refactor: Reorganize files

HepLean/PerturbationTheory/Wick/Signs/StaticWickCoef.lean

@@ -0,0 +1,91 @@
+/-
+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
+-/
+import Mathlib.Algebra.FreeAlgebra
+import Mathlib.Algebra.Lie.OfAssociative
+import Mathlib.Analysis.Complex.Basic
+import HepLean.PerturbationTheory.Wick.Signs.KoszulSign
+/-!
+
+# Koszul sign insert
+
+-/
+
+namespace Wick
+
+open HepLean.List
+
+def superCommuteCoefLE {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)
+
+lemma superCommuteCoefLE_eq_q {I : Type} (q : I → Fin 2) (le1 :I → I → Prop) (r : List I)
+    [DecidableRel le1] (i : I) (n : Fin r.length)
+    (hq : q i = q (r.get n)) :
+    superCommuteCoefLE 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 [superCommuteCoefLE, superCommuteCoef, grade, hq]
+
+
+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
+  | 0, r0 :: r, hn => by
+    simp
+  | n + 1, r0 :: r, hn => by
+    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]
+    exact insertIdx_eraseIdx n r _
+
+lemma superCommuteCoefLE_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)
+    (heq : q i = q (r.get n)) :
+    superCommuteCoefLE q le1 r i n = superCommuteCoef q [r.get n] (r.take n) := by
+  rw [superCommuteCoefLE_eq_q]
+  let r' := r.eraseIdx ↑n
+  have hr : List.insertIdx n (r.get n) (r.eraseIdx n) = r := by
+    exact insertIdx_eraseIdx n.1 r n.prop
+  conv_lhs =>
+    lhs
+    lhs
+    rw [← hr]
+    rw [koszulSign_insertIdx q le1 (r.get n) ((r.eraseIdx ↑n)) n (by
+      rw [List.length_eraseIdx]
+      simp only [Fin.is_lt, ↓reduceIte]
+      omega)]
+    rhs
+    rhs
+    rw [hr]
+  conv_lhs =>
+    lhs
+    lhs
+    rhs
+    enter [2, 1, 1]
+    rw [insertionSortEquiv_congr _ _ hr]
+  simp only [List.get_eq_getElem, Equiv.trans_apply, RelIso.coe_fn_toEquiv, Fin.castOrderIso_apply,
+    Fin.cast_mk, Fin.eta, Fin.coe_cast]
+  conv_lhs =>
+    lhs
+    rw [mul_assoc]
+    rhs
+    rw [insertSign]
+    rw [superCommuteCoef_mul_self]
+  simp only [mul_one]
+  rw [mul_assoc]
+  rw [koszulSign_mul_self]
+  simp only [mul_one]
+  rw [insertSign_eraseIdx]
+  rfl
+  exact heq
+
+end Wick