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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@ -19,8 +19,8 @@ open FieldStatistic
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namespace FieldOpAlgebra
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variable {𝓕 : FieldSpecification}
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lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnListF {φ1 φ2 φ3 : 𝓕.CrAnStates}
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(φs1 φs2 : List 𝓕.CrAnStates) (h :
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lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnListF {φ1 φ2 φ3 : 𝓕.CrAnFieldOp}
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(φs1 φs2 : List 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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@ -139,8 +139,8 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnListF {φ1 φ2 φ3
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map_smul, smul_sub]
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simp_all
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lemma ι_timeOrderF_superCommuteF_superCommuteF_ofCrAnListF {φ1 φ2 φ3 : 𝓕.CrAnStates}
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(φs1 φs2 : List 𝓕.CrAnStates) :
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lemma ι_timeOrderF_superCommuteF_superCommuteF_ofCrAnListF {φ1 φ2 φ3 : 𝓕.CrAnFieldOp}
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(φs1 φs2 : List 𝓕.CrAnFieldOp) :
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ι 𝓣ᶠ(ofCrAnListF φs1 * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca * ofCrAnListF φs2)
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= 0 := by
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by_cases h :
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@ -153,7 +153,7 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_ofCrAnListF {φ1 φ2 φ3 : 𝓕.
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simp
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@[simp]
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lemma ι_timeOrderF_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.FieldOpFreeAlgebra) :
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lemma ι_timeOrderF_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnFieldOp} (a b : 𝓕.FieldOpFreeAlgebra) :
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ι 𝓣ᶠ(a * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca * b) = 0 := by
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let pb (b : 𝓕.FieldOpFreeAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
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Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca * b) = 0
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@ -181,7 +181,7 @@ lemma ι_timeOrderF_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_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
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lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnFieldOp}
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(hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.FieldOpFreeAlgebra) :
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ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * b) =
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ι ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * 𝓣ᶠ(a * b)) := by
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@ -277,7 +277,7 @@ lemma ι_timeOrderF_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_timeOrderF_superCommuteF_neq_time {φ ψ : 𝓕.CrAnStates}
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lemma ι_timeOrderF_superCommuteF_neq_time {φ ψ : 𝓕.CrAnFieldOp}
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(hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) (a b : 𝓕.FieldOpFreeAlgebra) :
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ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * b) = 0 := by
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rw [timeOrderF_timeOrderF_mid]
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@ -393,18 +393,18 @@ scoped[FieldSpecification.FieldOpAlgebra] notation "𝓣(" a ")" => timeOrder a
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lemma timeOrder_eq_ι_timeOrderF (a : 𝓕.FieldOpFreeAlgebra) :
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𝓣(ι a) = ι 𝓣ᶠ(a) := rfl
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lemma timeOrder_ofFieldOp_ofFieldOp_ordered {φ ψ : 𝓕.States} (h : timeOrderRel φ ψ) :
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lemma timeOrder_ofFieldOp_ofFieldOp_ordered {φ ψ : 𝓕.FieldOp} (h : timeOrderRel φ ψ) :
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𝓣(ofFieldOp φ * ofFieldOp ψ) = ofFieldOp φ * ofFieldOp ψ := by
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rw [ofFieldOp, ofFieldOp, ← map_mul, timeOrder_eq_ι_timeOrderF,
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timeOrderF_ofFieldOpF_ofFieldOpF_ordered h]
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lemma timeOrder_ofFieldOp_ofFieldOp_not_ordered {φ ψ : 𝓕.States} (h : ¬ timeOrderRel φ ψ) :
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lemma timeOrder_ofFieldOp_ofFieldOp_not_ordered {φ ψ : 𝓕.FieldOp} (h : ¬ timeOrderRel φ ψ) :
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𝓣(ofFieldOp φ * ofFieldOp ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • ofFieldOp ψ * ofFieldOp φ := by
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rw [ofFieldOp, ofFieldOp, ← map_mul, timeOrder_eq_ι_timeOrderF,
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timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered h]
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simp
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lemma timeOrder_ofFieldOp_ofFieldOp_not_ordered_eq_timeOrder {φ ψ : 𝓕.States}
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lemma timeOrder_ofFieldOp_ofFieldOp_not_ordered_eq_timeOrder {φ ψ : 𝓕.FieldOp}
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(h : ¬ timeOrderRel φ ψ) :
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𝓣(ofFieldOp φ * ofFieldOp ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • 𝓣(ofFieldOp ψ * ofFieldOp φ) := by
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rw [ofFieldOp, ofFieldOp, ← map_mul, timeOrder_eq_ι_timeOrderF,
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@ -417,11 +417,11 @@ lemma timeOrder_ofFieldOpList_nil : 𝓣(ofFieldOpList (𝓕 := 𝓕) []) = 1 :=
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simp
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@[simp]
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lemma timeOrder_ofFieldOpList_singleton (φ : 𝓕.States) :
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lemma timeOrder_ofFieldOpList_singleton (φ : 𝓕.FieldOp) :
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𝓣(ofFieldOpList [φ]) = ofFieldOpList [φ] := by
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rw [ofFieldOpList, timeOrder_eq_ι_timeOrderF, timeOrderF_ofFieldOpListF_singleton]
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lemma timeOrder_eq_maxTimeField_mul_finset (φ : 𝓕.States) (φs : List 𝓕.States) :
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lemma timeOrder_eq_maxTimeField_mul_finset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
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𝓣(ofFieldOpList (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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@ -429,7 +429,7 @@ lemma timeOrder_eq_maxTimeField_mul_finset (φ : 𝓕.States) (φs : List 𝓕.S
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rw [ofFieldOpList, timeOrder_eq_ι_timeOrderF, timeOrderF_eq_maxTimeField_mul_finset]
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rfl
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lemma timeOrder_superCommute_eq_time_mid {φ ψ : 𝓕.CrAnStates}
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lemma timeOrder_superCommute_eq_time_mid {φ ψ : 𝓕.CrAnFieldOp}
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(hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.FieldOpAlgebra) :
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𝓣(a * [ofCrAnFieldOp φ, ofCrAnFieldOp ψ]ₛ * b) =
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[ofCrAnFieldOp φ, ofCrAnFieldOp ψ]ₛ * 𝓣(a * b) := by
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@ -443,7 +443,7 @@ lemma timeOrder_superCommute_eq_time_mid {φ ψ : 𝓕.CrAnStates}
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· simp_all
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· simp_all
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lemma timeOrder_superCommute_eq_time_left {φ ψ : 𝓕.CrAnStates}
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lemma timeOrder_superCommute_eq_time_left {φ ψ : 𝓕.CrAnFieldOp}
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(hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (b : 𝓕.FieldOpAlgebra) :
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𝓣([ofCrAnFieldOp φ, ofCrAnFieldOp ψ]ₛ * b) =
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[ofCrAnFieldOp φ, ofCrAnFieldOp ψ]ₛ * 𝓣(b) := by
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@ -452,7 +452,7 @@ lemma timeOrder_superCommute_eq_time_left {φ ψ : 𝓕.CrAnStates}
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rw [timeOrder_superCommute_eq_time_mid hφψ hψφ]
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simp
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lemma timeOrder_superCommute_neq_time {φ ψ : 𝓕.CrAnStates}
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lemma timeOrder_superCommute_neq_time {φ ψ : 𝓕.CrAnFieldOp}
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(hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) :
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𝓣([ofCrAnFieldOp φ, ofCrAnFieldOp ψ]ₛ) = 0 := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp]
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@ -463,7 +463,7 @@ lemma timeOrder_superCommute_neq_time {φ ψ : 𝓕.CrAnStates}
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rw [ι_timeOrderF_superCommuteF_neq_time]
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exact hφψ
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lemma timeOrder_superCommute_anPart_ofFieldOp_neq_time {φ ψ : 𝓕.States}
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lemma timeOrder_superCommute_anPart_ofFieldOp_neq_time {φ ψ : 𝓕.FieldOp}
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(hφψ : ¬ (timeOrderRel φ ψ ∧ timeOrderRel ψ φ)) :
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𝓣([anPart φ,ofFieldOp ψ]ₛ) = 0 := by
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rw [ofFieldOp_eq_sum]
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