Physlib.QFT.PerturbationTheory.FieldOpFreeAlgebra.NormalOrder

Imports

/-
Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
module

public import Physlib.QFT.PerturbationTheory.FieldSpecification.NormalOrder
public import Physlib.QFT.PerturbationTheory.FieldOpFreeAlgebra.SuperCommute

Normal Ordering in the FieldOpFreeAlgebra

In the module Physlib.QFT.PerturbationTheory.FieldSpecification.NormalOrder we defined the normal ordering of a list of CrAnFieldOp. In this module we extend the normal ordering to a linear map on FieldOpFreeAlgebra.

We derive properties of this normal ordering.

@[expose] public section

For a field specification 𝓕, normalOrderF is the linear map

FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕

defined by its action on the basis ofCrAnListF φs, taking ofCrAnListF φs to

normalOrderSign φs • ofCrAnListF (normalOrderList φs).

That is, normalOrderF normal-orders the field operators and multiplies by the sign of the normal order.

The notation 𝓝ᶠ(a) is used for normalOrderF a for a an element of FieldOpFreeAlgebra 𝓕.

def normalOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 :=
  Basis.constr ofCrAnListFBasis ℂ fun φs =>
  normalOrderSign φs • ofCrAnListF (normalOrderList φs)

              @[inherit_doc normalOrderF]
scoped[FieldSpecification.FieldOpFreeAlgebra] notation "𝓝ᶠ(" a ")" => normalOrderF a

lemma normalOrderF_ofCrAnListF (φs : List 𝓕.CrAnFieldOp) :
    𝓝ᶠ(ofCrAnListF φs) = normalOrderSign φs • ofCrAnListF (normalOrderList φs) := by𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs) = normalOrderSign φs • ofCrAnListF (normalOrderList φs)
  rw [← ofListBasis_eq_ofList,𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListFBasis φs) = normalOrderSign φs • ofCrAnListF (normalOrderList φs)All goals completed! 🐙 normalOrderF,𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOp⊢ ((ofCrAnListFBasis.constr ℂ) fun φs => normalOrderSign φs • ofCrAnListF (normalOrderList φs)) (ofCrAnListFBasis φs) =
  normalOrderSign φs • ofCrAnListF (normalOrderList φs)All goals completed! 🐙 Basis.constr_basis𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOp⊢ normalOrderSign φs • ofCrAnListF (normalOrderList φs) = normalOrderSign φs • ofCrAnListF (normalOrderList φs)All goals completed! 🐙]All goals completed! 🐙

lemma ofCrAnListF_eq_normalOrderF (φs : List 𝓕.CrAnFieldOp) :
    ofCrAnListF (normalOrderList φs) = normalOrderSign φs • 𝓝ᶠ(ofCrAnListF φs) := by𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOp⊢ ofCrAnListF (normalOrderList φs) = normalOrderSign φs • normalOrderF (ofCrAnListF φs)
  rw [normalOrderF_ofCrAnListF,𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOp⊢ ofCrAnListF (normalOrderList φs) = normalOrderSign φs • normalOrderSign φs • ofCrAnListF (normalOrderList φs)All goals completed! 🐙 normalOrderList,𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOp⊢ ofCrAnListF (List.insertionSort normalOrderRel φs) =
  normalOrderSign φs • normalOrderSign φs • ofCrAnListF (List.insertionSort normalOrderRel φs)All goals completed! 🐙 smul_smul,𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOp⊢ ofCrAnListF (List.insertionSort normalOrderRel φs) =
  (normalOrderSign φs * normalOrderSign φs) • ofCrAnListF (List.insertionSort normalOrderRel φs)All goals completed! 🐙 normalOrderSign,𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOp⊢ ofCrAnListF (List.insertionSort normalOrderRel φs) =
  (Wick.koszulSign 𝓕.crAnStatistics normalOrderRel φs * Wick.koszulSign 𝓕.crAnStatistics normalOrderRel φs) •
    ofCrAnListF (List.insertionSort normalOrderRel φs)All goals completed! 🐙
    Wick.koszulSign_mul_self,𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOp⊢ ofCrAnListF (List.insertionSort normalOrderRel φs) = 1 • ofCrAnListF (List.insertionSort normalOrderRel φs)All goals completed! 🐙 one_smul𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOp⊢ ofCrAnListF (List.insertionSort normalOrderRel φs) = ofCrAnListF (List.insertionSort normalOrderRel φs)All goals completed! 🐙]All goals completed! 🐙

lemma normalOrderF_one : normalOrderF (𝓕 := 𝓕) 1 = 1 := by𝓕:FieldSpecification⊢ normalOrderF 1 = 1
  rw [← ofCrAnListF_nil,𝓕:FieldSpecification⊢ normalOrderF (ofCrAnListF []) = ofCrAnListF []All goals completed! 🐙 normalOrderF_ofCrAnListF,𝓕:FieldSpecification⊢ normalOrderSign [] • ofCrAnListF (normalOrderList []) = ofCrAnListF []All goals completed! 🐙 normalOrderSign_nil,𝓕:FieldSpecification⊢ 1 • ofCrAnListF (normalOrderList []) = ofCrAnListF []All goals completed! 🐙 normalOrderList_nil,𝓕:FieldSpecification⊢ 1 • ofCrAnListF [] = ofCrAnListF []All goals completed! 🐙
    ofCrAnListF_nil,𝓕:FieldSpecification⊢ 1 • 1 = 1All goals completed! 🐙 one_smul𝓕:FieldSpecification⊢ 1 = 1All goals completed! 🐙]All goals completed! 🐙

lemma normalOrderF_normalOrderF_mid (a b c : 𝓕.FieldOpFreeAlgebra) :
    𝓝ᶠ(a * b * c) = 𝓝ᶠ(a * 𝓝ᶠ(b) * c) := by𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)
  let pc (c : 𝓕.FieldOpFreeAlgebra) (hc : c ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
    Prop := 𝓝ᶠ(a * b * c) = 𝓝ᶠ(a * 𝓝ᶠ(b) * c)𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)⊢ normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)
  change pc c (Basis.mem_span _ c)𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)⊢ pc c ⋯
  apply Submodule.span_inductionmem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ Set.range ⇑ofCrAnListFBasis), pc x ⋯zero𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)⊢ pc 0 ⋯add𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis))
  (hy : y ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)), pc x hx → pc y hy → pc (x + y) ⋯smul𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)), pc x hx → pc (a • x) ⋯
  ·mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ Set.range ⇑ofCrAnListFBasis), pc x ⋯ intro x hxmem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)x:𝓕.FieldOpFreeAlgebrahx:x ∈ Set.range ⇑ofCrAnListFBasis⊢ pc x ⋯
    obtain ⟨φs, rfl⟩ := hxmem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOp⊢ pc (ofCrAnListFBasis φs) ⋯
    simp only [ofListBasis_eq_ofList, pc]mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOp⊢ normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)
    let pb (b : 𝓕.FieldOpFreeAlgebra) (hb : b ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
      Prop := 𝓝ᶠ(a * b * ofCrAnListF φs) = 𝓝ᶠ(a * 𝓝ᶠ(b) * ofCrAnListF φs)mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)⊢ normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)
    change pb b (Basis.mem_span _ b)mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)⊢ pb b ⋯
    apply Submodule.span_inductionmem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ Set.range ⇑ofCrAnListFBasis), pb x ⋯mem.zero𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)⊢ pb 0 ⋯add𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis))
  (hy : y ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)), pb x hx → pb y hy → pb (x + y) ⋯smul𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)), pb x hx → pb (a • x) ⋯
    ·mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ Set.range ⇑ofCrAnListFBasis), pb x ⋯ intro x hxmem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)x:𝓕.FieldOpFreeAlgebrahx:x ∈ Set.range ⇑ofCrAnListFBasis⊢ pb x ⋯
      obtain ⟨φs', rfl⟩ := hxmem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOp⊢ pb (ofCrAnListFBasis φs') ⋯
      simp only [ofListBasis_eq_ofList, pb]mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOp⊢ normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) = normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)
      let pa (a : 𝓕.FieldOpFreeAlgebra) (ha : a ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
        Prop := 𝓝ᶠ(a * ofCrAnListF φs' * ofCrAnListF φs) =
        𝓝ᶠ(a * 𝓝ᶠ(ofCrAnListF φs') * ofCrAnListF φs)mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)⊢ normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) = normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)
      change pa a (Basis.mem_span _ a)mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)⊢ pa a ⋯
      apply Submodule.span_inductionmem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ Set.range ⇑ofCrAnListFBasis), pa x ⋯mem.mem.zero𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)⊢ pa 0 ⋯add𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis))
  (hy : y ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)), pa x hx → pa y hy → pa (x + y) ⋯smul𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)), pa x hx → pa (a • x) ⋯
      ·mem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ Set.range ⇑ofCrAnListFBasis), pa x ⋯ intro x hxmem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)x:𝓕.FieldOpFreeAlgebrahx:x ∈ Set.range ⇑ofCrAnListFBasis⊢ pa x ⋯
        obtain ⟨φs'', rfl⟩ := hxmem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ pa (ofCrAnListFBasis φs'') ⋯
        simp only [ofListBasis_eq_ofList, pa]mem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs'' * ofCrAnListF φs' * ofCrAnListF φs) =
  normalOrderF (ofCrAnListF φs'' * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)
        rw [normalOrderF_ofCrAnListFmem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs'' * ofCrAnListF φs' * ofCrAnListF φs) =
  normalOrderF (ofCrAnListF φs'' * normalOrderSign φs' • ofCrAnListF (normalOrderList φs') * ofCrAnListF φs)mem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs'' * ofCrAnListF φs' * ofCrAnListF φs) =
  normalOrderF (ofCrAnListF φs'' * normalOrderSign φs' • ofCrAnListF (normalOrderList φs') * ofCrAnListF φs)]mem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs'' * ofCrAnListF φs' * ofCrAnListF φs) =
  normalOrderF (ofCrAnListF φs'' * normalOrderSign φs' • ofCrAnListF (normalOrderList φs') * ofCrAnListF φs)
        simp only [← ofCrAnListF_append, Algebra.mul_smul_comm,
          Algebra.smul_mul_assoc, map_smul]mem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF (φs'' ++ φs' ++ φs)) =
  normalOrderSign φs' • normalOrderF (ofCrAnListF (φs'' ++ normalOrderList φs' ++ φs))
        rw [normalOrderF_ofCrAnListF,mem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs'' ++ φs' ++ φs) • ofCrAnListF (normalOrderList (φs'' ++ φs' ++ φs)) =
  normalOrderSign φs' • normalOrderF (ofCrAnListF (φs'' ++ normalOrderList φs' ++ φs))mem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs'' ++ φs' ++ φs) • ofCrAnListF (normalOrderList (φs'' ++ φs' ++ φs)) =
  (normalOrderSign φs' * normalOrderSign (φs'' ++ normalOrderList φs' ++ φs)) •
    ofCrAnListF (normalOrderList (φs'' ++ normalOrderList φs' ++ φs)) normalOrderF_ofCrAnListF,mem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs'' ++ φs' ++ φs) • ofCrAnListF (normalOrderList (φs'' ++ φs' ++ φs)) =
  normalOrderSign φs' •
    normalOrderSign (φs'' ++ normalOrderList φs' ++ φs) •
      ofCrAnListF (normalOrderList (φs'' ++ normalOrderList φs' ++ φs))mem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs'' ++ φs' ++ φs) • ofCrAnListF (normalOrderList (φs'' ++ φs' ++ φs)) =
  (normalOrderSign φs' * normalOrderSign (φs'' ++ normalOrderList φs' ++ φs)) •
    ofCrAnListF (normalOrderList (φs'' ++ normalOrderList φs' ++ φs)) smul_smulmem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs'' ++ φs' ++ φs) • ofCrAnListF (normalOrderList (φs'' ++ φs' ++ φs)) =
  (normalOrderSign φs' * normalOrderSign (φs'' ++ normalOrderList φs' ++ φs)) •
    ofCrAnListF (normalOrderList (φs'' ++ normalOrderList φs' ++ φs))mem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs'' ++ φs' ++ φs) • ofCrAnListF (normalOrderList (φs'' ++ φs' ++ φs)) =
  (normalOrderSign φs' * normalOrderSign (φs'' ++ normalOrderList φs' ++ φs)) •
    ofCrAnListF (normalOrderList (φs'' ++ normalOrderList φs' ++ φs))]mem.mem.mem𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs'' ++ φs' ++ φs) • ofCrAnListF (normalOrderList (φs'' ++ φs' ++ φs)) =
  (normalOrderSign φs' * normalOrderSign (φs'' ++ normalOrderList φs' ++ φs)) •
    ofCrAnListF (normalOrderList (φs'' ++ normalOrderList φs' ++ φs))
        congr 1mem.mem.mem.e_a𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs'' ++ φs' ++ φs) = normalOrderSign φs' * normalOrderSign (φs'' ++ normalOrderList φs' ++ φs)mem.mem.mem.e_a𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ ofCrAnListF (normalOrderList (φs'' ++ φs' ++ φs)) = ofCrAnListF (normalOrderList (φs'' ++ normalOrderList φs' ++ φs))
        ·mem.mem.mem.e_a𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs'' ++ φs' ++ φs) = normalOrderSign φs' * normalOrderSign (φs'' ++ normalOrderList φs' ++ φs) simp only [normalOrderSign, normalOrderList]mem.mem.mem.e_a𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ Wick.koszulSign 𝓕.crAnStatistics normalOrderRel (φs'' ++ φs' ++ φs) =
  Wick.koszulSign 𝓕.crAnStatistics normalOrderRel φs' *
    Wick.koszulSign 𝓕.crAnStatistics normalOrderRel (φs'' ++ List.insertionSort normalOrderRel φs' ++ φs)
          rw [Wick.koszulSign_of_append_eq_insertionSort,mem.mem.mem.e_a𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ Wick.koszulSign 𝓕.crAnStatistics normalOrderRel (φs'' ++ List.insertionSort normalOrderRel φs' ++ φs) *
    Wick.koszulSign 𝓕.crAnStatistics normalOrderRel φs' =
  Wick.koszulSign 𝓕.crAnStatistics normalOrderRel φs' *
    Wick.koszulSign 𝓕.crAnStatistics normalOrderRel (φs'' ++ List.insertionSort normalOrderRel φs' ++ φs)All goals completed! 🐙 mul_commmem.mem.mem.e_a𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ Wick.koszulSign 𝓕.crAnStatistics normalOrderRel φs' *
    Wick.koszulSign 𝓕.crAnStatistics normalOrderRel (φs'' ++ List.insertionSort normalOrderRel φs' ++ φs) =
  Wick.koszulSign 𝓕.crAnStatistics normalOrderRel φs' *
    Wick.koszulSign 𝓕.crAnStatistics normalOrderRel (φs'' ++ List.insertionSort normalOrderRel φs' ++ φs)All goals completed! 🐙]All goals completed! 🐙
        ·mem.mem.mem.e_a𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ ofCrAnListF (normalOrderList (φs'' ++ φs' ++ φs)) = ofCrAnListF (normalOrderList (φs'' ++ normalOrderList φs' ++ φs)) congr 1mem.mem.mem.e_a𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ normalOrderList (φs'' ++ φs' ++ φs) = normalOrderList (φs'' ++ normalOrderList φs' ++ φs)
          simp only [normalOrderList]mem.mem.mem.e_a𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ List.insertionSort normalOrderRel (φs'' ++ φs' ++ φs) =
  List.insertionSort normalOrderRel (φs'' ++ List.insertionSort normalOrderRel φs' ++ φs)
          rw [Physlib.List.insertionSort_append_insertionSort_appendmem.mem.mem.e_a𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)φs'':List 𝓕.CrAnFieldOp⊢ List.insertionSort normalOrderRel (φs'' ++ φs' ++ φs) = List.insertionSort normalOrderRel (φs'' ++ φs' ++ φs)All goals completed! 🐙]All goals completed! 🐙
      ·mem.mem.zero𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)⊢ pa 0 ⋯ simp [pa]All goals completed! 🐙
      ·add𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis))
  (hy : y ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)), pa x hx → pa y hy → pa (x + y) ⋯ intro x y hx hy h1 h2add𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)hy:y ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)h1:pa x hxh2:pa y hy⊢ pa (x + y) ⋯
        simp_all [pa, add_mul]All goals completed! 🐙
      ·smul𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)), pa x hx → pa (a • x) ⋯ intro x hx hsmul𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)φs':List 𝓕.CrAnFieldOppa:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := 
  fun a ha =>
    normalOrderF (a * ofCrAnListF φs' * ofCrAnListF φs) =
      normalOrderF (a * normalOrderF (ofCrAnListF φs') * ofCrAnListF φs)x:ℂhx:𝓕.FieldOpFreeAlgebrah:hx ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)⊢ pa hx h → pa (x • hx) ⋯
        simp_all [pa]All goals completed! 🐙
    ·mem.zero𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)⊢ pb 0 ⋯ simp [pb]All goals completed! 🐙
    ·add𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis))
  (hy : y ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)), pb x hx → pb y hy → pb (x + y) ⋯ intro x y hx hy h1 h2add𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)hy:y ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)h1:pb x hxh2:pb y hy⊢ pb (x + y) ⋯
      simp_all [pb, mul_add, add_mul]All goals completed! 🐙
    ·smul𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)), pb x hx → pb (a • x) ⋯ intro x hx hsmul𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)φs:List 𝓕.CrAnFieldOppb:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun b hb => normalOrderF (a * b * ofCrAnListF φs) = normalOrderF (a * normalOrderF b * ofCrAnListF φs)x:ℂhx:𝓕.FieldOpFreeAlgebrah:hx ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)⊢ pb hx h → pb (x • hx) ⋯
      simp_all [pb]All goals completed! 🐙
  ·zero𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)⊢ pc 0 ⋯ simp [pc]All goals completed! 🐙
  ·add𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis))
  (hy : y ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)), pc x hx → pc y hy → pc (x + y) ⋯ intro x y hx hy h1 h2add𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)hy:y ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)h1:pc x hxh2:pc y hy⊢ pc (x + y) ⋯
    simp_all [pc, mul_add]All goals completed! 🐙
  ·smul𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)), pc x hx → pc (a • x) ⋯ intro x hx h hpsmul𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrac:𝓕.FieldOpFreeAlgebrapc:(c : 𝓕.FieldOpFreeAlgebra) → c ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis) → Prop := fun c hc => normalOrderF (a * b * c) = normalOrderF (a * normalOrderF b * c)x:ℂhx:𝓕.FieldOpFreeAlgebrah:hx ∈ Submodule.span ℂ (Set.range ⇑ofCrAnListFBasis)hp:pc hx h⊢ pc (x • hx) ⋯
    simp_all [pc]All goals completed! 🐙lemma normalOrderF_normalOrderF_right (a b : 𝓕.FieldOpFreeAlgebra) :
    𝓝ᶠ(a * b) = 𝓝ᶠ(a * 𝓝ᶠ(b)) := by𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * b) = normalOrderF (a * normalOrderF b)
  simpa using normalOrderF_normalOrderF_mid a b 1All goals completed! 🐙lemma normalOrderF_normalOrderF_left (a b : 𝓕.FieldOpFreeAlgebra) :
    𝓝ᶠ(a * b) = 𝓝ᶠ(𝓝ᶠ(a) * b) := by𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * b) = normalOrderF (normalOrderF a * b)
  simpa using normalOrderF_normalOrderF_mid 1 a bAll goals completed! 🐙

Normal ordering with a creation operator on the left or annihilation on the right

              

lemma normalOrderF_ofCrAnListF_cons_create (φ : 𝓕.CrAnFieldOp)
    (hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.create) (φs : List 𝓕.CrAnFieldOp) :
    𝓝ᶠ(ofCrAnListF (φ :: φs)) = ofCrAnOpF φ * 𝓝ᶠ(ofCrAnListF φs) := by𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF (φ :: φs)) = ofCrAnOpF φ * normalOrderF (ofCrAnListF φs)
  rw [normalOrderF_ofCrAnListF,𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOp⊢ normalOrderSign (φ :: φs) • ofCrAnListF (normalOrderList (φ :: φs)) = ofCrAnOpF φ * normalOrderF (ofCrAnListF φs)All goals completed! 🐙 normalOrderSign_cons_create φ hφ,𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOp⊢ normalOrderSign φs • ofCrAnListF (normalOrderList (φ :: φs)) = ofCrAnOpF φ * normalOrderF (ofCrAnListF φs)All goals completed! 🐙
    normalOrderList_cons_create φ hφ φs,𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOp⊢ normalOrderSign φs • ofCrAnListF (φ :: normalOrderList φs) = ofCrAnOpF φ * normalOrderF (ofCrAnListF φs)All goals completed! 🐙 ofCrAnListF_cons,𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOp⊢ normalOrderSign φs • (ofCrAnOpF φ * ofCrAnListF (normalOrderList φs)) = ofCrAnOpF φ * normalOrderF (ofCrAnListF φs)All goals completed! 🐙 normalOrderF_ofCrAnListF,𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOp⊢ normalOrderSign φs • (ofCrAnOpF φ * ofCrAnListF (normalOrderList φs)) =
  ofCrAnOpF φ * normalOrderSign φs • ofCrAnListF (normalOrderList φs)All goals completed! 🐙
    mul_smul_comm𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOp⊢ normalOrderSign φs • (ofCrAnOpF φ * ofCrAnListF (normalOrderList φs)) =
  normalOrderSign φs • (ofCrAnOpF φ * ofCrAnListF (normalOrderList φs))All goals completed! 🐙]All goals completed! 🐙

lemma normalOrderF_create_mul (φ : 𝓕.CrAnFieldOp)
    (hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.create) (a : FieldOpFreeAlgebra 𝓕) :
    𝓝ᶠ(ofCrAnOpF φ * a) = ofCrAnOpF φ * 𝓝ᶠ(a) := by𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.createa:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (ofCrAnOpF φ * a) = ofCrAnOpF φ * normalOrderF a
  change (normalOrderF ∘ₗ mulLinearMap (ofCrAnOpF φ)) a =
    (mulLinearMap (ofCrAnOpF φ) ∘ₗ normalOrderF) a𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.createa:𝓕.FieldOpFreeAlgebra⊢ (normalOrderF ∘ₗ mulLinearMap (ofCrAnOpF φ)) a = (mulLinearMap (ofCrAnOpF φ) ∘ₗ normalOrderF) a
  refine LinearMap.congr_fun (ofCrAnListFBasis.ext fun l ↦ ?_) a𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.createa:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ (normalOrderF ∘ₗ mulLinearMap (ofCrAnOpF φ)) (ofCrAnListFBasis l) =
  (mulLinearMap (ofCrAnOpF φ) ∘ₗ normalOrderF) (ofCrAnListFBasis l)
  simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, ofListBasis_eq_ofList,
    LinearMap.coe_comp, Function.comp_apply]𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.createa:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnOpF φ * ofCrAnListF l) = ofCrAnOpF φ * normalOrderF (ofCrAnListF l)
  rw [← ofCrAnListF_cons,𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.createa:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF (φ :: l)) = ofCrAnOpF φ * normalOrderF (ofCrAnListF l)All goals completed! 🐙 normalOrderF_ofCrAnListF_cons_create φ hφ𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.createa:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ ofCrAnOpF φ * normalOrderF (ofCrAnListF l) = ofCrAnOpF φ * normalOrderF (ofCrAnListF l)All goals completed! 🐙]All goals completed! 🐙

lemma normalOrderF_ofCrAnListF_append_annihilate (φ : 𝓕.CrAnFieldOp)
    (hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.annihilate) (φs : List 𝓕.CrAnFieldOp) :
    𝓝ᶠ(ofCrAnListF (φs ++ [φ])) = 𝓝ᶠ(ofCrAnListF φs) * ofCrAnOpF φ := by𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF (φs ++ [φ])) = normalOrderF (ofCrAnListF φs) * ofCrAnOpF φ
  rw [normalOrderF_ofCrAnListF,𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ [φ]) • ofCrAnListF (normalOrderList (φs ++ [φ])) = normalOrderF (ofCrAnListF φs) * ofCrAnOpF φAll goals completed! 🐙 normalOrderSign_append_annihilate φ hφ φs,𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOp⊢ normalOrderSign φs • ofCrAnListF (normalOrderList (φs ++ [φ])) = normalOrderF (ofCrAnListF φs) * ofCrAnOpF φAll goals completed! 🐙
    normalOrderList_append_annihilate φ hφ φs,𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOp⊢ normalOrderSign φs • ofCrAnListF (normalOrderList φs ++ [φ]) = normalOrderF (ofCrAnListF φs) * ofCrAnOpF φAll goals completed! 🐙 ofCrAnListF_append,𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOp⊢ normalOrderSign φs • (ofCrAnListF (normalOrderList φs) * ofCrAnListF [φ]) = normalOrderF (ofCrAnListF φs) * ofCrAnOpF φAll goals completed! 🐙 ofCrAnListF_singleton,𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOp⊢ normalOrderSign φs • (ofCrAnListF (normalOrderList φs) * ofCrAnOpF φ) = normalOrderF (ofCrAnListF φs) * ofCrAnOpF φAll goals completed! 🐙
      normalOrderF_ofCrAnListF,𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOp⊢ normalOrderSign φs • (ofCrAnListF (normalOrderList φs) * ofCrAnOpF φ) =
  normalOrderSign φs • ofCrAnListF (normalOrderList φs) * ofCrAnOpF φAll goals completed! 🐙 smul_mul_assoc𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOp⊢ normalOrderSign φs • (ofCrAnListF (normalOrderList φs) * ofCrAnOpF φ) =
  normalOrderSign φs • (ofCrAnListF (normalOrderList φs) * ofCrAnOpF φ)All goals completed! 🐙]All goals completed! 🐙

lemma normalOrderF_mul_annihilate (φ : 𝓕.CrAnFieldOp)
    (hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.annihilate)
    (a : FieldOpFreeAlgebra 𝓕) : 𝓝ᶠ(a * ofCrAnOpF φ) = 𝓝ᶠ(a) * ofCrAnOpF φ := by𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * ofCrAnOpF φ) = normalOrderF a * ofCrAnOpF φ
  change (normalOrderF ∘ₗ mulLinearMap.flip (ofCrAnOpF φ)) a =
    (mulLinearMap.flip (ofCrAnOpF φ) ∘ₗ normalOrderF) a𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebra⊢ (normalOrderF ∘ₗ mulLinearMap.flip (ofCrAnOpF φ)) a = (mulLinearMap.flip (ofCrAnOpF φ) ∘ₗ normalOrderF) a
  refine LinearMap.congr_fun (ofCrAnListFBasis.ext fun l ↦ ?_) a𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ (normalOrderF ∘ₗ mulLinearMap.flip (ofCrAnOpF φ)) (ofCrAnListFBasis l) =
  (mulLinearMap.flip (ofCrAnOpF φ) ∘ₗ normalOrderF) (ofCrAnListFBasis l)
  simp only [mulLinearMap, ofListBasis_eq_ofList, LinearMap.coe_comp, Function.comp_apply,
    LinearMap.flip_apply, LinearMap.coe_mk, AddHom.coe_mk]𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF l * ofCrAnOpF φ) = normalOrderF (ofCrAnListF l) * ofCrAnOpF φ
  rw [← ofCrAnListF_singleton,𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF l * ofCrAnListF [φ]) = normalOrderF (ofCrAnListF l) * ofCrAnListF [φ]All goals completed! 🐙 ← ofCrAnListF_append,𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF (l ++ [φ])) = normalOrderF (ofCrAnListF l) * ofCrAnListF [φ]All goals completed! 🐙 ofCrAnListF_singleton,𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF (l ++ [φ])) = normalOrderF (ofCrAnListF l) * ofCrAnOpF φAll goals completed! 🐙
    normalOrderF_ofCrAnListF_append_annihilate φ hφ𝓕:FieldSpecificationφ:𝓕.CrAnFieldOphφ:𝓕|>ᶜφ = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF l) * ofCrAnOpF φ = normalOrderF (ofCrAnListF l) * ofCrAnOpF φAll goals completed! 🐙]All goals completed! 🐙lemma normalOrderF_crPartF_mul (φ : 𝓕.FieldOp) (a : FieldOpFreeAlgebra 𝓕) :
    𝓝ᶠ(crPartF φ * a) =
    crPartF φ * 𝓝ᶠ(a) := by𝓕:FieldSpecificationφ:𝓕.FieldOpa:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (crPartF φ * a) = crPartF φ * normalOrderF a
  match φ with
  | .inAsymp _ | .position _ =>𝓕:FieldSpecificationφ:𝓕.FieldOpa:𝓕.FieldOpFreeAlgebraa✝:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ normalOrderF (crPartF (FieldOp.position a✝) * a) = crPartF (FieldOp.position a✝) * normalOrderF a𝓕:FieldSpecificationφ:𝓕.FieldOpa:𝓕.FieldOpFreeAlgebraa✝:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ normalOrderF (crPartF (FieldOp.inAsymp a✝) * a) = crPartF (FieldOp.inAsymp a✝) * normalOrderF a exact normalOrderF_create_mul _ rfl _All goals completed! 🐙
  | .outAsymp _ =>𝓕:FieldSpecificationφ:𝓕.FieldOpa:𝓕.FieldOpFreeAlgebraa✝:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ normalOrderF (crPartF (FieldOp.outAsymp a✝) * a) = crPartF (FieldOp.outAsymp a✝) * normalOrderF a simpAll goals completed! 🐙lemma normalOrderF_mul_anPartF (φ : 𝓕.FieldOp) (a : FieldOpFreeAlgebra 𝓕) :
    𝓝ᶠ(a * anPartF φ) =
    𝓝ᶠ(a) * anPartF φ := by𝓕:FieldSpecificationφ:𝓕.FieldOpa:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * anPartF φ) = normalOrderF a * anPartF φ
  match φ with
  | .inAsymp _ =>𝓕:FieldSpecificationφ:𝓕.FieldOpa:𝓕.FieldOpFreeAlgebraa✝:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ normalOrderF (a * anPartF (FieldOp.inAsymp a✝)) = normalOrderF a * anPartF (FieldOp.inAsymp a✝) simpAll goals completed! 🐙
  | .position _ | .outAsymp _ =>𝓕:FieldSpecificationφ:𝓕.FieldOpa:𝓕.FieldOpFreeAlgebraa✝:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ normalOrderF (a * anPartF (FieldOp.outAsymp a✝)) = normalOrderF a * anPartF (FieldOp.outAsymp a✝)𝓕:FieldSpecificationφ:𝓕.FieldOpa:𝓕.FieldOpFreeAlgebraa✝:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ normalOrderF (a * anPartF (FieldOp.position a✝)) = normalOrderF a * anPartF (FieldOp.position a✝) exact normalOrderF_mul_annihilate _ rfl _All goals completed! 🐙

Normal ordering for an adjacent creation and annihilation state

The main result of this section is normalOrderF_superCommuteF_annihilate_create.

              

lemma normalOrderF_swap_create_annihilate_ofCrAnListF_ofCrAnListF (φc φa : 𝓕.CrAnFieldOp)
    (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
    (φs φs' : List 𝓕.CrAnFieldOp) :
    𝓝ᶠ(ofCrAnListF φs' * ofCrAnOpF φc * ofCrAnOpF φa * ofCrAnListF φs) = 𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
    𝓝ᶠ(ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs) := by𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs' * ofCrAnOpF φc * ofCrAnOpF φa * ofCrAnListF φs) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)
  rw [mul_assoc,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs' * ofCrAnOpF φc * (ofCrAnOpF φa * ofCrAnListF φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF (φs' ++ φc :: φa :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs) mul_assoc,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs' * (ofCrAnOpF φc * (ofCrAnOpF φa * ofCrAnListF φs))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF (φs' ++ φc :: φa :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs) ← ofCrAnListF_cons,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs' * (ofCrAnOpF φc * ofCrAnListF (φa :: φs))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF (φs' ++ φc :: φa :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs) ← ofCrAnListF_cons,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs' * ofCrAnListF (φc :: φa :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF (φs' ++ φc :: φa :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs) ← ofCrAnListF_append𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF (φs' ++ φc :: φa :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF (φs' ++ φc :: φa :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF (φs' ++ φc :: φa :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)
  rw [normalOrderF_ofCrAnListF,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs' ++ φc :: φa :: φs) • ofCrAnListF (normalOrderList (φs' ++ φc :: φa :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) * normalOrderSign (φs' ++ φa :: φc :: φs)) •
    ofCrAnListF (normalOrderList (φs' ++ φc :: φa :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs) normalOrderSign_swap_create_annihilate φc φa hφc hφa𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) * normalOrderSign (φs' ++ φa :: φc :: φs)) •
    ofCrAnListF (normalOrderList (φs' ++ φc :: φa :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) * normalOrderSign (φs' ++ φa :: φc :: φs)) •
    ofCrAnListF (normalOrderList (φs' ++ φc :: φa :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) * normalOrderSign (φs' ++ φa :: φc :: φs)) •
    ofCrAnListF (normalOrderList (φs' ++ φc :: φa :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)
  rw [normalOrderList_swap_create_annihilate φc φa hφc hφa,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) * normalOrderSign (φs' ++ φa :: φc :: φs)) •
    ofCrAnListF (normalOrderList (φs' ++ φa :: φc :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (ofCrAnListF (φs' ++ φa :: φc :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs) ← smul_smul,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderSign (φs' ++ φa :: φc :: φs) • ofCrAnListF (normalOrderList (φs' ++ φa :: φc :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (ofCrAnListF (φs' ++ φa :: φc :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs) ← normalOrderF_ofCrAnListF𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (ofCrAnListF (φs' ++ φa :: φc :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (ofCrAnListF (φs' ++ φa :: φc :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (ofCrAnListF (φs' ++ φa :: φc :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)
  rw [ofCrAnListF_append,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnListF (φa :: φc :: φs)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * (ofCrAnOpF φa * (ofCrAnOpF φc * ofCrAnListF φs))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs) ofCrAnListF_cons,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * (ofCrAnOpF φa * ofCrAnListF (φc :: φs))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * (ofCrAnOpF φa * (ofCrAnOpF φc * ofCrAnListF φs))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs) ofCrAnListF_cons𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * (ofCrAnOpF φa * (ofCrAnOpF φc * ofCrAnListF φs))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * (ofCrAnOpF φa * (ofCrAnOpF φc * ofCrAnListF φs))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * (ofCrAnOpF φa * (ofCrAnOpF φc * ofCrAnListF φs))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)
  noncomm_ringAll goals completed! 🐙

lemma normalOrderF_swap_create_annihilate_ofCrAnListF (φc φa : 𝓕.CrAnFieldOp)
    (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
    (φs : List 𝓕.CrAnFieldOp) (a : 𝓕.FieldOpFreeAlgebra) :
    𝓝ᶠ(ofCrAnListF φs * ofCrAnOpF φc * ofCrAnOpF φa * a) = 𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
    𝓝ᶠ(ofCrAnListF φs * ofCrAnOpF φa * ofCrAnOpF φc * a) := by𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpa:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (ofCrAnListF φs * ofCrAnOpF φc * ofCrAnOpF φa * a) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs * ofCrAnOpF φa * ofCrAnOpF φc * a)
  change (normalOrderF ∘ₗ mulLinearMap (ofCrAnListF φs * ofCrAnOpF φc * ofCrAnOpF φa)) a =
    (smulLinearMap _ ∘ₗ normalOrderF ∘ₗ
    mulLinearMap (ofCrAnListF φs * ofCrAnOpF φa * ofCrAnOpF φc)) a𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpa:𝓕.FieldOpFreeAlgebra⊢ (normalOrderF ∘ₗ mulLinearMap (ofCrAnListF φs * ofCrAnOpF φc * ofCrAnOpF φa)) a =
  (smulLinearMap ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa)) ∘ₗ
      normalOrderF ∘ₗ mulLinearMap (ofCrAnListF φs * ofCrAnOpF φa * ofCrAnOpF φc))
    a
  refine LinearMap.congr_fun (ofCrAnListFBasis.ext fun l ↦ ?_) a𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpa:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ (normalOrderF ∘ₗ mulLinearMap (ofCrAnListF φs * ofCrAnOpF φc * ofCrAnOpF φa)) (ofCrAnListFBasis l) =
  (smulLinearMap ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa)) ∘ₗ
      normalOrderF ∘ₗ mulLinearMap (ofCrAnListF φs * ofCrAnOpF φa * ofCrAnOpF φc))
    (ofCrAnListFBasis l)
  simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, ofListBasis_eq_ofList,
    LinearMap.coe_comp, Function.comp_apply]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpa:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs * ofCrAnOpF φc * ofCrAnOpF φa * ofCrAnListF l) =
  (smulLinearMap ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa)))
    (normalOrderF (ofCrAnListF φs * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF l))
  rw [normalOrderF_swap_create_annihilate_ofCrAnListF_ofCrAnListF φc φa hφc hφa𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpa:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF l) =
  (smulLinearMap ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa)))
    (normalOrderF (ofCrAnListF φs * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF l))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpa:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF l) =
  (smulLinearMap ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa)))
    (normalOrderF (ofCrAnListF φs * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF l))]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpa:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF φs * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF l) =
  (smulLinearMap ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa)))
    (normalOrderF (ofCrAnListF φs * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF l))
  rflAll goals completed! 🐙

lemma normalOrderF_swap_create_annihilate (φc φa : 𝓕.CrAnFieldOp)
    (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
    (a b : 𝓕.FieldOpFreeAlgebra) :
    𝓝ᶠ(a * ofCrAnOpF φc * ofCrAnOpF φa * b) = 𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
    𝓝ᶠ(a * ofCrAnOpF φa * ofCrAnOpF φc * b) := by𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * ofCrAnOpF φc * ofCrAnOpF φa * b) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * ofCrAnOpF φa * ofCrAnOpF φc * b)
  rw [mul_assoc,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * ofCrAnOpF φc * (ofCrAnOpF φa * b)) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * ofCrAnOpF φa * ofCrAnOpF φc * b)𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * (ofCrAnOpF φc * (ofCrAnOpF φa * b))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * (ofCrAnOpF φa * (ofCrAnOpF φc * b))) mul_assoc,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * (ofCrAnOpF φc * (ofCrAnOpF φa * b))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * ofCrAnOpF φa * ofCrAnOpF φc * b)𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * (ofCrAnOpF φc * (ofCrAnOpF φa * b))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * (ofCrAnOpF φa * (ofCrAnOpF φc * b))) mul_assoc,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * (ofCrAnOpF φc * (ofCrAnOpF φa * b))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * ofCrAnOpF φa * (ofCrAnOpF φc * b))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * (ofCrAnOpF φc * (ofCrAnOpF φa * b))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * (ofCrAnOpF φa * (ofCrAnOpF φc * b))) mul_assoc𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * (ofCrAnOpF φc * (ofCrAnOpF φa * b))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * (ofCrAnOpF φa * (ofCrAnOpF φc * b)))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * (ofCrAnOpF φc * (ofCrAnOpF φa * b))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * (ofCrAnOpF φa * (ofCrAnOpF φc * b)))]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * (ofCrAnOpF φc * (ofCrAnOpF φa * b))) =
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * (ofCrAnOpF φa * (ofCrAnOpF φc * b)))
  change (normalOrderF ∘ₗ mulLinearMap.flip (ofCrAnOpF φc * (ofCrAnOpF φa * b))) a =
    (smulLinearMap (𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa)) ∘ₗ
    normalOrderF ∘ₗ mulLinearMap.flip (ofCrAnOpF φa * (ofCrAnOpF φc * b))) a𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ (normalOrderF ∘ₗ mulLinearMap.flip (ofCrAnOpF φc * (ofCrAnOpF φa * b))) a =
  (smulLinearMap ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa)) ∘ₗ
      normalOrderF ∘ₗ mulLinearMap.flip (ofCrAnOpF φa * (ofCrAnOpF φc * b)))
    a
  refine LinearMap.congr_fun (ofCrAnListFBasis.ext fun l ↦ ?_) _𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ (normalOrderF ∘ₗ mulLinearMap.flip (ofCrAnOpF φc * (ofCrAnOpF φa * b))) (ofCrAnListFBasis l) =
  (smulLinearMap ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa)) ∘ₗ
      normalOrderF ∘ₗ mulLinearMap.flip (ofCrAnOpF φa * (ofCrAnOpF φc * b)))
    (ofCrAnListFBasis l)
  simp only [mulLinearMap, ofListBasis_eq_ofList, LinearMap.coe_comp, Function.comp_apply,
    LinearMap.flip_apply, LinearMap.coe_mk, AddHom.coe_mk, ← mul_assoc,
      normalOrderF_swap_create_annihilate_ofCrAnListF φc φa hφc hφa]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebral:List 𝓕.CrAnFieldOp⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) •
    normalOrderF (ofCrAnListF l * ofCrAnOpF φa * ofCrAnOpF φc * b) =
  (smulLinearMap ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa)))
    (normalOrderF (ofCrAnListF l * ofCrAnOpF φa * ofCrAnOpF φc * b))
  rflAll goals completed! 🐙

lemma normalOrderF_superCommuteF_create_annihilate (φc φa : 𝓕.CrAnFieldOp)
    (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
    (a b : 𝓕.FieldOpFreeAlgebra) :
    𝓝ᶠ(a * [ofCrAnOpF φc, ofCrAnOpF φa]ₛF * b) = 0 := by𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * (superCommuteF (ofCrAnOpF φc)) (ofCrAnOpF φa) * b) = 0
  simp only [superCommuteF_ofCrAnOpF_ofCrAnOpF, Algebra.smul_mul_assoc]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF
    (a *
        (ofCrAnOpF φc * ofCrAnOpF φa -
          (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • (ofCrAnOpF φa * ofCrAnOpF φc)) *
      b) =
  0
  rw [mul_sub,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF
    ((a * (ofCrAnOpF φc * ofCrAnOpF φa) -
        a * (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • (ofCrAnOpF φa * ofCrAnOpF φc)) *
      b) =
  0𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * ofCrAnOpF φa * ofCrAnOpF φc * b) -
    normalOrderF (a * (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • ofCrAnOpF φa * ofCrAnOpF φc * b) =
  0 sub_mul,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF
    (a * (ofCrAnOpF φc * ofCrAnOpF φa) * b -
      a * (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • (ofCrAnOpF φa * ofCrAnOpF φc) * b) =
  0𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * ofCrAnOpF φa * ofCrAnOpF φc * b) -
    normalOrderF (a * (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • ofCrAnOpF φa * ofCrAnOpF φc * b) =
  0 map_sub,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * (ofCrAnOpF φc * ofCrAnOpF φa) * b) -
    normalOrderF (a * (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • (ofCrAnOpF φa * ofCrAnOpF φc) * b) =
  0𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * ofCrAnOpF φa * ofCrAnOpF φc * b) -
    normalOrderF (a * (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • ofCrAnOpF φa * ofCrAnOpF φc * b) =
  0 ← smul_mul_assoc,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * (ofCrAnOpF φc * ofCrAnOpF φa) * b) -
    normalOrderF (a * ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • ofCrAnOpF φa * ofCrAnOpF φc) * b) =
  0𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * ofCrAnOpF φa * ofCrAnOpF φc * b) -
    normalOrderF (a * (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • ofCrAnOpF φa * ofCrAnOpF φc * b) =
  0 ← mul_assoc,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * ofCrAnOpF φc * ofCrAnOpF φa * b) -
    normalOrderF (a * ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • ofCrAnOpF φa * ofCrAnOpF φc) * b) =
  0𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * ofCrAnOpF φa * ofCrAnOpF φc * b) -
    normalOrderF (a * (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • ofCrAnOpF φa * ofCrAnOpF φc * b) =
  0 ← mul_assoc,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * ofCrAnOpF φc * ofCrAnOpF φa * b) -
    normalOrderF (a * (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • ofCrAnOpF φa * ofCrAnOpF φc * b) =
  0𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * ofCrAnOpF φa * ofCrAnOpF φc * b) -
    normalOrderF (a * (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • ofCrAnOpF φa * ofCrAnOpF φc * b) =
  0
    normalOrderF_swap_create_annihilate φc φa hφc hφa𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * ofCrAnOpF φa * ofCrAnOpF φc * b) -
    normalOrderF (a * (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • ofCrAnOpF φa * ofCrAnOpF φc * b) =
  0𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * ofCrAnOpF φa * ofCrAnOpF φc * b) -
    normalOrderF (a * (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • ofCrAnOpF φa * ofCrAnOpF φc * b) =
  0]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • normalOrderF (a * ofCrAnOpF φa * ofCrAnOpF φc * b) -
    normalOrderF (a * (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φa) • ofCrAnOpF φa * ofCrAnOpF φc * b) =
  0
  simpAll goals completed! 🐙

lemma normalOrderF_superCommuteF_annihilate_create (φc φa : 𝓕.CrAnFieldOp)
    (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
    (a b : 𝓕.FieldOpFreeAlgebra) :
    𝓝ᶠ(a * [ofCrAnOpF φa, ofCrAnOpF φc]ₛF * b) = 0 := by𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * (superCommuteF (ofCrAnOpF φa)) (ofCrAnOpF φc) * b) = 0
  rw [superCommuteF_ofCrAnOpF_ofCrAnOpF_symm𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF
    (a * -(exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φc) • (superCommuteF (ofCrAnOpF φc)) (ofCrAnOpF φa) *
      b) =
  0𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF
    (a * -(exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φc) • (superCommuteF (ofCrAnOpF φc)) (ofCrAnOpF φa) *
      b) =
  0]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφa:𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatea:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF
    (a * -(exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φc) • (superCommuteF (ofCrAnOpF φc)) (ofCrAnOpF φa) *
      b) =
  0
  simp [normalOrderF_superCommuteF_create_annihilate φc φa hφc hφa]All goals completed! 🐙lemma normalOrderF_swap_crPartF_anPartF (φ φ' : 𝓕.FieldOp) (a b : FieldOpFreeAlgebra 𝓕) :
    𝓝ᶠ(a * (crPartF φ) * (anPartF φ') * b) =
    𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
    𝓝ᶠ(a * (anPartF φ') * (crPartF φ) * b) := by𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * crPartF φ * anPartF φ' * b) =
  (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • normalOrderF (a * anPartF φ' * crPartF φ * b)
  match φ, φ' with
  | _, .inAsymp φ' =>𝓕:FieldSpecificationφ:𝓕.FieldOpφ'✝:𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrax✝:𝓕.FieldOpφ':((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ normalOrderF (a * crPartF x✝ * anPartF (FieldOp.inAsymp φ') * b) =
  (exchangeSign (𝓕|>ₛx✝)) (𝓕|>ₛFieldOp.inAsymp φ') • normalOrderF (a * anPartF (FieldOp.inAsymp φ') * crPartF x✝ * b) simpAll goals completed! 🐙
  | .outAsymp φ, _ =>𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentumx✝:𝓕.FieldOp⊢ normalOrderF (a * crPartF (FieldOp.outAsymp φ) * anPartF x✝ * b) =
  (exchangeSign (𝓕|>ₛFieldOp.outAsymp φ)) (𝓕|>ₛx✝) • normalOrderF (a * anPartF x✝ * crPartF (FieldOp.outAsymp φ) * b) simpAll goals completed! 🐙
  | .inAsymp _, .position _ | .inAsymp _, .outAsymp _
  | .position _, .position _ | .position _, .outAsymp _ =>𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraa✝¹:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTimea✝:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ normalOrderF (a * crPartF (FieldOp.position a✝¹) * anPartF (FieldOp.outAsymp a✝) * b) =
  (exchangeSign (𝓕|>ₛFieldOp.position a✝¹)) (𝓕|>ₛFieldOp.outAsymp a✝) •
    normalOrderF (a * anPartF (FieldOp.outAsymp a✝) * crPartF (FieldOp.position a✝¹) * b)𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraa✝¹:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTimea✝:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ normalOrderF (a * crPartF (FieldOp.position a✝¹) * anPartF (FieldOp.position a✝) * b) =
  (exchangeSign (𝓕|>ₛFieldOp.position a✝¹)) (𝓕|>ₛFieldOp.position a✝) •
    normalOrderF (a * anPartF (FieldOp.position a✝) * crPartF (FieldOp.position a✝¹) * b)𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraa✝¹:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentuma✝:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ normalOrderF (a * crPartF (FieldOp.inAsymp a✝¹) * anPartF (FieldOp.outAsymp a✝) * b) =
  (exchangeSign (𝓕|>ₛFieldOp.inAsymp a✝¹)) (𝓕|>ₛFieldOp.outAsymp a✝) •
    normalOrderF (a * anPartF (FieldOp.outAsymp a✝) * crPartF (FieldOp.inAsymp a✝¹) * b)𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraa✝¹:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentuma✝:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ normalOrderF (a * crPartF (FieldOp.inAsymp a✝¹) * anPartF (FieldOp.position a✝) * b) =
  (exchangeSign (𝓕|>ₛFieldOp.inAsymp a✝¹)) (𝓕|>ₛFieldOp.position a✝) •
    normalOrderF (a * anPartF (FieldOp.position a✝) * crPartF (FieldOp.inAsymp a✝¹) * b)
    exact normalOrderF_swap_create_annihilate _ _ rfl rfl ..All goals completed! 🐙

Normal ordering for an anPartF and crPartF

Using the results from above.

              lemma normalOrderF_swap_anPartF_crPartF (φ φ' : 𝓕.FieldOp) (a b : FieldOpFreeAlgebra 𝓕) :
    𝓝ᶠ(a * (anPartF φ) * (crPartF φ') * b) =
    𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • 𝓝ᶠ(a * (crPartF φ') *
      (anPartF φ) * b) := by𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * anPartF φ * crPartF φ' * b) =
  (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • normalOrderF (a * crPartF φ' * anPartF φ * b)
  simp [normalOrderF_swap_crPartF_anPartF, smul_smul]All goals completed! 🐙lemma normalOrderF_superCommuteF_crPartF_anPartF (φ φ' : 𝓕.FieldOp) (a b : FieldOpFreeAlgebra 𝓕) :
    𝓝ᶠ(a * superCommuteF
      (crPartF φ) (anPartF φ') * b) = 0 := by𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * (superCommuteF (crPartF φ)) (anPartF φ') * b) = 0
  match φ, φ' with
  | _, .inAsymp φ' =>𝓕:FieldSpecificationφ:𝓕.FieldOpφ'✝:𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrax✝:𝓕.FieldOpφ':((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ normalOrderF (a * (superCommuteF (crPartF x✝)) (anPartF (FieldOp.inAsymp φ')) * b) = 0 simpAll goals completed! 🐙
  | .outAsymp φ', _ =>𝓕:FieldSpecificationφ:𝓕.FieldOpφ'✝:𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraφ':((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentumx✝:𝓕.FieldOp⊢ normalOrderF (a * (superCommuteF (crPartF (FieldOp.outAsymp φ'))) (anPartF x✝) * b) = 0 simpAll goals completed! 🐙
  | .inAsymp _, .position _ | .inAsymp _, .outAsymp _
  | .position _, .position _ | .position _, .outAsymp _ =>𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraa✝¹:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTimea✝:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ normalOrderF (a * (superCommuteF (crPartF (FieldOp.position a✝¹))) (anPartF (FieldOp.outAsymp a✝)) * b) = 0𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraa✝¹:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTimea✝:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ normalOrderF (a * (superCommuteF (crPartF (FieldOp.position a✝¹))) (anPartF (FieldOp.position a✝)) * b) = 0𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraa✝¹:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentuma✝:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ normalOrderF (a * (superCommuteF (crPartF (FieldOp.inAsymp a✝¹))) (anPartF (FieldOp.outAsymp a✝)) * b) = 0𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraa✝¹:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentuma✝:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ normalOrderF (a * (superCommuteF (crPartF (FieldOp.inAsymp a✝¹))) (anPartF (FieldOp.position a✝)) * b) = 0
    exact normalOrderF_superCommuteF_create_annihilate _ _ rfl rfl ..All goals completed! 🐙lemma normalOrderF_superCommuteF_anPartF_crPartF (φ φ' : 𝓕.FieldOp) (a b : FieldOpFreeAlgebra 𝓕) :
    𝓝ᶠ(a * superCommuteF
    (anPartF φ) (crPartF φ') * b) = 0 := by𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ normalOrderF (a * (superCommuteF (anPartF φ)) (crPartF φ') * b) = 0
  match φ, φ' with
  | .inAsymp φ', _ =>𝓕:FieldSpecificationφ:𝓕.FieldOpφ'✝:𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraφ':((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentumx✝:𝓕.FieldOp⊢ normalOrderF (a * (superCommuteF (anPartF (FieldOp.inAsymp φ'))) (crPartF x✝) * b) = 0 simpAll goals completed! 🐙
  | _, .outAsymp φ' =>𝓕:FieldSpecificationφ:𝓕.FieldOpφ'✝:𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrax✝:𝓕.FieldOpφ':((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ normalOrderF (a * (superCommuteF (anPartF x✝)) (crPartF (FieldOp.outAsymp φ')) * b) = 0 simpAll goals completed! 🐙
  | .position _, .position _ | .position _, .inAsymp _
  | .outAsymp _, .position _ | .outAsymp _, .inAsymp _ =>𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraa✝¹:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentuma✝:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ normalOrderF (a * (superCommuteF (anPartF (FieldOp.outAsymp a✝¹))) (crPartF (FieldOp.inAsymp a✝)) * b) = 0𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraa✝¹:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentuma✝:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ normalOrderF (a * (superCommuteF (anPartF (FieldOp.outAsymp a✝¹))) (crPartF (FieldOp.position a✝)) * b) = 0𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraa✝¹:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTimea✝:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ normalOrderF (a * (superCommuteF (anPartF (FieldOp.position a✝¹))) (crPartF (FieldOp.inAsymp a✝)) * b) = 0𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOpa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraa✝¹:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTimea✝:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ normalOrderF (a * (superCommuteF (anPartF (FieldOp.position a✝¹))) (crPartF (FieldOp.position a✝)) * b) = 0
    exact normalOrderF_superCommuteF_annihilate_create _ _ rfl rfl ..All goals completed! 🐙

The normal ordering of a product of two states

              

@[simp]
lemma normalOrderF_crPartF_mul_crPartF (φ φ' : 𝓕.FieldOp) :
    𝓝ᶠ(crPartF φ * crPartF φ') =
    crPartF φ * crPartF φ' := by𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ normalOrderF (crPartF φ * crPartF φ') = crPartF φ * crPartF φ'
  rw [normalOrderF_crPartF_mul,𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ crPartF φ * normalOrderF (crPartF φ') = crPartF φ * crPartF φ'All goals completed! 🐙 ← mul_one (crPartF φ'),𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ crPartF φ * normalOrderF (crPartF φ' * 1) = crPartF φ * (crPartF φ' * 1)All goals completed! 🐙 normalOrderF_crPartF_mul,𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ crPartF φ * (crPartF φ' * normalOrderF 1) = crPartF φ * (crPartF φ' * 1)All goals completed! 🐙
    normalOrderF_one𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ crPartF φ * (crPartF φ' * 1) = crPartF φ * (crPartF φ' * 1)All goals completed! 🐙]All goals completed! 🐙

@[simp]
lemma normalOrderF_anPartF_mul_anPartF (φ φ' : 𝓕.FieldOp) :
    𝓝ᶠ(anPartF φ * anPartF φ') =
    anPartF φ * anPartF φ' := by𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ normalOrderF (anPartF φ * anPartF φ') = anPartF φ * anPartF φ'
  rw [normalOrderF_mul_anPartF,𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ normalOrderF (anPartF φ) * anPartF φ' = anPartF φ * anPartF φ'All goals completed! 🐙 ← one_mul (anPartF φ),𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ normalOrderF (1 * anPartF φ) * anPartF φ' = 1 * anPartF φ * anPartF φ'All goals completed! 🐙 normalOrderF_mul_anPartF,𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ normalOrderF 1 * anPartF φ * anPartF φ' = 1 * anPartF φ * anPartF φ'All goals completed! 🐙
    normalOrderF_one𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ 1 * anPartF φ * anPartF φ' = 1 * anPartF φ * anPartF φ'All goals completed! 🐙]All goals completed! 🐙

@[simp]
lemma normalOrderF_crPartF_mul_anPartF (φ φ' : 𝓕.FieldOp) :
    𝓝ᶠ(crPartF φ * anPartF φ') =
    crPartF φ * anPartF φ' := by𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ normalOrderF (crPartF φ * anPartF φ') = crPartF φ * anPartF φ'
  rw [normalOrderF_crPartF_mul,𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ crPartF φ * normalOrderF (anPartF φ') = crPartF φ * anPartF φ'All goals completed! 🐙 ← one_mul (anPartF φ'),𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ crPartF φ * normalOrderF (1 * anPartF φ') = crPartF φ * (1 * anPartF φ')All goals completed! 🐙 normalOrderF_mul_anPartF,𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ crPartF φ * (normalOrderF 1 * anPartF φ') = crPartF φ * (1 * anPartF φ')All goals completed! 🐙
    normalOrderF_one𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ crPartF φ * (1 * anPartF φ') = crPartF φ * (1 * anPartF φ')All goals completed! 🐙]All goals completed! 🐙@[simp]
lemma normalOrderF_anPartF_mul_crPartF (φ φ' : 𝓕.FieldOp) :
    𝓝ᶠ(anPartF φ * crPartF φ') =
    𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
    (crPartF φ' * anPartF φ) := by𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ normalOrderF (anPartF φ * crPartF φ') = (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • (crPartF φ' * anPartF φ)
  simpa using normalOrderF_swap_anPartF_crPartF φ φ' 1 1All goals completed! 🐙lemma normalOrderF_ofFieldOpF_mul_ofFieldOpF (φ φ' : 𝓕.FieldOp) :
    𝓝ᶠ(ofFieldOpF φ * ofFieldOpF φ') =
    crPartF φ * crPartF φ' +
    𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
    (crPartF φ' * anPartF φ) +
    crPartF φ * anPartF φ' +
    anPartF φ * anPartF φ' := by𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ normalOrderF (ofFieldOpF φ * ofFieldOpF φ') =
  crPartF φ * crPartF φ' + (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • (crPartF φ' * anPartF φ) + crPartF φ * anPartF φ' +
    anPartF φ * anPartF φ'
  simp only [ofFieldOpF_eq_crPartF_add_anPartF, mul_add, add_mul, map_add,
    normalOrderF_crPartF_mul_crPartF, normalOrderF_anPartF_mul_crPartF,
    normalOrderF_crPartF_mul_anPartF, normalOrderF_anPartF_mul_anPartF]𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ crPartF φ * crPartF φ' + (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • (crPartF φ' * anPartF φ) +
    (crPartF φ * anPartF φ' + anPartF φ * anPartF φ') =
  crPartF φ * crPartF φ' + (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • (crPartF φ' * anPartF φ) + crPartF φ * anPartF φ' +
    anPartF φ * anPartF φ'
  abelAll goals completed! 🐙

Normal order with super commutators

              TODO "Split the following two lemmas up into smaller parts."

lemma normalOrderF_superCommuteF_ofCrAnListF_create_create_ofCrAnListF
    (φc φc' : 𝓕.CrAnFieldOp) (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create)
    (hφc' : 𝓕 |>ᶜ φc' = CreateAnnihilate.create) (φs φs' : List 𝓕.CrAnFieldOp) :
    (𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φc, ofCrAnOpF φc']ₛF * ofCrAnListF φs')) =
      normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) * [ofCrAnOpF φc, ofCrAnOpF φc']ₛF *
      ofCrAnListF (createFilter φs') * ofCrAnListF (annihilateFilter (φs ++ φs'))) := by𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs * (superCommuteF (ofCrAnOpF φc)) (ofCrAnOpF φc') * ofCrAnListF φs') =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) * (superCommuteF (ofCrAnOpF φc)) (ofCrAnOpF φc') * ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))
  rw [superCommuteF_ofCrAnOpF_ofCrAnOpF,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF
    (ofCrAnListF φs *
        (ofCrAnOpF φc * ofCrAnOpF φc' -
          (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
      ofCrAnListF φs') =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs * (ofCrAnOpF φc * ofCrAnOpF φc') * ofCrAnListF φs') -
    normalOrderF
      (ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF φs') =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) mul_sub,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF
    ((ofCrAnListF φs * (ofCrAnOpF φc * ofCrAnOpF φc') -
        ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc)) *
      ofCrAnListF φs') =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs * (ofCrAnOpF φc * ofCrAnOpF φc') * ofCrAnListF φs') -
    normalOrderF
      (ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF φs') =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) sub_mul,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF
    (ofCrAnListF φs * (ofCrAnOpF φc * ofCrAnOpF φc') * ofCrAnListF φs' -
      ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF φs') =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs * (ofCrAnOpF φc * ofCrAnOpF φc') * ofCrAnListF φs') -
    normalOrderF
      (ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF φs') =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) map_sub𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs * (ofCrAnOpF φc * ofCrAnOpF φc') * ofCrAnListF φs') -
    normalOrderF
      (ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF φs') =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs * (ofCrAnOpF φc * ofCrAnOpF φc') * ofCrAnListF φs') -
    normalOrderF
      (ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF φs') =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs * (ofCrAnOpF φc * ofCrAnOpF φc') * ofCrAnListF φs') -
    normalOrderF
      (ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF φs') =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))
  conv_lhs =>
    lhs𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderF (ofCrAnListF φs * (ofCrAnOpF φc * ofCrAnOpF φc') * ofCrAnListF φs'); rhs𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF φs * (ofCrAnOpF φc * ofCrAnOpF φc') * ofCrAnListF φs'
    rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_append,
      ← ofCrAnListF_append, ← ofCrAnListF_append]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF (φs ++ ([φc] ++ [φc']) ++ φs')
  conv_lhs =>
    lhs𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderF (ofCrAnListF (φs ++ ([φc] ++ [φc']) ++ φs'))
    rw [normalOrderF_ofCrAnListF, normalOrderList_eq_createFilter_append_annihilateFilter]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φc] ++ [φc']) ++ φs') •
  ofCrAnListF (createFilter (φs ++ ([φc] ++ [φc']) ++ φs') ++ annihilateFilter (φs ++ ([φc] ++ [φc']) ++ φs'))
    rw [createFilter_append, createFilter_append, createFilter_append,
      createFilter_singleton_create _ hφc, createFilter_singleton_create _ hφc']𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φc] ++ [φc']) ++ φs') •
  ofCrAnListF
    (createFilter φs ++ ([φc] ++ [φc']) ++ createFilter φs' ++ annihilateFilter (φs ++ ([φc] ++ [φc']) ++ φs'))
    rw [annihilateFilter_append, annihilateFilter_append, annihilateFilter_append,
      annihilateFilter_singleton_create _ hφc, annihilateFilter_singleton_create _ hφc']𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φc] ++ [φc']) ++ φs') •
  ofCrAnListF
    (createFilter φs ++ ([φc] ++ [φc']) ++ createFilter φs' ++
      (annihilateFilter φs ++ ([] ++ []) ++ annihilateFilter φs'))
    enter [2, 1, 2]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| annihilateFilter φs ++ ([] ++ []) ++ annihilateFilter φs'
    simp only [List.singleton_append, List.append_assoc, List.cons_append, List.append_nil,
      Algebra.smul_mul_assoc, Algebra.mul_smul_comm, map_smul]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| annihilateFilter φs ++ annihilateFilter φs'
    rw [← annihilateFilter_append]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| annihilateFilter (φs ++ φs')
  conv_lhs =>
    rhs𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderF
  (ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
    ofCrAnListF φs'); rhs𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
  ofCrAnListF φs'
    rw [smul_mul_assoc, Algebra.mul_smul_comm, smul_mul_assoc]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
  (ofCrAnListF φs * (ofCrAnOpF φc' * ofCrAnOpF φc) * ofCrAnListF φs')
    rhs𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF φs * (ofCrAnOpF φc' * ofCrAnOpF φc) * ofCrAnListF φs'
    rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_append,
      ← ofCrAnListF_append, ← ofCrAnListF_append]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF (φs ++ ([φc'] ++ [φc]) ++ φs')
  conv_lhs =>
    rhs𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderF ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnListF (φs ++ ([φc'] ++ [φc]) ++ φs'))
    rw [map_smul]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • normalOrderF (ofCrAnListF (φs ++ ([φc'] ++ [φc]) ++ φs'))
    rhs𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderF (ofCrAnListF (φs ++ ([φc'] ++ [φc]) ++ φs'))
    rw [normalOrderF_ofCrAnListF, normalOrderList_eq_createFilter_append_annihilateFilter]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φc'] ++ [φc]) ++ φs') •
  ofCrAnListF (createFilter (φs ++ ([φc'] ++ [φc]) ++ φs') ++ annihilateFilter (φs ++ ([φc'] ++ [φc]) ++ φs'))
    rw [createFilter_append, createFilter_append, createFilter_append,
      createFilter_singleton_create _ hφc, createFilter_singleton_create _ hφc']𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φc'] ++ [φc]) ++ φs') •
  ofCrAnListF
    (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ ([φc'] ++ [φc]) ++ φs'))
    rw [annihilateFilter_append, annihilateFilter_append, annihilateFilter_append,
      annihilateFilter_singleton_create _ hφc, annihilateFilter_singleton_create _ hφc']𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φc'] ++ [φc]) ++ φs') •
  ofCrAnListF
    (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++
      (annihilateFilter φs ++ ([] ++ []) ++ annihilateFilter φs'))
    enter [2, 1, 2]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| annihilateFilter φs ++ ([] ++ []) ++ annihilateFilter φs'
    simp only [List.singleton_append, List.append_assoc, List.cons_append,
      List.append_nil, Algebra.smul_mul_assoc]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| annihilateFilter φs ++ annihilateFilter φs'
    rw [← annihilateFilter_append]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| annihilateFilter (φs ++ φs')
  conv_lhs =>
    lhs𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φc] ++ [φc']) ++ φs') •
  ofCrAnListF (createFilter φs ++ ([φc] ++ [φc']) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')); lhs𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φc] ++ [φc']) ++ φs')
    simp𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ φc :: φc' :: φs')
  conv_lhs =>
    rhs𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
  normalOrderSign (φs ++ ([φc'] ++ [φc]) ++ φs') •
    ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')); rhs𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φc'] ++ [φc]) ++ φs') •
  ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')); lhs𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φc'] ++ [φc]) ++ φs')
    simp𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ φc' :: φc :: φs')
  rw [normalOrderSign_swap_create_create φc φc' hφc hφc'𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
      ofCrAnListF (createFilter φs ++ ([φc] ++ [φc']) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) -
    (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
      normalOrderSign (φs ++ φc' :: φc :: φs') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
      ofCrAnListF (createFilter φs ++ ([φc] ++ [φc']) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) -
    (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
      normalOrderSign (φs ++ φc' :: φc :: φs') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
      ofCrAnListF (createFilter φs ++ ([φc] ++ [φc']) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) -
    (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
      normalOrderSign (φs ++ φc' :: φc :: φs') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))
  rw [smul_smul,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
      ofCrAnListF (createFilter φs ++ ([φc] ++ [φc']) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) -
    ((exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') * normalOrderSign (φs ++ φc' :: φc :: φs')) •
      ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
      ofCrAnListF (createFilter φs ++ ([φc] ++ [φc']) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) -
    normalOrderSign (φs ++ φc' :: φc :: φs') •
      (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) mul_comm,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
      ofCrAnListF (createFilter φs ++ ([φc] ++ [φc']) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) -
    (normalOrderSign (φs ++ φc' :: φc :: φs') * (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc')) •
      ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
      ofCrAnListF (createFilter φs ++ ([φc] ++ [φc']) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) -
    normalOrderSign (φs ++ φc' :: φc :: φs') •
      (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) ← smul_smul𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
      ofCrAnListF (createFilter φs ++ ([φc] ++ [φc']) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) -
    normalOrderSign (φs ++ φc' :: φc :: φs') •
      (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
      ofCrAnListF (createFilter φs ++ ([φc] ++ [φc']) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) -
    normalOrderSign (φs ++ φc' :: φc :: φs') •
      (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
      ofCrAnListF (createFilter φs ++ ([φc] ++ [φc']) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) -
    normalOrderSign (φs ++ φc' :: φc :: φs') •
      (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))
  rw [← smul_sub,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs ++ ([φc] ++ [φc']) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')) -
      (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) * ofCrAnListF ([φc] ++ [φc']) * ofCrAnListF (createFilter φs') *
        ofCrAnListF (annihilateFilter (φs ++ φs')) -
      (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) ofCrAnListF_append,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs ++ ([φc] ++ [φc']) ++ createFilter φs') * ofCrAnListF (annihilateFilter (φs ++ φs')) -
      (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) * ofCrAnListF ([φc] ++ [φc']) * ofCrAnListF (createFilter φs') *
        ofCrAnListF (annihilateFilter (φs ++ φs')) -
      (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) ofCrAnListF_append,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs ++ ([φc] ++ [φc'])) * ofCrAnListF (createFilter φs') *
        ofCrAnListF (annihilateFilter (φs ++ φs')) -
      (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) * ofCrAnListF ([φc] ++ [φc']) * ofCrAnListF (createFilter φs') *
        ofCrAnListF (annihilateFilter (φs ++ φs')) -
      (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) ofCrAnListF_append𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) * ofCrAnListF ([φc] ++ [φc']) * ofCrAnListF (createFilter φs') *
        ofCrAnListF (annihilateFilter (φs ++ φs')) -
      (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) * ofCrAnListF ([φc] ++ [φc']) * ofCrAnListF (createFilter φs') *
        ofCrAnListF (annihilateFilter (φs ++ φs')) -
      (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) * ofCrAnListF ([φc] ++ [φc']) * ofCrAnListF (createFilter φs') *
        ofCrAnListF (annihilateFilter (φs ++ φs')) -
      (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
        ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))
  conv_lhs =>
    rhs𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF (createFilter φs) * ofCrAnListF ([φc] ++ [φc']) * ofCrAnListF (createFilter φs') *
    ofCrAnListF (annihilateFilter (φs ++ φs')) -
  (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
    ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs')); rhs𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
  ofCrAnListF (createFilter φs ++ ([φc'] ++ [φc]) ++ createFilter φs' ++ annihilateFilter (φs ++ φs'))
    rw [ofCrAnListF_append, ofCrAnListF_append, ofCrAnListF_append]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') •
  (ofCrAnListF (createFilter φs) * ofCrAnListF ([φc'] ++ [φc]) * ofCrAnListF (createFilter φs') *
    ofCrAnListF (annihilateFilter (φs ++ φs')))
    rw [← smul_mul_assoc, ← smul_mul_assoc, ← Algebra.mul_smul_comm]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF (createFilter φs) *
      (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnListF ([φc'] ++ [φc]) *
    ofCrAnListF (createFilter φs') *
  ofCrAnListF (annihilateFilter (φs ++ φs'))
  rw [← sub_mul,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    ((ofCrAnListF (createFilter φs) * ofCrAnListF ([φc] ++ [φc']) * ofCrAnListF (createFilter φs') -
        ofCrAnListF (createFilter φs) *
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnListF ([φc'] ++ [φc]) *
          ofCrAnListF (createFilter φs')) *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnListF ([φc'] ++ [φc])) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) ← sub_mul,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    ((ofCrAnListF (createFilter φs) * ofCrAnListF ([φc] ++ [φc']) -
          ofCrAnListF (createFilter φs) *
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnListF ([φc'] ++ [φc])) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnListF ([φc'] ++ [φc])) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) ← mul_sub,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnListF ([φc] ++ [φc']) -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnListF ([φc'] ++ [φc])) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnListF ([φc'] ++ [φc])) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) ofCrAnListF_append,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnListF [φc] * ofCrAnListF [φc'] -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnListF ([φc'] ++ [φc])) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnListF ([φc'] ++ [φc])) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) ofCrAnListF_singleton,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnListF [φc'] -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnListF ([φc'] ++ [φc])) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnListF ([φc'] ++ [φc])) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))
    ofCrAnListF_singleton𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnListF ([φc'] ++ [φc])) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnListF ([φc'] ++ [φc])) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))]𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnListF ([φc'] ++ [φc])) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))
  rw [ofCrAnListF_append,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • (ofCrAnListF [φc'] * ofCrAnListF [φc])) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))All goals completed! 🐙 ofCrAnListF_singleton,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • (ofCrAnOpF φc' * ofCrAnListF [φc])) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))All goals completed! 🐙 ofCrAnListF_singleton,𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • (ofCrAnOpF φc' * ofCrAnOpF φc)) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • ofCrAnOpF φc' * ofCrAnOpF φc) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))All goals completed! 🐙 smul_mul_assoc𝓕:FieldSpecificationφc:𝓕.CrAnFieldOpφc':𝓕.CrAnFieldOphφc:𝓕|>ᶜφc = CreateAnnihilate.createhφc':𝓕|>ᶜφc' = CreateAnnihilate.createφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • (ofCrAnOpF φc' * ofCrAnOpF φc)) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs'))) =
  normalOrderSign (φs ++ φc' :: φc :: φs') •
    (ofCrAnListF (createFilter φs) *
          (ofCrAnOpF φc * ofCrAnOpF φc' -
            (exchangeSign (𝓕.crAnStatistics φc)) (𝓕.crAnStatistics φc') • (ofCrAnOpF φc' * ofCrAnOpF φc)) *
        ofCrAnListF (createFilter φs') *
      ofCrAnListF (annihilateFilter (φs ++ φs')))All goals completed! 🐙]All goals completed! 🐙

lemma normalOrderF_superCommuteF_ofCrAnListF_annihilate_annihilate_ofCrAnListF
    (φa φa' : 𝓕.CrAnFieldOp)
    (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
    (hφa' : 𝓕 |>ᶜ φa' = CreateAnnihilate.annihilate)
    (φs φs' : List 𝓕.CrAnFieldOp) :
    𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛF * ofCrAnListF φs') =
      normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs'))
      * ofCrAnListF (annihilateFilter φs) * [ofCrAnOpF φa, ofCrAnOpF φa']ₛF
      * ofCrAnListF (annihilateFilter φs')) := by𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs * (superCommuteF (ofCrAnOpF φa)) (ofCrAnOpF φa') * ofCrAnListF φs') =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (superCommuteF (ofCrAnOpF φa)) (ofCrAnOpF φa') *
      ofCrAnListF (annihilateFilter φs'))
  rw [superCommuteF_ofCrAnOpF_ofCrAnOpF,𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF
    (ofCrAnListF φs *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF φs') =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs * (ofCrAnOpF φa * ofCrAnOpF φa') * ofCrAnListF φs') -
    normalOrderF
      (ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
        ofCrAnListF φs') =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs')) mul_sub,𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF
    ((ofCrAnListF φs * (ofCrAnOpF φa * ofCrAnOpF φa') -
        ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa)) *
      ofCrAnListF φs') =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs * (ofCrAnOpF φa * ofCrAnOpF φa') * ofCrAnListF φs') -
    normalOrderF
      (ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
        ofCrAnListF φs') =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs')) sub_mul,𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF
    (ofCrAnListF φs * (ofCrAnOpF φa * ofCrAnOpF φa') * ofCrAnListF φs' -
      ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
        ofCrAnListF φs') =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs * (ofCrAnOpF φa * ofCrAnOpF φa') * ofCrAnListF φs') -
    normalOrderF
      (ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
        ofCrAnListF φs') =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs')) map_sub𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs * (ofCrAnOpF φa * ofCrAnOpF φa') * ofCrAnListF φs') -
    normalOrderF
      (ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
        ofCrAnListF φs') =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs * (ofCrAnOpF φa * ofCrAnOpF φa') * ofCrAnListF φs') -
    normalOrderF
      (ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
        ofCrAnListF φs') =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderF (ofCrAnListF φs * (ofCrAnOpF φa * ofCrAnOpF φa') * ofCrAnListF φs') -
    normalOrderF
      (ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
        ofCrAnListF φs') =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))
  conv_lhs =>
    lhs𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderF (ofCrAnListF φs * (ofCrAnOpF φa * ofCrAnOpF φa') * ofCrAnListF φs'); rhs𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF φs * (ofCrAnOpF φa * ofCrAnOpF φa') * ofCrAnListF φs'
    rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_append,
      ← ofCrAnListF_append, ← ofCrAnListF_append]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF (φs ++ ([φa] ++ [φa']) ++ φs')
  conv_lhs =>
    lhs𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderF (ofCrAnListF (φs ++ ([φa] ++ [φa']) ++ φs'))
    rw [normalOrderF_ofCrAnListF, normalOrderList_eq_createFilter_append_annihilateFilter]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φa] ++ [φa']) ++ φs') •
  ofCrAnListF (createFilter (φs ++ ([φa] ++ [φa']) ++ φs') ++ annihilateFilter (φs ++ ([φa] ++ [φa']) ++ φs'))
    rw [createFilter_append, createFilter_append, createFilter_append,
      createFilter_singleton_annihilate _ hφa, createFilter_singleton_annihilate _ hφa']𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φa] ++ [φa']) ++ φs') •
  ofCrAnListF (createFilter φs ++ ([] ++ []) ++ createFilter φs' ++ annihilateFilter (φs ++ ([φa] ++ [φa']) ++ φs'))
    rw [annihilateFilter_append, annihilateFilter_append, annihilateFilter_append,
      annihilateFilter_singleton_annihilate _ hφa, annihilateFilter_singleton_annihilate _ hφa']𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φa] ++ [φa']) ++ φs') •
  ofCrAnListF
    (createFilter φs ++ ([] ++ []) ++ createFilter φs' ++
      (annihilateFilter φs ++ ([φa] ++ [φa']) ++ annihilateFilter φs'))
    enter [2, 1, 1]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| createFilter φs ++ ([] ++ []) ++ createFilter φs'
    simp only [List.singleton_append, List.append_assoc, List.cons_append, List.append_nil,
      Algebra.smul_mul_assoc, Algebra.mul_smul_comm, map_smul]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| createFilter φs ++ createFilter φs'
    rw [← createFilter_append]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| createFilter (φs ++ φs')
  conv_lhs =>
    rhs𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderF
  (ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
    ofCrAnListF φs'); rhs𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF φs * ((exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
  ofCrAnListF φs'
    rw [smul_mul_assoc]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF φs * (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • (ofCrAnOpF φa' * ofCrAnOpF φa) *
  ofCrAnListF φs'
    rw [Algebra.mul_smul_comm, smul_mul_assoc]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
  (ofCrAnListF φs * (ofCrAnOpF φa' * ofCrAnOpF φa) * ofCrAnListF φs')
    rhs𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF φs * (ofCrAnOpF φa' * ofCrAnOpF φa) * ofCrAnListF φs'
    rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_append,
      ← ofCrAnListF_append, ← ofCrAnListF_append]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF (φs ++ ([φa'] ++ [φa]) ++ φs')
  conv_lhs =>
    rhs𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderF ((exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF (φs ++ ([φa'] ++ [φa]) ++ φs'))
    rw [map_smul]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • normalOrderF (ofCrAnListF (φs ++ ([φa'] ++ [φa]) ++ φs'))
    rhs𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderF (ofCrAnListF (φs ++ ([φa'] ++ [φa]) ++ φs'))
    rw [normalOrderF_ofCrAnListF, normalOrderList_eq_createFilter_append_annihilateFilter]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φa'] ++ [φa]) ++ φs') •
  ofCrAnListF (createFilter (φs ++ ([φa'] ++ [φa]) ++ φs') ++ annihilateFilter (φs ++ ([φa'] ++ [φa]) ++ φs'))
    rw [createFilter_append, createFilter_append, createFilter_append,
      createFilter_singleton_annihilate _ hφa, createFilter_singleton_annihilate _ hφa']𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φa'] ++ [φa]) ++ φs') •
  ofCrAnListF (createFilter φs ++ ([] ++ []) ++ createFilter φs' ++ annihilateFilter (φs ++ ([φa'] ++ [φa]) ++ φs'))
    rw [annihilateFilter_append, annihilateFilter_append, annihilateFilter_append,
      annihilateFilter_singleton_annihilate _ hφa, annihilateFilter_singleton_annihilate _ hφa']𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φa'] ++ [φa]) ++ φs') •
  ofCrAnListF
    (createFilter φs ++ ([] ++ []) ++ createFilter φs' ++
      (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs'))
    enter [2, 1, 1]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| createFilter φs ++ ([] ++ []) ++ createFilter φs'
    simp only [List.singleton_append, List.append_assoc, List.cons_append,
      List.append_nil, Algebra.smul_mul_assoc]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| createFilter φs ++ createFilter φs'
    rw [← createFilter_append]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| createFilter (φs ++ φs')
  conv_lhs =>
    lhs𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φa] ++ [φa']) ++ φs') •
  ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa] ++ [φa']) ++ annihilateFilter φs')); lhs𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φa] ++ [φa']) ++ φs')
    simp𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ φa :: φa' :: φs')
  conv_lhs =>
    rhs𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
  normalOrderSign (φs ++ ([φa'] ++ [φa]) ++ φs') •
    ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs')); rhs𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φa'] ++ [φa]) ++ φs') •
  ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs')); lhs𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ ([φa'] ++ [φa]) ++ φs')
    simp𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| normalOrderSign (φs ++ φa' :: φa :: φs')
  rw [normalOrderSign_swap_annihilate_annihilate φa φa' hφa hφa'𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
      ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa] ++ [φa']) ++ annihilateFilter φs')) -
    (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
      normalOrderSign (φs ++ φa' :: φa :: φs') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs')) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
      ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa] ++ [φa']) ++ annihilateFilter φs')) -
    (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
      normalOrderSign (φs ++ φa' :: φa :: φs') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs')) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
      ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa] ++ [φa']) ++ annihilateFilter φs')) -
    (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
      normalOrderSign (φs ++ φa' :: φa :: φs') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs')) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))
  rw [smul_smul,𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
      ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa] ++ [φa']) ++ annihilateFilter φs')) -
    ((exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') * normalOrderSign (φs ++ φa' :: φa :: φs')) •
      ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs')) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
      ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa] ++ [φa']) ++ annihilateFilter φs')) -
    normalOrderSign (φs ++ φa' :: φa :: φs') •
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs')) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs')) mul_comm,𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
      ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa] ++ [φa']) ++ annihilateFilter φs')) -
    (normalOrderSign (φs ++ φa' :: φa :: φs') * (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa')) •
      ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs')) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
      ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa] ++ [φa']) ++ annihilateFilter φs')) -
    normalOrderSign (φs ++ φa' :: φa :: φs') •
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs')) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs')) ← smul_smul𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
      ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa] ++ [φa']) ++ annihilateFilter φs')) -
    normalOrderSign (φs ++ φa' :: φa :: φs') •
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs')) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
      ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa] ++ [φa']) ++ annihilateFilter φs')) -
    normalOrderSign (φs ++ φa' :: φa :: φs') •
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs')) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
      ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa] ++ [φa']) ++ annihilateFilter φs')) -
    normalOrderSign (φs ++ φa' :: φa :: φs') •
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs')) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))
  rw [← smul_sub,𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa] ++ [φa']) ++ annihilateFilter φs')) -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) *
        (ofCrAnListF (annihilateFilter φs) * ofCrAnListF ([φa] ++ [φa']) * ofCrAnListF (annihilateFilter φs')) -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs')) ofCrAnListF_append,𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) *
        ofCrAnListF (annihilateFilter φs ++ ([φa] ++ [φa']) ++ annihilateFilter φs') -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) *
        (ofCrAnListF (annihilateFilter φs) * ofCrAnListF ([φa] ++ [φa']) * ofCrAnListF (annihilateFilter φs')) -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs')) ofCrAnListF_append,𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) *
        (ofCrAnListF (annihilateFilter φs ++ ([φa] ++ [φa'])) * ofCrAnListF (annihilateFilter φs')) -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) *
        (ofCrAnListF (annihilateFilter φs) * ofCrAnListF ([φa] ++ [φa']) * ofCrAnListF (annihilateFilter φs')) -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs')) ofCrAnListF_append𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) *
        (ofCrAnListF (annihilateFilter φs) * ofCrAnListF ([φa] ++ [φa']) * ofCrAnListF (annihilateFilter φs')) -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) *
        (ofCrAnListF (annihilateFilter φs) * ofCrAnListF ([φa] ++ [φa']) * ofCrAnListF (annihilateFilter φs')) -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) *
        (ofCrAnListF (annihilateFilter φs) * ofCrAnListF ([φa] ++ [φa']) * ofCrAnListF (annihilateFilter φs')) -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
        ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))
  conv_lhs =>
    rhs𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF (createFilter (φs ++ φs')) *
    (ofCrAnListF (annihilateFilter φs) * ofCrAnListF ([φa] ++ [φa']) * ofCrAnListF (annihilateFilter φs')) -
  (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
    ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs')); rhs𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
  ofCrAnListF (createFilter (φs ++ φs') ++ (annihilateFilter φs ++ ([φa'] ++ [φa]) ++ annihilateFilter φs'))
    rw [ofCrAnListF_append, ofCrAnListF_append, ofCrAnListF_append]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') •
  (ofCrAnListF (createFilter (φs ++ φs')) *
    (ofCrAnListF (annihilateFilter φs) * ofCrAnListF ([φa'] ++ [φa]) * ofCrAnListF (annihilateFilter φs')))
    rw [← Algebra.mul_smul_comm, ← smul_mul_assoc, ← Algebra.mul_smul_comm]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF (createFilter (φs ++ φs')) *
  (ofCrAnListF (annihilateFilter φs) *
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa]) *
    ofCrAnListF (annihilateFilter φs'))
  rw [← mul_sub,𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) *
      (ofCrAnListF (annihilateFilter φs) * ofCrAnListF ([φa] ++ [φa']) * ofCrAnListF (annihilateFilter φs') -
        ofCrAnListF (annihilateFilter φs) *
            (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa]) *
          ofCrAnListF (annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) *
      (ofCrAnListF (annihilateFilter φs) *
          (ofCrAnListF ([φa] ++ [φa']) -
            (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
        ofCrAnListF (annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs')) ← sub_mul,𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) *
      ((ofCrAnListF (annihilateFilter φs) * ofCrAnListF ([φa] ++ [φa']) -
          ofCrAnListF (annihilateFilter φs) *
            (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
        ofCrAnListF (annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) *
      (ofCrAnListF (annihilateFilter φs) *
          (ofCrAnListF ([φa] ++ [φa']) -
            (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
        ofCrAnListF (annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs')) ← mul_sub𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) *
      (ofCrAnListF (annihilateFilter φs) *
          (ofCrAnListF ([φa] ++ [φa']) -
            (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
        ofCrAnListF (annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) *
      (ofCrAnListF (annihilateFilter φs) *
          (ofCrAnListF ([φa] ++ [φa']) -
            (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
        ofCrAnListF (annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) *
      (ofCrAnListF (annihilateFilter φs) *
          (ofCrAnListF ([φa] ++ [φa']) -
            (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
        ofCrAnListF (annihilateFilter φs'))) =
  normalOrderSign (φs ++ φa' :: φa :: φs') •
    (ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
        (ofCrAnOpF φa * ofCrAnOpF φa' -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))
  apply congrArg𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ ofCrAnListF (createFilter (φs ++ φs')) *
    (ofCrAnListF (annihilateFilter φs) *
        (ofCrAnListF ([φa] ++ [φa']) -
          (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
      ofCrAnListF (annihilateFilter φs')) =
  ofCrAnListF (createFilter (φs ++ φs')) * ofCrAnListF (annihilateFilter φs) *
      (ofCrAnOpF φa * ofCrAnOpF φa' -
        (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
    ofCrAnListF (annihilateFilter φs')
  conv_rhs => rw [mul_assoc, mul_assoc]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ofCrAnListF (createFilter (φs ++ φs')) *
  (ofCrAnListF (annihilateFilter φs) *
    ((ofCrAnOpF φa * ofCrAnOpF φa' -
        (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs')))
  apply congrArg𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ ofCrAnListF (annihilateFilter φs) *
      (ofCrAnListF ([φa] ++ [φa']) -
        (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
    ofCrAnListF (annihilateFilter φs') =
  ofCrAnListF (annihilateFilter φs) *
    ((ofCrAnOpF φa * ofCrAnOpF φa' -
        (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))
  rw [mul_assoc𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ ofCrAnListF (annihilateFilter φs) *
    ((ofCrAnListF ([φa] ++ [φa']) -
        (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
      ofCrAnListF (annihilateFilter φs')) =
  ofCrAnListF (annihilateFilter φs) *
    ((ofCrAnOpF φa * ofCrAnOpF φa' -
        (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ ofCrAnListF (annihilateFilter φs) *
    ((ofCrAnListF ([φa] ++ [φa']) -
        (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
      ofCrAnListF (annihilateFilter φs')) =
  ofCrAnListF (annihilateFilter φs) *
    ((ofCrAnOpF φa * ofCrAnOpF φa' -
        (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ ofCrAnListF (annihilateFilter φs) *
    ((ofCrAnListF ([φa] ++ [φa']) -
        (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
      ofCrAnListF (annihilateFilter φs')) =
  ofCrAnListF (annihilateFilter φs) *
    ((ofCrAnOpF φa * ofCrAnOpF φa' -
        (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
      ofCrAnListF (annihilateFilter φs'))
  apply congrArg𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (ofCrAnListF ([φa] ++ [φa']) -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
    ofCrAnListF (annihilateFilter φs') =
  (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
    ofCrAnListF (annihilateFilter φs')
  rw [ofCrAnListF_append,𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (ofCrAnListF [φa] * ofCrAnListF [φa'] -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
    ofCrAnListF (annihilateFilter φs') =
  (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
    ofCrAnListF (annihilateFilter φs')𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
    ofCrAnListF (annihilateFilter φs') =
  (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
    ofCrAnListF (annihilateFilter φs') ofCrAnListF_singleton,𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (ofCrAnOpF φa * ofCrAnListF [φa'] -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
    ofCrAnListF (annihilateFilter φs') =
  (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
    ofCrAnListF (annihilateFilter φs')𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
    ofCrAnListF (annihilateFilter φs') =
  (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
    ofCrAnListF (annihilateFilter φs') ofCrAnListF_singleton𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
    ofCrAnListF (annihilateFilter φs') =
  (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
    ofCrAnListF (annihilateFilter φs')𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
    ofCrAnListF (annihilateFilter φs') =
  (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
    ofCrAnListF (annihilateFilter φs')]𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnListF ([φa'] ++ [φa])) *
    ofCrAnListF (annihilateFilter φs') =
  (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
    ofCrAnListF (annihilateFilter φs')
  rw [ofCrAnListF_append,𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • (ofCrAnListF [φa'] * ofCrAnListF [φa])) *
    ofCrAnListF (annihilateFilter φs') =
  (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
    ofCrAnListF (annihilateFilter φs')All goals completed! 🐙 ofCrAnListF_singleton,𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • (ofCrAnOpF φa' * ofCrAnListF [φa])) *
    ofCrAnListF (annihilateFilter φs') =
  (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
    ofCrAnListF (annihilateFilter φs')All goals completed! 🐙 ofCrAnListF_singleton,𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • (ofCrAnOpF φa' * ofCrAnOpF φa)) *
    ofCrAnListF (annihilateFilter φs') =
  (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • ofCrAnOpF φa' * ofCrAnOpF φa) *
    ofCrAnListF (annihilateFilter φs')All goals completed! 🐙 smul_mul_assoc𝓕:FieldSpecificationφa:𝓕.CrAnFieldOpφa':𝓕.CrAnFieldOphφa:𝓕|>ᶜφa = CreateAnnihilate.annihilatehφa':𝓕|>ᶜφa' = CreateAnnihilate.annihilateφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • (ofCrAnOpF φa' * ofCrAnOpF φa)) *
    ofCrAnListF (annihilateFilter φs') =
  (ofCrAnOpF φa * ofCrAnOpF φa' -
      (exchangeSign (𝓕.crAnStatistics φa)) (𝓕.crAnStatistics φa') • (ofCrAnOpF φa' * ofCrAnOpF φa)) *
    ofCrAnListF (annihilateFilter φs')All goals completed! 🐙]All goals completed! 🐙

Super commutators involving a normal order.

lemma ofCrAnListF_superCommuteF_normalOrderF_ofCrAnListF (φs φs' : List 𝓕.CrAnFieldOp) :
    [ofCrAnListF φs, 𝓝ᶠ(ofCrAnListF φs')]ₛF =
    ofCrAnListF φs * 𝓝ᶠ(ofCrAnListF φs') -
    𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofCrAnListF φs') * ofCrAnListF φs := by𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (superCommuteF (ofCrAnListF φs)) (normalOrderF (ofCrAnListF φs')) =
  ofCrAnListF φs * normalOrderF (ofCrAnListF φs') -
    (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics φs') • normalOrderF (ofCrAnListF φs') *
      ofCrAnListF φs
  simp only [normalOrderF_ofCrAnListF, map_smul, superCommuteF_ofCrAnListF_ofCrAnListF,
    ofCrAnListF_append, normalOrderList_statistics, smul_sub, smul_smul,
    Algebra.mul_smul_comm, mul_comm, Algebra.smul_mul_assoc]All goals completed! 🐙



lemma ofCrAnListF_superCommuteF_normalOrderF_ofFieldOpListF (φs : List 𝓕.CrAnFieldOp)
    (φs' : List 𝓕.FieldOp) : [ofCrAnListF φs, 𝓝ᶠ(ofFieldOpListF φs')]ₛF =
    ofCrAnListF φs * 𝓝ᶠ(ofFieldOpListF φs') -
    𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs') * ofCrAnListF φs := by𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ (superCommuteF (ofCrAnListF φs)) (normalOrderF (ofFieldOpListF φs')) =
  ofCrAnListF φs * normalOrderF (ofFieldOpListF φs') -
    (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofFieldOpListF φs') *
      ofCrAnListF φs
  rw [ofFieldOpListF_sum,𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ (superCommuteF (ofCrAnListF φs)) (normalOrderF (∑ s, ofCrAnListF ↑s)) =
  ofCrAnListF φs * normalOrderF (∑ s, ofCrAnListF ↑s) -
    (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (∑ s, ofCrAnListF ↑s) *
      ofCrAnListF φs𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ ∑ x, (superCommuteF (ofCrAnListF φs)) (normalOrderF (ofCrAnListF ↑x)) =
  ∑ x,
    (ofCrAnListF φs * normalOrderF (ofCrAnListF ↑x) -
      (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑x) *
        ofCrAnListF φs) map_sum,𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ (superCommuteF (ofCrAnListF φs)) (∑ x, normalOrderF (ofCrAnListF ↑x)) =
  ofCrAnListF φs * ∑ x, normalOrderF (ofCrAnListF ↑x) -
    ((exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • ∑ x, normalOrderF (ofCrAnListF ↑x)) *
      ofCrAnListF φs𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ ∑ x, (superCommuteF (ofCrAnListF φs)) (normalOrderF (ofCrAnListF ↑x)) =
  ∑ x,
    (ofCrAnListF φs * normalOrderF (ofCrAnListF ↑x) -
      (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑x) *
        ofCrAnListF φs) Finset.mul_sum,𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ (superCommuteF (ofCrAnListF φs)) (∑ x, normalOrderF (ofCrAnListF ↑x)) =
  ∑ i, ofCrAnListF φs * normalOrderF (ofCrAnListF ↑i) -
    ((exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • ∑ x, normalOrderF (ofCrAnListF ↑x)) *
      ofCrAnListF φs𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ ∑ x, (superCommuteF (ofCrAnListF φs)) (normalOrderF (ofCrAnListF ↑x)) =
  ∑ x,
    (ofCrAnListF φs * normalOrderF (ofCrAnListF ↑x) -
      (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑x) *
        ofCrAnListF φs) Finset.smul_sum,𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ (superCommuteF (ofCrAnListF φs)) (∑ x, normalOrderF (ofCrAnListF ↑x)) =
  ∑ i, ofCrAnListF φs * normalOrderF (ofCrAnListF ↑i) -
    (∑ x, (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑x)) *
      ofCrAnListF φs𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ ∑ x, (superCommuteF (ofCrAnListF φs)) (normalOrderF (ofCrAnListF ↑x)) =
  ∑ x,
    (ofCrAnListF φs * normalOrderF (ofCrAnListF ↑x) -
      (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑x) *
        ofCrAnListF φs) Finset.sum_mul,𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ (superCommuteF (ofCrAnListF φs)) (∑ x, normalOrderF (ofCrAnListF ↑x)) =
  ∑ i, ofCrAnListF φs * normalOrderF (ofCrAnListF ↑i) -
    ∑ i,
      (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑i) *
        ofCrAnListF φs𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ ∑ x, (superCommuteF (ofCrAnListF φs)) (normalOrderF (ofCrAnListF ↑x)) =
  ∑ x,
    (ofCrAnListF φs * normalOrderF (ofCrAnListF ↑x) -
      (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑x) *
        ofCrAnListF φs)
    ← Finset.sum_sub_distrib,𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ (superCommuteF (ofCrAnListF φs)) (∑ x, normalOrderF (ofCrAnListF ↑x)) =
  ∑ x,
    (ofCrAnListF φs * normalOrderF (ofCrAnListF ↑x) -
      (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑x) *
        ofCrAnListF φs)𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ ∑ x, (superCommuteF (ofCrAnListF φs)) (normalOrderF (ofCrAnListF ↑x)) =
  ∑ x,
    (ofCrAnListF φs * normalOrderF (ofCrAnListF ↑x) -
      (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑x) *
        ofCrAnListF φs) map_sum𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ ∑ x, (superCommuteF (ofCrAnListF φs)) (normalOrderF (ofCrAnListF ↑x)) =
  ∑ x,
    (ofCrAnListF φs * normalOrderF (ofCrAnListF ↑x) -
      (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑x) *
        ofCrAnListF φs)𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ ∑ x, (superCommuteF (ofCrAnListF φs)) (normalOrderF (ofCrAnListF ↑x)) =
  ∑ x,
    (ofCrAnListF φs * normalOrderF (ofCrAnListF ↑x) -
      (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑x) *
        ofCrAnListF φs)]𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ ∑ x, (superCommuteF (ofCrAnListF φs)) (normalOrderF (ofCrAnListF ↑x)) =
  ∑ x,
    (ofCrAnListF φs * normalOrderF (ofCrAnListF ↑x) -
      (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑x) *
        ofCrAnListF φs)
  congre_f𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ (fun x => (superCommuteF (ofCrAnListF φs)) (normalOrderF (ofCrAnListF ↑x))) = fun x =>
  ofCrAnListF φs * normalOrderF (ofCrAnListF ↑x) -
    (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑x) *
      ofCrAnListF φs
  funext ne_f𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOpn:CrAnSection φs'⊢ (superCommuteF (ofCrAnListF φs)) (normalOrderF (ofCrAnListF ↑n)) =
  ofCrAnListF φs * normalOrderF (ofCrAnListF ↑n) -
    (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑n) *
      ofCrAnListF φs
  rw [ofCrAnListF_superCommuteF_normalOrderF_ofCrAnListF,e_f𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOpn:CrAnSection φs'⊢ ofCrAnListF φs * normalOrderF (ofCrAnListF ↑n) -
    (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics ↑n) • normalOrderF (ofCrAnListF ↑n) *
      ofCrAnListF φs =
  ofCrAnListF φs * normalOrderF (ofCrAnListF ↑n) -
    (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑n) *
      ofCrAnListF φsAll goals completed! 🐙
    CrAnSection.statistics_eq_state_statisticse_f𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOpn:CrAnSection φs'⊢ ofCrAnListF φs * normalOrderF (ofCrAnListF ↑n) -
    (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑n) *
      ofCrAnListF φs =
  ofCrAnListF φs * normalOrderF (ofCrAnListF ↑n) -
    (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofCrAnListF ↑n) *
      ofCrAnListF φsAll goals completed! 🐙]All goals completed! 🐙

Multiplications with normal order written in terms of super commute.

              lemma ofCrAnListF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnFieldOp)
    (φs' : List 𝓕.FieldOp) :
    ofCrAnListF φs * 𝓝ᶠ(ofFieldOpListF φs') =
    𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs') * ofCrAnListF φs
    + [ofCrAnListF φs, 𝓝ᶠ(ofFieldOpListF φs')]ₛF := by𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ ofCrAnListF φs * normalOrderF (ofFieldOpListF φs') =
  (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofFieldOpListF φs') *
      ofCrAnListF φs +
    (superCommuteF (ofCrAnListF φs)) (normalOrderF (ofFieldOpListF φs'))
  simp [ofCrAnListF_superCommuteF_normalOrderF_ofFieldOpListF]All goals completed! 🐙lemma ofCrAnOpF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.CrAnFieldOp)
    (φs' : List 𝓕.FieldOp) : ofCrAnOpF φ * 𝓝ᶠ(ofFieldOpListF φs') =
    𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs') * ofCrAnOpF φ
    + [ofCrAnOpF φ, 𝓝ᶠ(ofFieldOpListF φs')]ₛF := by𝓕:FieldSpecificationφ:𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ ofCrAnOpF φ * normalOrderF (ofFieldOpListF φs') =
  (exchangeSign (𝓕.crAnStatistics φ)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofFieldOpListF φs') *
      ofCrAnOpF φ +
    (superCommuteF (ofCrAnOpF φ)) (normalOrderF (ofFieldOpListF φs'))
  simp [← ofCrAnListF_singleton, ofCrAnListF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF]All goals completed! 🐙

lemma anPartF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.FieldOp)
    (φs' : List 𝓕.FieldOp) :
    anPartF φ * 𝓝ᶠ(ofFieldOpListF φs') =
    𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs' * anPartF φ)
    + [anPartF φ, 𝓝ᶠ(ofFieldOpListF φs')]ₛF := by𝓕:FieldSpecificationφ:𝓕.FieldOpφs':List 𝓕.FieldOp⊢ anPartF φ * normalOrderF (ofFieldOpListF φs') =
  (exchangeSign (𝓕|>ₛφ)) (ofList 𝓕.fieldOpStatistic φs') • normalOrderF (ofFieldOpListF φs' * anPartF φ) +
    (superCommuteF (anPartF φ)) (normalOrderF (ofFieldOpListF φs'))
  rw [normalOrderF_mul_anPartF𝓕:FieldSpecificationφ:𝓕.FieldOpφs':List 𝓕.FieldOp⊢ anPartF φ * normalOrderF (ofFieldOpListF φs') =
  (exchangeSign (𝓕|>ₛφ)) (ofList 𝓕.fieldOpStatistic φs') • (normalOrderF (ofFieldOpListF φs') * anPartF φ) +
    (superCommuteF (anPartF φ)) (normalOrderF (ofFieldOpListF φs'))𝓕:FieldSpecificationφ:𝓕.FieldOpφs':List 𝓕.FieldOp⊢ anPartF φ * normalOrderF (ofFieldOpListF φs') =
  (exchangeSign (𝓕|>ₛφ)) (ofList 𝓕.fieldOpStatistic φs') • (normalOrderF (ofFieldOpListF φs') * anPartF φ) +
    (superCommuteF (anPartF φ)) (normalOrderF (ofFieldOpListF φs'))]𝓕:FieldSpecificationφ:𝓕.FieldOpφs':List 𝓕.FieldOp⊢ anPartF φ * normalOrderF (ofFieldOpListF φs') =
  (exchangeSign (𝓕|>ₛφ)) (ofList 𝓕.fieldOpStatistic φs') • (normalOrderF (ofFieldOpListF φs') * anPartF φ) +
    (superCommuteF (anPartF φ)) (normalOrderF (ofFieldOpListF φs'))
  match φ with
  | .inAsymp _ =>𝓕:FieldSpecificationφ:𝓕.FieldOpφs':List 𝓕.FieldOpa✝:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ anPartF (FieldOp.inAsymp a✝) * normalOrderF (ofFieldOpListF φs') =
  (exchangeSign (𝓕|>ₛFieldOp.inAsymp a✝)) (ofList 𝓕.fieldOpStatistic φs') •
      (normalOrderF (ofFieldOpListF φs') * anPartF (FieldOp.inAsymp a✝)) +
    (superCommuteF (anPartF (FieldOp.inAsymp a✝))) (normalOrderF (ofFieldOpListF φs')) simpAll goals completed! 🐙
  | .position _ | .outAsymp _ =>𝓕:FieldSpecificationφ:𝓕.FieldOpφs':List 𝓕.FieldOpa✝:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ anPartF (FieldOp.outAsymp a✝) * normalOrderF (ofFieldOpListF φs') =
  (exchangeSign (𝓕|>ₛFieldOp.outAsymp a✝)) (ofList 𝓕.fieldOpStatistic φs') •
      (normalOrderF (ofFieldOpListF φs') * anPartF (FieldOp.outAsymp a✝)) +
    (superCommuteF (anPartF (FieldOp.outAsymp a✝))) (normalOrderF (ofFieldOpListF φs'))𝓕:FieldSpecificationφ:𝓕.FieldOpφs':List 𝓕.FieldOpa✝:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ anPartF (FieldOp.position a✝) * normalOrderF (ofFieldOpListF φs') =
  (exchangeSign (𝓕|>ₛFieldOp.position a✝)) (ofList 𝓕.fieldOpStatistic φs') •
      (normalOrderF (ofFieldOpListF φs') * anPartF (FieldOp.position a✝)) +
    (superCommuteF (anPartF (FieldOp.position a✝))) (normalOrderF (ofFieldOpListF φs'))
    simp [ofCrAnOpF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF, crAnStatistics]All goals completed! 🐙