refactor: Rename States to FieldOps
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jstoobysmith
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171e80fc04
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8f41de5785
36 changed files with 946 additions and 946 deletions
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@ -34,7 +34,7 @@ def timeOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 :=
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@[inherit_doc timeOrderF]
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scoped[FieldSpecification.FieldOpFreeAlgebra] notation "𝓣ᶠ(" a ")" => timeOrderF a
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lemma timeOrderF_ofCrAnListF (φs : List 𝓕.CrAnStates) :
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lemma timeOrderF_ofCrAnListF (φs : List 𝓕.CrAnFieldOp) :
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𝓣ᶠ(ofCrAnListF φs) = crAnTimeOrderSign φs • ofCrAnListF (crAnTimeOrderList φs) := by
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rw [← ofListBasis_eq_ofList]
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simp only [timeOrderF, Basis.constr_basis]
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@ -99,7 +99,7 @@ lemma timeOrderF_timeOrderF_left (a b : 𝓕.FieldOpFreeAlgebra) : 𝓣ᶠ(a * b
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· rw [timeOrderF_timeOrderF_mid]
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simp
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lemma timeOrderF_ofFieldOpListF (φs : List 𝓕.States) :
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lemma timeOrderF_ofFieldOpListF (φs : List 𝓕.FieldOp) :
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𝓣ᶠ(ofFieldOpListF φs) = timeOrderSign φs • ofFieldOpListF (timeOrderList φs) := by
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conv_lhs =>
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rw [ofFieldOpListF_sum, map_sum]
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@ -116,10 +116,10 @@ lemma timeOrderF_ofFieldOpListF_nil : timeOrderF (𝓕 := 𝓕) (ofFieldOpListF
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simp [timeOrderSign, Wick.koszulSign, timeOrderList]
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@[simp]
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lemma timeOrderF_ofFieldOpListF_singleton (φ : 𝓕.States) : 𝓣ᶠ(ofFieldOpListF [φ]) = ofFieldOpListF [φ] := by
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lemma timeOrderF_ofFieldOpListF_singleton (φ : 𝓕.FieldOp) : 𝓣ᶠ(ofFieldOpListF [φ]) = ofFieldOpListF [φ] := by
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simp [timeOrderF_ofFieldOpListF, timeOrderSign, timeOrderList]
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lemma timeOrderF_ofFieldOpF_ofFieldOpF_ordered {φ ψ : 𝓕.States} (h : timeOrderRel φ ψ) :
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lemma timeOrderF_ofFieldOpF_ofFieldOpF_ordered {φ ψ : 𝓕.FieldOp} (h : timeOrderRel φ ψ) :
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𝓣ᶠ(ofFieldOpF φ * ofFieldOpF ψ) = ofFieldOpF φ * ofFieldOpF ψ := by
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rw [← ofFieldOpListF_singleton, ← ofFieldOpListF_singleton, ← ofFieldOpListF_append,
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timeOrderF_ofFieldOpListF]
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@ -127,7 +127,7 @@ lemma timeOrderF_ofFieldOpF_ofFieldOpF_ordered {φ ψ : 𝓕.States} (h : timeOr
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rw [timeOrderSign_pair_ordered h, timeOrderList_pair_ordered h]
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simp
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lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered {φ ψ : 𝓕.States} (h : ¬ timeOrderRel φ ψ) :
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lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered {φ ψ : 𝓕.FieldOp} (h : ¬ timeOrderRel φ ψ) :
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𝓣ᶠ(ofFieldOpF φ * ofFieldOpF ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • ofFieldOpF ψ * ofFieldOpF φ := by
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rw [← ofFieldOpListF_singleton, ← ofFieldOpListF_singleton,
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← ofFieldOpListF_append, timeOrderF_ofFieldOpListF]
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@ -135,7 +135,7 @@ lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered {φ ψ : 𝓕.States} (h : ¬
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rw [timeOrderSign_pair_not_ordered h, timeOrderList_pair_not_ordered h]
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simp [← ofFieldOpListF_append]
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lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered_eq_timeOrderF {φ ψ : 𝓕.States}
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lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered_eq_timeOrderF {φ ψ : 𝓕.FieldOp}
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(h : ¬ timeOrderRel φ ψ) :
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𝓣ᶠ(ofFieldOpF φ * ofFieldOpF ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • 𝓣ᶠ(ofFieldOpF ψ * ofFieldOpF φ) := by
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rw [timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered h]
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@ -145,7 +145,7 @@ lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered_eq_timeOrderF {φ ψ : 𝓕.S
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simp_all
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lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) :
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{φ ψ : 𝓕.CrAnFieldOp} (h : ¬ crAnTimeOrderRel φ ψ) :
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𝓣ᶠ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca) = 0 := by
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rw [superCommuteF_ofCrAnOpF_ofCrAnOpF]
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simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, map_sub, map_smul]
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@ -163,28 +163,28 @@ lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel
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simp_all
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lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel_right
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.FieldOpFreeAlgebra) :
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{φ ψ : 𝓕.CrAnFieldOp} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.FieldOpFreeAlgebra) :
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𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca) = 0 := by
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rw [timeOrderF_timeOrderF_right,
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timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel h]
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simp
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lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel_left
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.FieldOpFreeAlgebra) :
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{φ ψ : 𝓕.CrAnFieldOp} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.FieldOpFreeAlgebra) :
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𝓣ᶠ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * a) = 0 := by
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rw [timeOrderF_timeOrderF_left,
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timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel h]
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simp
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lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel_mid
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a b : 𝓕.FieldOpFreeAlgebra) :
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{φ ψ : 𝓕.CrAnFieldOp} (h : ¬ crAnTimeOrderRel φ ψ) (a b : 𝓕.FieldOpFreeAlgebra) :
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𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * b) = 0 := by
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rw [timeOrderF_timeOrderF_mid,
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timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel h]
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simp
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lemma timeOrderF_superCommuteF_superCommuteF_ofCrAnOpF_not_crAnTimeOrderRel
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{φ1 φ2 : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ1 φ2) (a : 𝓕.FieldOpFreeAlgebra) :
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{φ1 φ2 : 𝓕.CrAnFieldOp} (h : ¬ crAnTimeOrderRel φ1 φ2) (a : 𝓕.FieldOpFreeAlgebra) :
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𝓣ᶠ([a, [ofCrAnOpF φ1, ofCrAnOpF 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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@ -207,7 +207,7 @@ lemma timeOrderF_superCommuteF_superCommuteF_ofCrAnOpF_not_crAnTimeOrderRel
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simp
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lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_not_crAnTimeOrderRel
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{φ1 φ2 φ3 : 𝓕.CrAnStates} (h12 : ¬ crAnTimeOrderRel φ1 φ2)
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{φ1 φ2 φ3 : 𝓕.CrAnFieldOp} (h12 : ¬ crAnTimeOrderRel φ1 φ2)
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(h13 : ¬ crAnTimeOrderRel φ1 φ3) :
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𝓣ᶠ([ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca) = 0 := by
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
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@ -223,7 +223,7 @@ lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_not_crAnTimeOrderRel
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simp
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lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_not_crAnTimeOrderRel'
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{φ1 φ2 φ3 : 𝓕.CrAnStates} (h12 : ¬ crAnTimeOrderRel φ2 φ1)
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{φ1 φ2 φ3 : 𝓕.CrAnFieldOp} (h12 : ¬ crAnTimeOrderRel φ2 φ1)
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(h13 : ¬ crAnTimeOrderRel φ3 φ1) :
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𝓣ᶠ([ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca) = 0 := by
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
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@ -239,7 +239,7 @@ lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_not_crAnTimeOrderRel'
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simp
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lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_all_not_crAnTimeOrderRel
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(φ1 φ2 φ3 : 𝓕.CrAnStates) (h : ¬
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(φ1 φ2 φ3 : 𝓕.CrAnFieldOp) (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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@ -277,7 +277,7 @@ lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_all_not_crAnTimeOrderRel
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exact IsTrans.trans φ3 φ2 φ1 h32 h21
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lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_eq_time
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{φ ψ : 𝓕.CrAnStates} (h1 : crAnTimeOrderRel φ ψ) (h2 : crAnTimeOrderRel ψ φ) :
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{φ ψ : 𝓕.CrAnFieldOp} (h1 : crAnTimeOrderRel φ ψ) (h2 : crAnTimeOrderRel ψ φ) :
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𝓣ᶠ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca) = [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca := by
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rw [superCommuteF_ofCrAnOpF_ofCrAnOpF]
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simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, map_sub, map_smul]
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@ -297,7 +297,7 @@ lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_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
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which has maximum time and `s` is the exchange sign of `φᵢ` and `φ₀φ₁…φᵢ₋₁`. -/
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lemma timeOrderF_eq_maxTimeField_mul (φ : 𝓕.States) (φs : List 𝓕.States) :
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lemma timeOrderF_eq_maxTimeField_mul (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
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𝓣ᶠ(ofFieldOpListF (φ :: φs)) =
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𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ (φ :: φs).take (maxTimeFieldPos φ φs)) •
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ofFieldOpF (maxTimeField φ φs) * 𝓣ᶠ(ofFieldOpListF (eraseMaxTimeField φ φs)) := by
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@ -312,7 +312,7 @@ lemma timeOrderF_eq_maxTimeField_mul (φ : 𝓕.States) (φs : List 𝓕.States)
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where `φᵢ` is the state
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which has maximum time and `s` is the exchange sign of `φᵢ` and `φ₀φ₁…φᵢ₋₁`.
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Here `s` is written using finite sets. -/
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lemma timeOrderF_eq_maxTimeField_mul_finset (φ : 𝓕.States) (φs : List 𝓕.States) :
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lemma timeOrderF_eq_maxTimeField_mul_finset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
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𝓣ᶠ(ofFieldOpListF (φ :: φs)) = 𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ ⟨(eraseMaxTimeField φ φs).get,
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(Finset.filter (fun x =>
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(maxTimeFieldPosFin φ φs).succAbove x < maxTimeFieldPosFin φ φs) Finset.univ)⟩) •
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