_fermionic (Submodule.coe_mem (fermionicProj a)) h']
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simp only [ofCrAnList_singleton, map_add]
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rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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rw [timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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simp
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lemma timeOrder_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel
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lemma timeOrderF_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel
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{φ1 φ2 φ3 : 𝓕.CrAnStates} (h12 : ¬ crAnTimeOrderRel φ1 φ2)
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(h13 : ¬ crAnTimeOrderRel φ1 φ3) :
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𝓣ᶠ([ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca) = 0 := by
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@ -213,14 +213,14 @@ lemma timeOrder_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel
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simp only [instCommGroup.eq_1, ofList_singleton, ofCrAnList_singleton, neg_smul, map_smul,
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map_sub, map_neg, smul_eq_zero]
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right
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rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h12]
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rw [timeOrderF_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h12]
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rw [superCommuteF_ofCrAnState_ofCrAnState_symm φ3]
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simp only [smul_zero, neg_zero, instCommGroup.eq_1, neg_smul, map_neg, map_smul, smul_neg,
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sub_neg_eq_add, zero_add, smul_eq_zero]
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rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h13]
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rw [timeOrderF_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h13]
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simp
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lemma timeOrder_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel'
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lemma timeOrderF_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel'
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{φ1 φ2 φ3 : 𝓕.CrAnStates} (h12 : ¬ crAnTimeOrderRel φ2 φ1)
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(h13 : ¬ crAnTimeOrderRel φ3 φ1) :
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𝓣ᶠ([ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca) = 0 := by
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@ -231,12 +231,12 @@ lemma timeOrder_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel'
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right
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rw [superCommuteF_ofCrAnState_ofCrAnState_symm φ1]
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simp only [instCommGroup.eq_1, neg_smul, map_neg, map_smul, smul_neg, neg_neg]
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rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h12]
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rw [timeOrderF_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h12]
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simp only [smul_zero, zero_sub, neg_eq_zero, smul_eq_zero]
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rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h13]
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rw [timeOrderF_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h13]
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simp
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lemma timeOrder_superCommuteF_ofCrAnState_superCommuteF_all_not_crAnTimeOrderRel
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lemma timeOrderF_superCommuteF_ofCrAnState_superCommuteF_all_not_crAnTimeOrderRel
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(φ1 φ2 φ3 : 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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@ -245,13 +245,13 @@ lemma timeOrder_superCommuteF_ofCrAnState_superCommuteF_all_not_crAnTimeOrderRel
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simp only [not_and] at h
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by_cases h23 : ¬ crAnTimeOrderRel φ2 φ3
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· simp_all only [IsEmpty.forall_iff, implies_true]
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rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h23]
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rw [timeOrderF_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h23]
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simp_all only [Decidable.not_not, forall_const]
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by_cases h32 : ¬ crAnTimeOrderRel φ3 φ2
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· simp_all only [not_false_eq_true, implies_true]
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rw [superCommuteF_ofCrAnState_ofCrAnState_symm]
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simp only [instCommGroup.eq_1, neg_smul, map_neg, map_smul, neg_eq_zero, smul_eq_zero]
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rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h32]
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rw [timeOrderF_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h32]
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simp
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simp_all only [imp_false, Decidable.not_not]
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by_cases h12 : ¬ crAnTimeOrderRel φ1 φ2
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@ -259,7 +259,7 @@ lemma timeOrder_superCommuteF_ofCrAnState_superCommuteF_all_not_crAnTimeOrderRel
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intro h13
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apply h12
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exact IsTrans.trans φ1 φ3 φ2 h13 h32
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rw [timeOrder_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel h12 h13]
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rw [timeOrderF_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel h12 h13]
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simp_all only [Decidable.not_not, forall_const]
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have h13 : crAnTimeOrderRel φ1 φ3 := IsTrans.trans φ1 φ2 φ3 h12 h23
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simp_all only [forall_const]
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@ -269,18 +269,18 @@ lemma timeOrder_superCommuteF_ofCrAnState_superCommuteF_all_not_crAnTimeOrderRel
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intro h31
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apply h21
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exact IsTrans.trans φ2 φ3 φ1 h23 h31
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rw [timeOrder_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel' h21 h31]
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rw [timeOrderF_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel' h21 h31]
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simp_all only [Decidable.not_not, forall_const]
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refine False.elim (h ?_)
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exact IsTrans.trans φ3 φ2 φ1 h32 h21
|
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|
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lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_eq_time
|
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lemma timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_eq_time
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{φ ψ : 𝓕.CrAnStates} (h1 : crAnTimeOrderRel φ ψ) (h2 : crAnTimeOrderRel ψ φ) :
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𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca) = [ofCrAnState φ, ofCrAnState ψ]ₛca := by
|
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rw [superCommuteF_ofCrAnState_ofCrAnState]
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simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, map_sub, map_smul]
|
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton,
|
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← ofCrAnList_append, ← ofCrAnList_append, timeOrder_ofCrAnList, timeOrder_ofCrAnList]
|
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← ofCrAnList_append, ← ofCrAnList_append, timeOrderF_ofCrAnList, timeOrderF_ofCrAnList]
|
||||
simp only [List.singleton_append]
|
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rw [crAnTimeOrderSign_pair_ordered h1, crAnTimeOrderList_pair_ordered h1,
|
||||
crAnTimeOrderSign_pair_ordered h2, crAnTimeOrderList_pair_ordered h2]
|
||||
|
|
@ -295,12 +295,12 @@ lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_eq_time
|
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/-- In the state algebra time, ordering obeys `T(φ₀φ₁…φₙ) = s * φᵢ * T(φ₀φ₁…φᵢ₋₁φᵢ₊₁…φₙ)`
|
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where `φᵢ` is the state
|
||||
which has maximum time and `s` is the exchange sign of `φᵢ` and `φ₀φ₁…φᵢ₋₁`. -/
|
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lemma timeOrder_eq_maxTimeField_mul (φ : 𝓕.States) (φs : List 𝓕.States) :
|
||||
lemma timeOrderF_eq_maxTimeField_mul (φ : 𝓕.States) (φs : List 𝓕.States) :
|
||||
𝓣ᶠ(ofStateList (φ :: φs)) =
|
||||
𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ (φ :: φs).take (maxTimeFieldPos φ φs)) •
|
||||
ofState (maxTimeField φ φs) * 𝓣ᶠ(ofStateList (eraseMaxTimeField φ φs)) := by
|
||||
rw [timeOrder_ofStateList, timeOrderList_eq_maxTimeField_timeOrderList]
|
||||
rw [ofStateList_cons, timeOrder_ofStateList]
|
||||
rw [timeOrderF_ofStateList, timeOrderList_eq_maxTimeField_timeOrderList]
|
||||
rw [ofStateList_cons, timeOrderF_ofStateList]
|
||||
simp only [instCommGroup.eq_1, Algebra.mul_smul_comm, Algebra.smul_mul_assoc, smul_smul]
|
||||
congr
|
||||
rw [timerOrderSign_of_eraseMaxTimeField, mul_assoc]
|
||||
|
|
@ -310,12 +310,12 @@ lemma timeOrder_eq_maxTimeField_mul (φ : 𝓕.States) (φs : List 𝓕.States)
|
|||
where `φᵢ` is the state
|
||||
which has maximum time and `s` is the exchange sign of `φᵢ` and `φ₀φ₁…φᵢ₋₁`.
|
||||
Here `s` is written using finite sets. -/
|
||||
lemma timeOrder_eq_maxTimeField_mul_finset (φ : 𝓕.States) (φs : List 𝓕.States) :
|
||||
lemma timeOrderF_eq_maxTimeField_mul_finset (φ : 𝓕.States) (φs : List 𝓕.States) :
|
||||
𝓣ᶠ(ofStateList (φ :: φs)) = 𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ ⟨(eraseMaxTimeField φ φs).get,
|
||||
(Finset.filter (fun x =>
|
||||
(maxTimeFieldPosFin φ φs).succAbove x < maxTimeFieldPosFin φ φs) Finset.univ)⟩) •
|
||||
ofState (maxTimeField φ φs) * 𝓣ᶠ(ofStateList (eraseMaxTimeField φ φs)) := by
|
||||
rw [timeOrder_eq_maxTimeField_mul]
|
||||
rw [timeOrderF_eq_maxTimeField_mul]
|
||||
congr 3
|
||||
apply FieldStatistic.ofList_perm
|
||||
nth_rewrite 1 [← List.finRange_map_get (φ :: φs)]
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue