refactor: Insert Sign
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jstoobysmith
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3 changed files with 62 additions and 42 deletions
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@ -7,17 +7,21 @@ import HepLean.Mathematics.List
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import HepLean.PerturbationTheory.Wick.Signs.SuperCommuteCoef
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/-!
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# Koszul signs and ordering for lists and algebras
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# Insert 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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/-- The sign associated with inserting `r0` into `r` at the position `n`.
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That is the sign associated with commuting `r0` with `List.take n r`. -/
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def insertSign {I : Type} (q : I → Fin 2) (n : ℕ) (r0 : I) (r : List I) : ℂ :=
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superCommuteCoef q [r0] (List.take n r)
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section
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/-!
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## Basic properties of lists
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To be replaced with Mathlib or Lean definitions when/where appropraite.
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-/
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lemma take_insert_same {I : Type} (i : I) :
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(n : ℕ) → (r : List I) →
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@ -28,12 +32,6 @@ lemma take_insert_same {I : Type} (i : I) :
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simp only [List.insertIdx_succ_cons, List.take_succ_cons, List.cons.injEq, true_and]
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exact take_insert_same i n as
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lemma insertSign_insert {I : Type} (q : I → Fin 2) (n : ℕ)
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(r0 : I) (r : List I) : insertSign q n r0 r = insertSign q n r0 (List.insertIdx n r0 r) := by
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simp only [insertSign]
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congr 1
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rw [take_insert_same]
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lemma take_eraseIdx_same {I : Type} :
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(n : ℕ) → (r : List I) →
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List.take n (List.eraseIdx r n) = List.take n r
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@ -43,22 +41,6 @@ lemma take_eraseIdx_same {I : Type} :
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simp only [List.eraseIdx_cons_succ, List.take_succ_cons, List.cons.injEq, true_and]
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exact take_eraseIdx_same n as
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lemma insertSign_eraseIdx {I : Type} (q : I → Fin 2) (n : ℕ)
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(r0 : I) (r : List I) : insertSign q n r0 (r.eraseIdx n) = insertSign q n r0 r := by
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simp only [insertSign]
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congr 1
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rw [take_eraseIdx_same]
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lemma insertSign_zero {I : Type} (q : I → Fin 2) (r0 : I) (r : List I) :
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insertSign q 0 r0 r = 1 := by
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simp [insertSign, superCommuteCoef]
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lemma insertSign_succ_cons {I : Type} (q : I → Fin 2) (n : ℕ)
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(r0 r1 : I) (r : List I) : insertSign q (n + 1) r0 (r1 :: r) =
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superCommuteCoef q [r0] [r1] * insertSign q n r0 r := by
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simp only [insertSign, List.take_succ_cons]
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rw [superCommuteCoef_cons]
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lemma take_insert_gt {I : Type} (i : I) :
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(n m : ℕ) → (h : n < m) → (r : List I) →
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List.take n (List.insertIdx m i r) = List.take n r
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@ -69,13 +51,6 @@ lemma take_insert_gt {I : Type} (i : I) :
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simp only [List.insertIdx_succ_cons, List.take_succ_cons, List.cons.injEq, true_and]
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refine take_insert_gt i n m (Nat.succ_lt_succ_iff.mp h) as
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lemma insertSign_insert_gt {I : Type} (q : I → Fin 2) (n m : ℕ)
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(r0 r1 : I) (r : List I) (hn : n < m) :
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insertSign q n r0 (List.insertIdx m r1 r) = insertSign q n r0 r := by
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rw [insertSign, insertSign]
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congr 1
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exact take_insert_gt r1 n m hn r
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lemma take_insert_let {I : Type} (i : I) :
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(n m : ℕ) → (h : m ≤ n) → (r : List I) → (hm : m ≤ r.length) →
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(List.take (n + 1) (List.insertIdx m i r)).Perm (i :: List.take n r)
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@ -91,20 +66,65 @@ lemma take_insert_let {I : Type} (i : I) :
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refine List.Perm.cons a ?_
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exact take_insert_let i n m (Nat.le_of_succ_le_succ h) as (Nat.le_of_succ_le_succ hm)
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lemma insertSign_insert_lt_eq_insertSort {I : Type} (q : I → Fin 2) (n m : ℕ)
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(r0 r1 : I) (r : List I) (hn : m ≤ n) (hm : m ≤ r.length) :
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end
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/-!
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## Insert sign
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-/
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section InsertSign
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variable {𝓕 : Type} (q : 𝓕 → FieldStatistic)
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/-- The sign associated with inserting `r0` into `r` at the position `n`.
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That is the sign associated with commuting `r0` with `List.take n r`. -/
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def insertSign (n : ℕ) (r0 : 𝓕) (r : List 𝓕) : ℂ :=
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superCommuteCoef q [r0] (List.take n r)
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lemma insertSign_insert (n : ℕ) (r0 : 𝓕) (r : List 𝓕) :
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insertSign q n r0 r = insertSign q n r0 (List.insertIdx n r0 r) := by
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simp only [insertSign]
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congr 1
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rw [take_insert_same]
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lemma insertSign_eraseIdx (n : ℕ) (r0 : 𝓕) (r : List 𝓕) :
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insertSign q n r0 (r.eraseIdx n) = insertSign q n r0 r := by
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simp only [insertSign]
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congr 1
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rw [take_eraseIdx_same]
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lemma insertSign_zero (r0 : 𝓕) (r : List 𝓕) : insertSign q 0 r0 r = 1 := by
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simp [insertSign, superCommuteCoef]
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lemma insertSign_succ_cons (n : ℕ) (r0 r1 : 𝓕) (r : List 𝓕) : insertSign q (n + 1) r0 (r1 :: r) =
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superCommuteCoef q [r0] [r1] * insertSign q n r0 r := by
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simp only [insertSign, List.take_succ_cons]
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rw [superCommuteCoef_cons]
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lemma insertSign_insert_gt (n m : ℕ) (r0 r1 : 𝓕) (r : List 𝓕) (hn : n < m) :
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insertSign q n r0 (List.insertIdx m r1 r) = insertSign q n r0 r := by
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rw [insertSign, insertSign]
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congr 1
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exact take_insert_gt r1 n m hn r
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lemma insertSign_insert_lt_eq_insertSort (n m : ℕ) (r0 r1 : 𝓕) (r : List 𝓕) (hn : m ≤ n)
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(hm : m ≤ r.length) :
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insertSign q (n + 1) r0 (List.insertIdx m r1 r) = insertSign q (n + 1) r0 (r1 :: r) := by
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rw [insertSign, insertSign]
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apply superCommuteCoef_perm_snd
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simp only [List.take_succ_cons]
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refine take_insert_let r1 n m hn r hm
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lemma insertSign_insert_lt {I : Type} (q : I → Fin 2) (n m : ℕ)
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(r0 r1 : I) (r : List I) (hn : m ≤ n) (hm : m ≤ r.length) :
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lemma insertSign_insert_lt (n m : ℕ) (r0 r1 : 𝓕) (r : List 𝓕) (hn : m ≤ n) (hm : m ≤ r.length) :
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insertSign q (n + 1) r0 (List.insertIdx m r1 r) = superCommuteCoef q [r0] [r1] *
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insertSign q n r0 r := by
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rw [insertSign_insert_lt_eq_insertSort, insertSign_succ_cons]
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exact hn
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exact hm
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end InsertSign
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end Wick
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@ -9,7 +9,7 @@ import Mathlib.Analysis.Complex.Basic
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import HepLean.PerturbationTheory.Wick.Signs.KoszulSign
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/-!
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# Koszul sign insert
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# Static wick coefficent
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-/
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@ -3,16 +3,16 @@ 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.Mathematics.List
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import HepLean.PerturbationTheory.FieldStatistics
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/-!
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# Koszul signs and ordering for lists and algebras
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# Super commutation coefficent.
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This is a complex number which is `-1` when commuting two fermionic operators and `1` otherwise.
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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)
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