445 lines
18 KiB
Text
445 lines
18 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.PerturbationTheory.Koszul.KoszulSignInsert
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import HepLean.Mathematics.List.InsertionSort
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/-!
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# Koszul sign
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-/
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namespace Wick
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open HepLean.List
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open FieldStatistic
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variable {𝓕 : Type} (q : 𝓕 → FieldStatistic) (le : 𝓕 → 𝓕 → Prop) [DecidableRel le]
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/-- Gives a factor of `- 1` for every fermion-fermion (`q` is `1`) crossing that occurs when sorting
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a list of based on `r`. -/
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def koszulSign (q : 𝓕 → FieldStatistic) (le : 𝓕 → 𝓕 → Prop) [DecidableRel le] :
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List 𝓕 → ℂ
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| [] => 1
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| a :: l => koszulSignInsert q le a l * koszulSign q le l
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@[simp]
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lemma koszulSign_singleton (q : 𝓕 → FieldStatistic) (le : 𝓕 → 𝓕 → Prop) [DecidableRel le] (φ : 𝓕) :
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koszulSign q le [φ] = 1 := by
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simp [koszulSign, koszulSignInsert]
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lemma koszulSign_mul_self (l : List 𝓕) : koszulSign q le l * koszulSign q le l = 1 := by
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induction l with
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| nil => simp [koszulSign]
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| cons a l ih =>
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simp only [koszulSign]
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trans (koszulSignInsert q le a l * koszulSignInsert q le a l) *
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(koszulSign q le l * koszulSign q le l)
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· ring
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· rw [ih, koszulSignInsert_mul_self, mul_one]
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@[simp]
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lemma koszulSign_freeMonoid_of (φ : 𝓕) : koszulSign q le (FreeMonoid.of φ) = 1 := by
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change koszulSign q le [φ] = 1
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simp only [koszulSign, mul_one]
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rfl
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lemma koszulSignInsert_erase_boson {𝓕 : Type} (q : 𝓕 → FieldStatistic)
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(le : 𝓕 → 𝓕 → Prop) [DecidableRel le] (φ : 𝓕) :
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(φs : List 𝓕) → (n : Fin φs.length) → (heq : q (φs.get n) = bosonic) →
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koszulSignInsert q le φ (φs.eraseIdx n) = koszulSignInsert q le φ φs
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| [], _, _ => by
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simp
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| r1 :: r, ⟨0, h⟩, hr => by
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simp only [List.eraseIdx_zero, List.tail_cons]
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simp only [List.length_cons, Fin.zero_eta, List.get_eq_getElem, Fin.val_zero,
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List.getElem_cons_zero] at hr
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rw [koszulSignInsert]
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simp [hr]
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| r1 :: r, ⟨n + 1, h⟩, hr => by
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simp only [List.eraseIdx_cons_succ]
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rw [koszulSignInsert, koszulSignInsert]
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rw [koszulSignInsert_erase_boson q le φ r ⟨n, Nat.succ_lt_succ_iff.mp h⟩ hr]
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lemma koszulSign_erase_boson {𝓕 : Type} (q : 𝓕 → FieldStatistic) (le : 𝓕 → 𝓕 → Prop)
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[DecidableRel le] :
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(φs : List 𝓕) → (n : Fin φs.length) → (heq : q (φs.get n) = bosonic) →
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koszulSign q le (φs.eraseIdx n) = koszulSign q le φs
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| [], _ => by
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simp
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| φ :: φs, ⟨0, h⟩ => by
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simp only [List.length_cons, Fin.zero_eta, List.get_eq_getElem, Fin.val_zero,
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List.getElem_cons_zero, Fin.isValue, List.eraseIdx_zero, List.tail_cons, koszulSign]
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intro h
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rw [koszulSignInsert_boson]
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simp only [one_mul]
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exact h
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| φ :: φs, ⟨n + 1, h⟩ => by
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simp only [List.length_cons, List.get_eq_getElem, List.getElem_cons_succ, Fin.isValue,
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List.eraseIdx_cons_succ]
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intro h'
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rw [koszulSign, koszulSign, koszulSign_erase_boson q le φs ⟨n, Nat.succ_lt_succ_iff.mp h⟩]
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congr 1
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rw [koszulSignInsert_erase_boson q le φ φs ⟨n, Nat.succ_lt_succ_iff.mp h⟩ h']
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exact h'
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lemma koszulSign_insertIdx [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φ : 𝓕) :
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(φs : List 𝓕) → (n : ℕ) → (hn : n ≤ φs.length) →
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koszulSign q le (List.insertIdx n φ φs) = 𝓢(q φ, ofList q (φs.take n)) * koszulSign q le φs *
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𝓢(q φ, ofList q ((List.insertionSort le (List.insertIdx n φ φs)).take
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(insertionSortEquiv le (List.insertIdx n φ φs) ⟨n, by
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rw [List.length_insertIdx _ _]
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simp only [hn, ↓reduceIte]
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omega⟩)))
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| [], 0, h => by
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simp [koszulSign, koszulSignInsert]
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| [], n + 1, h => by
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simp at h
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| φ1 :: φs, 0, h => by
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simp only [List.insertIdx_zero, List.insertionSort, List.length_cons, Fin.zero_eta]
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rw [koszulSign]
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trans koszulSign q le (φ1 :: φs) * koszulSignInsert q le φ (φ1 :: φs)
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ring
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simp only [insertionSortEquiv, List.length_cons, Nat.succ_eq_add_one, List.insertionSort,
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orderedInsertEquiv, OrderIso.toEquiv_symm, Fin.symm_castOrderIso, HepLean.Fin.equivCons_trans,
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Equiv.trans_apply, HepLean.Fin.equivCons_zero, HepLean.Fin.finExtractOne_apply_eq,
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Fin.isValue, HepLean.Fin.finExtractOne_symm_inl_apply, RelIso.coe_fn_toEquiv,
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Fin.castOrderIso_apply, Fin.cast_mk, Fin.eta]
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conv_rhs =>
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enter [2,2, 2, 2]
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rw [orderedInsert_eq_insertIdx_orderedInsertPos]
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conv_rhs =>
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rhs
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rw [← ofList_take_insert]
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change 𝓢(q φ, ofList q ((List.insertionSort le (φ1 :: φs)).take
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(↑(orderedInsertPos le ((List.insertionSort le (φ1 :: φs))) φ))))
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rw [← koszulSignInsert_eq_exchangeSign_take q le]
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rw [ofList_take_zero]
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simp
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| φ1 :: φs, n + 1, h => by
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conv_lhs =>
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rw [List.insertIdx_succ_cons]
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rw [koszulSign]
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rw [koszulSign_insertIdx]
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conv_rhs =>
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rhs
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simp only [List.insertIdx_succ_cons]
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simp only [List.insertionSort, List.length_cons, insertionSortEquiv, Nat.succ_eq_add_one,
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Equiv.trans_apply, HepLean.Fin.equivCons_succ]
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erw [orderedInsertEquiv_fin_succ]
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simp only [Fin.eta, Fin.coe_cast]
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rhs
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simp [orderedInsert_eq_insertIdx_orderedInsertPos]
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conv_rhs =>
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lhs
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rw [ofList_take_succ_cons, map_mul, koszulSign]
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ring_nf
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conv_lhs =>
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lhs
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rw [mul_assoc, mul_comm]
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rw [mul_assoc]
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conv_rhs =>
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rw [mul_assoc, mul_assoc]
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congr 1
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let rs := (List.insertionSort le (List.insertIdx n φ φs))
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have hnsL : n < (List.insertIdx n φ φs).length := by
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rw [List.length_insertIdx _ _]
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simp only [List.length_cons, add_le_add_iff_right] at h
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simp only [h, ↓reduceIte]
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omega
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let ni : Fin rs.length := (insertionSortEquiv le (List.insertIdx n φ φs))
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⟨n, hnsL⟩
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let nro : Fin (rs.length + 1) :=
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⟨↑(orderedInsertPos le rs φ1), orderedInsertPos_lt_length le rs φ1⟩
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rw [koszulSignInsert_insertIdx, koszulSignInsert_cons]
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trans koszulSignInsert q le φ1 φs * (koszulSignCons q le φ1 φ *
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𝓢(q φ, ofList q (rs.take ni)))
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· simp only [rs, ni]
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ring
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trans koszulSignInsert q le φ1 φs * (𝓢(q φ, q φ1) *
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𝓢(q φ, ofList q ((List.insertIdx nro φ1 rs).take (nro.succAbove ni))))
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swap
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· simp only [rs, nro, ni]
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ring
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congr 1
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simp only [Fin.succAbove]
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have hns : rs.get ni = φ := by
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simp only [Fin.eta, rs]
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rw [← insertionSortEquiv_get]
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simp only [Function.comp_apply, Equiv.symm_apply_apply, List.get_eq_getElem, ni]
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simp_all only [List.length_cons, add_le_add_iff_right, List.getElem_insertIdx_self]
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have hc1 (hninro : ni.castSucc < nro) : ¬ le φ1 φ := by
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rw [← hns]
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exact lt_orderedInsertPos_rel le φ1 rs ni hninro
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have hc2 (hninro : ¬ ni.castSucc < nro) : le φ1 φ := by
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rw [← hns]
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refine gt_orderedInsertPos_rel le φ1 rs ?_ ni hninro
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exact List.sorted_insertionSort le (List.insertIdx n φ φs)
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by_cases hn : ni.castSucc < nro
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· simp only [hn, ↓reduceIte, Fin.coe_castSucc]
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rw [ofList_take_insertIdx_gt]
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swap
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· exact hn
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congr 1
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rw [koszulSignCons_eq_exchangeSign]
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simp only [hc1 hn, ↓reduceIte]
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rw [exchangeSign_symm]
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· simp only [hn, ↓reduceIte, Fin.val_succ]
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rw [ofList_take_insertIdx_le, map_mul, ← mul_assoc]
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congr 1
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rw [exchangeSign_mul_self, koszulSignCons]
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simp only [hc2 hn, ↓reduceIte]
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exact Nat.le_of_not_lt hn
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exact Nat.le_of_lt_succ (orderedInsertPos_lt_length le rs φ1)
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· exact Nat.le_of_lt_succ h
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· exact Nat.le_of_lt_succ h
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lemma insertIdx_eraseIdx {I : Type} : (n : ℕ) → (r : List I) → (hn : n < r.length) →
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List.insertIdx n (r.get ⟨n, hn⟩) (r.eraseIdx n) = r
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| n, [], hn => by
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simp at hn
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| 0, r0 :: r, hn => by
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simp
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| n + 1, r0 :: r, hn => by
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simp only [List.length_cons, List.get_eq_getElem, List.getElem_cons_succ,
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List.eraseIdx_cons_succ, List.insertIdx_succ_cons, List.cons.injEq, true_and]
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exact insertIdx_eraseIdx n r _
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lemma koszulSign_eraseIdx [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φs : List 𝓕) (n : Fin φs.length) :
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koszulSign q le (φs.eraseIdx n) = koszulSign q le φs * 𝓢(q (φs.get n), ofList q (φs.take n)) *
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𝓢(q (φs.get n), ofList q (List.take (↑(insertionSortEquiv le φs n))
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(List.insertionSort le φs))) := by
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let φs' := φs.eraseIdx ↑n
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have hφs : List.insertIdx n (φs.get n) φs' = φs := by
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exact insertIdx_eraseIdx n.1 φs n.prop
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conv_rhs =>
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lhs
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lhs
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rw [← hφs]
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rw [koszulSign_insertIdx q le (φs.get n) ((φs.eraseIdx ↑n)) n (by
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rw [List.length_eraseIdx]
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simp only [Fin.is_lt, ↓reduceIte]
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omega)]
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rhs
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enter [2, 2, 2]
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rw [hφs]
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conv_rhs =>
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enter [1, 1, 2, 2, 2, 1, 1]
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rw [insertionSortEquiv_congr _ _ hφs]
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simp only [instCommGroup.eq_1, List.get_eq_getElem, Equiv.trans_apply, RelIso.coe_fn_toEquiv,
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Fin.castOrderIso_apply, Fin.cast_mk, Fin.eta, Fin.coe_cast]
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trans koszulSign q le (φs.eraseIdx ↑n) *
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(𝓢(q φs[↑n], ofList q ((φs.eraseIdx ↑n).take n)) * 𝓢(q φs[↑n], ofList q (List.take (↑n) φs))) *
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(𝓢(q φs[↑n], ofList q ((List.insertionSort le φs).take (↑((insertionSortEquiv le φs) n)))) *
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𝓢(q φs[↑n], ofList q (List.take (↑((insertionSortEquiv le φs) n)) (List.insertionSort le φs))))
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swap
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· simp only [Fin.getElem_fin]
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rw [Equiv.trans_apply, Equiv.trans_apply]
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simp only [instCommGroup.eq_1, mul_one, Fin.castOrderIso,
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Equiv.coe_fn_mk, Fin.cast_mk, Fin.eta, Fin.coe_cast]
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ring
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conv_rhs =>
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rhs
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rw [exchangeSign_mul_self]
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simp only [instCommGroup.eq_1, Fin.getElem_fin, mul_one]
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conv_rhs =>
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rhs
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rw [ofList_take_eraseIdx, exchangeSign_mul_self]
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simp
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lemma koszulSign_eraseIdx_insertionSortMinPos [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φ : 𝓕) (φs : List 𝓕) :
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koszulSign q le ((φ :: φs).eraseIdx (insertionSortMinPos le φ φs)) = koszulSign q le (φ :: φs)
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* 𝓢(q (insertionSortMin le φ φs), ofList q ((φ :: φs).take (insertionSortMinPos le φ φs))) := by
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rw [koszulSign_eraseIdx]
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conv_lhs =>
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rhs
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rhs
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lhs
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simp [insertionSortMinPos]
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erw [Equiv.apply_symm_apply]
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simp only [instCommGroup.eq_1, List.get_eq_getElem, List.length_cons, List.insertionSort,
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List.take_zero, ofList_empty, exchangeSign_bosonic, mul_one, mul_eq_mul_left_iff]
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apply Or.inl
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rfl
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lemma koszulSign_swap_eq_rel_cons {ψ φ : 𝓕}
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(h1 : le φ ψ) (h2 : le ψ φ) (φs' : List 𝓕) :
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koszulSign q le (φ :: ψ :: φs') = koszulSign q le (ψ :: φ :: φs') := by
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simp only [Wick.koszulSign, ← mul_assoc, mul_eq_mul_right_iff]
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left
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rw [mul_comm]
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simp [Wick.koszulSignInsert, h1, h2]
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lemma koszulSign_swap_eq_rel {ψ φ : 𝓕} (h1 : le φ ψ) (h2 : le ψ φ) : (φs φs' : List 𝓕) →
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koszulSign q le (φs ++ φ :: ψ :: φs') = koszulSign q le (φs ++ ψ :: φ :: φs')
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| [], φs' => by
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simp only [List.nil_append]
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exact koszulSign_swap_eq_rel_cons q le h1 h2 φs'
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| φ'' :: φs, φs' => by
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simp only [List.cons_append, koszulSign]
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rw [koszulSign_swap_eq_rel h1 h2]
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congr 1
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apply Wick.koszulSignInsert_eq_perm
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exact List.Perm.append_left φs (List.Perm.swap ψ φ φs')
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lemma koszulSign_eq_rel_eq_stat_append {ψ φ : 𝓕} [IsTrans 𝓕 le]
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(h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs : List 𝓕) →
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koszulSign q le (φ :: ψ :: φs) = koszulSign q le φs := by
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intro φs
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simp only [koszulSign, ← mul_assoc]
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trans 1 * koszulSign q le φs
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swap
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simp only [one_mul]
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congr
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simp only [koszulSignInsert, ite_mul, neg_mul]
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simp_all only [and_self, ite_true]
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rw [koszulSignInsert_eq_rel_eq_stat q le h1 h2 hq]
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simp
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lemma koszulSign_eq_rel_eq_stat {ψ φ : 𝓕} [IsTrans 𝓕 le]
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(h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs' φs : List 𝓕) →
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koszulSign q le (φs' ++ φ :: ψ :: φs) = koszulSign q le (φs' ++ φs)
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| [], φs => by
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simp only [List.nil_append]
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exact koszulSign_eq_rel_eq_stat_append q le h1 h2 hq φs
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| φ'' :: φs', φs => by
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simp only [List.cons_append, koszulSign]
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rw [koszulSign_eq_rel_eq_stat h1 h2 hq φs' φs]
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simp only [mul_eq_mul_right_iff]
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left
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trans koszulSignInsert q le φ'' (φ :: ψ :: (φs' ++ φs))
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apply koszulSignInsert_eq_perm
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refine List.Perm.symm (List.perm_cons_append_cons φ ?_)
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exact List.Perm.symm List.perm_middle
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rw [koszulSignInsert_eq_remove_same_stat_append q le]
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exact h1
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exact h2
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exact hq
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lemma koszulSign_of_sorted : (φs : List 𝓕)
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→ (hs : List.Sorted le φs) → koszulSign q le φs = 1
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| [], _ => by
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simp [koszulSign]
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| φ :: φs, h => by
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simp only [koszulSign]
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simp only [List.sorted_cons] at h
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rw [koszulSign_of_sorted φs h.2]
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simp only [mul_one]
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exact koszulSignInsert_of_le_mem _ _ _ _ h.1
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@[simp]
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lemma koszulSign_of_insertionSort [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φs : List 𝓕) :
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koszulSign q le (List.insertionSort le φs) = 1 := by
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apply koszulSign_of_sorted
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exact List.sorted_insertionSort le φs
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lemma koszulSign_of_append_eq_insertionSort_left [IsTotal 𝓕 le] [IsTrans 𝓕 le] :
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(φs φs' : List 𝓕) → koszulSign q le (φs ++ φs') =
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koszulSign q le (List.insertionSort le φs ++ φs') * koszulSign q le φs
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| φs, [] => by
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simp
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| φs, φ :: φs' => by
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have h1 : (φs ++ φ :: φs') = List.insertIdx φs.length φ (φs ++ φs') := by
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rw [insertIdx_length_fst_append]
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have h2 : (List.insertionSort le φs ++ φ :: φs') =
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List.insertIdx (List.insertionSort le φs).length φ (List.insertionSort le φs ++ φs') := by
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rw [insertIdx_length_fst_append]
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rw [h1, h2]
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rw [koszulSign_insertIdx]
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simp only [instCommGroup.eq_1, List.take_left', List.length_insertionSort]
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rw [koszulSign_insertIdx]
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||
simp only [mul_assoc, instCommGroup.eq_1, List.length_insertionSort, List.take_left',
|
||
ofList_insertionSort, mul_eq_mul_left_iff]
|
||
left
|
||
rw [koszulSign_of_append_eq_insertionSort_left φs φs']
|
||
simp only [mul_assoc, mul_eq_mul_left_iff]
|
||
left
|
||
simp only [mul_comm, mul_eq_mul_left_iff]
|
||
left
|
||
congr 3
|
||
· have h2 : (List.insertionSort le φs ++ φ :: φs') =
|
||
List.insertIdx φs.length φ (List.insertionSort le φs ++ φs') := by
|
||
rw [← insertIdx_length_fst_append]
|
||
simp
|
||
rw [insertionSortEquiv_congr _ _ h2.symm]
|
||
simp only [Equiv.trans_apply, RelIso.coe_fn_toEquiv, Fin.castOrderIso_apply, Fin.cast_mk,
|
||
Fin.coe_cast]
|
||
rw [insertionSortEquiv_insertionSort_append]
|
||
simp only [finCongr_apply, Fin.coe_cast]
|
||
rw [insertionSortEquiv_congr _ _ h1.symm]
|
||
simp
|
||
· rw [insertIdx_length_fst_append]
|
||
rw [show φs.length = (List.insertionSort le φs).length by simp]
|
||
rw [insertIdx_length_fst_append]
|
||
symm
|
||
apply insertionSort_insertionSort_append
|
||
· simp
|
||
· simp
|
||
|
||
lemma koszulSign_of_append_eq_insertionSort [IsTotal 𝓕 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'=> by
|
||
simp only [List.nil_append]
|
||
exact koszulSign_of_append_eq_insertionSort_left q le φs φs'
|
||
| φ'' :: φs'', φs, φs' => by
|
||
simp only [List.cons_append, koszulSign]
|
||
rw [koszulSign_of_append_eq_insertionSort φs'' φs φs', ← mul_assoc]
|
||
congr 2
|
||
apply koszulSignInsert_eq_perm
|
||
refine (List.perm_append_right_iff φs').mpr ?_
|
||
refine List.Perm.append_left φs'' ?_
|
||
exact List.Perm.symm (List.perm_insertionSort le φs)
|
||
|
||
/-!
|
||
|
||
# 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) := by
|
||
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)
|
||
change motive φs φs' hp
|
||
apply List.Perm.recOn
|
||
· simp [motive]
|
||
· intro x l1 l2 h ih hxφ
|
||
simp_all only [List.mem_cons, or_true, and_self, implies_true, nonempty_prop, forall_const,
|
||
forall_eq_or_imp, List.cons_append, motive]
|
||
simp only [koszulSign, ih, mul_eq_mul_right_iff]
|
||
left
|
||
apply koszulSignInsert_eq_perm
|
||
exact (List.perm_append_right_iff φs2).mpr h
|
||
· intro x y l h
|
||
simp_all only [List.mem_cons, forall_eq_or_imp, List.cons_append]
|
||
apply Wick.koszulSign_swap_eq_rel_cons
|
||
exact IsTrans.trans y φ x h.1.2 h.2.1.1
|
||
exact IsTrans.trans x φ y h.2.1.2 h.1.1
|
||
· intro l1 l2 l3 h1 h2 ih1 ih2 h
|
||
simp_all only [and_self, implies_true, nonempty_prop, forall_const, motive]
|
||
refine (ih2 ?_)
|
||
intro φ' hφ
|
||
refine h φ' ?_
|
||
exact (List.Perm.mem_iff (id (List.Perm.symm h1))).mp hφ
|
||
|
||
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 => by
|
||
simp only [List.nil_append]
|
||
exact koszulSign_perm_eq_append q le φ φs φs' φs2 hp h
|
||
| φ1 :: φs1, φs, φs', φs2, h, hp => by
|
||
simp only [List.cons_append, koszulSign]
|
||
have ih := koszulSign_perm_eq φ φs1 φs φs' φs2 h hp
|
||
rw [ih]
|
||
congr 1
|
||
apply koszulSignInsert_eq_perm
|
||
refine (List.perm_append_right_iff φs2).mpr ?_
|
||
exact List.Perm.append_left φs1 hp
|
||
|
||
end Wick
|