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refactor: Static Wick coef

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jstoobysmith 2024-12-20 12:37:44 +00:00
parent 3aa69ac44e
commit 3073aa0ab3
2 changed files with 27 additions and 32 deletions
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19
HepLean/PerturbationTheory/Wick/Signs/KoszulSign.lean
View file

@ -61,7 +61,7 @@ lemma koszulSignInsert_erase_boson {𝓕 : Type} (q : 𝓕 → FieldStatistic)
rw [koszulSignInsert, koszulSignInsert]
rw [koszulSignInsert_erase_boson q le r0 r ⟨n, Nat.succ_lt_succ_iff.mp h⟩ hr]
lemma koszulSign_erase_boson {𝓕 : Type} (q : 𝓕 → FieldStatistic) (le :𝓕 → 𝓕 → Prop)
lemma koszulSign_erase_boson {𝓕 : Type} (q : 𝓕 → FieldStatistic) (le : 𝓕 → 𝓕 → Prop)
[DecidableRel le] :
(r : List 𝓕) → (n : Fin r.length) → (heq : q (r.get n) = bosonic) →
koszulSign q le (r.eraseIdx n) = koszulSign q le r
@ -78,20 +78,17 @@ lemma koszulSign_erase_boson {𝓕 : Type} (q : 𝓕 → FieldStatistic) (le :
simp only [List.length_cons, List.get_eq_getElem, List.getElem_cons_succ, Fin.isValue,
List.eraseIdx_cons_succ]
intro h'
rw [koszulSign, koszulSign]
rw [koszulSign_erase_boson q le r ⟨n, Nat.succ_lt_succ_iff.mp h⟩]
rw [koszulSign, koszulSign, koszulSign_erase_boson q le r ⟨n, Nat.succ_lt_succ_iff.mp h⟩]
congr 1
rw [koszulSignInsert_erase_boson q le r0 r ⟨n, Nat.succ_lt_succ_iff.mp h⟩ h']
exact h'
lemma koszulSign_insertIdx [IsTotal 𝓕 le] [IsTrans 𝓕 le] (i : 𝓕) :
(r : List 𝓕) → (n : ℕ) → (hn : n ≤ r.length) →
koszulSign q le (List.insertIdx n i r) = insertSign q n i r
* koszulSign q le r
* insertSign q (insertionSortEquiv le (List.insertIdx n i r) ⟨n, by
koszulSign q le (List.insertIdx n i r) = insertSign q n i r * koszulSign q le r *
insertSign q (insertionSortEquiv le (List.insertIdx n i r) ⟨n, by
rw [List.length_insertIdx _ _ hn]
omega⟩) i
(List.insertionSort le (List.insertIdx n i r))
omega⟩) i (List.insertionSort le (List.insertIdx n i r))
| [], 0, h => by
simp [koszulSign, insertSign, superCommuteCoef, koszulSignInsert]
| [], n + 1, h => by
@ -188,11 +185,9 @@ lemma koszulSign_insertIdx [IsTotal 𝓕 le] [IsTrans 𝓕 le] (i : 𝓕) :
simp only [hc1 hn, ↓reduceIte]
rw [superCommuteCoef_comm]
· simp only [hn, ↓reduceIte, Fin.val_succ]
rw [insertSign_insert_lt]
rw [← mul_assoc]
rw [insertSign_insert_lt, ← mul_assoc]
congr 1
rw [superCommuteCoef_mul_self]
rw [koszulSignCons]
rw [superCommuteCoef_mul_self, koszulSignCons]
simp only [hc2 hn, ↓reduceIte]
exact Nat.le_of_not_lt hn
exact Nat.le_of_lt_succ (orderedInsertPos_lt_length le rs r0)

40
HepLean/PerturbationTheory/Wick/Signs/StaticWickCoef.lean
View file

@ -17,28 +17,29 @@ namespace Wick
open HepLean.List
open FieldStatistic
variable {𝓕 : Type} (q : 𝓕 → FieldStatistic) (le :𝓕 → 𝓕 → Prop) [DecidableRel le]
/-- The sign that appears in the static version of Wicks theorem.
This is actually equal to `superCommuteCoef q [r.get n] (r.take n)`, something
which will be proved in a lemma. -/
def staticWickCoef {I : Type} (q : I → Fin 2) (le1 :I → I → Prop) (r : List I)
[DecidableRel le1] (i : I) (n : Fin r.length) : ℂ :=
koszulSign le1 q r *
superCommuteCoef q [i] (List.take (↑((HepLean.List.insertionSortEquiv le1 r) n))
(List.insertionSort le1 r)) *
koszulSign le1 q (r.eraseIdx ↑n)
def staticWickCoef (r : List 𝓕) (i : 𝓕) (n : Fin r.length) : ℂ :=
koszulSign q le r *
superCommuteCoef q [i] (List.take (↑((HepLean.List.insertionSortEquiv le r) n))
(List.insertionSort le r)) *
koszulSign q le (r.eraseIdx ↑n)
lemma staticWickCoef_eq_q {I : Type} (q : I → Fin 2) (le1 :I → I → Prop) (r : List I)
[DecidableRel le1] (i : I) (n : Fin r.length)
lemma staticWickCoef_eq_q (r : List 𝓕) (i : 𝓕) (n : Fin r.length)
(hq : q i = q (r.get n)) :
staticWickCoef q le1 r i n =
koszulSign le1 q r *
superCommuteCoef q [r.get n] (List.take (↑(insertionSortEquiv le1 r n))
(List.insertionSort le1 r)) *
koszulSign le1 q (r.eraseIdx ↑n) := by
simp [staticWickCoef, superCommuteCoef, grade, hq]
staticWickCoef q le r i n =
koszulSign q le r *
superCommuteCoef q [r.get n] (List.take (↑(insertionSortEquiv le r n))
(List.insertionSort le r)) *
koszulSign q le (r.eraseIdx ↑n) := by
simp [staticWickCoef, superCommuteCoef, ofList, hq]
lemma insertIdx_eraseIdx {I : Type} :
(n : ℕ) → (r : List I) → (hn : n < r.length) →
lemma insertIdx_eraseIdx {I : Type} : (n : ℕ) → (r : List I) → (hn : n < r.length) →
List.insertIdx n (r.get ⟨n, hn⟩) (r.eraseIdx n) = r
| n, [], hn => by
simp at hn
@ -49,10 +50,9 @@ lemma insertIdx_eraseIdx {I : Type} :
List.eraseIdx_cons_succ, List.insertIdx_succ_cons, List.cons.injEq, true_and]
exact insertIdx_eraseIdx n r _
lemma staticWickCoef_eq_get {I : Type} (q : I → Fin 2) (le1 :I → I → Prop) (r : List I)
[DecidableRel le1] [IsTotal I le1] [IsTrans I le1] (i : I) (n : Fin r.length)
lemma staticWickCoef_eq_get [IsTotal 𝓕 le] [IsTrans 𝓕 le] (r : List 𝓕) (i : 𝓕) (n : Fin r.length)
(heq : q i = q (r.get n)) :
staticWickCoef q le1 r i n = superCommuteCoef q [r.get n] (r.take n) := by
staticWickCoef q le r i n = superCommuteCoef q [r.get n] (r.take n) := by
rw [staticWickCoef_eq_q]
let r' := r.eraseIdx ↑n
have hr : List.insertIdx n (r.get n) (r.eraseIdx n) = r := by
@ -61,7 +61,7 @@ lemma staticWickCoef_eq_get {I : Type} (q : I → Fin 2) (le1 :I → I → Prop)
lhs
lhs
rw [← hr]
rw [koszulSign_insertIdx q le1 (r.get n) ((r.eraseIdx ↑n)) n (by
rw [koszulSign_insertIdx q le (r.get n) ((r.eraseIdx ↑n)) n (by
rw [List.length_eraseIdx]
simp only [Fin.is_lt, ↓reduceIte]
omega)]