Physlib.QFT.PerturbationTheory.Koszul.KoszulSign

Imports

/-
Copyright (c) 2024 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.Koszul.KoszulSignInsert
public import Physlib.Mathematics.List.InsertionSort
import all Physlib.Mathematics.List

Koszul sign

@[expose] public section

Gives a factor of - 1 for every fermion-fermion (q is 1) crossing that occurs when sorting a list of based on r.

def koszulSign (q : 𝓕 → FieldStatistic) (le : 𝓕 → 𝓕 → Prop) [DecidableRel le] :
    List 𝓕 → ℂ
  | [] => 1
  | a :: l => koszulSignInsert q le a l * koszulSign q le l

              @[simp]
lemma koszulSign_singleton (q : 𝓕 → FieldStatistic) (le : 𝓕 → 𝓕 → Prop) [DecidableRel le] (φ : 𝓕) :
    koszulSign q le [φ] = 1 := by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕⊢ koszulSign q le [φ] = 1
  simp [koszulSign, koszulSignInsert]All goals completed! 🐙

lemma koszulSign_mul_self (l : List 𝓕) : koszulSign q le l * koszulSign q le l = 1 := by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel lel:List 𝓕⊢ koszulSign q le l * koszulSign q le l = 1
  induction l with
  | nil =>nil𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel le⊢ koszulSign q le [] * koszulSign q le [] = 1 simp [koszulSign]All goals completed! 🐙
  | cons a l ih =>cons𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel lea:𝓕l:List 𝓕ih:koszulSign q le l * koszulSign q le l = 1⊢ koszulSign q le (a :: l) * koszulSign q le (a :: l) = 1
    simp only [koszulSign]cons𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel lea:𝓕l:List 𝓕ih:koszulSign q le l * koszulSign q le l = 1⊢ koszulSignInsert q le a l * koszulSign q le l * (koszulSignInsert q le a l * koszulSign q le l) = 1
    trans (koszulSignInsert q le a l * koszulSignInsert q le a l) *
      (koszulSign q le l * koszulSign q le l)𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel lea:𝓕l:List 𝓕ih:koszulSign q le l * koszulSign q le l = 1⊢ koszulSignInsert q le a l * koszulSign q le l * (koszulSignInsert q le a l * koszulSign q le l) =
  koszulSignInsert q le a l * koszulSignInsert q le a l * (koszulSign q le l * koszulSign q le l)𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel lea:𝓕l:List 𝓕ih:koszulSign q le l * koszulSign q le l = 1⊢ koszulSignInsert q le a l * koszulSignInsert q le a l * (koszulSign q le l * koszulSign q le l) = 1
    ·𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel lea:𝓕l:List 𝓕ih:koszulSign q le l * koszulSign q le l = 1⊢ koszulSignInsert q le a l * koszulSign q le l * (koszulSignInsert q le a l * koszulSign q le l) =
  koszulSignInsert q le a l * koszulSignInsert q le a l * (koszulSign q le l * koszulSign q le l) ringAll goals completed! 🐙
    ·𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel lea:𝓕l:List 𝓕ih:koszulSign q le l * koszulSign q le l = 1⊢ koszulSignInsert q le a l * koszulSignInsert q le a l * (koszulSign q le l * koszulSign q le l) = 1 rw [ih,𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel lea:𝓕l:List 𝓕ih:koszulSign q le l * koszulSign q le l = 1⊢ koszulSignInsert q le a l * koszulSignInsert q le a l * 1 = 1All goals completed! 🐙 koszulSignInsert_mul_self,𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel lea:𝓕l:List 𝓕ih:koszulSign q le l * koszulSign q le l = 1⊢ 1 * 1 = 1All goals completed! 🐙 mul_one𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel lea:𝓕l:List 𝓕ih:koszulSign q le l * koszulSign q le l = 1⊢ 1 = 1All goals completed! 🐙]All goals completed! 🐙@[simp]
lemma koszulSign_freeMonoid_of (φ : 𝓕) : koszulSign q le (FreeMonoid.of φ) = 1 := by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕⊢ koszulSign q le (FreeMonoid.of φ) = 1
  change koszulSign q le [φ] = 1𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕⊢ koszulSign q le [φ] = 1
  simp only [koszulSign, mul_one]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕⊢ koszulSignInsert q le φ [] = 1
  rflAll goals completed! 🐙

lemma koszulSignInsert_erase_boson {𝓕 : Type} (q : 𝓕 → FieldStatistic)
    (le : 𝓕 → 𝓕 → Prop) [DecidableRel le] (φ : 𝓕) :
    (φs : List 𝓕) → (n : Fin φs.length) → (heq : q (φs.get n) = bosonic) →
    koszulSignInsert q le φ (φs.eraseIdx n) = koszulSignInsert q le φ φs
  | [], _, _ =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕x✝¹:Fin [].lengthx✝:q ([].get x✝¹) = bosonic⊢ koszulSignInsert q le φ ([].eraseIdx ↑x✝¹) = koszulSignInsert q le φ [] by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕x✝¹:Fin [].lengthx✝:q ([].get x✝¹) = bosonic⊢ koszulSignInsert q le φ ([].eraseIdx ↑x✝¹) = koszulSignInsert q le φ []
    simpAll goals completed! 🐙
  | r1 :: r, ⟨0, h⟩, hr =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕h:0 < (r1 :: r).lengthhr:q ((r1 :: r).get ⟨0, h⟩) = bosonic⊢ koszulSignInsert q le φ ((r1 :: r).eraseIdx ↑⟨0, h⟩) = koszulSignInsert q le φ (r1 :: r) by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕h:0 < (r1 :: r).lengthhr:q ((r1 :: r).get ⟨0, h⟩) = bosonic⊢ koszulSignInsert q le φ ((r1 :: r).eraseIdx ↑⟨0, h⟩) = koszulSignInsert q le φ (r1 :: r)
    simp only [List.eraseIdx_zero, List.tail_cons]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕h:0 < (r1 :: r).lengthhr:q ((r1 :: r).get ⟨0, h⟩) = bosonic⊢ koszulSignInsert q le φ r = koszulSignInsert q le φ (r1 :: r)
    simp only [List.length_cons, Fin.zero_eta, List.get_eq_getElem, Fin.val_zero,
      List.getElem_cons_zero] at hr𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕h:0 < (r1 :: r).lengthhr:q r1 = bosonic⊢ koszulSignInsert q le φ r = koszulSignInsert q le φ (r1 :: r)
    rw [koszulSignInsert𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕h:0 < (r1 :: r).lengthhr:q r1 = bosonic⊢ koszulSignInsert q le φ r =
  if le φ r1 then koszulSignInsert q le φ r
  else if q φ = fermionic ∧ q r1 = fermionic then -koszulSignInsert q le φ r else koszulSignInsert q le φ r𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕h:0 < (r1 :: r).lengthhr:q r1 = bosonic⊢ koszulSignInsert q le φ r =
  if le φ r1 then koszulSignInsert q le φ r
  else if q φ = fermionic ∧ q r1 = fermionic then -koszulSignInsert q le φ r else koszulSignInsert q le φ r]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕h:0 < (r1 :: r).lengthhr:q r1 = bosonic⊢ koszulSignInsert q le φ r =
  if le φ r1 then koszulSignInsert q le φ r
  else if q φ = fermionic ∧ q r1 = fermionic then -koszulSignInsert q le φ r else koszulSignInsert q le φ r
    simp [hr]All goals completed! 🐙
  | r1 :: r, ⟨n + 1, h⟩, hr =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕n:ℕh:n + 1 < (r1 :: r).lengthhr:q ((r1 :: r).get ⟨n + 1, h⟩) = bosonic⊢ koszulSignInsert q le φ ((r1 :: r).eraseIdx ↑⟨n + 1, h⟩) = koszulSignInsert q le φ (r1 :: r) by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕n:ℕh:n + 1 < (r1 :: r).lengthhr:q ((r1 :: r).get ⟨n + 1, h⟩) = bosonic⊢ koszulSignInsert q le φ ((r1 :: r).eraseIdx ↑⟨n + 1, h⟩) = koszulSignInsert q le φ (r1 :: r)
    simp only [List.eraseIdx_cons_succ]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕n:ℕh:n + 1 < (r1 :: r).lengthhr:q ((r1 :: r).get ⟨n + 1, h⟩) = bosonic⊢ koszulSignInsert q le φ (r1 :: r.eraseIdx n) = koszulSignInsert q le φ (r1 :: r)
    rw [koszulSignInsert,𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕n:ℕh:n + 1 < (r1 :: r).lengthhr:q ((r1 :: r).get ⟨n + 1, h⟩) = bosonic⊢ (if le φ r1 then koszulSignInsert q le φ (r.eraseIdx n)
  else
    if q φ = fermionic ∧ q r1 = fermionic then -koszulSignInsert q le φ (r.eraseIdx n)
    else koszulSignInsert q le φ (r.eraseIdx n)) =
  koszulSignInsert q le φ (r1 :: r)𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕n:ℕh:n + 1 < (r1 :: r).lengthhr:q ((r1 :: r).get ⟨n + 1, h⟩) = bosonic⊢ (if le φ r1 then koszulSignInsert q le φ (r.eraseIdx n)
  else
    if q φ = fermionic ∧ q r1 = fermionic then -koszulSignInsert q le φ (r.eraseIdx n)
    else koszulSignInsert q le φ (r.eraseIdx n)) =
  if le φ r1 then koszulSignInsert q le φ r
  else if q φ = fermionic ∧ q r1 = fermionic then -koszulSignInsert q le φ r else koszulSignInsert q le φ r koszulSignInsert𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕n:ℕh:n + 1 < (r1 :: r).lengthhr:q ((r1 :: r).get ⟨n + 1, h⟩) = bosonic⊢ (if le φ r1 then koszulSignInsert q le φ (r.eraseIdx n)
  else
    if q φ = fermionic ∧ q r1 = fermionic then -koszulSignInsert q le φ (r.eraseIdx n)
    else koszulSignInsert q le φ (r.eraseIdx n)) =
  if le φ r1 then koszulSignInsert q le φ r
  else if q φ = fermionic ∧ q r1 = fermionic then -koszulSignInsert q le φ r else koszulSignInsert q le φ r𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕n:ℕh:n + 1 < (r1 :: r).lengthhr:q ((r1 :: r).get ⟨n + 1, h⟩) = bosonic⊢ (if le φ r1 then koszulSignInsert q le φ (r.eraseIdx n)
  else
    if q φ = fermionic ∧ q r1 = fermionic then -koszulSignInsert q le φ (r.eraseIdx n)
    else koszulSignInsert q le φ (r.eraseIdx n)) =
  if le φ r1 then koszulSignInsert q le φ r
  else if q φ = fermionic ∧ q r1 = fermionic then -koszulSignInsert q le φ r else koszulSignInsert q le φ r]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕n:ℕh:n + 1 < (r1 :: r).lengthhr:q ((r1 :: r).get ⟨n + 1, h⟩) = bosonic⊢ (if le φ r1 then koszulSignInsert q le φ (r.eraseIdx n)
  else
    if q φ = fermionic ∧ q r1 = fermionic then -koszulSignInsert q le φ (r.eraseIdx n)
    else koszulSignInsert q le φ (r.eraseIdx n)) =
  if le φ r1 then koszulSignInsert q le φ r
  else if q φ = fermionic ∧ q r1 = fermionic then -koszulSignInsert q le φ r else koszulSignInsert q le φ r
    rw [koszulSignInsert_erase_boson q le φ r ⟨n, Nat.succ_lt_succ_iff.mp h⟩ hr𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕r1:𝓕r:List 𝓕n:ℕh:n + 1 < (r1 :: r).lengthhr:q ((r1 :: r).get ⟨n + 1, h⟩) = bosonic⊢ (if le φ r1 then koszulSignInsert q le φ r
  else if q φ = fermionic ∧ q r1 = fermionic then -koszulSignInsert q le φ r else koszulSignInsert q le φ r) =
  if le φ r1 then koszulSignInsert q le φ r
  else if q φ = fermionic ∧ q r1 = fermionic then -koszulSignInsert q le φ r else koszulSignInsert q le φ rAll goals completed! 🐙]All goals completed! 🐙

lemma koszulSign_erase_boson {𝓕 : Type} (q : 𝓕 → FieldStatistic) (le : 𝓕 → 𝓕 → Prop)
    [DecidableRel le] :
    (φs : List 𝓕) → (n : Fin φs.length) → (heq : q (φs.get n) = bosonic) →
    koszulSign q le (φs.eraseIdx n) = koszulSign q le φs
  | [], _ =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel lex✝:Fin [].length⊢ q ([].get x✝) = bosonic → koszulSign q le ([].eraseIdx ↑x✝) = koszulSign q le [] by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel lex✝:Fin [].length⊢ q ([].get x✝) = bosonic → koszulSign q le ([].eraseIdx ↑x✝) = koszulSign q le []
    simpAll goals completed! 🐙
  | φ :: φs, ⟨0, h⟩ =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕h:0 < (φ :: φs).length⊢ q ((φ :: φs).get ⟨0, h⟩) = bosonic → koszulSign q le ((φ :: φs).eraseIdx ↑⟨0, h⟩) = koszulSign q le (φ :: φs) by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕h:0 < (φ :: φs).length⊢ q ((φ :: φs).get ⟨0, h⟩) = bosonic → koszulSign q le ((φ :: φs).eraseIdx ↑⟨0, h⟩) = koszulSign q le (φ :: φs)
    simp only [List.length_cons, Fin.zero_eta, List.get_eq_getElem, Fin.val_zero,
      List.getElem_cons_zero, List.eraseIdx_zero, List.tail_cons, koszulSign]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕h:0 < (φ :: φs).length⊢ q φ = bosonic → koszulSign q le φs = koszulSignInsert q le φ φs * koszulSign q le φs
    intro h𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕h✝:0 < (φ :: φs).lengthh:q φ = bosonic⊢ koszulSign q le φs = koszulSignInsert q le φ φs * koszulSign q le φs
    rw [koszulSignInsert_boson _ _ _ h𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕h✝:0 < (φ :: φs).lengthh:q φ = bosonic⊢ koszulSign q le φs = 1 * koszulSign q le φs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕h✝:0 < (φ :: φs).lengthh:q φ = bosonic⊢ koszulSign q le φs = 1 * koszulSign q le φs]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕h✝:0 < (φ :: φs).lengthh:q φ = bosonic⊢ koszulSign q le φs = 1 * koszulSign q le φs
    simp only [one_mul]All goals completed! 🐙
  | φ :: φs, ⟨n + 1, h⟩ =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕n:ℕh:n + 1 < (φ :: φs).length⊢ q ((φ :: φs).get ⟨n + 1, h⟩) = bosonic → koszulSign q le ((φ :: φs).eraseIdx ↑⟨n + 1, h⟩) = koszulSign q le (φ :: φs) by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕n:ℕh:n + 1 < (φ :: φs).length⊢ q ((φ :: φs).get ⟨n + 1, h⟩) = bosonic → koszulSign q le ((φ :: φs).eraseIdx ↑⟨n + 1, h⟩) = koszulSign q le (φ :: φs)
    simp only [List.length_cons, List.get_eq_getElem, List.getElem_cons_succ,
      List.eraseIdx_cons_succ]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕n:ℕh:n + 1 < (φ :: φs).length⊢ q φs[n] = bosonic → koszulSign q le (φ :: φs.eraseIdx n) = koszulSign q le (φ :: φs)
    intro h'𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕n:ℕh:n + 1 < (φ :: φs).lengthh':q φs[n] = bosonic⊢ koszulSign q le (φ :: φs.eraseIdx n) = koszulSign q le (φ :: φs)
    rw [koszulSign,𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕n:ℕh:n + 1 < (φ :: φs).lengthh':q φs[n] = bosonic⊢ koszulSignInsert q le φ (φs.eraseIdx n) * koszulSign q le (φs.eraseIdx n) = koszulSign q le (φ :: φs)𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕n:ℕh:n + 1 < (φ :: φs).lengthh':q φs[n] = bosonic⊢ koszulSignInsert q le φ (φs.eraseIdx n) * koszulSign q le φs = koszulSignInsert q le φ φs * koszulSign q le φs koszulSign,𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕n:ℕh:n + 1 < (φ :: φs).lengthh':q φs[n] = bosonic⊢ koszulSignInsert q le φ (φs.eraseIdx n) * koszulSign q le (φs.eraseIdx n) =
  koszulSignInsert q le φ φs * koszulSign q le φs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕n:ℕh:n + 1 < (φ :: φs).lengthh':q φs[n] = bosonic⊢ koszulSignInsert q le φ (φs.eraseIdx n) * koszulSign q le φs = koszulSignInsert q le φ φs * koszulSign q le φs koszulSign_erase_boson q le φs ⟨n, Nat.succ_lt_succ_iff.mp h⟩ h'𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕n:ℕh:n + 1 < (φ :: φs).lengthh':q φs[n] = bosonic⊢ koszulSignInsert q le φ (φs.eraseIdx n) * koszulSign q le φs = koszulSignInsert q le φ φs * koszulSign q le φs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕n:ℕh:n + 1 < (φ :: φs).lengthh':q φs[n] = bosonic⊢ koszulSignInsert q le φ (φs.eraseIdx n) * koszulSign q le φs = koszulSignInsert q le φ φs * koszulSign q le φs]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕n:ℕh:n + 1 < (φ :: φs).lengthh':q φs[n] = bosonic⊢ koszulSignInsert q le φ (φs.eraseIdx n) * koszulSign q le φs = koszulSignInsert q le φ φs * koszulSign q le φs
    congr 1e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕n:ℕh:n + 1 < (φ :: φs).lengthh':q φs[n] = bosonic⊢ koszulSignInsert q le φ (φs.eraseIdx n) = koszulSignInsert q le φ φs
    rw [koszulSignInsert_erase_boson q le φ φs ⟨n, Nat.succ_lt_succ_iff.mp h⟩ h'e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕n:ℕh:n + 1 < (φ :: φs).lengthh':q φs[n] = bosonic⊢ koszulSignInsert q le φ φs = koszulSignInsert q le φ φsAll goals completed! 🐙]All goals completed! 🐙

set_option backward.isDefEq.respectTransparency false in
lemma koszulSign_insertIdx [Std.Total le] [IsTrans 𝓕 le] (φ : 𝓕) :
    (φs : List 𝓕) → (n : ℕ) → (hn : n ≤ φs.length) →
    koszulSign q le (List.insertIdx φs n φ) = 𝓢(q φ, ofList q (φs.take n)) * koszulSign q le φs *
      𝓢(q φ, ofList q ((List.insertionSort le (List.insertIdx φs n φ)).take
      (insertionSortEquiv le (List.insertIdx φs n φ) ⟨n, by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕n:ℕhn:n ≤ φs.length⊢ n < (φs.insertIdx n φ).length
        rw [List.length_insertIdx𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕n:ℕhn:n ≤ φs.length⊢ n < if n ≤ φs.length then φs.length + 1 else φs.length𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕n:ℕhn:n ≤ φs.length⊢ n < if n ≤ φs.length then φs.length + 1 else φs.length]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕n:ℕhn:n ≤ φs.length⊢ n < if n ≤ φs.length then φs.length + 1 else φs.length
        simp only [hn, ↓reduceIte]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕n:ℕhn:n ≤ φs.length⊢ n < φs.length + 1
        omegaAll goals completed! 🐙⟩)))
  | [], 0, h =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕h:0 ≤ [].length⊢ koszulSign q le ([].insertIdx 0 φ) =
  (exchangeSign (q φ)) (ofList q (List.take 0 [])) * koszulSign q le [] *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ([].insertIdx 0 φ)) ⟨0, ⋯⟩)) (List.insertionSort le ([].insertIdx 0 φ)))) by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕h:0 ≤ [].length⊢ koszulSign q le ([].insertIdx 0 φ) =
  (exchangeSign (q φ)) (ofList q (List.take 0 [])) * koszulSign q le [] *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ([].insertIdx 0 φ)) ⟨0, ⋯⟩)) (List.insertionSort le ([].insertIdx 0 φ))))
    simp [koszulSign]All goals completed! 🐙
  | [], n + 1, h =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕n:ℕh:n + 1 ≤ [].length⊢ koszulSign q le ([].insertIdx (n + 1) φ) =
  (exchangeSign (q φ)) (ofList q (List.take (n + 1) [])) * koszulSign q le [] *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ([].insertIdx (n + 1) φ)) ⟨n + 1, ⋯⟩))
          (List.insertionSort le ([].insertIdx (n + 1) φ)))) by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕n:ℕh:n + 1 ≤ [].length⊢ koszulSign q le ([].insertIdx (n + 1) φ) =
  (exchangeSign (q φ)) (ofList q (List.take (n + 1) [])) * koszulSign q le [] *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ([].insertIdx (n + 1) φ)) ⟨n + 1, ⋯⟩))
          (List.insertionSort le ([].insertIdx (n + 1) φ))))
    simp at hAll goals completed! 🐙
  | φ1 :: φs, 0, h =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length⊢ koszulSign q le ((φ1 :: φs).insertIdx 0 φ) =
  (exchangeSign (q φ)) (ofList q (List.take 0 (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φ1 :: φs).insertIdx 0 φ)) ⟨0, ⋯⟩))
          (List.insertionSort le ((φ1 :: φs).insertIdx 0 φ)))) by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length⊢ koszulSign q le ((φ1 :: φs).insertIdx 0 φ) =
  (exchangeSign (q φ)) (ofList q (List.take 0 (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φ1 :: φs).insertIdx 0 φ)) ⟨0, ⋯⟩))
          (List.insertionSort le ((φ1 :: φs).insertIdx 0 φ))))
    simp only [List.insertIdx_zero, List.insertionSort_cons, List.length_cons, Fin.zero_eta]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length⊢ koszulSign q le (φ :: φ1 :: φs) =
  (exchangeSign (q φ)) (ofList q (List.take 0 (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φ :: φ1 :: φs)) 0))
          (List.orderedInsert le φ (List.orderedInsert le φ1 (List.insertionSort le φs)))))
    rw [koszulSign𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length⊢ koszulSignInsert q le φ (φ1 :: φs) * koszulSign q le (φ1 :: φs) =
  (exchangeSign (q φ)) (ofList q (List.take 0 (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φ :: φ1 :: φs)) 0))
          (List.orderedInsert le φ (List.orderedInsert le φ1 (List.insertionSort le φs)))))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length⊢ koszulSignInsert q le φ (φ1 :: φs) * koszulSign q le (φ1 :: φs) =
  (exchangeSign (q φ)) (ofList q (List.take 0 (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φ :: φ1 :: φs)) 0))
          (List.orderedInsert le φ (List.orderedInsert le φ1 (List.insertionSort le φs)))))]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length⊢ koszulSignInsert q le φ (φ1 :: φs) * koszulSign q le (φ1 :: φs) =
  (exchangeSign (q φ)) (ofList q (List.take 0 (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φ :: φ1 :: φs)) 0))
          (List.orderedInsert le φ (List.orderedInsert le φ1 (List.insertionSort le φs)))))
    trans koszulSign q le (φ1 :: φs) * koszulSignInsert q le φ (φ1 :: φs)𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length⊢ koszulSignInsert q le φ (φ1 :: φs) * koszulSign q le (φ1 :: φs) =
  koszulSign q le (φ1 :: φs) * koszulSignInsert q le φ (φ1 :: φs)𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length⊢ koszulSign q le (φ1 :: φs) * koszulSignInsert q le φ (φ1 :: φs) =
  (exchangeSign (q φ)) (ofList q (List.take 0 (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φ :: φ1 :: φs)) 0))
          (List.orderedInsert le φ (List.orderedInsert le φ1 (List.insertionSort le φs)))))
    ·𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length⊢ koszulSignInsert q le φ (φ1 :: φs) * koszulSign q le (φ1 :: φs) =
  koszulSign q le (φ1 :: φs) * koszulSignInsert q le φ (φ1 :: φs) ringAll goals completed! 🐙
    simp only [insertionSortEquiv, List.length_cons, Nat.succ_eq_add_one, orderedInsertEquiv,
      Physlib.Fin.equivCons_trans, Equiv.trans_apply, Physlib.Fin.equivCons_zero]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length⊢ koszulSign q le (φ1 :: φs) * koszulSignInsert q le φ (φ1 :: φs) =
  (exchangeSign (q φ)) (ofList q (List.take 0 (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take
          (↑(((Physlib.Fin.finExtractOne 0).trans
                ((Physlib.Fin.finExtractOne ⟨↑(orderedInsertPos le (List.insertionSort le (φ1 :: φs)) φ), ⋯⟩).symm.trans
                  (Fin.castOrderIso ⋯).symm))
              0))
          (List.orderedInsert le φ (List.orderedInsert le φ1 (List.insertionSort le φs)))))
    conv_rhs =>
      enter [2,2, 2, 2]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length| List.orderedInsert le φ (List.orderedInsert le φ1 (List.insertionSort le φs))
      rw [orderedInsert_eq_insertIdx_orderedInsertPos]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length| (List.orderedInsert le φ1 (List.insertionSort le φs)).insertIdx
  (↑(orderedInsertPos le (List.orderedInsert le φ1 (List.insertionSort le φs)) φ)) φ
    conv_rhs =>
      rhs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length| (exchangeSign (q φ))
  (ofList q
    (List.take
      (↑(((Physlib.Fin.finExtractOne 0).trans
            ((Physlib.Fin.finExtractOne ⟨↑(orderedInsertPos le (List.insertionSort le (φ1 :: φs)) φ), ⋯⟩).symm.trans
              (Fin.castOrderIso ⋯).symm))
          0))
      ((List.orderedInsert le φ1 (List.insertionSort le φs)).insertIdx
        (↑(orderedInsertPos le (List.orderedInsert le φ1 (List.insertionSort le φs)) φ)) φ)))
      erw [← ofList_take_insert]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length| (exchangeSign (q φ))
  (ofList q
    (List.take
      (↑(((Physlib.Fin.finExtractOne 0).trans
            ((Physlib.Fin.finExtractOne ⟨↑(orderedInsertPos le (List.insertionSort le (φ1 :: φs)) φ), ⋯⟩).symm.trans
              (Fin.castOrderIso ⋯).symm))
          0))
      (List.orderedInsert le φ1 (List.insertionSort le φs))))
      change 𝓢(q φ, ofList q ((List.insertionSort le (φ1 :: φs)).take
        (↑(orderedInsertPos le ((List.insertionSort le (φ1 :: φs))) φ))))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length| (exchangeSign (q φ))
  (ofList q
    (List.take (↑(orderedInsertPos le (List.insertionSort le (φ1 :: φs)) φ)) (List.insertionSort le (φ1 :: φs))))
      rw [← koszulSignInsert_eq_exchangeSign_take q le]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length| koszulSignInsert q le φ (φ1 :: φs)
    rw [ofList_take_zero𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length⊢ koszulSign q le (φ1 :: φs) * koszulSignInsert q le φ (φ1 :: φs) =
  (exchangeSign (q φ)) 1 * koszulSign q le (φ1 :: φs) * koszulSignInsert q le φ (φ1 :: φs)𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length⊢ koszulSign q le (φ1 :: φs) * koszulSignInsert q le φ (φ1 :: φs) =
  (exchangeSign (q φ)) 1 * koszulSign q le (φ1 :: φs) * koszulSignInsert q le φ (φ1 :: φs)]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕h:0 ≤ (φ1 :: φs).length⊢ koszulSign q le (φ1 :: φs) * koszulSignInsert q le φ (φ1 :: φs) =
  (exchangeSign (q φ)) 1 * koszulSign q le (φ1 :: φs) * koszulSignInsert q le φ (φ1 :: φs)
    simpAll goals completed! 🐙
  | φ1 :: φs, n + 1, h =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length⊢ koszulSign q le ((φ1 :: φs).insertIdx (n + 1) φ) =
  (exchangeSign (q φ)) (ofList q (List.take (n + 1) (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φ1 :: φs).insertIdx (n + 1) φ)) ⟨n + 1, ⋯⟩))
          (List.insertionSort le ((φ1 :: φs).insertIdx (n + 1) φ)))) by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length⊢ koszulSign q le ((φ1 :: φs).insertIdx (n + 1) φ) =
  (exchangeSign (q φ)) (ofList q (List.take (n + 1) (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φ1 :: φs).insertIdx (n + 1) φ)) ⟨n + 1, ⋯⟩))
          (List.insertionSort le ((φ1 :: φs).insertIdx (n + 1) φ))))
    conv_lhs =>
      rw [List.insertIdx_succ_cons]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length| koszulSign q le (φ1 :: φs.insertIdx n φ)
      rw [koszulSign]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length| koszulSignInsert q le φ1 (φs.insertIdx n φ) * koszulSign q le (φs.insertIdx n φ)
    rw [koszulSign_insertIdx _ _ _ (Nat.le_of_lt_succ h)𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length⊢ koszulSignInsert q le φ1 (φs.insertIdx n φ) *
    ((exchangeSign (q φ)) (ofList q (List.take n φs)) * koszulSign q le φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩))
            (List.insertionSort le (φs.insertIdx n φ))))) =
  (exchangeSign (q φ)) (ofList q (List.take (n + 1) (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φ1 :: φs).insertIdx (n + 1) φ)) ⟨n + 1, ⋯⟩))
          (List.insertionSort le ((φ1 :: φs).insertIdx (n + 1) φ))))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length⊢ koszulSignInsert q le φ1 (φs.insertIdx n φ) *
    ((exchangeSign (q φ)) (ofList q (List.take n φs)) * koszulSign q le φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩))
            (List.insertionSort le (φs.insertIdx n φ))))) =
  (exchangeSign (q φ)) (ofList q (List.take (n + 1) (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φ1 :: φs).insertIdx (n + 1) φ)) ⟨n + 1, ⋯⟩))
          (List.insertionSort le ((φ1 :: φs).insertIdx (n + 1) φ))))]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length⊢ koszulSignInsert q le φ1 (φs.insertIdx n φ) *
    ((exchangeSign (q φ)) (ofList q (List.take n φs)) * koszulSign q le φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩))
            (List.insertionSort le (φs.insertIdx n φ))))) =
  (exchangeSign (q φ)) (ofList q (List.take (n + 1) (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φ1 :: φs).insertIdx (n + 1) φ)) ⟨n + 1, ⋯⟩))
          (List.insertionSort le ((φ1 :: φs).insertIdx (n + 1) φ))))
    conv_rhs =>
      rhs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length| (exchangeSign (q φ))
  (ofList q
    (List.take (↑((insertionSortEquiv le ((φ1 :: φs).insertIdx (n + 1) φ)) ⟨n + 1, ⋯⟩))
      (List.insertionSort le ((φ1 :: φs).insertIdx (n + 1) φ))))
      simp only [List.insertIdx_succ_cons]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length| (exchangeSign (q φ))
  (ofList q
    (List.take (↑((insertionSortEquiv le (φ1 :: φs.insertIdx n φ)) ⟨n + 1, ⋯⟩))
      (List.insertionSort le (φ1 :: φs.insertIdx n φ))))
      simp only [List.insertionSort_cons, List.length_cons, insertionSortEquiv, Nat.succ_eq_add_one,
        Equiv.trans_apply, Physlib.Fin.equivCons_succ]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length| (exchangeSign (q φ))
  (ofList q
    (List.take
      (↑((orderedInsertEquiv le (List.insertionSort le (φs.insertIdx n φ)) φ1)
          ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩).succ))
      (List.orderedInsert le φ1 (List.insertionSort le (φs.insertIdx n φ)))))
      erw [orderedInsertEquiv_fin_succ]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length| (exchangeSign (q φ))
  (ofList q
    (List.take
      (↑(Fin.cast ⋯
          (⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
            ⟨↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩), ⋯⟩)))
      (List.orderedInsert le φ1 (List.insertionSort le (φs.insertIdx n φ)))))
      simp only [Fin.eta, Fin.val_cast]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length| (exchangeSign (q φ))
  (ofList q
    (List.take
      (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
          ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
      (List.orderedInsert le φ1 (List.insertionSort le (φs.insertIdx n φ)))))
      rhs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length| ofList q
  (List.take
    (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
        ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
    (List.orderedInsert le φ1 (List.insertionSort le (φs.insertIdx n φ))))
      simp [orderedInsert_eq_insertIdx_orderedInsertPos]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length| ofList q
  (List.take
    (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
        ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
    ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
      (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))
    conv_rhs =>
      lhs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length| (exchangeSign (q φ)) (ofList q (List.take (n + 1) (φ1 :: φs))) * koszulSign q le (φ1 :: φs)
      rw [ofList_take_succ_cons, map_mul, koszulSign]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length| (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (ofList q (List.take n φs)) *
  (koszulSignInsert q le φ1 φs * koszulSign q le φs)
    ring_nf𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length⊢ koszulSignInsert q le φ1 (φs.insertIdx n φ) * (exchangeSign (q φ)) (ofList q (List.take n φs)) * koszulSign q le φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (ofList q (List.take n φs)) * koszulSign q le φs * (exchangeSign (q φ)) (q φ1) *
      koszulSignInsert q le φ1 φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take
          (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
              ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
          ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
            (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1)))
    conv_lhs =>
      lhs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length| koszulSignInsert q le φ1 (φs.insertIdx n φ) * (exchangeSign (q φ)) (ofList q (List.take n φs)) * koszulSign q le φs
      rw [mul_assoc, mul_comm]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length| (exchangeSign (q φ)) (ofList q (List.take n φs)) * koszulSign q le φs * koszulSignInsert q le φ1 (φs.insertIdx n φ)
    rw [mul_assoc𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length⊢ (exchangeSign (q φ)) (ofList q (List.take n φs)) * koszulSign q le φs *
    (koszulSignInsert q le φ1 (φs.insertIdx n φ) *
      (exchangeSign (q φ))
        (ofList q
          (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩))
            (List.insertionSort le (φs.insertIdx n φ))))) =
  (exchangeSign (q φ)) (ofList q (List.take n φs)) * koszulSign q le φs * (exchangeSign (q φ)) (q φ1) *
      koszulSignInsert q le φ1 φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take
          (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
              ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
          ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
            (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1)))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length⊢ (exchangeSign (q φ)) (ofList q (List.take n φs)) * koszulSign q le φs *
    (koszulSignInsert q le φ1 (φs.insertIdx n φ) *
      (exchangeSign (q φ))
        (ofList q
          (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩))
            (List.insertionSort le (φs.insertIdx n φ))))) =
  (exchangeSign (q φ)) (ofList q (List.take n φs)) * koszulSign q le φs * (exchangeSign (q φ)) (q φ1) *
      koszulSignInsert q le φ1 φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take
          (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
              ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
          ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
            (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1)))]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length⊢ (exchangeSign (q φ)) (ofList q (List.take n φs)) * koszulSign q le φs *
    (koszulSignInsert q le φ1 (φs.insertIdx n φ) *
      (exchangeSign (q φ))
        (ofList q
          (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩))
            (List.insertionSort le (φs.insertIdx n φ))))) =
  (exchangeSign (q φ)) (ofList q (List.take n φs)) * koszulSign q le φs * (exchangeSign (q φ)) (q φ1) *
      koszulSignInsert q le φ1 φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take
          (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
              ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
          ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
            (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1)))
    conv_rhs =>
      rw [mul_assoc, mul_assoc]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length| (exchangeSign (q φ)) (ofList q (List.take n φs)) * koszulSign q le φs *
  ((exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1)))))
    congr 1e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length⊢ koszulSignInsert q le φ1 (φs.insertIdx n φ) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))
    let rs := (List.insertionSort le (List.insertIdx φs n φ))e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)⊢ koszulSignInsert q le φ1 (φs.insertIdx n φ) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))
    have hnsL : n < (List.insertIdx φs n φ).length := by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length⊢ koszulSign q le ((φ1 :: φs).insertIdx (n + 1) φ) =
  (exchangeSign (q φ)) (ofList q (List.take (n + 1) (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φ1 :: φs).insertIdx (n + 1) φ)) ⟨n + 1, ⋯⟩))
          (List.insertionSort le ((φ1 :: φs).insertIdx (n + 1) φ))))e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).length⊢ koszulSignInsert q le φ1 (φs.insertIdx n φ) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))
      rw [List.length_insertIdx𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)⊢ n < if n ≤ φs.length then φs.length + 1 else φs.length𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)⊢ n < if n ≤ φs.length then φs.length + 1 else φs.lengthe_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).length⊢ koszulSignInsert q le φ1 (φs.insertIdx n φ) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)⊢ n < if n ≤ φs.length then φs.length + 1 else φs.lengthe_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).length⊢ koszulSignInsert q le φ1 (φs.insertIdx n φ) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))
      simp only [List.length_cons, add_le_add_iff_right] at h𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)h:n ≤ φs.length⊢ n < if n ≤ φs.length then φs.length + 1 else φs.lengthe_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).length⊢ koszulSignInsert q le φ1 (φs.insertIdx n φ) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))
      simp only [h, ↓reduceIte]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)h:n ≤ φs.length⊢ n < φs.length + 1e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).length⊢ koszulSignInsert q le φ1 (φs.insertIdx n φ) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))
      omegae_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).length⊢ koszulSignInsert q le φ1 (φs.insertIdx n φ) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).length⊢ koszulSignInsert q le φ1 (φs.insertIdx n φ) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))
    let ni : Fin rs.length := (insertionSortEquiv le (List.insertIdx φs n φ))
      ⟨n, hnsL⟩e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩⊢ koszulSignInsert q le φ1 (φs.insertIdx n φ) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))
    let nro : Fin (rs.length + 1) :=
      ⟨↑(orderedInsertPos le rs φ1), orderedInsertPos_lt_length le rs φ1⟩e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignInsert q le φ1 (φs.insertIdx n φ) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))
    rw [koszulSignInsert_insertIdx _ _ _ _ _ _ (Nat.le_of_lt_succ h),e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignInsert q le φ1 (φ :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignCons q le φ1 φ * koszulSignInsert q le φ1 φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1)))) koszulSignInsert_conse_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignCons q le φ1 φ * koszulSignInsert q le φ1 φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignCons q le φ1 φ * koszulSignInsert q le φ1 φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))]e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignCons q le φ1 φ * koszulSignInsert q le φ1 φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))
    trans koszulSignInsert q le φ1 φs * (koszulSignCons q le φ1 φ *
      𝓢(q φ, ofList q (rs.take ni)))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignCons q le φ1 φ * koszulSignInsert q le φ1 φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  koszulSignInsert q le φ1 φs * (koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignInsert q le φ1 φs * (koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))
    ·𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignCons q le φ1 φ * koszulSignInsert q le φ1 φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  koszulSignInsert q le φ1 φs * (koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs))) simp only [rs, ni]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignCons q le φ1 φ * koszulSignInsert q le φ1 φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)) (List.insertionSort le (φs.insertIdx n φ)))) =
  koszulSignInsert q le φ1 φs *
    (koszulSignCons q le φ1 φ *
      (exchangeSign (q φ))
        (ofList q
          (List.take (↑((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩))
            (List.insertionSort le (φs.insertIdx n φ)))))
      ringAll goals completed! 🐙
    trans koszulSignInsert q le φ1 φs * (𝓢(q φ, q φ1) *
          𝓢(q φ, ofList q ((List.insertIdx rs nro φ1).take (nro.succAbove ni))))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignInsert q le φ1 φs * (koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs))) =
  koszulSignInsert q le φ1 φs *
    ((exchangeSign (q φ)) (q φ1) *
      (exchangeSign (q φ)) (ofList q (List.take (↑(nro.succAbove ni)) (rs.insertIdx (↑nro) φ1))))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignInsert q le φ1 φs *
    ((exchangeSign (q φ)) (q φ1) *
      (exchangeSign (q φ)) (ofList q (List.take (↑(nro.succAbove ni)) (rs.insertIdx (↑nro) φ1)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))
    swap𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignInsert q le φ1 φs *
    ((exchangeSign (q φ)) (q φ1) *
      (exchangeSign (q φ)) (ofList q (List.take (↑(nro.succAbove ni)) (rs.insertIdx (↑nro) φ1)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignInsert q le φ1 φs * (koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs))) =
  koszulSignInsert q le φ1 φs *
    ((exchangeSign (q φ)) (q φ1) *
      (exchangeSign (q φ)) (ofList q (List.take (↑(nro.succAbove ni)) (rs.insertIdx (↑nro) φ1))))
    ·𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignInsert q le φ1 φs *
    ((exchangeSign (q φ)) (q φ1) *
      (exchangeSign (q φ)) (ofList q (List.take (↑(nro.succAbove ni)) (rs.insertIdx (↑nro) φ1)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1)))) simp only [rs, nro, ni]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignInsert q le φ1 φs *
    ((exchangeSign (q φ)) (q φ1) *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1)))) =
  (exchangeSign (q φ)) (q φ1) *
    (koszulSignInsert q le φ1 φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑(⟨↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1), ⋯⟩.succAbove
                ((insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, ⋯⟩)))
            ((List.insertionSort le (φs.insertIdx n φ)).insertIdx
              (↑(orderedInsertPos le (List.insertionSort le (φs.insertIdx n φ)) φ1)) φ1))))
      ringAll goals completed! 🐙
    congr 1e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ)) (ofList q (List.take (↑(nro.succAbove ni)) (rs.insertIdx (↑nro) φ1)))
    simp only [Fin.succAbove]e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))
    have hns : rs.get ni = φ := by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length⊢ koszulSign q le ((φ1 :: φs).insertIdx (n + 1) φ) =
  (exchangeSign (q φ)) (ofList q (List.take (n + 1) (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φ1 :: φs).insertIdx (n + 1) φ)) ⟨n + 1, ⋯⟩))
          (List.insertionSort le ((φ1 :: φs).insertIdx (n + 1) φ))))e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))
      simp only [rs]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ (List.insertionSort le (φs.insertIdx n φ)).get ni = φe_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))
      rw [← insertionSortEquiv_get𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ ((φs.insertIdx n φ).get ∘ ⇑(insertionSortEquiv le (φs.insertIdx n φ)).symm) ni = φ𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ ((φs.insertIdx n φ).get ∘ ⇑(insertionSortEquiv le (φs.insertIdx n φ)).symm) ni = φe_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ ((φs.insertIdx n φ).get ∘ ⇑(insertionSortEquiv le (φs.insertIdx n φ)).symm) ni = φe_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))
      simp only [Function.comp_apply, Equiv.symm_apply_apply, List.get_eq_getElem, ni]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩⊢ (φs.insertIdx n φ)[n] = φe_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))
      simp_all only [List.length_cons, add_le_add_iff_right, List.getElem_insertIdx_self]e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))
    have hc1 (hninro : ni.castSucc < nro) : ¬ le φ1 φ := by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length⊢ koszulSign q le ((φ1 :: φs).insertIdx (n + 1) φ) =
  (exchangeSign (q φ)) (ofList q (List.take (n + 1) (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φ1 :: φs).insertIdx (n + 1) φ)) ⟨n + 1, ⋯⟩))
          (List.insertionSort le ((φ1 :: φs).insertIdx (n + 1) φ))))e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))
      rw [← hns𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhninro:ni.castSucc < nro⊢ ¬le φ1 (rs.get ni)𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhninro:ni.castSucc < nro⊢ ¬le φ1 (rs.get ni)e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhninro:ni.castSucc < nro⊢ ¬le φ1 (rs.get ni)e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))
      exact lt_orderedInsertPos_rel le φ1 rs ni hninroe_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))
    have hc2 (hninro : ¬ ni.castSucc < nro) : le φ1 φ := by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).length⊢ koszulSign q le ((φ1 :: φs).insertIdx (n + 1) φ) =
  (exchangeSign (q φ)) (ofList q (List.take (n + 1) (φ1 :: φs))) * koszulSign q le (φ1 :: φs) *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φ1 :: φs).insertIdx (n + 1) φ)) ⟨n + 1, ⋯⟩))
          (List.insertionSort le ((φ1 :: φs).insertIdx (n + 1) φ))))e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))
      rw [← hns𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhninro:¬ni.castSucc < nro⊢ le φ1 (rs.get ni)𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhninro:¬ni.castSucc < nro⊢ le φ1 (rs.get ni)e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhninro:¬ni.castSucc < nro⊢ le φ1 (rs.get ni)e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))
      refine gt_orderedInsertPos_rel le φ1 rs ?_ ni hninro𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhninro:¬ni.castSucc < nro⊢ List.Pairwise le rse_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))
      exact List.pairwise_insertionSort le (List.insertIdx φs n φ)e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φ⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))
    by_cases hn : ni.castSucc < nropos𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))neg𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1)))
    ·pos𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1))) simp only [hn, ↓reduceIte, Fin.val_castSucc]pos𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (ofList q (List.take (↑ni) (rs.insertIdx (↑nro) φ1)))
      rw [ofList_take_insertIdx_gtpos𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs))pos.hn𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ ↑ni < ↑nropos𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs))pos.hn𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ ↑ni < ↑nro]pos𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs))pos.hn𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ ↑ni < ↑nro
      swappos.hn𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ ↑ni < ↑nropos𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs))
      ·pos.hn𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ ↑ni < ↑nro exact hnAll goals completed! 🐙
      congr 1pos.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ koszulSignCons q le φ1 φ = (exchangeSign (q φ)) (q φ1)
      rw [koszulSignCons_eq_exchangeSignpos.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ (if le φ1 φ then 1 else (exchangeSign (q φ1)) (q φ)) = (exchangeSign (q φ)) (q φ1)pos.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ (if le φ1 φ then 1 else (exchangeSign (q φ1)) (q φ)) = (exchangeSign (q φ)) (q φ1)]pos.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ (if le φ1 φ then 1 else (exchangeSign (q φ1)) (q φ)) = (exchangeSign (q φ)) (q φ1)
      simp only [hc1 hn, ↓reduceIte]pos.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ (exchangeSign (q φ1)) (q φ) = (exchangeSign (q φ)) (q φ1)
      rw [exchangeSign_symmpos.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:ni.castSucc < nro⊢ (exchangeSign (q φ)) (q φ1) = (exchangeSign (q φ)) (q φ1)All goals completed! 🐙]All goals completed! 🐙
    ·neg𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) *
    (exchangeSign (q φ))
      (ofList q (List.take (↑(if ni.castSucc < nro then ni.castSucc else ni.succ)) (rs.insertIdx (↑nro) φ1))) simp only [hn, ↓reduceIte, Fin.val_succ]neg𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (ofList q (List.take (↑ni + 1) (rs.insertIdx (↑nro) φ1)))
      rw [ofList_take_insertIdx_le,neg𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (q φ1 * ofList q (List.take (↑ni) rs))neg.hn𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ ↑nineg.hm𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ rs.lengthneg𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs))neg.hn𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ ↑nineg.hm𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ rs.length map_mul,neg𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) * ((exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)))neg.hn𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ ↑nineg.hm𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ rs.lengthneg𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs))neg.hn𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ ↑nineg.hm𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ rs.length ← mul_assocneg𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs))neg.hn𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ ↑nineg.hm𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ rs.lengthneg𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs))neg.hn𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ ↑nineg.hm𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ rs.length]neg𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs))neg.hn𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ ↑nineg.hm𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ rs.length
      ·neg𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ koszulSignCons q le φ1 φ * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) =
  (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (ofList q (List.take (↑ni) rs)) congr 1neg.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ koszulSignCons q le φ1 φ = (exchangeSign (q φ)) (q φ1) * (exchangeSign (q φ)) (q φ1)
        rw [exchangeSign_mul_self,neg.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ koszulSignCons q le φ1 φ = 1neg.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ (if le φ1 φ then 1 else if q φ1 = fermionic ∧ q φ = fermionic then -1 else 1) = 1 koszulSignConsneg.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ (if le φ1 φ then 1 else if q φ1 = fermionic ∧ q φ = fermionic then -1 else 1) = 1neg.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ (if le φ1 φ then 1 else if q φ1 = fermionic ∧ q φ = fermionic then -1 else 1) = 1]neg.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ (if le φ1 φ then 1 else if q φ1 = fermionic ∧ q φ = fermionic then -1 else 1) = 1
        simp only [hc2 hn, ↓reduceIte]All goals completed! 🐙
      ·neg.hn𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ ↑ni exact Nat.le_of_not_lt hnAll goals completed! 🐙
      ·neg.hm𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs:List 𝓕n:ℕh:n + 1 ≤ (φ1 :: φs).lengthrs:List 𝓕 := List.insertionSort le (φs.insertIdx n φ)hnsL:n < (φs.insertIdx n φ).lengthni:Fin rs.length := (insertionSortEquiv le (φs.insertIdx n φ)) ⟨n, hnsL⟩nro:Fin (rs.length + 1) := ⟨↑(orderedInsertPos le rs φ1), ⋯⟩hns:rs.get ni = φhc1:ni.castSucc < nro → ¬le φ1 φhc2:¬ni.castSucc < nro → le φ1 φhn:¬ni.castSucc < nro⊢ ↑nro ≤ rs.length exact Nat.le_of_lt_succ (orderedInsertPos_lt_length le rs φ1)All goals completed! 🐙lemma insertIdx_eraseIdx {I : Type} : (n : ℕ) → (r : List I) → (hn : n < r.length) →
    List.insertIdx (r.eraseIdx n) n (r.get ⟨n, hn⟩) = r
  | n, [], hn =>I:Typen:ℕhn:n < [].length⊢ ([].eraseIdx n).insertIdx n ([].get ⟨n, hn⟩) = [] byI:Typen:ℕhn:n < [].length⊢ ([].eraseIdx n).insertIdx n ([].get ⟨n, hn⟩) = []
    simp at hnAll goals completed! 🐙
  | 0, r0 :: r, hn =>I:Typer0:Ir:List Ihn:0 < (r0 :: r).length⊢ ((r0 :: r).eraseIdx 0).insertIdx 0 ((r0 :: r).get ⟨0, hn⟩) = r0 :: r byI:Typer0:Ir:List Ihn:0 < (r0 :: r).length⊢ ((r0 :: r).eraseIdx 0).insertIdx 0 ((r0 :: r).get ⟨0, hn⟩) = r0 :: r
    simpAll goals completed! 🐙
  | n + 1, r0 :: r, hn =>I:Typen:ℕr0:Ir:List Ihn:n + 1 < (r0 :: r).length⊢ ((r0 :: r).eraseIdx (n + 1)).insertIdx (n + 1) ((r0 :: r).get ⟨n + 1, hn⟩) = r0 :: r byI:Typen:ℕr0:Ir:List Ihn:n + 1 < (r0 :: r).length⊢ ((r0 :: r).eraseIdx (n + 1)).insertIdx (n + 1) ((r0 :: r).get ⟨n + 1, hn⟩) = r0 :: r
    simp only [List.length_cons, List.get_eq_getElem, List.getElem_cons_succ,
      List.eraseIdx_cons_succ, List.insertIdx_succ_cons, List.cons.injEq, true_and]I:Typen:ℕr0:Ir:List Ihn:n + 1 < (r0 :: r).length⊢ (r.eraseIdx n).insertIdx n r[n] = r
    exact insertIdx_eraseIdx n r _All goals completed! 🐙

lemma koszulSign_eraseIdx [Std.Total le] [IsTrans 𝓕 le] (φs : List 𝓕) (n : Fin φs.length) :
    koszulSign q le (φs.eraseIdx n) = koszulSign q le φs * 𝓢(q (φs.get n), ofList q (φs.take n)) *
    𝓢(q (φs.get n), ofList q (List.take (↑(insertionSortEquiv le φs n))
    (List.insertionSort le φs))) := by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.length⊢ koszulSign q le (φs.eraseIdx ↑n) =
  koszulSign q le φs * (exchangeSign (q (φs.get n))) (ofList q (List.take (↑n) φs)) *
    (exchangeSign (q (φs.get n))) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))
  let φs' := φs.eraseIdx ↑n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑n⊢ koszulSign q le (φs.eraseIdx ↑n) =
  koszulSign q le φs * (exchangeSign (q (φs.get n))) (ofList q (List.take (↑n) φs)) *
    (exchangeSign (q (φs.get n))) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))
  have hφs : List.insertIdx φs' n (φs.get n) = φs := by
    exact insertIdx_eraseIdx n.1 φs n.prop𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) =
  koszulSign q le φs * (exchangeSign (q (φs.get n))) (ofList q (List.take (↑n) φs)) *
    (exchangeSign (q (φs.get n))) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) =
  koszulSign q le φs * (exchangeSign (q (φs.get n))) (ofList q (List.take (↑n) φs)) *
    (exchangeSign (q (φs.get n))) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))
  conv_rhs =>
    lhs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs| koszulSign q le φs * (exchangeSign (q (φs.get n))) (ofList q (List.take (↑n) φs))
    lhs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs| koszulSign q le φs
    rw [← hφs]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs| koszulSign q le (φs'.insertIdx (↑n) (φs.get n))
    rw [koszulSign_insertIdx q le (φs.get n) ((φs.eraseIdx ↑n)) n (by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ ↑n ≤ (φs.eraseIdx ↑n).length
      rw [List.length_eraseIdx𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ ↑n ≤ if ↑n < φs.length then φs.length - 1 else φs.length𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ ↑n ≤ if ↑n < φs.length then φs.length - 1 else φs.length]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ ↑n ≤ if ↑n < φs.length then φs.length - 1 else φs.length
      simp only [Fin.is_lt, ↓reduceIte]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ ↑n ≤ φs.length - 1
      omegaAll goals completed! 🐙)]
    rhs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs| (exchangeSign (q (φs.get n)))
  (ofList q
    (List.take (↑((insertionSortEquiv le ((φs.eraseIdx ↑n).insertIdx (↑n) (φs.get n))) ⟨↑n, ⋯⟩))
      (List.insertionSort le ((φs.eraseIdx ↑n).insertIdx (↑n) (φs.get n)))))
    enter [2, 2, 2]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs| List.insertionSort le ((φs.eraseIdx ↑n).insertIdx (↑n) (φs.get n))
    rw [hφs]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs| List.insertionSort le φs
  conv_rhs =>
    enter [1, 1, 2, 2, 2, 1, 1]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs| (insertionSortEquiv le ((φs.eraseIdx ↑n).insertIdx (↑n) (φs.get n))) ⟨↑n, ⋯⟩
    rw [insertionSortEquiv_congr _ _ hφs]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs| ((Fin.castOrderIso ⋯).trans ((insertionSortEquiv le φs).trans (Fin.castOrderIso ⋯).toEquiv)) ⟨↑n, ⋯⟩
  simp only [List.get_eq_getElem]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) =
  (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) * koszulSign q le (φs.eraseIdx ↑n) *
        (exchangeSign (q φs[↑n]))
          (ofList q
            (List.take
              (↑(((Fin.castOrderIso ⋯).trans ((insertionSortEquiv le φs).trans (Fin.castOrderIso ⋯).toEquiv)) ⟨↑n, ⋯⟩))
              (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs)) *
    (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))
  trans koszulSign q le (φs.eraseIdx ↑n) *
    (𝓢(q φs[↑n], ofList q ((φs.eraseIdx ↑n).take n)) * 𝓢(q φs[↑n], ofList q (List.take (↑n) φs))) *
    (𝓢(q φs[↑n], ofList q ((List.insertionSort le φs).take (↑((insertionSortEquiv le φs) n)))) *
    𝓢(q φs[↑n], ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) =
  koszulSign q le (φs.eraseIdx ↑n) *
      ((exchangeSign (q φs[n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) *
        (exchangeSign (q φs[n])) (ofList q (List.take (↑n) φs))) *
    ((exchangeSign (q φs[n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) *
      (exchangeSign (q φs[n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) *
      ((exchangeSign (q φs[n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) *
        (exchangeSign (q φs[n])) (ofList q (List.take (↑n) φs))) *
    ((exchangeSign (q φs[n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) *
      (exchangeSign (q φs[n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))) =
  (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) * koszulSign q le (φs.eraseIdx ↑n) *
        (exchangeSign (q φs[↑n]))
          (ofList q
            (List.take
              (↑(((Fin.castOrderIso ⋯).trans ((insertionSortEquiv le φs).trans (Fin.castOrderIso ⋯).toEquiv)) ⟨↑n, ⋯⟩))
              (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs)) *
    (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))
  swap𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) *
      ((exchangeSign (q φs[n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) *
        (exchangeSign (q φs[n])) (ofList q (List.take (↑n) φs))) *
    ((exchangeSign (q φs[n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) *
      (exchangeSign (q φs[n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))) =
  (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) * koszulSign q le (φs.eraseIdx ↑n) *
        (exchangeSign (q φs[↑n]))
          (ofList q
            (List.take
              (↑(((Fin.castOrderIso ⋯).trans ((insertionSortEquiv le φs).trans (Fin.castOrderIso ⋯).toEquiv)) ⟨↑n, ⋯⟩))
              (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs)) *
    (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) =
  koszulSign q le (φs.eraseIdx ↑n) *
      ((exchangeSign (q φs[n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) *
        (exchangeSign (q φs[n])) (ofList q (List.take (↑n) φs))) *
    ((exchangeSign (q φs[n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) *
      (exchangeSign (q φs[n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))))
  ·𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) *
      ((exchangeSign (q φs[n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) *
        (exchangeSign (q φs[n])) (ofList q (List.take (↑n) φs))) *
    ((exchangeSign (q φs[n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) *
      (exchangeSign (q φs[n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))) =
  (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) * koszulSign q le (φs.eraseIdx ↑n) *
        (exchangeSign (q φs[↑n]))
          (ofList q
            (List.take
              (↑(((Fin.castOrderIso ⋯).trans ((insertionSortEquiv le φs).trans (Fin.castOrderIso ⋯).toEquiv)) ⟨↑n, ⋯⟩))
              (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs)) *
    (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) simp only [Fin.getElem_fin]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) *
      ((exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) *
        (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs))) *
    ((exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))) =
  (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) * koszulSign q le (φs.eraseIdx ↑n) *
        (exchangeSign (q φs[↑n]))
          (ofList q
            (List.take
              (↑(((Fin.castOrderIso ⋯).trans ((insertionSortEquiv le φs).trans (Fin.castOrderIso ⋯).toEquiv)) ⟨↑n, ⋯⟩))
              (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs)) *
    (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))
    rw [Equiv.trans_apply,𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) *
      ((exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) *
        (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs))) *
    ((exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))) =
  (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) * koszulSign q le (φs.eraseIdx ↑n) *
        (exchangeSign (q φs[↑n]))
          (ofList q
            (List.take
              (↑(((insertionSortEquiv le φs).trans (Fin.castOrderIso ⋯).toEquiv)
                  ((Fin.castOrderIso ⋯).toEquiv ⟨↑n, ⋯⟩)))
              (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs)) *
    (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) *
      ((exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) *
        (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs))) *
    ((exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))) =
  (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) * koszulSign q le (φs.eraseIdx ↑n) *
        (exchangeSign (q φs[↑n]))
          (ofList q
            (List.take
              (↑((Fin.castOrderIso ⋯).toEquiv ((insertionSortEquiv le φs) ((Fin.castOrderIso ⋯).toEquiv ⟨↑n, ⋯⟩))))
              (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs)) *
    (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) Equiv.trans_apply𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) *
      ((exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) *
        (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs))) *
    ((exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))) =
  (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) * koszulSign q le (φs.eraseIdx ↑n) *
        (exchangeSign (q φs[↑n]))
          (ofList q
            (List.take
              (↑((Fin.castOrderIso ⋯).toEquiv ((insertionSortEquiv le φs) ((Fin.castOrderIso ⋯).toEquiv ⟨↑n, ⋯⟩))))
              (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs)) *
    (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) *
      ((exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) *
        (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs))) *
    ((exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))) =
  (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) * koszulSign q le (φs.eraseIdx ↑n) *
        (exchangeSign (q φs[↑n]))
          (ofList q
            (List.take
              (↑((Fin.castOrderIso ⋯).toEquiv ((insertionSortEquiv le φs) ((Fin.castOrderIso ⋯).toEquiv ⟨↑n, ⋯⟩))))
              (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs)) *
    (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) *
      ((exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) *
        (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs))) *
    ((exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))) =
  (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) * koszulSign q le (φs.eraseIdx ↑n) *
        (exchangeSign (q φs[↑n]))
          (ofList q
            (List.take
              (↑((Fin.castOrderIso ⋯).toEquiv ((insertionSortEquiv le φs) ((Fin.castOrderIso ⋯).toEquiv ⟨↑n, ⋯⟩))))
              (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs)) *
    (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))
    simp only [Fin.castOrderIso,
      Equiv.coe_fn_mk, Fin.cast_mk, Fin.eta, Fin.val_cast]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) *
      ((exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) *
        (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs))) *
    ((exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))) =
  (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) * koszulSign q le (φs.eraseIdx ↑n) *
        (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs)) *
    (exchangeSign (q φs[↑n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))
    ringAll goals completed! 🐙
  conv_rhs =>
    rhs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs| (exchangeSign (q φs[n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))) *
  (exchangeSign (q φs[n])) (ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs)))
    rw [exchangeSign_mul_self]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs| 1
  simp only [Fin.getElem_fin, mul_one]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs⊢ koszulSign q le (φs.eraseIdx ↑n) =
  koszulSign q le (φs.eraseIdx ↑n) *
    ((exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) *
      (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs)))
  conv_rhs =>
    rhs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs| (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) (φs.eraseIdx ↑n))) *
  (exchangeSign (q φs[↑n])) (ofList q (List.take (↑n) φs))
    rw [ofList_take_eraseIdx, exchangeSign_mul_self]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕n:Fin φs.lengthφs':List 𝓕 := φs.eraseIdx ↑nhφs:φs'.insertIdx (↑n) (φs.get n) = φs| 1
  simpAll goals completed! 🐙

lemma koszulSign_eraseIdx_insertionSortMinPos [Std.Total le] [IsTrans 𝓕 le] (φ : 𝓕) (φs : List 𝓕) :
    koszulSign q le ((φ :: φs).eraseIdx (insertionSortMinPos le φ φs)) = koszulSign q le (φ :: φs)
    * 𝓢(q (insertionSortMin le φ φs), ofList q ((φ :: φs).take (insertionSortMinPos le φ φs))) := by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕⊢ koszulSign q le ((φ :: φs).eraseIdx ↑(insertionSortMinPos le φ φs)) =
  koszulSign q le (φ :: φs) *
    (exchangeSign (q (insertionSortMin le φ φs))) (ofList q (List.take (↑(insertionSortMinPos le φ φs)) (φ :: φs)))
  rw [koszulSign_eraseIdx𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕⊢ koszulSign q le (φ :: φs) *
      (exchangeSign (q ((φ :: φs).get (insertionSortMinPos le φ φs))))
        (ofList q (List.take (↑(insertionSortMinPos le φ φs)) (φ :: φs))) *
    (exchangeSign (q ((φ :: φs).get (insertionSortMinPos le φ φs))))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φ :: φs)) (insertionSortMinPos le φ φs)))
          (List.insertionSort le (φ :: φs)))) =
  koszulSign q le (φ :: φs) *
    (exchangeSign (q (insertionSortMin le φ φs))) (ofList q (List.take (↑(insertionSortMinPos le φ φs)) (φ :: φs)))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕⊢ koszulSign q le (φ :: φs) *
      (exchangeSign (q ((φ :: φs).get (insertionSortMinPos le φ φs))))
        (ofList q (List.take (↑(insertionSortMinPos le φ φs)) (φ :: φs))) *
    (exchangeSign (q ((φ :: φs).get (insertionSortMinPos le φ φs))))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φ :: φs)) (insertionSortMinPos le φ φs)))
          (List.insertionSort le (φ :: φs)))) =
  koszulSign q le (φ :: φs) *
    (exchangeSign (q (insertionSortMin le φ φs))) (ofList q (List.take (↑(insertionSortMinPos le φ φs)) (φ :: φs)))]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕⊢ koszulSign q le (φ :: φs) *
      (exchangeSign (q ((φ :: φs).get (insertionSortMinPos le φ φs))))
        (ofList q (List.take (↑(insertionSortMinPos le φ φs)) (φ :: φs))) *
    (exchangeSign (q ((φ :: φs).get (insertionSortMinPos le φ φs))))
      (ofList q
        (List.take (↑((insertionSortEquiv le (φ :: φs)) (insertionSortMinPos le φ φs)))
          (List.insertionSort le (φ :: φs)))) =
  koszulSign q le (φ :: φs) *
    (exchangeSign (q (insertionSortMin le φ φs))) (ofList q (List.take (↑(insertionSortMinPos le φ φs)) (φ :: φs)))
  conv_lhs =>
    rhs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕| (exchangeSign (q ((φ :: φs).get (insertionSortMinPos le φ φs))))
  (ofList q
    (List.take (↑((insertionSortEquiv le (φ :: φs)) (insertionSortMinPos le φ φs))) (List.insertionSort le (φ :: φs))))
    rhs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕| ofList q
  (List.take (↑((insertionSortEquiv le (φ :: φs)) (insertionSortMinPos le φ φs))) (List.insertionSort le (φ :: φs)))
    lhs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕| q
    simp [insertionSortMinPos]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕| q
  erw [Equiv.apply_symm_apply𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕⊢ koszulSign q le (φ :: φs) *
      (exchangeSign (q ((φ :: φs).get (insertionSortMinPos le φ φs))))
        (ofList q (List.take (↑(insertionSortMinPos le φ φs)) (φ :: φs))) *
    (exchangeSign (q ((φ :: φs).get (insertionSortMinPos le φ φs))))
      (ofList q (List.take (↑⟨0, ⋯⟩) (List.insertionSort le (φ :: φs)))) =
  koszulSign q le (φ :: φs) *
    (exchangeSign (q (insertionSortMin le φ φs))) (ofList q (List.take (↑(insertionSortMinPos le φ φs)) (φ :: φs)))]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕⊢ koszulSign q le (φ :: φs) *
      (exchangeSign (q ((φ :: φs).get (insertionSortMinPos le φ φs))))
        (ofList q (List.take (↑(insertionSortMinPos le φ φs)) (φ :: φs))) *
    (exchangeSign (q ((φ :: φs).get (insertionSortMinPos le φ φs))))
      (ofList q (List.take (↑⟨0, ⋯⟩) (List.insertionSort le (φ :: φs)))) =
  koszulSign q le (φ :: φs) *
    (exchangeSign (q (insertionSortMin le φ φs))) (ofList q (List.take (↑(insertionSortMinPos le φ φs)) (φ :: φs)))
  simp only [List.get_eq_getElem, List.length_cons, List.insertionSort,
    List.take_zero, ofList_empty, exchangeSign_bosonic, mul_one, mul_eq_mul_left_iff]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕⊢ (exchangeSign (q (φ :: φs)[↑(insertionSortMinPos le φ φs)]))
      (ofList q (List.take (↑(insertionSortMinPos le φ φs)) (φ :: φs))) =
    (exchangeSign (q (insertionSortMin le φ φs))) (ofList q (List.take (↑(insertionSortMinPos le φ φs)) (φ :: φs))) ∨
  koszulSign q le (φ :: φs) = 0
  apply Or.inl𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕⊢ (exchangeSign (q (φ :: φs)[↑(insertionSortMinPos le φ φs)]))
    (ofList q (List.take (↑(insertionSortMinPos le φ φs)) (φ :: φs))) =
  (exchangeSign (q (insertionSortMin le φ φs))) (ofList q (List.take (↑(insertionSortMinPos le φ φs)) (φ :: φs)))
  rflAll goals completed! 🐙

lemma koszulSign_swap_eq_rel_cons {ψ φ : 𝓕}
    (h1 : le φ ψ) (h2 : le ψ φ) (φs' : List 𝓕) :
    koszulSign q le (φ :: ψ :: φs') = koszulSign q le (ψ :: φ :: φs') := by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφs':List 𝓕⊢ koszulSign q le (φ :: ψ :: φs') = koszulSign q le (ψ :: φ :: φs')
  simp only [Wick.koszulSign, ← mul_assoc, mul_eq_mul_right_iff]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφs':List 𝓕⊢ koszulSignInsert q le φ (ψ :: φs') * koszulSignInsert q le ψ φs' =
    koszulSignInsert q le ψ (φ :: φs') * koszulSignInsert q le φ φs' ∨
  koszulSign q le φs' = 0
  left𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφs':List 𝓕⊢ koszulSignInsert q le φ (ψ :: φs') * koszulSignInsert q le ψ φs' =
  koszulSignInsert q le ψ (φ :: φs') * koszulSignInsert q le φ φs'
  rw [mul_comm𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφs':List 𝓕⊢ koszulSignInsert q le ψ φs' * koszulSignInsert q le φ (ψ :: φs') =
  koszulSignInsert q le ψ (φ :: φs') * koszulSignInsert q le φ φs'𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφs':List 𝓕⊢ koszulSignInsert q le ψ φs' * koszulSignInsert q le φ (ψ :: φs') =
  koszulSignInsert q le ψ (φ :: φs') * koszulSignInsert q le φ φs']𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφs':List 𝓕⊢ koszulSignInsert q le ψ φs' * koszulSignInsert q le φ (ψ :: φs') =
  koszulSignInsert q le ψ (φ :: φs') * koszulSignInsert q le φ φs'
  simp [Wick.koszulSignInsert, h1, h2]All goals completed! 🐙

lemma koszulSign_swap_eq_rel {ψ φ : 𝓕} (h1 : le φ ψ) (h2 : le ψ φ) : (φs φs' : List 𝓕) →
    koszulSign q le (φs ++ φ :: ψ :: φs') = koszulSign q le (φs ++ ψ :: φ :: φs')
  | [], φs' =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφs':List 𝓕⊢ koszulSign q le ([] ++ φ :: ψ :: φs') = koszulSign q le ([] ++ ψ :: φ :: φs') by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφs':List 𝓕⊢ koszulSign q le ([] ++ φ :: ψ :: φs') = koszulSign q le ([] ++ ψ :: φ :: φs')
    simp only [List.nil_append]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφs':List 𝓕⊢ koszulSign q le (φ :: ψ :: φs') = koszulSign q le (ψ :: φ :: φs')
    exact koszulSign_swap_eq_rel_cons q le h1 h2 φs'All goals completed! 🐙
  | φ'' :: φs, φs' =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφ'':𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSign q le (φ'' :: φs ++ φ :: ψ :: φs') = koszulSign q le (φ'' :: φs ++ ψ :: φ :: φs') by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφ'':𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSign q le (φ'' :: φs ++ φ :: ψ :: φs') = koszulSign q le (φ'' :: φs ++ ψ :: φ :: φs')
    simp only [List.cons_append, koszulSign]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφ'':𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSignInsert q le φ'' (φs ++ φ :: ψ :: φs') * koszulSign q le (φs ++ φ :: ψ :: φs') =
  koszulSignInsert q le φ'' (φs ++ ψ :: φ :: φs') * koszulSign q le (φs ++ ψ :: φ :: φs')
    rw [koszulSign_swap_eq_rel h1 h2𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφ'':𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSignInsert q le φ'' (φs ++ φ :: ψ :: φs') * koszulSign q le (φs ++ ψ :: φ :: φs') =
  koszulSignInsert q le φ'' (φs ++ ψ :: φ :: φs') * koszulSign q le (φs ++ ψ :: φ :: φs')𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφ'':𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSignInsert q le φ'' (φs ++ φ :: ψ :: φs') * koszulSign q le (φs ++ ψ :: φ :: φs') =
  koszulSignInsert q le φ'' (φs ++ ψ :: φ :: φs') * koszulSign q le (φs ++ ψ :: φ :: φs')]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφ'':𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSignInsert q le φ'' (φs ++ φ :: ψ :: φs') * koszulSign q le (φs ++ ψ :: φ :: φs') =
  koszulSignInsert q le φ'' (φs ++ ψ :: φ :: φs') * koszulSign q le (φs ++ ψ :: φ :: φs')
    congr 1e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφ'':𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSignInsert q le φ'' (φs ++ φ :: ψ :: φs') = koszulSignInsert q le φ'' (φs ++ ψ :: φ :: φs')
    apply Wick.koszulSignInsert_eq_perme_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leψ:𝓕φ:𝓕h1:le φ ψh2:le ψ φφ'':𝓕φs:List 𝓕φs':List 𝓕⊢ (φs ++ φ :: ψ :: φs').Perm (φs ++ ψ :: φ :: φs')
    exact List.Perm.append_left φs (List.Perm.swap ψ φ φs')All goals completed! 🐙

lemma koszulSign_eq_rel_eq_stat_append {ψ φ : 𝓕} [IsTrans 𝓕 le]
    (h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs : List 𝓕) →
    koszulSign q le (φ :: ψ :: φs) = koszulSign q le φs := by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φ⊢ ∀ (φs : List 𝓕), koszulSign q le (φ :: ψ :: φs) = koszulSign q le φs
  intro φs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ koszulSign q le (φ :: ψ :: φs) = koszulSign q le φs
  simp only [koszulSign, ← mul_assoc]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ koszulSignInsert q le φ (ψ :: φs) * koszulSignInsert q le ψ φs * koszulSign q le φs = koszulSign q le φs
  trans 1 * koszulSign q le φs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ koszulSignInsert q le φ (ψ :: φs) * koszulSignInsert q le ψ φs * koszulSign q le φs = 1 * koszulSign q le φs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ 1 * koszulSign q le φs = koszulSign q le φs
  swap𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ 1 * koszulSign q le φs = koszulSign q le φs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ koszulSignInsert q le φ (ψ :: φs) * koszulSignInsert q le ψ φs * koszulSign q le φs = 1 * koszulSign q le φs
  ·𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ 1 * koszulSign q le φs = koszulSign q le φs simp only [one_mul]All goals completed! 🐙
  congre_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ koszulSignInsert q le φ (ψ :: φs) * koszulSignInsert q le ψ φs = 1
  simp only [koszulSignInsert, ite_mul, neg_mul]e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ (if le φ ψ then koszulSignInsert q le φ φs * koszulSignInsert q le ψ φs
  else
    if q φ = fermionic ∧ q ψ = fermionic then -(koszulSignInsert q le φ φs * koszulSignInsert q le ψ φs)
    else koszulSignInsert q le φ φs * koszulSignInsert q le ψ φs) =
  1
  simp_all only [and_self, ite_true]e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ koszulSignInsert q le φ φs * koszulSignInsert q le ψ φs = 1
  rw [koszulSignInsert_eq_rel_eq_stat q le h1 h2 hqe_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ koszulSignInsert q le ψ φs * koszulSignInsert q le ψ φs = 1e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ koszulSignInsert q le ψ φs * koszulSignInsert q le ψ φs = 1]e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ koszulSignInsert q le ψ φs * koszulSignInsert q le ψ φs = 1
  simpAll goals completed! 🐙

lemma koszulSign_eq_rel_eq_stat {ψ φ : 𝓕} [IsTrans 𝓕 le]
    (h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs' φs : List 𝓕) →
    koszulSign q le (φs' ++ φ :: ψ :: φs) = koszulSign q le (φs' ++ φs)
  | [], φs =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ koszulSign q le ([] ++ φ :: ψ :: φs) = koszulSign q le ([] ++ φs) by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ koszulSign q le ([] ++ φ :: ψ :: φs) = koszulSign q le ([] ++ φs)
    simp only [List.nil_append]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφs:List 𝓕⊢ koszulSign q le (φ :: ψ :: φs) = koszulSign q le φs
    exact koszulSign_eq_rel_eq_stat_append q le h1 h2 hq φsAll goals completed! 🐙
  | φ'' :: φs', φs =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφ'':𝓕φs':List 𝓕φs:List 𝓕⊢ koszulSign q le (φ'' :: φs' ++ φ :: ψ :: φs) = koszulSign q le (φ'' :: φs' ++ φs) by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφ'':𝓕φs':List 𝓕φs:List 𝓕⊢ koszulSign q le (φ'' :: φs' ++ φ :: ψ :: φs) = koszulSign q le (φ'' :: φs' ++ φs)
    simp only [List.cons_append, koszulSign]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφ'':𝓕φs':List 𝓕φs:List 𝓕⊢ koszulSignInsert q le φ'' (φs' ++ φ :: ψ :: φs) * koszulSign q le (φs' ++ φ :: ψ :: φs) =
  koszulSignInsert q le φ'' (φs' ++ φs) * koszulSign q le (φs' ++ φs)
    rw [koszulSign_eq_rel_eq_stat h1 h2 hq φs' φs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφ'':𝓕φs':List 𝓕φs:List 𝓕⊢ koszulSignInsert q le φ'' (φs' ++ φ :: ψ :: φs) * koszulSign q le (φs' ++ φs) =
  koszulSignInsert q le φ'' (φs' ++ φs) * koszulSign q le (φs' ++ φs)𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφ'':𝓕φs':List 𝓕φs:List 𝓕⊢ koszulSignInsert q le φ'' (φs' ++ φ :: ψ :: φs) * koszulSign q le (φs' ++ φs) =
  koszulSignInsert q le φ'' (φs' ++ φs) * koszulSign q le (φs' ++ φs)]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφ'':𝓕φs':List 𝓕φs:List 𝓕⊢ koszulSignInsert q le φ'' (φs' ++ φ :: ψ :: φs) * koszulSign q le (φs' ++ φs) =
  koszulSignInsert q le φ'' (φs' ++ φs) * koszulSign q le (φs' ++ φs)
    simp only [mul_eq_mul_right_iff]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφ'':𝓕φs':List 𝓕φs:List 𝓕⊢ koszulSignInsert q le φ'' (φs' ++ φ :: ψ :: φs) = koszulSignInsert q le φ'' (φs' ++ φs) ∨
  koszulSign q le (φs' ++ φs) = 0
    left𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφ'':𝓕φs':List 𝓕φs:List 𝓕⊢ koszulSignInsert q le φ'' (φs' ++ φ :: ψ :: φs) = koszulSignInsert q le φ'' (φs' ++ φs)
    trans koszulSignInsert q le φ'' (φ :: ψ :: (φs' ++ φs))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφ'':𝓕φs':List 𝓕φs:List 𝓕⊢ koszulSignInsert q le φ'' (φs' ++ φ :: ψ :: φs) = koszulSignInsert q le φ'' (φ :: ψ :: (φs' ++ φs))𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφ'':𝓕φs':List 𝓕φs:List 𝓕⊢ koszulSignInsert q le φ'' (φ :: ψ :: (φs' ++ φs)) = koszulSignInsert q le φ'' (φs' ++ φs)
    ·𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφ'':𝓕φs':List 𝓕φs:List 𝓕⊢ koszulSignInsert q le φ'' (φs' ++ φ :: ψ :: φs) = koszulSignInsert q le φ'' (φ :: ψ :: (φs' ++ φs)) apply koszulSignInsert_eq_perm𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφ'':𝓕φs':List 𝓕φs:List 𝓕⊢ (φs' ++ φ :: ψ :: φs).Perm (φ :: ψ :: (φs' ++ φs))
      exact (List.perm_cons_append_cons φ List.perm_middle.symm).symmAll goals completed! 🐙
    ·𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφ'':𝓕φs':List 𝓕φs:List 𝓕⊢ koszulSignInsert q le φ'' (φ :: ψ :: (φs' ++ φs)) = koszulSignInsert q le φ'' (φs' ++ φs) rw [koszulSignInsert_eq_remove_same_stat_append q le h1 h2 hq𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leψ:𝓕φ:𝓕inst✝:IsTrans 𝓕 leh1:le φ ψh2:le ψ φhq:q ψ = q φφ'':𝓕φs':List 𝓕φs:List 𝓕⊢ koszulSignInsert q le φ'' (φs' ++ φs) = koszulSignInsert q le φ'' (φs' ++ φs)All goals completed! 🐙]All goals completed! 🐙

lemma koszulSign_of_sorted : (φs : List 𝓕)
    → (hs : List.Pairwise le φs) → koszulSign q le φs = 1
  | [], _ =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel lex✝:List.Pairwise le []⊢ koszulSign q le [] = 1 by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel lex✝:List.Pairwise le []⊢ koszulSign q le [] = 1
    simp [koszulSign]All goals completed! 🐙
  | φ :: φs, h =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕h:List.Pairwise le (φ :: φs)⊢ koszulSign q le (φ :: φs) = 1 by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕h:List.Pairwise le (φ :: φs)⊢ koszulSign q le (φ :: φs) = 1
    simp only [koszulSign]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕h:List.Pairwise le (φ :: φs)⊢ koszulSignInsert q le φ φs * koszulSign q le φs = 1
    simp only [List.pairwise_cons] at h𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕h:(∀ a' ∈ φs, le φ a') ∧ List.Pairwise le φs⊢ koszulSignInsert q le φ φs * koszulSign q le φs = 1
    rw [koszulSign_of_sorted φs h.2𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕h:(∀ a' ∈ φs, le φ a') ∧ List.Pairwise le φs⊢ koszulSignInsert q le φ φs * 1 = 1𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕h:(∀ a' ∈ φs, le φ a') ∧ List.Pairwise le φs⊢ koszulSignInsert q le φ φs * 1 = 1]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕h:(∀ a' ∈ φs, le φ a') ∧ List.Pairwise le φs⊢ koszulSignInsert q le φ φs * 1 = 1
    simp only [mul_one]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝:DecidableRel leφ:𝓕φs:List 𝓕h:(∀ a' ∈ φs, le φ a') ∧ List.Pairwise le φs⊢ koszulSignInsert q le φ φs = 1
    exact koszulSignInsert_of_le_mem _ _ _ _ h.1All goals completed! 🐙@[simp]
lemma koszulSign_of_insertionSort [Std.Total le] [IsTrans 𝓕 le] (φs : List 𝓕) :
    koszulSign q le (List.insertionSort le φs) = 1 := by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕⊢ koszulSign q le (List.insertionSort le φs) = 1
  apply koszulSign_of_sorted𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕⊢ List.Pairwise le (List.insertionSort le φs)
  exact List.pairwise_insertionSort le φsAll goals completed! 🐙

lemma koszulSign_of_append_eq_insertionSort_left [Std.Total le] [IsTrans 𝓕 le] :
    (φs φs' : List 𝓕) → koszulSign q le (φs ++ φs') =
    koszulSign q le (List.insertionSort le φs ++ φs') * koszulSign q le φs
  | φs, [] =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕⊢ koszulSign q le (φs ++ []) = koszulSign q le (List.insertionSort le φs ++ []) * koszulSign q le φs by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕⊢ koszulSign q le (φs ++ []) = koszulSign q le (List.insertionSort le φs ++ []) * koszulSign q le φs
    simpAll goals completed! 🐙
  | φs, φ :: φs' =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕⊢ koszulSign q le (φs ++ φ :: φs') = koszulSign q le (List.insertionSort le φs ++ φ :: φs') * koszulSign q le φs by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕⊢ koszulSign q le (φs ++ φ :: φs') = koszulSign q le (List.insertionSort le φs ++ φ :: φs') * koszulSign q le φs
    have h1 : (φs ++ φ :: φs') = List.insertIdx (φs ++ φs') φs.length φ := by
      rw [insertIdx_length_fst_append𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕⊢ φs ++ φ :: φs' = φs ++ φ :: φs'𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φ⊢ koszulSign q le (φs ++ φ :: φs') = koszulSign q le (List.insertionSort le φs ++ φ :: φs') * koszulSign q le φs]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φ⊢ koszulSign q le (φs ++ φ :: φs') = koszulSign q le (List.insertionSort le φs ++ φ :: φs') * koszulSign q le φs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φ⊢ koszulSign q le (φs ++ φ :: φs') = koszulSign q le (List.insertionSort le φs ++ φ :: φs') * koszulSign q le φs
    have h2 : (List.insertionSort le φs ++ φ :: φs') =
        List.insertIdx (List.insertionSort le φs ++ φs') (List.insertionSort le φs).length φ := by
      rw [insertIdx_length_fst_append𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φ⊢ List.insertionSort le φs ++ φ :: φs' = List.insertionSort le φs ++ φ :: φs'𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ koszulSign q le (φs ++ φ :: φs') = koszulSign q le (List.insertionSort le φs ++ φ :: φs') * koszulSign q le φs]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ koszulSign q le (φs ++ φ :: φs') = koszulSign q le (List.insertionSort le φs ++ φ :: φs') * koszulSign q le φs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ koszulSign q le (φs ++ φ :: φs') = koszulSign q le (List.insertionSort le φs ++ φ :: φs') * koszulSign q le φs
    rw [h1,𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ koszulSign q le ((φs ++ φs').insertIdx φs.length φ) =
  koszulSign q le (List.insertionSort le φs ++ φ :: φs') * koszulSign q le φs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ koszulSign q le ((φs ++ φs').insertIdx φs.length φ) =
  koszulSign q le ((List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ) * koszulSign q le φs h2𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ koszulSign q le ((φs ++ φs').insertIdx φs.length φ) =
  koszulSign q le ((List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ) * koszulSign q le φs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ koszulSign q le ((φs ++ φs').insertIdx φs.length φ) =
  koszulSign q le ((List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ) * koszulSign q le φs]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ koszulSign q le ((φs ++ φs').insertIdx φs.length φ) =
  koszulSign q le ((List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ) * koszulSign q le φs
    rw [koszulSign_insertIdx _ _ _ _ _ (by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ φs.length ≤ (φs ++ φs').length𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ (exchangeSign (q φ)) (ofList q (List.take φs.length (φs ++ φs'))) * koszulSign q le (φs ++ φs') *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
          (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
  koszulSign q le ((List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ) * koszulSign q le φs simpAll goals completed! 🐙𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ (exchangeSign (q φ)) (ofList q (List.take φs.length (φs ++ φs'))) * koszulSign q le (φs ++ φs') *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
          (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
  koszulSign q le ((List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ) * koszulSign q le φs)]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ (exchangeSign (q φ)) (ofList q (List.take φs.length (φs ++ φs'))) * koszulSign q le (φs ++ φs') *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
          (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
  koszulSign q le ((List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ) * koszulSign q le φs
    simp only [List.take_left', List.length_insertionSort]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ (exchangeSign (q φ)) (ofList q φs) * koszulSign q le (φs ++ φs') *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
          (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
  koszulSign q le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ) * koszulSign q le φs
    rw [koszulSign_insertIdx _ _ _ _ _ (by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ φs.length ≤ (List.insertionSort le φs ++ φs').length𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ (exchangeSign (q φ)) (ofList q φs) * koszulSign q le (φs ++ φs') *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
          (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
  (exchangeSign (q φ)) (ofList q (List.take φs.length (List.insertionSort le φs ++ φs'))) *
        koszulSign q le (List.insertionSort le φs ++ φs') *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
            (List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)))) *
    koszulSign q le φs simpAll goals completed! 🐙𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ (exchangeSign (q φ)) (ofList q φs) * koszulSign q le (φs ++ φs') *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
          (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
  (exchangeSign (q φ)) (ofList q (List.take φs.length (List.insertionSort le φs ++ φs'))) *
        koszulSign q le (List.insertionSort le φs ++ φs') *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
            (List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)))) *
    koszulSign q le φs)]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ (exchangeSign (q φ)) (ofList q φs) * koszulSign q le (φs ++ φs') *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
          (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
  (exchangeSign (q φ)) (ofList q (List.take φs.length (List.insertionSort le φs ++ φs'))) *
        koszulSign q le (List.insertionSort le φs ++ φs') *
      (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
            (List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)))) *
    koszulSign q le φs
    simp only [mul_assoc, List.length_insertionSort, List.take_left',
      ofList_insertionSort, mul_eq_mul_left_iff]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ koszulSign q le (φs ++ φs') *
      (exchangeSign (q φ))
        (ofList q
          (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
            (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
    koszulSign q le (List.insertionSort le φs ++ φs') *
      ((exchangeSign (q φ))
          (ofList q
            (List.take
              (↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
              (List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)))) *
        koszulSign q le φs) ∨
  (exchangeSign (q φ)) (ofList q φs) = 0
    left𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ koszulSign q le (φs ++ φs') *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
          (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
  koszulSign q le (List.insertionSort le φs ++ φs') *
    ((exchangeSign (q φ))
        (ofList q
          (List.take
            (↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
            (List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)))) *
      koszulSign q le φs)
    rw [koszulSign_of_append_eq_insertionSort_left φs φs'𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ koszulSign q le (List.insertionSort le φs ++ φs') * koszulSign q le φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
          (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
  koszulSign q le (List.insertionSort le φs ++ φs') *
    ((exchangeSign (q φ))
        (ofList q
          (List.take
            (↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
            (List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)))) *
      koszulSign q le φs)𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ koszulSign q le (List.insertionSort le φs ++ φs') * koszulSign q le φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
          (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
  koszulSign q le (List.insertionSort le φs ++ φs') *
    ((exchangeSign (q φ))
        (ofList q
          (List.take
            (↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
            (List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)))) *
      koszulSign q le φs)]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ koszulSign q le (List.insertionSort le φs ++ φs') * koszulSign q le φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
          (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
  koszulSign q le (List.insertionSort le φs ++ φs') *
    ((exchangeSign (q φ))
        (ofList q
          (List.take
            (↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
            (List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)))) *
      koszulSign q le φs)
    simp only [mul_assoc, mul_eq_mul_left_iff]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ koszulSign q le φs *
      (exchangeSign (q φ))
        (ofList q
          (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
            (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
    (exchangeSign (q φ))
        (ofList q
          (List.take
            (↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
            (List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)))) *
      koszulSign q le φs ∨
  koszulSign q le (List.insertionSort le φs ++ φs') = 0
    left𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ koszulSign q le φs *
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
          (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
  (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
          (List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)))) *
    koszulSign q le φs
    simp only [mul_comm, mul_eq_mul_left_iff]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
          (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
    (exchangeSign (q φ))
      (ofList q
        (List.take (↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
          (List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)))) ∨
  koszulSign q le φs = 0
    left𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ (exchangeSign (q φ))
    (ofList q
      (List.take (↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
        (List.insertionSort le ((φs ++ φs').insertIdx φs.length φ)))) =
  (exchangeSign (q φ))
    (ofList q
      (List.take (↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩))
        (List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ))))
    congr 3e_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ ↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) =
  ↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩)e_6.e_φs.e_xs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ List.insertionSort le ((φs ++ φs').insertIdx φs.length φ) =
  List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)
    ·e_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ ↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) =
  ↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) have h2 : (List.insertionSort le φs ++ φ :: φs') =
          List.insertIdx (List.insertionSort le φs ++ φs') φs.length φ := by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕⊢ koszulSign q le (φs ++ φ :: φs') = koszulSign q le (List.insertionSort le φs ++ φ :: φs') * koszulSign q le φse_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) =
  ↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩)
        rw [← insertIdx_length_fst_append𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ =
  (List.insertionSort le φs ++ φs').insertIdx φs.length φ𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ =
  (List.insertionSort le φs ++ φs').insertIdx φs.length φe_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) =
  ↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩)]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ =
  (List.insertionSort le φs ++ φs').insertIdx φs.length φe_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) =
  ↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩)
        simpe_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) =
  ↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩)e_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) =
  ↑((insertionSortEquiv le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩)
      rw [insertionSortEquiv_congr _ _ h2.symme_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) =
  ↑(((Fin.castOrderIso ⋯).trans
        ((insertionSortEquiv le (List.insertionSort le φs ++ φ :: φs')).trans (Fin.castOrderIso ⋯).toEquiv))
      ⟨φs.length, ⋯⟩)e_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) =
  ↑(((Fin.castOrderIso ⋯).trans
        ((insertionSortEquiv le (List.insertionSort le φs ++ φ :: φs')).trans (Fin.castOrderIso ⋯).toEquiv))
      ⟨φs.length, ⋯⟩)]e_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) =
  ↑(((Fin.castOrderIso ⋯).trans
        ((insertionSortEquiv le (List.insertionSort le φs ++ φ :: φs')).trans (Fin.castOrderIso ⋯).toEquiv))
      ⟨φs.length, ⋯⟩)
      simp only [Equiv.trans_apply, RelIso.coe_fn_toEquiv, Fin.castOrderIso_apply, Fin.cast_mk,
        Fin.val_cast]e_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) =
  ↑((insertionSortEquiv le (List.insertionSort le φs ++ φ :: φs')) ⟨φs.length, ⋯⟩)
      rw [insertionSortEquiv_insertionSort_appende_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) =
  ↑((finCongr ⋯) ((insertionSortEquiv le (φs ++ φ :: φs')) ⟨φs.length, ⋯⟩))e_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) =
  ↑((finCongr ⋯) ((insertionSortEquiv le (φs ++ φ :: φs')) ⟨φs.length, ⋯⟩))]e_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) =
  ↑((finCongr ⋯) ((insertionSortEquiv le (φs ++ φ :: φs')) ⟨φs.length, ⋯⟩))
      simp only [finCongr_apply, Fin.val_cast]e_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑((insertionSortEquiv le ((φs ++ φs').insertIdx φs.length φ)) ⟨φs.length, ⋯⟩) =
  ↑((insertionSortEquiv le (φs ++ φ :: φs')) ⟨φs.length, ⋯⟩)
      rw [insertionSortEquiv_congr _ _ h1.symme_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑(((Fin.castOrderIso ⋯).trans ((insertionSortEquiv le (φs ++ φ :: φs')).trans (Fin.castOrderIso ⋯).toEquiv))
      ⟨φs.length, ⋯⟩) =
  ↑((insertionSortEquiv le (φs ++ φ :: φs')) ⟨φs.length, ⋯⟩)e_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑(((Fin.castOrderIso ⋯).trans ((insertionSortEquiv le (φs ++ φ :: φs')).trans (Fin.castOrderIso ⋯).toEquiv))
      ⟨φs.length, ⋯⟩) =
  ↑((insertionSortEquiv le (φs ++ φ :: φs')) ⟨φs.length, ⋯⟩)]e_6.e_φs.e_n𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2✝:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx φs.length φ⊢ ↑(((Fin.castOrderIso ⋯).trans ((insertionSortEquiv le (φs ++ φ :: φs')).trans (Fin.castOrderIso ⋯).toEquiv))
      ⟨φs.length, ⋯⟩) =
  ↑((insertionSortEquiv le (φs ++ φ :: φs')) ⟨φs.length, ⋯⟩)
      simpAll goals completed! 🐙
    ·e_6.e_φs.e_xs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ List.insertionSort le ((φs ++ φs').insertIdx φs.length φ) =
  List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ) rw [insertIdx_length_fst_appende_6.e_φs.e_xs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ List.insertionSort le (φs ++ φ :: φs') = List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)e_6.e_φs.e_xs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ List.insertionSort le (φs ++ φ :: φs') = List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)]e_6.e_φs.e_xs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ List.insertionSort le (φs ++ φ :: φs') = List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx φs.length φ)
      rw [show φs.length = (List.insertionSort le φs).length by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕⊢ koszulSign q le (φs ++ φ :: φs') = koszulSign q le (List.insertionSort le φs ++ φ :: φs') * koszulSign q le φse_6.e_φs.e_xs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ List.insertionSort le (φs ++ φ :: φs') =
  List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ) simpAll goals completed! 🐙e_6.e_φs.e_xs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ List.insertionSort le (φs ++ φ :: φs') =
  List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ)]e_6.e_φs.e_xs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ List.insertionSort le (φs ++ φ :: φs') =
  List.insertionSort le ((List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ)
      rw [insertIdx_length_fst_appende_6.e_φs.e_xs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ List.insertionSort le (φs ++ φ :: φs') = List.insertionSort le (List.insertionSort le φs ++ φ :: φs')e_6.e_φs.e_xs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ List.insertionSort le (φs ++ φ :: φs') = List.insertionSort le (List.insertionSort le φs ++ φ :: φs')]e_6.e_φs.e_xs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ List.insertionSort le (φs ++ φ :: φs') = List.insertionSort le (List.insertionSort le φs ++ φ :: φs')
      symme_6.e_φs.e_xs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φ:𝓕φs':List 𝓕h1:φs ++ φ :: φs' = (φs ++ φs').insertIdx φs.length φh2:List.insertionSort le φs ++ φ :: φs' = (List.insertionSort le φs ++ φs').insertIdx (List.insertionSort le φs).length φ⊢ List.insertionSort le (List.insertionSort le φs ++ φ :: φs') = List.insertionSort le (φs ++ φ :: φs')
      apply insertionSort_insertionSort_appendAll goals completed! 🐙

lemma koszulSign_of_append_eq_insertionSort [Std.Total le] [IsTrans 𝓕 le] : (φs'' φs φs' : List 𝓕) →
    koszulSign q le (φs'' ++ φs ++ φs') =
    koszulSign q le (φs'' ++ List.insertionSort le φs ++ φs') * koszulSign q le φs
  | [], φs, φs'=>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φs':List 𝓕⊢ koszulSign q le ([] ++ φs ++ φs') = koszulSign q le ([] ++ List.insertionSort le φs ++ φs') * koszulSign q le φs by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φs':List 𝓕⊢ koszulSign q le ([] ++ φs ++ φs') = koszulSign q le ([] ++ List.insertionSort le φs ++ φs') * koszulSign q le φs
    simp only [List.nil_append]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφs:List 𝓕φs':List 𝓕⊢ koszulSign q le (φs ++ φs') = koszulSign q le (List.insertionSort le φs ++ φs') * koszulSign q le φs
    exact koszulSign_of_append_eq_insertionSort_left q le φs φs'All goals completed! 🐙
  | φ'' :: φs'', φs, φs' =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ'':𝓕φs'':List 𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSign q le (φ'' :: φs'' ++ φs ++ φs') =
  koszulSign q le (φ'' :: φs'' ++ List.insertionSort le φs ++ φs') * koszulSign q le φs by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ'':𝓕φs'':List 𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSign q le (φ'' :: φs'' ++ φs ++ φs') =
  koszulSign q le (φ'' :: φs'' ++ List.insertionSort le φs ++ φs') * koszulSign q le φs
    simp only [List.cons_append, koszulSign]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ'':𝓕φs'':List 𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSignInsert q le φ'' (φs'' ++ φs ++ φs') * koszulSign q le (φs'' ++ φs ++ φs') =
  koszulSignInsert q le φ'' (φs'' ++ List.insertionSort le φs ++ φs') *
      koszulSign q le (φs'' ++ List.insertionSort le φs ++ φs') *
    koszulSign q le φs
    rw [koszulSign_of_append_eq_insertionSort φs'' φs φs',𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ'':𝓕φs'':List 𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSignInsert q le φ'' (φs'' ++ φs ++ φs') *
    (koszulSign q le (φs'' ++ List.insertionSort le φs ++ φs') * koszulSign q le φs) =
  koszulSignInsert q le φ'' (φs'' ++ List.insertionSort le φs ++ φs') *
      koszulSign q le (φs'' ++ List.insertionSort le φs ++ φs') *
    koszulSign q le φs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ'':𝓕φs'':List 𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSignInsert q le φ'' (φs'' ++ φs ++ φs') * koszulSign q le (φs'' ++ List.insertionSort le φs ++ φs') *
    koszulSign q le φs =
  koszulSignInsert q le φ'' (φs'' ++ List.insertionSort le φs ++ φs') *
      koszulSign q le (φs'' ++ List.insertionSort le φs ++ φs') *
    koszulSign q le φs ← mul_assoc𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ'':𝓕φs'':List 𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSignInsert q le φ'' (φs'' ++ φs ++ φs') * koszulSign q le (φs'' ++ List.insertionSort le φs ++ φs') *
    koszulSign q le φs =
  koszulSignInsert q le φ'' (φs'' ++ List.insertionSort le φs ++ φs') *
      koszulSign q le (φs'' ++ List.insertionSort le φs ++ φs') *
    koszulSign q le φs𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ'':𝓕φs'':List 𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSignInsert q le φ'' (φs'' ++ φs ++ φs') * koszulSign q le (φs'' ++ List.insertionSort le φs ++ φs') *
    koszulSign q le φs =
  koszulSignInsert q le φ'' (φs'' ++ List.insertionSort le φs ++ φs') *
      koszulSign q le (φs'' ++ List.insertionSort le φs ++ φs') *
    koszulSign q le φs]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ'':𝓕φs'':List 𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSignInsert q le φ'' (φs'' ++ φs ++ φs') * koszulSign q le (φs'' ++ List.insertionSort le φs ++ φs') *
    koszulSign q le φs =
  koszulSignInsert q le φ'' (φs'' ++ List.insertionSort le φs ++ φs') *
      koszulSign q le (φs'' ++ List.insertionSort le φs ++ φs') *
    koszulSign q le φs
    congr 2e_a.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ'':𝓕φs'':List 𝓕φs:List 𝓕φs':List 𝓕⊢ koszulSignInsert q le φ'' (φs'' ++ φs ++ φs') = koszulSignInsert q le φ'' (φs'' ++ List.insertionSort le φs ++ φs')
    apply koszulSignInsert_eq_perme_a.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ'':𝓕φs'':List 𝓕φs:List 𝓕φs':List 𝓕⊢ (φs'' ++ φs ++ φs').Perm (φs'' ++ List.insertionSort le φs ++ φs')
    refine (List.perm_append_right_iff φs').mpr ?_e_a.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ'':𝓕φs'':List 𝓕φs:List 𝓕φs':List 𝓕⊢ (φs'' ++ φs).Perm (φs'' ++ List.insertionSort le φs)
    refine List.Perm.append_left φs'' ?_e_a.e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝²:DecidableRel leinst✝¹:Std.Total leinst✝:IsTrans 𝓕 leφ'':𝓕φs'':List 𝓕φs:List 𝓕φs':List 𝓕⊢ φs.Perm (List.insertionSort le φs)
    exact List.Perm.symm (List.perm_insertionSort le φs)All goals completed! 🐙

koszulSign with permutations

lemma koszulSign_perm_eq_append [IsTrans 𝓕 le] (φ : 𝓕) (φs φs' φs2 : List 𝓕)
    (hp : φs.Perm φs') : (h : ∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) →
    koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2) := by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'⊢ (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)
  let motive (φs φs' : List 𝓕) (hp : φs.Perm φs') : Prop :=
    (h : ∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) →
    koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)⊢ (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)
  change motive φs φs' hp𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)⊢ motive φs φs' hp
  apply List.Perm.recOnnil𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)⊢ motive [] [] ⋯cons𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)⊢ ∀ (x : 𝓕) {l₁ l₂ : List 𝓕} (a : l₁.Perm l₂), motive l₁ l₂ a → motive (x :: l₁) (x :: l₂) ⋯swap𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)⊢ ∀ (x y : 𝓕) (l : List 𝓕), motive (y :: x :: l) (x :: y :: l) ⋯trans𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)⊢ ∀ {l₁ l₂ l₃ : List 𝓕} (a : l₁.Perm l₂) (a_1 : l₂.Perm l₃), motive l₁ l₂ a → motive l₂ l₃ a_1 → motive l₁ l₃ ⋯
  ·nil𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)⊢ motive [] [] ⋯ simp [motive]All goals completed! 🐙
  ·cons𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)⊢ ∀ (x : 𝓕) {l₁ l₂ : List 𝓕} (a : l₁.Perm l₂), motive l₁ l₂ a → motive (x :: l₁) (x :: l₂) ⋯ intro x l1 l2 h ih hxφcons𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)x:𝓕l1:List 𝓕l2:List 𝓕h:l1.Perm l2ih:motive l1 l2 hhxφ:∀ φ' ∈ x :: l1, le φ φ' ∧ le φ' φ⊢ koszulSign q le (x :: l1 ++ φs2) = koszulSign q le (x :: l2 ++ φs2)
    simp_all only [List.mem_cons, or_true, and_self, implies_true, nonempty_prop, forall_const,
      forall_eq_or_imp, List.cons_append, motive]cons𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)x:𝓕l1:List 𝓕l2:List 𝓕h:l1.Perm l2ih:koszulSign q le (l1 ++ φs2) = koszulSign q le (l2 ++ φs2)hxφ:(le φ x ∧ le x φ) ∧ ∀ a ∈ l1, le φ a ∧ le a φ⊢ koszulSign q le (x :: (l1 ++ φs2)) = koszulSign q le (x :: (l2 ++ φs2))
    simp only [koszulSign, ih, mul_eq_mul_right_iff]cons𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)x:𝓕l1:List 𝓕l2:List 𝓕h:l1.Perm l2ih:koszulSign q le (l1 ++ φs2) = koszulSign q le (l2 ++ φs2)hxφ:(le φ x ∧ le x φ) ∧ ∀ a ∈ l1, le φ a ∧ le a φ⊢ koszulSignInsert q le x (l1 ++ φs2) = koszulSignInsert q le x (l2 ++ φs2) ∨ koszulSign q le (l2 ++ φs2) = 0
    leftcons𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)x:𝓕l1:List 𝓕l2:List 𝓕h:l1.Perm l2ih:koszulSign q le (l1 ++ φs2) = koszulSign q le (l2 ++ φs2)hxφ:(le φ x ∧ le x φ) ∧ ∀ a ∈ l1, le φ a ∧ le a φ⊢ koszulSignInsert q le x (l1 ++ φs2) = koszulSignInsert q le x (l2 ++ φs2)
    apply koszulSignInsert_eq_permcons𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)x:𝓕l1:List 𝓕l2:List 𝓕h:l1.Perm l2ih:koszulSign q le (l1 ++ φs2) = koszulSign q le (l2 ++ φs2)hxφ:(le φ x ∧ le x φ) ∧ ∀ a ∈ l1, le φ a ∧ le a φ⊢ (l1 ++ φs2).Perm (l2 ++ φs2)
    exact (List.perm_append_right_iff φs2).mpr hAll goals completed! 🐙
  ·swap𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)⊢ ∀ (x y : 𝓕) (l : List 𝓕), motive (y :: x :: l) (x :: y :: l) ⋯ intro x y l hswap𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)x:𝓕y:𝓕l:List 𝓕h:∀ φ' ∈ y :: x :: l, le φ φ' ∧ le φ' φ⊢ koszulSign q le (y :: x :: l ++ φs2) = koszulSign q le (x :: y :: l ++ φs2)
    simp_all only [List.mem_cons, forall_eq_or_imp, List.cons_append]swap𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)x:𝓕y:𝓕l:List 𝓕h:(le φ y ∧ le y φ) ∧ (le φ x ∧ le x φ) ∧ ∀ a ∈ l, le φ a ∧ le a φ⊢ koszulSign q le (y :: x :: (l ++ φs2)) = koszulSign q le (x :: y :: (l ++ φs2))
    apply Wick.koszulSign_swap_eq_rel_consswap.h1𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)x:𝓕y:𝓕l:List 𝓕h:(le φ y ∧ le y φ) ∧ (le φ x ∧ le x φ) ∧ ∀ a ∈ l, le φ a ∧ le a φ⊢ le y xswap.h2𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)x:𝓕y:𝓕l:List 𝓕h:(le φ y ∧ le y φ) ∧ (le φ x ∧ le x φ) ∧ ∀ a ∈ l, le φ a ∧ le a φ⊢ le x y
    ·swap.h1𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)x:𝓕y:𝓕l:List 𝓕h:(le φ y ∧ le y φ) ∧ (le φ x ∧ le x φ) ∧ ∀ a ∈ l, le φ a ∧ le a φ⊢ le y x exact IsTrans.trans y φ x h.1.2 h.2.1.1All goals completed! 🐙
    ·swap.h2𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)x:𝓕y:𝓕l:List 𝓕h:(le φ y ∧ le y φ) ∧ (le φ x ∧ le x φ) ∧ ∀ a ∈ l, le φ a ∧ le a φ⊢ le x y exact IsTrans.trans x φ y h.2.1.2 h.1.1All goals completed! 🐙
  ·trans𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)⊢ ∀ {l₁ l₂ l₃ : List 𝓕} (a : l₁.Perm l₂) (a_1 : l₂.Perm l₃), motive l₁ l₂ a → motive l₂ l₃ a_1 → motive l₁ l₃ ⋯ intro l1 l2 l3 h1 h2 ih1 ih2 htrans𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)l1:List 𝓕l2:List 𝓕l3:List 𝓕h1:l1.Perm l2h2:l2.Perm l3ih1:motive l1 l2 h1ih2:motive l2 l3 h2h:∀ φ' ∈ l1, le φ φ' ∧ le φ' φ⊢ koszulSign q le (l1 ++ φs2) = koszulSign q le (l3 ++ φs2)
    simp_all only [and_self, implies_true, nonempty_prop, forall_const, motive]trans𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)l1:List 𝓕l2:List 𝓕l3:List 𝓕h1:l1.Perm l2h2:l2.Perm l3ih1:koszulSign q le (l1 ++ φs2) = koszulSign q le (l2 ++ φs2)ih2:(∀ φ' ∈ l2, le φ φ' ∧ le φ' φ) → koszulSign q le (l2 ++ φs2) = koszulSign q le (l3 ++ φs2)h:∀ φ' ∈ l1, le φ φ' ∧ le φ' φ⊢ koszulSign q le (l2 ++ φs2) = koszulSign q le (l3 ++ φs2)
    refine (ih2 ?_)trans𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)l1:List 𝓕l2:List 𝓕l3:List 𝓕h1:l1.Perm l2h2:l2.Perm l3ih1:koszulSign q le (l1 ++ φs2) = koszulSign q le (l2 ++ φs2)ih2:(∀ φ' ∈ l2, le φ φ' ∧ le φ' φ) → koszulSign q le (l2 ++ φs2) = koszulSign q le (l3 ++ φs2)h:∀ φ' ∈ l1, le φ φ' ∧ le φ' φ⊢ ∀ φ' ∈ l2, le φ φ' ∧ le φ' φ
    intro φ' hφtrans𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)l1:List 𝓕l2:List 𝓕l3:List 𝓕h1:l1.Perm l2h2:l2.Perm l3ih1:koszulSign q le (l1 ++ φs2) = koszulSign q le (l2 ++ φs2)ih2:(∀ φ' ∈ l2, le φ φ' ∧ le φ' φ) → koszulSign q le (l2 ++ φs2) = koszulSign q le (l3 ++ φs2)h:∀ φ' ∈ l1, le φ φ' ∧ le φ' φφ':𝓕hφ:φ' ∈ l2⊢ le φ φ' ∧ le φ' φ
    refine h φ' ?_trans𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕hp:φs.Perm φs'motive:(φs φs' : List 𝓕) → φs.Perm φs' → Prop := fun φs φs' hp => (∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)l1:List 𝓕l2:List 𝓕l3:List 𝓕h1:l1.Perm l2h2:l2.Perm l3ih1:koszulSign q le (l1 ++ φs2) = koszulSign q le (l2 ++ φs2)ih2:(∀ φ' ∈ l2, le φ φ' ∧ le φ' φ) → koszulSign q le (l2 ++ φs2) = koszulSign q le (l3 ++ φs2)h:∀ φ' ∈ l1, le φ φ' ∧ le φ' φφ':𝓕hφ:φ' ∈ l2⊢ φ' ∈ l1
    exact (List.Perm.mem_iff (id (List.Perm.symm h1))).mp hφAll goals completed! 🐙



lemma koszulSign_perm_eq [IsTrans 𝓕 le] (φ : 𝓕) : (φs1 φs φs' φs2 : List 𝓕) →
    (h : ∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → (hp : φs.Perm φs') →
    koszulSign q le (φs1 ++ φs ++ φs2) = koszulSign q le (φs1 ++ φs' ++ φs2)
  | [], φs, φs', φs2, h, hp =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕h:∀ φ' ∈ φs, le φ φ' ∧ le φ' φhp:φs.Perm φs'⊢ koszulSign q le ([] ++ φs ++ φs2) = koszulSign q le ([] ++ φs' ++ φs2) by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕h:∀ φ' ∈ φs, le φ φ' ∧ le φ' φhp:φs.Perm φs'⊢ koszulSign q le ([] ++ φs ++ φs2) = koszulSign q le ([] ++ φs' ++ φs2)
    simp only [List.nil_append]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕h:∀ φ' ∈ φs, le φ φ' ∧ le φ' φhp:φs.Perm φs'⊢ koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)
    exact koszulSign_perm_eq_append q le φ φs φs' φs2 hp hAll goals completed! 🐙
  | φ1 :: φs1, φs, φs', φs2, h, hp =>𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs1:List 𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕h:∀ φ' ∈ φs, le φ φ' ∧ le φ' φhp:φs.Perm φs'⊢ koszulSign q le (φ1 :: φs1 ++ φs ++ φs2) = koszulSign q le (φ1 :: φs1 ++ φs' ++ φs2) by𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs1:List 𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕h:∀ φ' ∈ φs, le φ φ' ∧ le φ' φhp:φs.Perm φs'⊢ koszulSign q le (φ1 :: φs1 ++ φs ++ φs2) = koszulSign q le (φ1 :: φs1 ++ φs' ++ φs2)
    simp only [List.cons_append, koszulSign]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs1:List 𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕h:∀ φ' ∈ φs, le φ φ' ∧ le φ' φhp:φs.Perm φs'⊢ koszulSignInsert q le φ1 (φs1 ++ φs ++ φs2) * koszulSign q le (φs1 ++ φs ++ φs2) =
  koszulSignInsert q le φ1 (φs1 ++ φs' ++ φs2) * koszulSign q le (φs1 ++ φs' ++ φs2)
    have ih := koszulSign_perm_eq φ φs1 φs φs' φs2 h hp𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs1:List 𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕h:∀ φ' ∈ φs, le φ φ' ∧ le φ' φhp:φs.Perm φs'ih:koszulSign q le (φs1 ++ φs ++ φs2) = koszulSign q le (φs1 ++ φs' ++ φs2)⊢ koszulSignInsert q le φ1 (φs1 ++ φs ++ φs2) * koszulSign q le (φs1 ++ φs ++ φs2) =
  koszulSignInsert q le φ1 (φs1 ++ φs' ++ φs2) * koszulSign q le (φs1 ++ φs' ++ φs2)
    rw [ih𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs1:List 𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕h:∀ φ' ∈ φs, le φ φ' ∧ le φ' φhp:φs.Perm φs'ih:koszulSign q le (φs1 ++ φs ++ φs2) = koszulSign q le (φs1 ++ φs' ++ φs2)⊢ koszulSignInsert q le φ1 (φs1 ++ φs ++ φs2) * koszulSign q le (φs1 ++ φs' ++ φs2) =
  koszulSignInsert q le φ1 (φs1 ++ φs' ++ φs2) * koszulSign q le (φs1 ++ φs' ++ φs2)𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs1:List 𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕h:∀ φ' ∈ φs, le φ φ' ∧ le φ' φhp:φs.Perm φs'ih:koszulSign q le (φs1 ++ φs ++ φs2) = koszulSign q le (φs1 ++ φs' ++ φs2)⊢ koszulSignInsert q le φ1 (φs1 ++ φs ++ φs2) * koszulSign q le (φs1 ++ φs' ++ φs2) =
  koszulSignInsert q le φ1 (φs1 ++ φs' ++ φs2) * koszulSign q le (φs1 ++ φs' ++ φs2)]𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs1:List 𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕h:∀ φ' ∈ φs, le φ φ' ∧ le φ' φhp:φs.Perm φs'ih:koszulSign q le (φs1 ++ φs ++ φs2) = koszulSign q le (φs1 ++ φs' ++ φs2)⊢ koszulSignInsert q le φ1 (φs1 ++ φs ++ φs2) * koszulSign q le (φs1 ++ φs' ++ φs2) =
  koszulSignInsert q le φ1 (φs1 ++ φs' ++ φs2) * koszulSign q le (φs1 ++ φs' ++ φs2)
    congr 1e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs1:List 𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕h:∀ φ' ∈ φs, le φ φ' ∧ le φ' φhp:φs.Perm φs'ih:koszulSign q le (φs1 ++ φs ++ φs2) = koszulSign q le (φs1 ++ φs' ++ φs2)⊢ koszulSignInsert q le φ1 (φs1 ++ φs ++ φs2) = koszulSignInsert q le φ1 (φs1 ++ φs' ++ φs2)
    apply koszulSignInsert_eq_perme_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs1:List 𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕h:∀ φ' ∈ φs, le φ φ' ∧ le φ' φhp:φs.Perm φs'ih:koszulSign q le (φs1 ++ φs ++ φs2) = koszulSign q le (φs1 ++ φs' ++ φs2)⊢ (φs1 ++ φs ++ φs2).Perm (φs1 ++ φs' ++ φs2)
    refine (List.perm_append_right_iff φs2).mpr ?_e_a𝓕:Typeq:𝓕 → FieldStatisticle:𝓕 → 𝓕 → Propinst✝¹:DecidableRel leinst✝:IsTrans 𝓕 leφ:𝓕φ1:𝓕φs1:List 𝓕φs:List 𝓕φs':List 𝓕φs2:List 𝓕h:∀ φ' ∈ φs, le φ φ' ∧ le φ' φhp:φs.Perm φs'ih:koszulSign q le (φs1 ++ φs ++ φs2) = koszulSign q le (φs1 ++ φs' ++ φs2)⊢ (φs1 ++ φs).Perm (φs1 ++ φs')
    exact List.Perm.append_left φs1 hpAll goals completed! 🐙