251 lines
9.1 KiB
Text
251 lines
9.1 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.PerturbationTheory.Wick.Koszul.OperatorMap
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/-!
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# Koszul signs and ordering for lists and algebras
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-/
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namespace Wick
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noncomputable section
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open HepLean.List
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inductive ContractionsAux {I : Type} : (l : List I) → (aux : List I) → Type
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| nil : ContractionsAux [] []
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| cons {l : List I} {aux : List I} {a : I} (i : Option (Fin aux.length)) :
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ContractionsAux l aux → ContractionsAux (a :: l) (optionEraseZ aux a i)
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def Contractions {I : Type} (l : List I) : Type := Σ aux, ContractionsAux l aux
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namespace Contractions
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variable {I : Type} {l : List I} (c : Contractions l)
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def normalize : List I := c.1
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lemma contractions_nil (a : Contractions ([] : List I)) : a = ⟨[], ContractionsAux.nil⟩ := by
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cases a
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rename_i aux c
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cases c
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rfl
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lemma contractions_single {i : I} (a : Contractions [i]) : a =
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⟨[i], ContractionsAux.cons none ContractionsAux.nil⟩ := by
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cases a
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rename_i aux c
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cases c
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rename_i aux' c'
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cases c'
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cases aux'
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simp only [List.length_nil, optionEraseZ]
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rename_i x
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exact Fin.elim0 x
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def nilEquiv : Contractions ([] : List I) ≃ Unit where
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toFun _ := ()
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invFun _ := ⟨[], ContractionsAux.nil⟩
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left_inv a := Eq.symm (contractions_nil a)
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right_inv _ := rfl
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def consEquiv {a : I} {l : List I} :
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Contractions (a :: l) ≃ (c : Contractions l) × Option (Fin (c.normalize).length) where
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toFun c :=
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match c with
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| ⟨aux, c⟩ =>
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match c with
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| ContractionsAux.cons (aux := aux') i c => ⟨⟨aux', c⟩, i⟩
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invFun ci :=
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⟨(optionEraseZ (ci.fst.normalize) a ci.2), ContractionsAux.cons (a := a) ci.2 ci.1.2⟩
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left_inv c := by
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match c with
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| ⟨aux, c⟩ =>
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match c with
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| ContractionsAux.cons (aux := aux') i c => rfl
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right_inv ci := by rfl
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instance decidable : (l : List I) → DecidableEq (Contractions l)
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| [] => fun a b =>
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match a, b with
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| ⟨_, a⟩, ⟨_, b⟩ =>
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match a, b with
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| ContractionsAux.nil, ContractionsAux.nil => isTrue rfl
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| _ :: l =>
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haveI : DecidableEq (Contractions l) := decidable l
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haveI : DecidableEq ((c : Contractions l) × Option (Fin (c.normalize).length)) :=
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Sigma.instDecidableEqSigma
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Equiv.decidableEq consEquiv
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instance fintype : (l : List I) → Fintype (Contractions l)
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| [] => {
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elems := {⟨[], ContractionsAux.nil⟩}
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complete := by
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intro a
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rw [Finset.mem_singleton]
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exact contractions_nil a}
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| a :: l =>
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haveI : Fintype (Contractions l) := fintype l
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haveI : Fintype ((c : Contractions l) × Option (Fin (c.normalize).length)) :=
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Sigma.instFintype
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Fintype.ofEquiv _ consEquiv.symm
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structure Splitting {I : Type} (f : I → Type) [∀ i, Fintype (f i)]
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(le1 : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le1] where
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𝓑n : I → (Σ i, f i)
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𝓑p : I → (Σ i, f i)
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𝓧n : I → ℂ
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𝓧p : I → ℂ
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h𝓑 : ∀ i, ofListM f [i] 1 = ofList [𝓑n i] (𝓧n i) + ofList [𝓑p i] (𝓧p i)
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h𝓑n : ∀ i j, le1 (𝓑n i) j
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h𝓑p : ∀ i j, le1 j (𝓑p i)
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def toCenterTerm {I : Type} (f : I → Type) [∀ i, Fintype (f i)]
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(q : I → Fin 2)
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(le1 : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le1]
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ (Σ i, f i) →ₐ A) [OperatorMap (fun i => q i.1) le1 F] :
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{r : List I} → (c : Contractions r) → (S : Splitting f le1) → A
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| [], ⟨[], .nil⟩, _ => 1
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| _ :: _, ⟨_, .cons (aux := aux') none c⟩, S => toCenterTerm f q le1 F ⟨aux', c⟩ S
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| a :: _, ⟨_, .cons (aux := aux') (some n) c⟩, S => toCenterTerm f q le1 F ⟨aux', c⟩ S *
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superCommuteCoef q [aux'.get n] (List.take (↑n) aux') •
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F (((superCommute fun i => q i.fst) (ofList [S.𝓑p a] (S.𝓧p a))) (ofListM f [aux'.get n] 1))
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lemma toCenterTerm_none {I : Type} (f : I → Type) [∀ i, Fintype (f i)]
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(q : I → Fin 2) {r : List I}
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(le1 : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le1]
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ (Σ i, f i) →ₐ A) [OperatorMap (fun i => q i.1) le1 F]
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(S : Splitting f le1) (a : I) (c : Contractions r) :
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toCenterTerm (r := a :: r) f q le1 F (Contractions.consEquiv.symm ⟨c, none⟩) S =
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toCenterTerm f q le1 F c S := by
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rw [consEquiv]
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simp only [Equiv.coe_fn_symm_mk]
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dsimp [toCenterTerm]
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rfl
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lemma toCenterTerm_center {I : Type} (f : I → Type) [∀ i, Fintype (f i)]
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(q : I → Fin 2)
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(le1 : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le1]
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ (Σ i, f i) →ₐ A) [OperatorMap (fun i => q i.1) le1 F] :
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{r : List I} → (c : Contractions r) → (S : Splitting f le1) →
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(c.toCenterTerm f q le1 F S) ∈ Subalgebra.center ℂ A
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| [], ⟨[], .nil⟩, _ => by
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dsimp [toCenterTerm]
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exact Subalgebra.one_mem (Subalgebra.center ℂ A)
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| _ :: _, ⟨_, .cons (aux := aux') none c⟩, S => by
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dsimp [toCenterTerm]
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exact toCenterTerm_center f q le1 F ⟨aux', c⟩ S
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| a :: _, ⟨_, .cons (aux := aux') (some n) c⟩, S => by
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dsimp [toCenterTerm]
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refine Subalgebra.mul_mem (Subalgebra.center ℂ A) ?hx ?hy
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exact toCenterTerm_center f q le1 F ⟨aux', c⟩ S
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apply Subalgebra.smul_mem
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rw [ofListM_expand]
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rw [map_sum, map_sum]
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refine Subalgebra.sum_mem (Subalgebra.center ℂ A) ?hy.hx.h
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intro x _
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simp only [CreatAnnilateSect.toList]
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rw [ofList_singleton]
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exact OperatorMap.superCommute_ofList_singleton_ι_center (q := fun i => q i.1)
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(le1 := le1) F (S.𝓑p a) ⟨aux'[↑n], x.head⟩
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end Contractions
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lemma static_wick_nil {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
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(q : I → Fin 2)
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(le1 : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le1]
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ (Σ i, f i) →ₐ A) [OperatorMap (fun i => q i.1) le1 F]
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(S : Contractions.Splitting f le1) :
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F (ofListM f [] 1) = ∑ c : Contractions [],
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c.toCenterTerm f q le1 F S *
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F (koszulOrder le1 (fun i => q i.fst) (ofListM f c.normalize 1)) := by
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rw [← Contractions.nilEquiv.symm.sum_comp]
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simp only [Finset.univ_unique, PUnit.default_eq_unit, Contractions.nilEquiv, Equiv.coe_fn_symm_mk,
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Finset.sum_const, Finset.card_singleton, one_smul]
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dsimp [Contractions.normalize, Contractions.toCenterTerm]
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simp [ofListM_empty]
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lemma static_wick_cons {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
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(q : I → Fin 2)
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(le1 : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le1]
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[IsTrans ((i : I) × f i) le1] [IsTotal ((i : I) × f i) le1]
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{A : Type} [Semiring A] [Algebra ℂ A] (r : List I) (a : I)
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(F : FreeAlgebra ℂ (Σ i, f i) →ₐ A) [OperatorMap (fun i => q i.1) le1 F]
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(S : Contractions.Splitting f le1)
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(ih : F (ofListM f r 1) =
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∑ c : Contractions r, c.toCenterTerm f q le1 F S * F (koszulOrder le1 (fun i => q i.fst)
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(ofListM f c.normalize 1))) :
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F (ofListM f (a :: r) 1) = ∑ c : Contractions (a :: r),
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c.toCenterTerm f q le1 F S *
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F (koszulOrder le1 (fun i => q i.fst) (ofListM f c.normalize 1)) := by
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rw [ofListM_cons_eq_ofListM, map_mul, ih, Finset.mul_sum,
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← Contractions.consEquiv.symm.sum_comp]
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erw [Finset.sum_sigma]
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congr
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funext c
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have hb := S.h𝓑 a
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rw [← mul_assoc]
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have hi := c.toCenterTerm_center f q le1 F S
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rw [Subalgebra.mem_center_iff] at hi
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rw [hi, mul_assoc, ← map_mul, hb, add_mul, map_add]
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conv_lhs =>
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rhs
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lhs
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rw [ofList_eq_smul_one]
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rw [Algebra.smul_mul_assoc]
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rw [ofList_singleton]
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rw [mul_koszulOrder_le]
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conv_lhs =>
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rhs
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lhs
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rw [← map_smul, ← Algebra.smul_mul_assoc]
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rw [← ofList_singleton, ← ofList_eq_smul_one]
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conv_lhs =>
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rhs
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rhs
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rw [ofList_eq_smul_one, Algebra.smul_mul_assoc, map_smul]
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rw [le_all_mul_koszulOrder_ofListM_expand]
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conv_lhs =>
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rhs
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rhs
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rw [smul_add, Finset.smul_sum]
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rw [← map_smul, ← map_smul, ← Algebra.smul_mul_assoc, ← ofList_eq_smul_one]
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rhs
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rhs
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intro n
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rw [← Algebra.smul_mul_assoc, smul_comm, ← map_smul, ← LinearMap.map_smul₂,
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← ofList_eq_smul_one]
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rw [← add_assoc, ← map_add, ← map_add, ← add_mul, ← hb, ← ofListM_cons_eq_ofListM, mul_add]
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rw [Fintype.sum_option]
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congr 1
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rw [Finset.mul_sum]
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congr
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funext n
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rw [← mul_assoc]
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rfl
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exact S.h𝓑p a
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exact S.h𝓑n a
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theorem static_wick_theorem {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
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(q : I → Fin 2)
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(le1 : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le1] [IsTrans ((i : I) × f i) le1]
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[IsTotal ((i : I) × f i) le1]
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{A : Type} [Semiring A] [Algebra ℂ A] (r : List I)
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(F : FreeAlgebra ℂ (Σ i, f i) →ₐ A) [OperatorMap (fun i => q i.1) le1 F]
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(S : Contractions.Splitting f le1) :
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F (ofListM f r 1) = ∑ c : Contractions r, c.toCenterTerm f q le1 F S *
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F (koszulOrder le1 (fun i => q i.fst) (ofListM f c.normalize 1)) := by
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induction r with
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| nil => exact static_wick_nil q le1 F S
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| cons a r ih => exact static_wick_cons q le1 r a F S ih
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end
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end Wick
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