feat: Fill in sorries
This commit is contained in:
jstoobysmith
parent
3123e831d8
commit
ab7da149c6
5 changed files with 1432 additions and 454 deletions
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@ -20,6 +20,7 @@ import HepLean.PerturbationTheory.Wick.Koszul.Order
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-/
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namespace Wick
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open HepLean.List
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def grade {I : Type} (q : I → Fin 2) : (l : List I) → Fin 2
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| [] => 0
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@ -58,13 +59,13 @@ lemma grade_orderedInsert {I : Type} (q : I → Fin 2) (le1 : I → I → Prop)
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induction l with
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| nil => simp
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| cons j l ih =>
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simp
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simp only [List.orderedInsert]
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by_cases hij : le1 i j
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· simp [hij]
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· simp [hij]
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· simp only [hij, ↓reduceIte]
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rw [grade]
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rw [ih]
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simp [grade]
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simp only [grade, Fin.isValue]
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have h1 (a b c : Fin 2) : (if a = if b = c then 0 else 1 then (0 : Fin 2) else 1) = if b = if a = c then 0 else 1 then 0 else 1 := by
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fin_cases a <;> fin_cases b <;> fin_cases c <;> rfl
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exact h1 _ _ _
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@ -75,14 +76,75 @@ lemma grade_insertionSort {I : Type} (q : I → Fin 2) (le1 : I → I → Prop)
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induction l with
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| nil => simp
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| cons j l ih =>
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simp [grade]
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simp only [List.insertionSort, grade, Fin.isValue]
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rw [grade_orderedInsert]
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simp [grade]
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simp only [grade, Fin.isValue]
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rw [ih]
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lemma grade_count {I : Type} (q : I → Fin 2) (l : List I) :
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grade q l = if Nat.mod (List.countP (fun i => decide (q i = 1)) l) 2 = 0 then 0 else 1 := by
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induction l with
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| nil => simp
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| cons r0 r ih =>
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simp only [grade, Fin.isValue]
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rw [List.countP_cons]
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simp only [Fin.isValue, decide_eq_true_eq]
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rw [ih]
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by_cases h: q r0 = 1
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· simp only [h, Fin.isValue, ↓reduceIte]
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split
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next h1 =>
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simp_all only [Fin.isValue, ↓reduceIte, one_ne_zero]
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split
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next h2 =>
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simp_all only [Fin.isValue, one_ne_zero]
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have ha (a : ℕ) (ha : a % 2 = 0) : (a + 1) % 2 ≠ 0 := by
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omega
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exact ha (List.countP (fun i => decide (q i = 1)) r) h1 h2
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next h2 => simp_all only [Fin.isValue]
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next h1 =>
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simp_all only [Fin.isValue, ↓reduceIte]
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split
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next h2 => simp_all only [Fin.isValue]
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next h2 =>
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simp_all only [Fin.isValue, zero_ne_one]
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have ha (a : ℕ) (ha : ¬ a % 2 = 0) : (a + 1) % 2 = 0 := by
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omega
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exact h2 (ha (List.countP (fun i => decide (q i = 1)) r) h1)
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· have h0 : q r0 = 0 := by omega
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simp only [h0, Fin.isValue, zero_ne_one, ↓reduceIte, add_zero]
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by_cases hn : (List.countP (fun i => decide (q i = 1)) r).mod 2 = 0
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· simp [hn]
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· simp [hn]
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lemma grade_perm {I : Type} (q : I → Fin 2) {l l' : List I} (h : l.Perm l') :
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grade q l = grade q l' := by
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rw [grade_count, grade_count, h.countP_eq]
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def superCommuteCoef {I : Type} (q : I → Fin 2) (la lb : List I) : ℂ :=
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if grade q la = 1 ∧ grade q lb = 1 then - 1 else 1
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lemma superCommuteCoef_comm {I : Type} (q : I → Fin 2) (la lb : List I) :
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superCommuteCoef q la lb = superCommuteCoef q lb la := by
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simp only [superCommuteCoef, Fin.isValue]
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congr 1
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exact Eq.propIntro (fun a => id (And.symm a)) fun a => id (And.symm a)
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lemma superCommuteCoef_perm_snd {I : Type} (q : I → Fin 2) (la lb lb' : List I)
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(h : lb.Perm lb') :
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superCommuteCoef q la lb = superCommuteCoef q la lb' := by
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rw [superCommuteCoef, superCommuteCoef, grade_perm q h ]
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lemma superCommuteCoef_mul_self {I : Type} (q : I → Fin 2) (l lb : List I) :
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superCommuteCoef q l lb * superCommuteCoef q l lb = 1 := by
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simp only [superCommuteCoef, Fin.isValue, mul_ite, mul_neg, mul_one]
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have ha (a b : Fin 2) : (if a = 1 ∧ b = 1 then -if a = 1 ∧ b = 1 then -1 else 1
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else if a = 1 ∧ b = 1 then -1 else 1) = (1 : ℂ) := by
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fin_cases a <;> fin_cases b
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any_goals rfl
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simp
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exact ha (grade q l) (grade q lb)
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lemma superCommuteCoef_empty {I : Type} (q : I → Fin 2) (la : List I) :
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superCommuteCoef q la [] = 1 := by
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simp only [superCommuteCoef, Fin.isValue, grade_empty, zero_ne_one, and_false, ↓reduceIte]
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@ -109,6 +171,12 @@ lemma superCommuteCoef_append {I : Type} (q : I → Fin 2) (la lb lc : List I)
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omega
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simp [ha]
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lemma superCommuteCoef_cons {I : Type} (q : I → Fin 2) (i : I) (la lb : List I) :
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superCommuteCoef q la (i :: lb) = superCommuteCoef q la [i] * superCommuteCoef q la lb := by
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trans superCommuteCoef q la ([i] ++ lb)
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simp only [List.singleton_append]
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rw [superCommuteCoef_append]
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def superCommuteCoefM {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
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(q : I → Fin 2) (l : List (Σ i, f i)) (r : List I) : ℂ :=
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(if grade (fun i => q i.fst) l = 1 ∧ grade q r = 1 then -1 else 1)
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@ -118,13 +186,6 @@ lemma superCommuteCoefM_empty {I : Type} {f : I → Type} [∀ i, Fintype (f i)
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superCommuteCoefM q l [] = 1 := by
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simp [superCommuteCoefM]
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lemma koszulSign_first_remove {I : Type} (q : I → Fin 2) (le1 :I → I → Prop) (l : List I)
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[DecidableRel le1] (a : I) :
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let n0 := ((HepLean.List.insertionSortEquiv le1 (a :: l)).symm ⟨0, by
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rw [List.length_insertionSort]; simp⟩)
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koszulSign le1 q (a :: l) = superCommuteCoef q [(a :: l).get n0] (List.take n0 (a :: l)) *
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koszulSign le1 q ((a :: l).eraseIdx n0) := by sorry
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def superCommuteCoefLE {I : Type} (q : I → Fin 2) (le1 :I → I → Prop) (r : List I)
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[DecidableRel le1] (i : I) (n : Fin r.length) : ℂ :=
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koszulSign le1 q r *
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@ -132,124 +193,37 @@ def superCommuteCoefLE {I : Type} (q : I → Fin 2) (le1 :I → I → Prop) (r
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(List.insertionSort le1 r)) *
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koszulSign le1 q (r.eraseIdx ↑n)
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lemma superCommuteCoefLE_eq_q {I : Type} (q : I → Fin 2) (le1 :I → I → Prop) (r : List I)
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[DecidableRel le1] (i : I) (n : Fin r.length)
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(hq : q i = q (r.get n)) :
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superCommuteCoefLE q le1 r i n =
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koszulSign le1 q r *
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superCommuteCoef q [r.get n] (List.take (↑(insertionSortEquiv le1 r n))
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(List.insertionSort le1 r)) *
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koszulSign le1 q (r.eraseIdx ↑n) := by
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simp [superCommuteCoefLE, superCommuteCoef, grade, hq]
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def natTestQ' : ℕ → Fin 2 := fun n => if n % 2 = 0 then 0 else 1
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#eval superCommuteCoefLE (natTestQ') (fun i j => i ≤ j) [5, 4, 4, 3, 0] 0 ⟨0, by simp⟩
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#eval koszulSign (fun i j => i ≤ j) (natTestQ') [5, 4, 4, 3, 0]
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lemma koszulSignInsert_eq_superCommuteCoef{I : Type} (q : I → Fin 2) (le1 : I → I → Prop) (r : List I)
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(a : I) [DecidableRel le1] (i : I) :
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koszulSignInsert le1 q a r = superCommuteCoef q [i]
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(List.take (↑((HepLean.List.insertionSortEquiv le1 (a :: r)) ⟨0, by simp⟩))
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(List.orderedInsert le1 a (List.insertionSort le1 r))) := by
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sorry
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lemma superCommuteCoefLE_zero {I : Type} (q : I → Fin 2) (le1 : I → I → Prop) (r : List I)
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(a : I)
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[DecidableRel le1] (i : I) :
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superCommuteCoefLE q le1 (a :: r) i ⟨0, Nat.zero_lt_succ r.length⟩ = 1 := by
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simp [superCommuteCoefLE]
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simp [koszulSign]
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trans koszulSignInsert le1 q a r * (koszulSign le1 q r * koszulSign le1 q r) *
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superCommuteCoef q [i]
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(List.take (↑((HepLean.List.insertionSortEquiv le1 (a :: r)) ⟨0, Nat.zero_lt_succ r.length⟩))
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(List.orderedInsert le1 a (List.insertionSort le1 r)))
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· ring_nf
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rfl
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rw [koszulSign_mul_self]
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simp
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sorry
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lemma superCommuteCoefLE_eq_get_boson {I : Type} (q : I → Fin 2) (le1 :I → I → Prop) (r : List I)
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[DecidableRel le1] (i : I) (hi : q i = 0 ) (n : Fin r.length)
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(heq : q i = q (r.get n)) :
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superCommuteCoefLE q le1 r i n = superCommuteCoef q [r.get n] (r.take n) := by
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simp [superCommuteCoefLE, superCommuteCoef]
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simp [grade, hi]
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simp [hi] at heq
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simp [heq]
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rw [koszulSign_erase_boson]
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rw [koszulSign_mul_self]
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exact id (Eq.symm heq)
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lemma koszulSignInsert_eq_filter {I : Type} (q : I → Fin 2) (le1 : I → I → Prop) [DecidableRel le1] (r0 : I)
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: (r : List I) →
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koszulSignInsert le1 q r0 r = koszulSignInsert le1 q r0 (List.filter (fun i => decide (¬ le1 r0 i)) r)
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| [] => by
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simp [koszulSignInsert]
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| r1 :: r => by
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dsimp [koszulSignInsert]
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simp [List.filter]
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dsimp only [koszulSignInsert, Fin.isValue]
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simp only [Fin.isValue, List.filter, decide_not]
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by_cases h : le1 r0 r1
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· simp [h]
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· simp only [h, ↓reduceIte, decide_True, Bool.not_true]
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rw [koszulSignInsert_eq_filter]
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congr
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simp
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· simp [h]
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dsimp [koszulSignInsert]
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simp [h]
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· simp only [h, ↓reduceIte, Fin.isValue, decide_False, Bool.not_false]
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dsimp only [Fin.isValue, koszulSignInsert]
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simp only [Fin.isValue, h, ↓reduceIte]
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rw [koszulSignInsert_eq_filter]
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congr
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simp only [decide_not]
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simp
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simp
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lemma filter_le_orderedInsert {I : Type} (le1 : I → I → Prop)
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(le_trans : ∀ {i j k}, le1 i j → le1 j k → le1 i k)
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(le_connected : ∀ {i j}, ¬ le1 i j → le1 j i) [DecidableRel le1] (rn r0 : I)
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(r : List I) (hr : (List.filter (fun i => decide (¬ le1 rn i)) r) =
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List.takeWhile (fun i => decide (¬ le1 rn i)) r) :
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(List.filter (fun i => decide (¬ le1 rn i)) (List.orderedInsert le1 r0 r)) =
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List.takeWhile (fun i => decide (¬ le1 rn i)) (List.orderedInsert le1 r0 r) := by
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induction r with
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| nil =>
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simp [List.filter, List.takeWhile]
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| cons r1 r ih =>
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simp
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by_cases hr1 : le1 r0 r1
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· simp [hr1]
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rw [List.filter, List.takeWhile]
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by_cases hrn : le1 rn r0
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· simp [hrn]
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apply And.intro
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· exact le_trans hrn hr1
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· intro rp hr'
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have hrn1 := le_trans hrn hr1
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simp [List.filter, List.takeWhile, hrn1] at hr
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exact hr rp hr'
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· simp [hrn]
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simpa using hr
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· simp [hr1, List.filter, List.takeWhile]
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by_cases hrn : le1 rn r1
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· simp [hrn]
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apply And.intro
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· exact le_trans hrn (le_connected hr1)
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· simp [List.filter, List.takeWhile, hrn] at hr
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exact hr
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· simp [hrn]
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have hr' : List.filter (fun i => decide ¬le1 rn i) r = List.takeWhile (fun i => decide ¬le1 rn i) r := by
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simp [List.filter, List.takeWhile, hrn] at hr
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simpa using hr
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simpa only [decide_not] using ih hr'
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lemma filter_le_sort {I : Type} (le1 : I → I → Prop) [DecidableRel le1]
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(le_trans : ∀ {i j k}, le1 i j → le1 j k → le1 i k)
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(le_connected : ∀ {i j}, ¬ le1 i j → le1 j i) (r0 : I) (r : List I) :
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List.filter (fun i => decide (¬ le1 r0 i)) (List.insertionSort le1 r) =
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List.takeWhile (fun i => decide (¬ le1 r0 i)) (List.insertionSort le1 r) := by
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induction r with
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| nil =>
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simp
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| cons r1 r ih =>
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simp only [ List.insertionSort]
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rw [filter_le_orderedInsert]
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exact fun {i j k} a a_1 => le_trans a a_1
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exact fun {i j} a => le_connected a
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exact ih
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lemma koszulSignInsert_eq_grade {I : Type} (q : I → Fin 2) (le1 : I → I → Prop) [DecidableRel le1] (r0 : I)
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(r : List I) :
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@ -262,10 +236,10 @@ lemma koszulSignInsert_eq_grade {I : Type} (q : I → Fin 2) (le1 : I → I →
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rw [koszulSignInsert_eq_filter]
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by_cases hr1 : ¬ le1 r0 r1
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· rw [List.filter_cons_of_pos]
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· dsimp [koszulSignInsert]
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· dsimp only [koszulSignInsert, Fin.isValue, decide_not]
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rw [if_neg hr1]
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dsimp [grade]
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simp [grade]
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dsimp only [Fin.isValue, grade, ite_eq_right_iff, zero_ne_one, imp_false, decide_not]
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simp only [Fin.isValue, decide_not, ite_eq_right_iff, zero_ne_one, imp_false]
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have ha (a b c : Fin 2) : (if a = 1 ∧ b = 1 then -if ¬a = 0 ∧
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c = 1 then -1 else (1 : ℂ)
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else if ¬a = 0 ∧ c = 1 then -1 else 1) =
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@ -281,48 +255,392 @@ lemma koszulSignInsert_eq_grade {I : Type} (q : I → Fin 2) (le1 : I → I →
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simpa [grade] using ih
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· simp [hr1]
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· rw [List.filter_cons_of_neg]
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simp
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simp only [decide_not, Fin.isValue]
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rw [koszulSignInsert_eq_filter] at ih
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simpa [grade] using ih
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simpa using hr1
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lemma koszulSignInsert_eq_perm {I : Type} (q : I → Fin 2) (le1 : I → I → Prop) (r r' : List I)
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(a : I) [DecidableRel le1] (h : r.Perm r') :
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koszulSignInsert le1 q a r = koszulSignInsert le1 q a r' := by
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rw [koszulSignInsert_eq_grade]
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rw [koszulSignInsert_eq_grade]
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congr 1
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simp only [Fin.isValue, decide_not, eq_iff_iff, and_congr_right_iff]
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intro h'
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have hg : grade q (List.filter (fun i => !decide (le1 a i)) r) =
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grade q (List.filter (fun i => !decide (le1 a i)) r') := by
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rw [grade_count, grade_count]
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rw [List.countP_filter, List.countP_filter]
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rw [h.countP_eq]
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rw [hg]
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lemma koszulSignInsert_eq_sort {I : Type} (q : I → Fin 2) (le1 : I → I → Prop) (r : List I)
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(a : I) [DecidableRel le1] :
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koszulSignInsert le1 q a r = koszulSignInsert le1 q a (List.insertionSort le1 r) := by
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apply koszulSignInsert_eq_perm
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exact List.Perm.symm (List.perm_insertionSort le1 r)
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lemma koszulSign_erase_fermion {I : Type} (q : I → Fin 2) (le1 :I → I → Prop)
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[DecidableRel le1] :
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(r : List I) → (n : Fin r.length) → (heq : q (r.get n) = 1) →
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koszulSign le1 q (r.eraseIdx n) = koszulSign le1 q r *
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if grade q (r.take n) = grade q (List.take (↑((HepLean.List.insertionSortEquiv le1 r) n)) (List.insertionSort le1 r))
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then 1 else -1
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| [], _ => by
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lemma koszulSignInsert_eq_cons {I : Type} (q : I → Fin 2) (le1 : I → I → Prop) [DecidableRel le1]
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[IsTotal I le1] [IsTrans I le1] (r0 : I)
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(r : List I) :
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koszulSignInsert le1 q r0 r = koszulSignInsert le1 q r0 (r0 :: r):= by
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simp only [koszulSignInsert, Fin.isValue, and_self]
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have h1 : le1 r0 r0 := by
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simpa using IsTotal.total (r := le1) r0 r0
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simp [h1]
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def insertSign {I : Type} (q : I → Fin 2) (n : ℕ)
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(r0 : I) (r : List I) : ℂ :=
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superCommuteCoef q [r0] (List.take n r)
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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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List.take n (List.insertIdx n i r) = List.take n r
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| 0, _ => by simp
|
||||
| _+1, [] => by simp
|
||||
| n+1, a::as => by
|
||||
simp only [List.insertIdx_succ_cons, List.take_succ_cons, List.cons.injEq, true_and]
|
||||
exact take_insert_same i n as
|
||||
|
||||
lemma insertSign_insert {I : Type} (q : I → Fin 2) (n : ℕ)
|
||||
(r0 : I) (r : List I) : insertSign q n r0 r = insertSign q n r0 (List.insertIdx n r0 r) := by
|
||||
simp only [insertSign]
|
||||
congr 1
|
||||
rw [take_insert_same]
|
||||
|
||||
lemma take_eraseIdx_same {I : Type} :
|
||||
(n : ℕ) → (r : List I) →
|
||||
List.take n (List.eraseIdx r n) = List.take n r
|
||||
| 0, _ => by simp
|
||||
| _+1, [] => by simp
|
||||
| n+1, a::as => by
|
||||
simp only [List.eraseIdx_cons_succ, List.take_succ_cons, List.cons.injEq, true_and]
|
||||
exact take_eraseIdx_same n as
|
||||
|
||||
lemma insertSign_eraseIdx {I : Type} (q : I → Fin 2) (n : ℕ)
|
||||
(r0 : I) (r : List I) : insertSign q n r0 (r.eraseIdx n) = insertSign q n r0 r := by
|
||||
simp only [insertSign]
|
||||
congr 1
|
||||
rw [take_eraseIdx_same]
|
||||
|
||||
lemma insertSign_zero {I : Type} (q : I → Fin 2) (r0 : I) (r : List I) :
|
||||
insertSign q 0 r0 r = 1 := by
|
||||
simp [insertSign, superCommuteCoef]
|
||||
|
||||
lemma insertSign_succ_cons {I : Type} (q : I → Fin 2) (n : ℕ)
|
||||
(r0 r1 : I) (r : List I) : insertSign q (n + 1) r0 (r1 :: r) =
|
||||
superCommuteCoef q [r0] [r1] * insertSign q n r0 r := by
|
||||
simp only [insertSign, List.take_succ_cons]
|
||||
rw [superCommuteCoef_cons]
|
||||
|
||||
lemma take_insert_gt {I : Type} (i : I) :
|
||||
(n m : ℕ) → (h : n < m) → (r : List I) →
|
||||
List.take n (List.insertIdx m i r) = List.take n r
|
||||
| 0, 0, _, _ => by simp
|
||||
| 0, m + 1, _, _ => by simp
|
||||
| n+1, m + 1, _, [] => by simp
|
||||
| n+1, m + 1, h, a::as => by
|
||||
simp only [List.insertIdx_succ_cons, List.take_succ_cons, List.cons.injEq, true_and]
|
||||
refine take_insert_gt i n m (Nat.succ_lt_succ_iff.mp h) as
|
||||
|
||||
|
||||
lemma insertSign_insert_gt {I : Type} (q : I → Fin 2) (n m : ℕ)
|
||||
(r0 r1 : I) (r : List I) (hn : n < m) :
|
||||
insertSign q n r0 (List.insertIdx m r1 r) = insertSign q n r0 r := by
|
||||
rw [insertSign, insertSign]
|
||||
congr 1
|
||||
exact take_insert_gt r1 n m hn r
|
||||
|
||||
lemma take_insert_let {I : Type} (i : I) :
|
||||
(n m : ℕ) → (h : m ≤ n) → (r : List I) → (hm : m ≤ r.length) →
|
||||
(List.take (n + 1) (List.insertIdx m i r)).Perm (i :: List.take n r)
|
||||
| 0, 0, h, _, _ => by simp
|
||||
| m + 1, 0, h, r, _ => by simp
|
||||
| n + 1, m + 1, h, [], hm => by
|
||||
simp at hm
|
||||
| n + 1, m + 1, h, a::as, hm => by
|
||||
simp only [List.insertIdx_succ_cons, List.take_succ_cons]
|
||||
have hp : (i :: a :: List.take n as).Perm (a :: i :: List.take n as) := by
|
||||
exact List.Perm.swap a i (List.take n as)
|
||||
refine List.Perm.trans ?_ hp.symm
|
||||
refine List.Perm.cons a ?_
|
||||
exact take_insert_let i n m (Nat.le_of_succ_le_succ h) as (Nat.le_of_succ_le_succ hm)
|
||||
|
||||
lemma insertSign_insert_lt_eq_insertSort {I : Type} (q : I → Fin 2) (n m : ℕ)
|
||||
(r0 r1 : I) (r : List I) (hn : m ≤ n) (hm : m ≤ r.length):
|
||||
insertSign q (n + 1) r0 (List.insertIdx m r1 r) = insertSign q (n + 1) r0 (r1 :: r) := by
|
||||
rw [insertSign, insertSign]
|
||||
apply superCommuteCoef_perm_snd
|
||||
simp only [List.take_succ_cons]
|
||||
refine take_insert_let r1 n m hn r hm
|
||||
|
||||
lemma insertSign_insert_lt {I : Type} (q : I → Fin 2) (n m : ℕ)
|
||||
(r0 r1 : I) (r : List I) (hn : m ≤ n) (hm : m ≤ r.length):
|
||||
insertSign q (n + 1) r0 (List.insertIdx m r1 r) = superCommuteCoef q [r0] [r1] * insertSign q n r0 r := by
|
||||
rw [insertSign_insert_lt_eq_insertSort, insertSign_succ_cons]
|
||||
exact hn
|
||||
exact hm
|
||||
|
||||
|
||||
|
||||
|
||||
def koszulSignCons {I : Type} (q : I → Fin 2) (le1 : I → I → Prop) [DecidableRel le1] (r0 r1 : I) :
|
||||
ℂ :=
|
||||
if le1 r0 r1 then 1 else
|
||||
if q r0 = 1 ∧ q r1 = 1 then -1 else 1
|
||||
|
||||
lemma koszulSignCons_eq_superComuteCoef {I : Type} (q : I → Fin 2) (le1 : I → I → Prop) [DecidableRel le1]
|
||||
(r0 r1 : I) : koszulSignCons q le1 r0 r1 =
|
||||
if le1 r0 r1 then 1 else superCommuteCoef q [r0] [r1] := by
|
||||
simp only [koszulSignCons, Fin.isValue, superCommuteCoef, grade, ite_eq_right_iff, zero_ne_one,
|
||||
imp_false]
|
||||
congr 1
|
||||
by_cases h0 : q r0 = 1
|
||||
· by_cases h1 : q r1 = 1
|
||||
· simp [h0, h1]
|
||||
· have h1 : q r1 = 0 := by omega
|
||||
simp [h0, h1]
|
||||
· have h0 : q r0 = 0 := by omega
|
||||
by_cases h1 : q r1 = 1
|
||||
· simp [h0, h1]
|
||||
· have h1 : q r1 = 0 := by omega
|
||||
simp [h0, h1]
|
||||
|
||||
lemma koszulSignInsert_cons {I : Type} (q : I → Fin 2) (le1 : I → I → Prop) [DecidableRel le1]
|
||||
[IsTotal I le1] [IsTrans I le1] (r0 r1 : I) (r : List I) :
|
||||
koszulSignInsert le1 q r0 (r1 :: r) = koszulSignCons q le1 r0 r1 * koszulSignInsert le1 q r0 r := by
|
||||
simp [koszulSignInsert, koszulSignCons]
|
||||
|
||||
lemma koszulSignInsert_eq_insertSign {I : Type} (q : I → Fin 2) (le1 : I → I → Prop) [DecidableRel le1]
|
||||
[IsTotal I le1] [IsTrans I le1] (r0 : I) (r : List I) :
|
||||
koszulSignInsert le1 q r0 r = insertSign q (orderedInsertPos le1 (List.insertionSort le1 r) r0)
|
||||
r0 (List.insertionSort le1 r) := by
|
||||
rw [koszulSignInsert_eq_cons, koszulSignInsert_eq_sort, koszulSignInsert_eq_filter,
|
||||
koszulSignInsert_eq_grade, insertSign, superCommuteCoef]
|
||||
congr
|
||||
simp only [List.filter_filter, Bool.and_self]
|
||||
rw [List.insertionSort]
|
||||
nth_rewrite 1 [List.orderedInsert_eq_take_drop]
|
||||
rw [List.filter_append]
|
||||
have h1 : List.filter (fun a => decide ¬le1 r0 a) (List.takeWhile (fun b => decide ¬le1 r0 b) (List.insertionSort le1 r))
|
||||
= (List.takeWhile (fun b => decide ¬le1 r0 b) (List.insertionSort le1 r)) := by
|
||||
induction r with
|
||||
| nil => simp
|
||||
| cons r1 r ih =>
|
||||
simp only [decide_not, List.insertionSort, List.filter_eq_self, Bool.not_eq_eq_eq_not,
|
||||
Bool.not_true, decide_eq_false_iff_not]
|
||||
intro a ha
|
||||
have ha' := List.mem_takeWhile_imp ha
|
||||
simp_all
|
||||
rw [h1]
|
||||
rw [List.filter_cons]
|
||||
simp only [decide_not, (IsTotal.to_isRefl le1).refl r0, not_true_eq_false, decide_False,
|
||||
Bool.false_eq_true, ↓reduceIte]
|
||||
rw [orderedInsertPos_take]
|
||||
simp only [decide_not, List.append_right_eq_self, List.filter_eq_nil_iff, Bool.not_eq_eq_eq_not,
|
||||
Bool.not_true, decide_eq_false_iff_not, Decidable.not_not]
|
||||
intro a ha
|
||||
refine List.Sorted.rel_of_mem_take_of_mem_drop
|
||||
(k := (orderedInsertPos le1 (List.insertionSort le1 r) r0).1 + 1 )
|
||||
(List.sorted_insertionSort le1 (r0 :: r)) ?_ ?_
|
||||
· simp only [List.insertionSort, List.orderedInsert_eq_take_drop, decide_not]
|
||||
rw [List.take_append_eq_append_take]
|
||||
rw [List.take_of_length_le]
|
||||
· simp [orderedInsertPos]
|
||||
· simp [orderedInsertPos]
|
||||
· simp only [List.insertionSort, List.orderedInsert_eq_take_drop, decide_not]
|
||||
rw [List.drop_append_eq_append_drop]
|
||||
rw [List.drop_of_length_le]
|
||||
· simpa [orderedInsertPos] using ha
|
||||
· simp [orderedInsertPos]
|
||||
|
||||
lemma koszulSignInsert_insertIdx {I : Type} (q : I → Fin 2) (le1 : I → I → Prop) [DecidableRel le1]
|
||||
(i j : I) [IsTotal I le1] [IsTrans I le1] (r : List I) (n : ℕ) (hn : n ≤ r.length) :
|
||||
koszulSignInsert le1 q j (List.insertIdx n i r) = koszulSignInsert le1 q j (i :: r) := by
|
||||
apply koszulSignInsert_eq_perm
|
||||
exact List.perm_insertIdx i r hn
|
||||
|
||||
lemma take_insertIdx {I : Type} (i : I) : (r : List I) → (n : ℕ) → (hn : n ≤ r.length) →
|
||||
List.take n (List.insertIdx n i r) = List.take n r
|
||||
| [], 0, h => by
|
||||
simp
|
||||
| r0 :: r, ⟨0, h⟩ => by
|
||||
simp [koszulSign]
|
||||
intro h
|
||||
rw [koszulSignInsert_boson]
|
||||
| [], n + 1, h => by
|
||||
simp at h
|
||||
| r0 :: r, 0, h => by
|
||||
simp
|
||||
sorry
|
||||
sorry
|
||||
| r0 :: r, ⟨n + 1, h⟩ => by
|
||||
intro hn
|
||||
simp only [List.eraseIdx_cons_succ, List.take_succ_cons, List.insertionSort, List.length_cons,
|
||||
mul_one, mul_neg]
|
||||
sorry
|
||||
| r0 :: r, n + 1, h => by
|
||||
simp only [List.insertIdx_succ_cons, List.take_succ_cons, List.cons.injEq, true_and]
|
||||
exact take_insertIdx i r n (Nat.le_of_lt_succ h)
|
||||
|
||||
lemma superCommuteCoefLE_eq_get_fermion {I : Type} (q : I → Fin 2) (le1 :I → I → Prop) (r : List I)
|
||||
[DecidableRel le1] (i : I) (hi : q i = 1 ) (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
|
||||
simp [superCommuteCoefLE, superCommuteCoef]
|
||||
simp [grade, hi]
|
||||
simp [hi] at heq
|
||||
simp [← heq]
|
||||
sorry
|
||||
|
||||
lemma koszulSign_insertIdx {I : Type} (q : I → Fin 2) (le1 : I → I → Prop) [DecidableRel le1]
|
||||
(i : I) [IsTotal I le1] [IsTrans I le1] : (r : List I) → (n : ℕ) → (hn : n ≤ r.length) →
|
||||
koszulSign le1 q (List.insertIdx n i r) = insertSign q n i r
|
||||
* koszulSign le1 q r
|
||||
* insertSign q (insertionSortEquiv le1 (List.insertIdx n i r) ⟨n, by
|
||||
rw [List.length_insertIdx _ _ hn]
|
||||
omega⟩) i
|
||||
(List.insertionSort le1 (List.insertIdx n i r))
|
||||
| [], 0, h => by
|
||||
simp [koszulSign, insertSign, superCommuteCoef, koszulSignInsert]
|
||||
| [], n + 1, h => by
|
||||
simp at h
|
||||
| r0 :: r, 0, h => by
|
||||
simp only [List.insertIdx_zero, List.insertionSort, List.length_cons, Fin.zero_eta]
|
||||
rw [koszulSign]
|
||||
trans koszulSign le1 q (r0 :: r) * koszulSignInsert le1 q i (r0 :: r)
|
||||
ring
|
||||
simp only [insertionSortEquiv, List.length_cons, Nat.succ_eq_add_one, List.insertionSort,
|
||||
orderedInsertEquiv, OrderIso.toEquiv_symm, Fin.symm_castOrderIso, HepLean.Fin.equivCons_trans,
|
||||
Equiv.trans_apply, HepLean.Fin.equivCons_zero, HepLean.Fin.finExtractOne_apply_eq,
|
||||
Fin.isValue, HepLean.Fin.finExtractOne_symm_inl_apply, RelIso.coe_fn_toEquiv,
|
||||
Fin.castOrderIso_apply, Fin.cast_mk, Fin.eta]
|
||||
conv_rhs =>
|
||||
rhs
|
||||
rhs
|
||||
rw [orderedInsert_eq_insertIdx_orderedInsertPos]
|
||||
conv_rhs =>
|
||||
rhs
|
||||
rw [← insertSign_insert]
|
||||
change insertSign q (↑(orderedInsertPos le1 ((List.insertionSort le1 (r0 :: r))) i)) i
|
||||
(List.insertionSort le1 (r0 :: r))
|
||||
rw [← koszulSignInsert_eq_insertSign q le1]
|
||||
rw [insertSign_zero]
|
||||
simp
|
||||
| r0 :: r, n + 1, h => by
|
||||
conv_lhs =>
|
||||
rw [List.insertIdx_succ_cons]
|
||||
rw [koszulSign]
|
||||
rw [koszulSign_insertIdx]
|
||||
conv_rhs =>
|
||||
rhs
|
||||
simp only [List.insertIdx_succ_cons]
|
||||
simp only [List.insertionSort, List.length_cons, insertionSortEquiv, Nat.succ_eq_add_one,
|
||||
Equiv.trans_apply, HepLean.Fin.equivCons_succ]
|
||||
erw [orderedInsertEquiv_fin_succ]
|
||||
simp only [Fin.eta, Fin.coe_cast]
|
||||
rhs
|
||||
rw [orderedInsert_eq_insertIdx_orderedInsertPos]
|
||||
conv_rhs =>
|
||||
lhs
|
||||
rw [insertSign_succ_cons, koszulSign]
|
||||
ring_nf
|
||||
conv_lhs =>
|
||||
lhs
|
||||
rw [mul_assoc, mul_comm]
|
||||
rw [mul_assoc]
|
||||
conv_rhs =>
|
||||
rw [mul_assoc, mul_assoc]
|
||||
congr 1
|
||||
let rs := (List.insertionSort le1 (List.insertIdx n i r))
|
||||
have hnsL : n < (List.insertIdx n i r).length := by
|
||||
rw [List.length_insertIdx _ _]
|
||||
simp only [List.length_cons, add_le_add_iff_right] at h
|
||||
omega
|
||||
exact Nat.le_of_lt_succ h
|
||||
let ni : Fin rs.length := (insertionSortEquiv le1 (List.insertIdx n i r))
|
||||
⟨n, hnsL⟩
|
||||
let nro : Fin (rs.length + 1) := ⟨↑(orderedInsertPos le1 rs r0), orderedInsertPos_lt_length le1 rs r0⟩
|
||||
rw [koszulSignInsert_insertIdx, koszulSignInsert_cons]
|
||||
trans koszulSignInsert le1 q r0 r * (koszulSignCons q le1 r0 i *insertSign q ni i rs)
|
||||
· simp only [rs, ni]
|
||||
ring
|
||||
trans koszulSignInsert le1 q r0 r * (superCommuteCoef q [i] [r0] *
|
||||
insertSign q (nro.succAbove ni) i (List.insertIdx nro r0 rs))
|
||||
swap
|
||||
· simp only [rs, nro, ni]
|
||||
ring
|
||||
congr 1
|
||||
simp only [Fin.succAbove]
|
||||
have hns : rs.get ni = i := by
|
||||
simp only [Fin.eta, rs]
|
||||
rw [← insertionSortEquiv_get]
|
||||
simp only [Function.comp_apply, Equiv.symm_apply_apply, List.get_eq_getElem, ni]
|
||||
simp_all only [List.length_cons, add_le_add_iff_right, List.getElem_insertIdx_self]
|
||||
have hms : (List.orderedInsert le1 r0 rs).get ⟨nro, by simp⟩ = r0 := by
|
||||
simp [nro]
|
||||
have hc1 : ni.castSucc < nro → ¬ le1 r0 i := by
|
||||
intro hninro
|
||||
rw [← hns]
|
||||
exact lt_orderedInsertPos_rel le1 r0 rs ni hninro
|
||||
have hc2 : ¬ ni.castSucc < nro → le1 r0 i := by
|
||||
intro hninro
|
||||
rw [← hns]
|
||||
refine gt_orderedInsertPos_rel le1 r0 rs ?_ ni hninro
|
||||
exact List.sorted_insertionSort le1 (List.insertIdx n i r)
|
||||
by_cases hn : ni.castSucc < nro
|
||||
· simp only [hn, ↓reduceIte, Fin.coe_castSucc]
|
||||
rw [insertSign_insert_gt]
|
||||
swap
|
||||
· exact hn
|
||||
congr 1
|
||||
rw [koszulSignCons_eq_superComuteCoef]
|
||||
simp only [hc1 hn, ↓reduceIte]
|
||||
rw [superCommuteCoef_comm]
|
||||
· simp only [hn, ↓reduceIte, Fin.val_succ]
|
||||
rw [insertSign_insert_lt]
|
||||
rw [← mul_assoc]
|
||||
congr 1
|
||||
rw [superCommuteCoef_mul_self]
|
||||
rw [koszulSignCons]
|
||||
simp only [hc2 hn, ↓reduceIte]
|
||||
exact Nat.le_of_not_lt hn
|
||||
exact Nat.le_of_lt_succ (orderedInsertPos_lt_length le1 rs r0)
|
||||
· exact Nat.le_of_lt_succ h
|
||||
· exact Nat.le_of_lt_succ h
|
||||
|
||||
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] (i : I) (n : Fin r.length) (heq : q i = q (r.get n)) :
|
||||
[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
|
||||
sorry
|
||||
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
|
||||
|
|
|
|||
|
|
@ -20,6 +20,7 @@ import HepLean.PerturbationTheory.Wick.Koszul.Grade
|
|||
-/
|
||||
|
||||
namespace Wick
|
||||
open HepLean.List
|
||||
|
||||
noncomputable section
|
||||
|
||||
|
|
@ -136,6 +137,393 @@ lemma ofListM_cons_eq_ofListM {I : Type} (f : I → Type) [∀ i, Fintype (f i)]
|
|||
rw [ofListM_cons, ofListM_singleton]
|
||||
simp only [one_smul]
|
||||
|
||||
def CreatAnnilateSect {I : Type} (f : I → Type) [∀ i, Fintype (f i)] (l : List I) : Type :=
|
||||
Π i, f (l.get i)
|
||||
|
||||
namespace CreatAnnilateSect
|
||||
|
||||
variable {I : Type} {f : I → Type} [∀ i, Fintype (f i)] {l : List I} (a : CreatAnnilateSect f l)
|
||||
|
||||
instance fintype : Fintype (CreatAnnilateSect f l) := Pi.fintype
|
||||
|
||||
def tail : {l : List I} → (a : CreatAnnilateSect f l) → CreatAnnilateSect f l.tail
|
||||
| [], a => a
|
||||
| _ :: _, a => fun i => a (Fin.succ i)
|
||||
|
||||
def head {i : I} (a : CreatAnnilateSect f (i :: l)) : f i := a ⟨0, Nat.zero_lt_succ l.length⟩
|
||||
|
||||
def toList : {l : List I} → (a : CreatAnnilateSect f l) → List (Σ i, f i)
|
||||
| [], _ => []
|
||||
| i :: _, a => ⟨i, a.head⟩ :: toList a.tail
|
||||
|
||||
@[simp]
|
||||
lemma toList_length : (toList a).length = l.length := by
|
||||
induction l with
|
||||
| nil => rfl
|
||||
| cons i l ih =>
|
||||
simp only [toList, List.length_cons, Fin.zero_eta]
|
||||
rw [ih]
|
||||
|
||||
lemma toList_tail : {l : List I} → (a : CreatAnnilateSect f l) → toList a.tail = (toList a).tail
|
||||
| [], _ => rfl
|
||||
| i :: l, a => by
|
||||
simp [toList]
|
||||
|
||||
lemma toList_cons {i : I} (a : CreatAnnilateSect f (i :: l)) :
|
||||
(toList a) = ⟨i, a.head⟩ :: toList a.tail := by
|
||||
rfl
|
||||
|
||||
lemma toList_get (a : CreatAnnilateSect f l) :
|
||||
(toList a).get = (fun i => ⟨l.get i, a i⟩) ∘ Fin.cast (by simp) := by
|
||||
induction l with
|
||||
| nil =>
|
||||
funext i
|
||||
exact Fin.elim0 i
|
||||
| cons i l ih =>
|
||||
simp only [toList_cons, List.get_eq_getElem, Fin.zero_eta, List.getElem_cons_succ,
|
||||
Function.comp_apply, Fin.cast_mk]
|
||||
funext x
|
||||
match x with
|
||||
| ⟨0, h⟩ => rfl
|
||||
| ⟨x + 1, h⟩ =>
|
||||
simp only [List.get_eq_getElem, Prod.mk.eta, List.getElem_cons_succ, Function.comp_apply]
|
||||
change (toList a.tail).get _ = _
|
||||
rw [ih]
|
||||
simp [tail]
|
||||
|
||||
@[simp]
|
||||
lemma toList_grade (q : I → Fin 2) :
|
||||
grade (fun i => q i.fst) a.toList = 1 ↔ grade q l = 1 := by
|
||||
induction l with
|
||||
| nil =>
|
||||
simp [toList]
|
||||
| cons i r ih =>
|
||||
simp only [grade, Fin.isValue, ite_eq_right_iff, zero_ne_one, imp_false]
|
||||
have ih' := ih (fun i => a i.succ)
|
||||
have h1 : grade (fun i => q i.fst) a.tail.toList = grade q r := by
|
||||
by_cases h : grade q r = 1
|
||||
· simp_all
|
||||
· have h0 : grade q r = 0 := by
|
||||
omega
|
||||
rw [h0] at ih'
|
||||
simp only [Fin.isValue, zero_ne_one, iff_false] at ih'
|
||||
have h0' : grade (fun i => q i.fst) a.tail.toList = 0 := by
|
||||
simp [tail]
|
||||
omega
|
||||
rw [h0, h0']
|
||||
rw [h1]
|
||||
|
||||
def extractEquiv {I : Type} {f : I → Type} [(i : I) → Fintype (f i)] {l : List I} (n : Fin l.length) : CreatAnnilateSect f l ≃
|
||||
f (l.get n) × CreatAnnilateSect f (l.eraseIdx n) := by
|
||||
match l with
|
||||
| [] => exact Fin.elim0 n
|
||||
| l0 :: l =>
|
||||
let e1 : CreatAnnilateSect f ((l0 :: l).eraseIdx n) ≃ Π i, f ((l0 :: l).get (n.succAbove i)) :=
|
||||
Equiv.piCongr (Fin.castOrderIso (by rw [eraseIdx_cons_length])).toEquiv
|
||||
fun x => Equiv.cast (congrArg f (by
|
||||
rw [HepLean.List.eraseIdx_get]
|
||||
simp
|
||||
congr 1
|
||||
simp [Fin.succAbove]
|
||||
split
|
||||
next h =>
|
||||
simp_all only [Fin.coe_castSucc]
|
||||
split
|
||||
next h_1 => simp_all only [Fin.coe_castSucc, Fin.coe_cast]
|
||||
next h_1 =>
|
||||
simp_all only [not_lt, Fin.val_succ, Fin.coe_cast, self_eq_add_right, one_ne_zero]
|
||||
simp [Fin.le_def] at h_1
|
||||
simp [Fin.lt_def] at h
|
||||
omega
|
||||
next h =>
|
||||
simp_all only [not_lt, Fin.val_succ]
|
||||
split
|
||||
next h_1 =>
|
||||
simp_all only [Fin.coe_castSucc, Fin.coe_cast, add_right_eq_self, one_ne_zero]
|
||||
simp [Fin.lt_def] at h_1
|
||||
simp [Fin.le_def] at h
|
||||
omega
|
||||
next h_1 => simp_all only [not_lt, Fin.val_succ, Fin.coe_cast]))
|
||||
exact (Fin.insertNthEquiv _ _).symm.trans (Equiv.prodCongr (Equiv.refl _) e1.symm)
|
||||
|
||||
def eraseIdx (n : Fin l.length) : CreatAnnilateSect f (l.eraseIdx n) :=
|
||||
(extractEquiv n a).2
|
||||
|
||||
@[simp]
|
||||
lemma eraseIdx_zero_tail {i : I} {l : List I} (a : CreatAnnilateSect f (i :: l)) :
|
||||
(eraseIdx a (@OfNat.ofNat (Fin (l.length + 1)) 0 Fin.instOfNat : Fin (l.length + 1))) =
|
||||
a.tail := by
|
||||
simp [eraseIdx, extractEquiv]
|
||||
rfl
|
||||
|
||||
lemma eraseIdx_succ_head {i : I} {l : List I} (n : ℕ) (hn : n + 1 < (i :: l).length) (a : CreatAnnilateSect f (i :: l)) :
|
||||
(eraseIdx a ⟨n + 1, hn⟩).head = a.head := by
|
||||
rw [eraseIdx, extractEquiv]
|
||||
simp
|
||||
conv_lhs =>
|
||||
rhs
|
||||
rhs
|
||||
rhs
|
||||
erw [Fin.insertNthEquiv_symm_apply]
|
||||
simp [head, Equiv.piCongr, Equiv.piCongrRight, Equiv.piCongrLeft, Equiv.piCongrLeft']
|
||||
simp [Fin.removeNth, Fin.succAbove]
|
||||
refine cast_eq_iff_heq.mpr ?_
|
||||
congr
|
||||
simp [Fin.ext_iff]
|
||||
|
||||
lemma eraseIdx_succ_tail {i : I} {l : List I} (n : ℕ) (hn : n + 1 < (i :: l).length) (a : CreatAnnilateSect f (i :: l)) :
|
||||
(eraseIdx a ⟨n + 1, hn⟩).tail = eraseIdx a.tail ⟨n , Nat.succ_lt_succ_iff.mp hn⟩ := by
|
||||
match l with
|
||||
| [] =>
|
||||
simp at hn
|
||||
| r0 :: r =>
|
||||
rw [eraseIdx, extractEquiv]
|
||||
simp
|
||||
conv_lhs =>
|
||||
rhs
|
||||
rhs
|
||||
rhs
|
||||
erw [Fin.insertNthEquiv_symm_apply]
|
||||
rw [eraseIdx]
|
||||
conv_rhs =>
|
||||
rhs
|
||||
rw [extractEquiv]
|
||||
simp
|
||||
erw [Fin.insertNthEquiv_symm_apply]
|
||||
simp [tail, Equiv.piCongr, Equiv.piCongrRight, Equiv.piCongrLeft, Equiv.piCongrLeft']
|
||||
funext i
|
||||
simp
|
||||
have hcast {α β : Type} (h : α = β) (a : α) (b : β) : cast h a = b ↔ a = cast (Eq.symm h) b := by
|
||||
cases h
|
||||
simp
|
||||
rw [hcast]
|
||||
simp
|
||||
refine eq_cast_iff_heq.mpr ?_
|
||||
simp [Fin.removeNth, Fin.succAbove]
|
||||
congr
|
||||
simp [Fin.ext_iff]
|
||||
split
|
||||
next h =>
|
||||
simp_all only [Fin.coe_castSucc, Fin.val_succ, Fin.coe_cast, add_left_inj]
|
||||
split
|
||||
next h_1 => simp_all only [Fin.coe_castSucc, Fin.coe_cast]
|
||||
next h_1 =>
|
||||
simp_all only [not_lt, Fin.val_succ, Fin.coe_cast, self_eq_add_right, one_ne_zero]
|
||||
simp [Fin.lt_def] at h
|
||||
simp [Fin.le_def] at h_1
|
||||
omega
|
||||
next h =>
|
||||
simp_all only [not_lt, Fin.val_succ, Fin.coe_cast, add_left_inj]
|
||||
split
|
||||
next h_1 =>
|
||||
simp_all only [Fin.coe_castSucc, Fin.coe_cast, add_right_eq_self, one_ne_zero]
|
||||
simp [Fin.le_def] at h
|
||||
simp [Fin.lt_def] at h_1
|
||||
omega
|
||||
next h_1 => simp_all only [not_lt, Fin.val_succ, Fin.coe_cast]
|
||||
|
||||
lemma eraseIdx_toList : {l : List I} → {n : Fin l.length} → (a : CreatAnnilateSect f l) →
|
||||
(eraseIdx a n).toList = a.toList.eraseIdx n
|
||||
| [], n, _ => Fin.elim0 n
|
||||
| r0 :: r, ⟨0, h⟩, a => by
|
||||
simp [toList_tail]
|
||||
| r0 :: r, ⟨n + 1, h⟩, a => by
|
||||
simp [toList]
|
||||
apply And.intro
|
||||
· rw [eraseIdx_succ_head]
|
||||
· conv_rhs => rw [← eraseIdx_toList (l := r) (n := ⟨n, Nat.succ_lt_succ_iff.mp h⟩) a.tail]
|
||||
rw [eraseIdx_succ_tail]
|
||||
|
||||
lemma toList_koszulSignInsert {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
||||
(q : I → Fin 2) (le1 : I → I → Prop) [DecidableRel le1]
|
||||
(l : List I) (a : CreatAnnilateSect f l) (x : (i : I) × f i):
|
||||
koszulSignInsert (fun i j => le1 i.fst j.fst) (fun i => q i.fst) x a.toList =
|
||||
koszulSignInsert le1 q x.1 l := by
|
||||
induction l with
|
||||
| nil => simp [koszulSignInsert]
|
||||
| cons b l ih =>
|
||||
simp [koszulSignInsert]
|
||||
rw [ih]
|
||||
|
||||
lemma toList_koszulSign {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
||||
(q : I → Fin 2) (le1 : I → I → Prop) [DecidableRel le1]
|
||||
(l : List I) (a : CreatAnnilateSect f l) :
|
||||
koszulSign (fun i j => le1 i.fst j.fst) (fun i => q i.fst) a.toList =
|
||||
koszulSign le1 q l := by
|
||||
induction l with
|
||||
| nil => simp [koszulSign]
|
||||
| cons i l ih =>
|
||||
simp [koszulSign, liftM]
|
||||
rw [ih]
|
||||
congr 1
|
||||
rw [toList_koszulSignInsert]
|
||||
|
||||
|
||||
lemma insertionSortEquiv_toList {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
||||
(le1 : I → I → Prop) [DecidableRel le1](l : List I)
|
||||
(a : CreatAnnilateSect f l) :
|
||||
insertionSortEquiv (fun i j => le1 i.fst j.fst) a.toList =
|
||||
(Fin.castOrderIso (by simp)).toEquiv.trans ((insertionSortEquiv le1 l).trans
|
||||
(Fin.castOrderIso (by simp)).toEquiv) := by
|
||||
induction l with
|
||||
| nil =>
|
||||
simp [liftM, HepLean.List.insertionSortEquiv]
|
||||
| cons i l ih =>
|
||||
simp only [liftM, List.length_cons, Fin.zero_eta, List.insertionSort]
|
||||
conv_lhs => simp [HepLean.List.insertionSortEquiv]
|
||||
erw [orderedInsertEquiv_sigma]
|
||||
rw [ih]
|
||||
simp only [HepLean.Fin.equivCons_trans, Nat.succ_eq_add_one,
|
||||
HepLean.Fin.equivCons_castOrderIso, List.length_cons, Nat.add_zero, Nat.zero_eq,
|
||||
Fin.zero_eta]
|
||||
ext x
|
||||
conv_rhs => simp [HepLean.List.insertionSortEquiv]
|
||||
simp only [Equiv.trans_apply, RelIso.coe_fn_toEquiv, Fin.castOrderIso_apply, Fin.cast_trans,
|
||||
Fin.coe_cast]
|
||||
have h2' (i : Σ i, f i) (l' : List ( Σ i, f i)) :
|
||||
List.map (fun i => i.1) (List.orderedInsert (fun i j => le1 i.fst j.fst) i l') =
|
||||
List.orderedInsert le1 i.1 (List.map (fun i => i.1) l') := by
|
||||
induction l' with
|
||||
| nil =>
|
||||
simp [HepLean.List.orderedInsertEquiv]
|
||||
| cons j l' ih' =>
|
||||
by_cases hij : (fun i j => le1 i.fst j.fst) i j
|
||||
· rw [List.orderedInsert_of_le]
|
||||
· erw [List.orderedInsert_of_le]
|
||||
· simp
|
||||
· exact hij
|
||||
· exact hij
|
||||
· simp only [List.orderedInsert, hij, ↓reduceIte, List.unzip_snd, List.map_cons]
|
||||
have hn : ¬ le1 i.1 j.1 := hij
|
||||
simp only [hn, ↓reduceIte, List.cons.injEq, true_and]
|
||||
simpa using ih'
|
||||
have h2 (l' : List ( Σ i, f i)) :
|
||||
List.map (fun i => i.1) (List.insertionSort (fun i j => le1 i.fst j.fst) l') =
|
||||
List.insertionSort le1 (List.map (fun i => i.1) l') := by
|
||||
induction l' with
|
||||
| nil =>
|
||||
simp [HepLean.List.orderedInsertEquiv]
|
||||
| cons i l' ih' =>
|
||||
simp only [List.insertionSort, List.unzip_snd]
|
||||
simp only [List.unzip_snd] at h2'
|
||||
rw [h2']
|
||||
congr
|
||||
rw [HepLean.List.orderedInsertEquiv_congr _ _ _ (h2 _)]
|
||||
simp only [List.length_cons, Equiv.trans_apply, RelIso.coe_fn_toEquiv, Fin.castOrderIso_apply,
|
||||
Fin.cast_trans, Fin.coe_cast]
|
||||
have h3 : (List.insertionSort le1 (List.map (fun i => i.1) a.tail.toList)) =
|
||||
List.insertionSort le1 l := by
|
||||
congr
|
||||
have h3' (l : List I) (a : CreatAnnilateSect f l) :
|
||||
List.map (fun i => i.1) a.toList = l := by
|
||||
induction l with
|
||||
| nil => rfl
|
||||
| cons i l ih' =>
|
||||
simp only [toList, List.length_cons, Fin.zero_eta, Prod.mk.eta,
|
||||
List.unzip_snd, List.map_cons, List.cons.injEq, true_and]
|
||||
simpa using ih' _
|
||||
rw [h3']
|
||||
rfl
|
||||
rw [HepLean.List.orderedInsertEquiv_congr _ _ _ h3]
|
||||
simp only [List.length_cons, Equiv.trans_apply, RelIso.coe_fn_toEquiv, Fin.castOrderIso_apply,
|
||||
Fin.cast_trans, Fin.cast_eq_self, Fin.coe_cast]
|
||||
rfl
|
||||
|
||||
|
||||
def sort (le1 : I → I → Prop) [DecidableRel le1] : CreatAnnilateSect f (List.insertionSort le1 l) :=
|
||||
Equiv.piCongr (HepLean.List.insertionSortEquiv le1 l) (fun i => (Equiv.cast (by
|
||||
congr 1
|
||||
rw [← HepLean.List.insertionSortEquiv_get]
|
||||
simp))) a
|
||||
|
||||
lemma sort_toList {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
||||
(le1 : I → I → Prop) [DecidableRel le1](l : List I) (a : CreatAnnilateSect f l) :
|
||||
(a.sort le1).toList = List.insertionSort (fun i j => le1 i.fst j.fst) a.toList := by
|
||||
let l1 := List.insertionSort (fun i j => le1 i.fst j.fst) a.toList
|
||||
let l2 := (a.sort le1).toList
|
||||
symm
|
||||
change l1 = l2
|
||||
have hlen : l1.length = l2.length := by
|
||||
simp [l1, l2]
|
||||
have hget : l1.get = l2.get ∘ Fin.cast hlen := by
|
||||
rw [← HepLean.List.insertionSortEquiv_get]
|
||||
rw [toList_get, toList_get]
|
||||
funext i
|
||||
rw [insertionSortEquiv_toList]
|
||||
simp only [ Function.comp_apply, Equiv.symm_trans_apply,
|
||||
OrderIso.toEquiv_symm, Fin.symm_castOrderIso, RelIso.coe_fn_toEquiv, Fin.castOrderIso_apply,
|
||||
Fin.cast_trans, Fin.cast_eq_self, id_eq, eq_mpr_eq_cast, Fin.coe_cast, Sigma.mk.inj_iff]
|
||||
apply And.intro
|
||||
· have h1 := congrFun (HepLean.List.insertionSortEquiv_get (r := le1) l) (Fin.cast (by simp) i)
|
||||
rw [← h1]
|
||||
simp
|
||||
· simp [Equiv.piCongr, sort]
|
||||
exact (cast_heq _ _).symm
|
||||
apply List.ext_get hlen
|
||||
rw [hget]
|
||||
simp
|
||||
|
||||
end CreatAnnilateSect
|
||||
|
||||
|
||||
lemma ofListM_expand {I : Type} (f : I → Type) [∀ i, Fintype (f i)] (x : ℂ) :
|
||||
(l : List I) → ofListM f l x = ∑ (a : CreatAnnilateSect f l), ofList a.toList x
|
||||
| [] => by
|
||||
simp only [ofListM, CreatAnnilateSect, List.length_nil, List.get_eq_getElem, Finset.univ_unique,
|
||||
CreatAnnilateSect.toList, Finset.sum_const, Finset.card_singleton, one_smul]
|
||||
rw [ofList_eq_smul_one, map_smul, ofList_empty, ofList_eq_smul_one, ofList_empty, map_one]
|
||||
| i :: l => by
|
||||
rw [ofListM_cons, ofListM_expand f x l]
|
||||
conv_rhs => rw [← (CreatAnnilateSect.extractEquiv
|
||||
⟨0, by exact Nat.zero_lt_succ l.length⟩).symm.sum_comp (α := FreeAlgebra ℂ _)]
|
||||
erw [Finset.sum_product]
|
||||
rw [Finset.sum_mul]
|
||||
conv_lhs =>
|
||||
rhs
|
||||
intro n
|
||||
rw [Finset.mul_sum]
|
||||
congr
|
||||
funext j
|
||||
congr
|
||||
funext n
|
||||
rw [← ofList_singleton, ← ofList_pair, one_mul]
|
||||
rfl
|
||||
|
||||
lemma koszulOrder_ofListM {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
||||
(q : I → Fin 2) (le1 : I → I → Prop) [DecidableRel le1]
|
||||
(l : List I) (x : ℂ) : koszulOrder (fun i j => le1 i.1 j.1) (fun i => q i.fst) (ofListM f l x) =
|
||||
freeAlgebraMap f (koszulOrder le1 q (ofList l x)) := by
|
||||
rw [koszulOrder_ofList]
|
||||
rw [map_smul]
|
||||
change _ = _ • ofListM _ _ _
|
||||
rw [ofListM_expand]
|
||||
rw [map_sum]
|
||||
conv_lhs =>
|
||||
rhs
|
||||
intro a
|
||||
rw [koszulOrder_ofList]
|
||||
rw [CreatAnnilateSect.toList_koszulSign]
|
||||
rw [← Finset.smul_sum]
|
||||
apply congrArg
|
||||
conv_lhs =>
|
||||
rhs
|
||||
intro n
|
||||
rw [← CreatAnnilateSect.sort_toList]
|
||||
rw [ofListM_expand]
|
||||
refine Fintype.sum_equiv ((HepLean.List.insertionSortEquiv le1 l).piCongr fun i => Equiv.cast ?_) _ _ ?_
|
||||
congr 1
|
||||
· rw [← HepLean.List.insertionSortEquiv_get]
|
||||
simp
|
||||
· intro x
|
||||
rfl
|
||||
|
||||
lemma koszulOrder_ofListM_eq_ofListM {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
||||
(q : I → Fin 2) (le1 : I → I → Prop) [DecidableRel le1]
|
||||
(l : List I) (x : ℂ) : koszulOrder (fun i j => le1 i.1 j.1) (fun i => q i.fst) (ofListM f l x) =
|
||||
koszulSign le1 q l • ofListM f (List.insertionSort le1 l) x := by
|
||||
rw [koszulOrder_ofListM, koszulOrder_ofList, map_smul]
|
||||
rfl
|
||||
|
||||
def liftM {I : Type} (f : I → Type) [∀ i, Fintype (f i)] :
|
||||
(l : List I) → (a : Π i, f (l.get i)) → List (Σ i, f i)
|
||||
| [], _ => []
|
||||
|
|
@ -150,6 +538,10 @@ lemma liftM_length {I : Type} (f : I → Type) [∀ i, Fintype (f i)] (r : List
|
|||
simp only [liftM, List.length_cons, Fin.zero_eta, add_left_inj]
|
||||
rw [ih]
|
||||
|
||||
lemma liftM_cons {I : Type} (f : I → Type) [∀ i, Fintype (f i)] (r0 : I) (r : List I) (a : Π i, f ((r0 :: r).get i)) :
|
||||
liftM f (r0 :: r) a = ⟨r0, a ⟨0, Nat.zero_lt_succ r.length⟩⟩ :: liftM f r (fun i => a (Fin.succ i)) := by
|
||||
simp [liftM, List.length_cons, Fin.zero_eta]
|
||||
|
||||
lemma liftM_get {I : Type} (f : I → Type) [∀ i, Fintype (f i)] (r : List I) (a : Π i, f (r.get i)) :
|
||||
(liftM f r a).get = (fun i => ⟨r.get i, a i⟩) ∘ Fin.cast (by simp) := by
|
||||
induction r with
|
||||
|
|
@ -172,6 +564,22 @@ def liftMCongrEquiv {I : Type} (f : I → Type) [∀ i, Fintype (f i)] (r0 : I)
|
|||
(Π i, f ((r0 :: r).get i)) ≃ f ((r0 :: r).get n) × Π i, f ((r0 :: r).get (n.succAbove i)) :=
|
||||
(Fin.insertNthEquiv _ _).symm
|
||||
|
||||
lemma liftMCongrEquiv_symm_succAbove {I : Type} (f : I → Type) [∀ i, Fintype (f i)] (r0 : I) (r : List I)
|
||||
(n : Fin (r0 :: r).length) (a0 : f ((r0 :: r).get n) ) (a : Π i, f ((r0 :: r).get (n.succAbove i)))
|
||||
(i : Fin r.length) :
|
||||
(liftMCongrEquiv f r0 r n).symm (a0, a) (n.succAbove i) = a i := by
|
||||
simp [liftMCongrEquiv]
|
||||
|
||||
@[simp]
|
||||
lemma liftMCongrEquiv_symm_zero_succ {I : Type} (f : I → Type) [∀ i, Fintype (f i)] (r0 : I) (r : List I)
|
||||
(a0 : f ((r0 :: r).get ⟨0, by simp⟩) ) (a : Π i, f ((r0 :: r).get ( i.succ)))
|
||||
(i : Fin r.length) :
|
||||
(liftMCongrEquiv f r0 r ⟨0, by simp⟩).symm (a0, a) i.succ = a i := by
|
||||
trans (liftMCongrEquiv f r0 r ⟨0, by simp⟩).symm (a0, a)
|
||||
((⟨0, by simp⟩ : Fin (r0 :: r).length).succAbove i)
|
||||
rfl
|
||||
rw [liftMCongrEquiv_symm_succAbove]
|
||||
|
||||
lemma ofListM_expand {I : Type} (f : I → Type) [∀ i, Fintype (f i)] (x : ℂ) :
|
||||
(l : List I) → ofListM f l x = ∑ (a : Π i, f (l.get i)), ofList (liftM f l a) x
|
||||
| [] => by
|
||||
|
|
@ -233,11 +641,10 @@ lemma liftM_grade_take {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
|||
refine (liftM_grade_take q r (fun i => a i.succ) n)
|
||||
rw [ih]
|
||||
|
||||
open HepLean.List
|
||||
|
||||
def listMEraseEquiv {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
||||
(q : I → Fin 2) {r0 : I} {r : List I} (n : Fin (r0 :: r).length) :
|
||||
(Π (i :Fin ((r0 :: r).eraseIdx ↑n).length) , f (((r0 :: r).eraseIdx ↑n).get i))
|
||||
{r0 : I} {r : List I} (n : Fin (r0 :: r).length) :
|
||||
(Π (i : Fin ((r0 :: r).eraseIdx ↑n).length) , f (((r0 :: r).eraseIdx ↑n).get i))
|
||||
≃ Π (i : Fin r.length), f ((r0 :: r).get (n.succAbove i)) :=
|
||||
Equiv.piCongr (Fin.castOrderIso (by rw [eraseIdx_cons_length])).toEquiv
|
||||
fun x => Equiv.cast (congrArg f (by
|
||||
|
|
@ -245,10 +652,54 @@ def listMEraseEquiv {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
|||
simp
|
||||
congr 1
|
||||
simp [Fin.succAbove]
|
||||
split
|
||||
next h =>
|
||||
simp_all only [Fin.coe_castSucc]
|
||||
split
|
||||
next h_1 => simp_all only [Fin.coe_castSucc, Fin.coe_cast]
|
||||
next h_1 =>
|
||||
simp_all only [not_lt, Fin.val_succ, Fin.coe_cast, self_eq_add_right, one_ne_zero]
|
||||
simp [Fin.le_def] at h_1
|
||||
simp [Fin.lt_def] at h
|
||||
omega
|
||||
next h =>
|
||||
simp_all only [not_lt, Fin.val_succ]
|
||||
split
|
||||
next h_1 =>
|
||||
simp_all only [Fin.coe_castSucc, Fin.coe_cast, add_right_eq_self, one_ne_zero]
|
||||
simp [Fin.lt_def] at h_1
|
||||
simp [Fin.le_def] at h
|
||||
omega
|
||||
next h_1 => simp_all only [not_lt, Fin.val_succ, Fin.coe_cast]))
|
||||
|
||||
lemma liftM_eraseIdx {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
||||
(r0 : I) :
|
||||
(r : List I) → (n : Fin (r0 :: r).length) →
|
||||
(a0 : f (r0 :: r)[↑n]) → (a : (i : Fin r.length) → f (r0 :: r)[↑(n.succAbove i)]) →
|
||||
(liftM f (r0 :: r) ((liftMCongrEquiv f r0 r n).symm (a0, a))).eraseIdx ↑n =
|
||||
liftM f ((r0 :: r).eraseIdx ↑n) ((listMEraseEquiv n).symm a) := by
|
||||
intro r n a0 a
|
||||
match n with
|
||||
| ⟨0, h0⟩ =>
|
||||
simp
|
||||
rw [liftM_cons]
|
||||
simp
|
||||
conv_lhs =>
|
||||
rhs
|
||||
intro n
|
||||
erw [liftMCongrEquiv_symm_zero_succ]
|
||||
simp [listMEraseEquiv]
|
||||
| ⟨n + 1, hn⟩ =>
|
||||
simp
|
||||
rw [liftM_cons, liftM_cons]
|
||||
simp
|
||||
apply And.intro
|
||||
· sorry
|
||||
·
|
||||
|
||||
|
||||
|
||||
sorry
|
||||
|
||||
))
|
||||
/-
|
||||
lemma liftM_eraseIdx {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
||||
(q : I → Fin 2) (r0 : I): (r : List I) → (n : Fin (r0 :: r).length) → (a : Π i, f ((r0 :: r).get i)) →
|
||||
|
|
@ -350,42 +801,7 @@ lemma insertionSortEquiv_liftM {I : Type} {f : I → Type} [∀ i, Fintype (f i)
|
|||
| cons i l ih =>
|
||||
simp only [liftM, List.length_cons, Fin.zero_eta, List.insertionSort]
|
||||
conv_lhs => simp [HepLean.List.insertionSortEquiv]
|
||||
have h1 (l' : List (Σ i, f i)) :
|
||||
(HepLean.List.insertEquiv (fun i j => le1 i.fst j.fst) ⟨i, a ⟨0, by simp⟩⟩ l') =
|
||||
(Fin.castOrderIso (by simp)).toEquiv.trans
|
||||
((HepLean.List.insertEquiv le1 i (List.map (fun i => i.1) l')).trans
|
||||
(Fin.castOrderIso (by simp [List.orderedInsert_length])).toEquiv) := by
|
||||
induction l' with
|
||||
| nil =>
|
||||
simp only [List.length_cons, Nat.add_zero, Nat.zero_eq, Fin.zero_eta, List.length_singleton,
|
||||
List.orderedInsert, HepLean.List.insertEquiv, Fin.castOrderIso_refl,
|
||||
OrderIso.refl_toEquiv, Equiv.trans_refl]
|
||||
rfl
|
||||
| cons j l' ih' =>
|
||||
by_cases hr : (fun (i j : Σ i, f i) => le1 i.fst j.fst) ⟨i, a ⟨0, by simp⟩⟩ j
|
||||
· rw [HepLean.List.insertEquiv_cons_pos]
|
||||
· erw [HepLean.List.insertEquiv_cons_pos]
|
||||
· rfl
|
||||
· exact hr
|
||||
· exact hr
|
||||
· rw [HepLean.List.insertEquiv_cons_neg]
|
||||
· erw [HepLean.List.insertEquiv_cons_neg]
|
||||
· simp only [List.length_cons, Nat.add_zero, Nat.zero_eq, Fin.zero_eta,
|
||||
List.orderedInsert, Prod.mk.eta, Fin.mk_one]
|
||||
erw [ih']
|
||||
ext x
|
||||
simp only [Prod.mk.eta, List.length_cons, Nat.add_zero, Nat.zero_eq, Fin.zero_eta,
|
||||
HepLean.Fin.equivCons_trans, Nat.succ_eq_add_one,
|
||||
HepLean.Fin.equivCons_castOrderIso, Equiv.trans_apply, RelIso.coe_fn_toEquiv,
|
||||
Fin.castOrderIso_apply, Fin.cast_trans, Fin.coe_cast]
|
||||
congr 2
|
||||
match x with
|
||||
| ⟨0, h⟩ => rfl
|
||||
| ⟨1, h⟩ => rfl
|
||||
| ⟨Nat.succ (Nat.succ x), h⟩ => rfl
|
||||
· exact hr
|
||||
· exact hr
|
||||
erw [h1]
|
||||
erw [orderedInsertEquiv_sigma]
|
||||
rw [ih]
|
||||
simp only [HepLean.Fin.equivCons_trans, Nat.succ_eq_add_one,
|
||||
HepLean.Fin.equivCons_castOrderIso, List.length_cons, Nat.add_zero, Nat.zero_eq,
|
||||
|
|
@ -399,7 +815,7 @@ lemma insertionSortEquiv_liftM {I : Type} {f : I → Type} [∀ i, Fintype (f i)
|
|||
List.orderedInsert le1 i.1 (List.map (fun i => i.1) l') := by
|
||||
induction l' with
|
||||
| nil =>
|
||||
simp [HepLean.List.insertEquiv]
|
||||
simp [HepLean.List.orderedInsertEquiv]
|
||||
| cons j l' ih' =>
|
||||
by_cases hij : (fun i j => le1 i.fst j.fst) i j
|
||||
· rw [List.orderedInsert_of_le]
|
||||
|
|
@ -416,13 +832,13 @@ lemma insertionSortEquiv_liftM {I : Type} {f : I → Type} [∀ i, Fintype (f i)
|
|||
List.insertionSort le1 (List.map (fun i => i.1) l') := by
|
||||
induction l' with
|
||||
| nil =>
|
||||
simp [HepLean.List.insertEquiv]
|
||||
simp [HepLean.List.orderedInsertEquiv]
|
||||
| cons i l' ih' =>
|
||||
simp only [List.insertionSort, List.unzip_snd]
|
||||
simp only [List.unzip_snd] at h2'
|
||||
rw [h2']
|
||||
congr
|
||||
rw [HepLean.List.insertEquiv_congr _ _ _ (h2 _)]
|
||||
rw [HepLean.List.orderedInsertEquiv_congr _ _ _ (h2 _)]
|
||||
simp only [List.length_cons, Equiv.trans_apply, RelIso.coe_fn_toEquiv, Fin.castOrderIso_apply,
|
||||
Fin.cast_trans, Fin.coe_cast]
|
||||
have h3 : (List.insertionSort le1 (List.map (fun i => i.1) (liftM f l (fun i => a i.succ)))) =
|
||||
|
|
@ -437,7 +853,7 @@ lemma insertionSortEquiv_liftM {I : Type} {f : I → Type} [∀ i, Fintype (f i)
|
|||
List.unzip_snd, List.map_cons, List.cons.injEq, true_and]
|
||||
simpa using ih' _
|
||||
rw [h3']
|
||||
rw [HepLean.List.insertEquiv_congr _ _ _ h3]
|
||||
rw [HepLean.List.orderedInsertEquiv_congr _ _ _ h3]
|
||||
simp only [List.length_cons, Equiv.trans_apply, RelIso.coe_fn_toEquiv, Fin.castOrderIso_apply,
|
||||
Fin.cast_trans, Fin.cast_eq_self, Fin.coe_cast]
|
||||
|
||||
|
|
@ -477,40 +893,6 @@ lemma insertionSort_liftM {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
|||
rw [hget]
|
||||
simp
|
||||
|
||||
lemma koszulOrder_ofListM {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
||||
(q : I → Fin 2) (le1 : I → I → Prop) [DecidableRel le1]
|
||||
(l : List I) (x : ℂ) : koszulOrder (fun i j => le1 i.1 j.1) (fun i => q i.fst) (ofListM f l x) =
|
||||
freeAlgebraMap f (koszulOrder le1 q (ofList l x)) := by
|
||||
rw [koszulOrder_ofList]
|
||||
rw [map_smul]
|
||||
change _ = _ • ofListM _ _ _
|
||||
rw [ofListM_expand]
|
||||
rw [map_sum]
|
||||
conv_lhs =>
|
||||
rhs
|
||||
intro a
|
||||
rw [koszulOrder_ofList]
|
||||
rw [koszulSign_liftM]
|
||||
rw [← Finset.smul_sum]
|
||||
apply congrArg
|
||||
conv_lhs =>
|
||||
rhs
|
||||
intro n
|
||||
rw [insertionSort_liftM]
|
||||
rw [ofListM_expand]
|
||||
refine Fintype.sum_equiv ((HepLean.List.insertionSortEquiv le1 l).piCongr fun i => Equiv.cast ?_) _ _ ?_
|
||||
congr 1
|
||||
· rw [← HepLean.List.insertionSortEquiv_get]
|
||||
simp
|
||||
· intro x
|
||||
rfl
|
||||
|
||||
lemma koszulOrder_ofListM_eq_ofListM {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
||||
(q : I → Fin 2) (le1 : I → I → Prop) [DecidableRel le1]
|
||||
(l : List I) (x : ℂ) : koszulOrder (fun i j => le1 i.1 j.1) (fun i => q i.fst) (ofListM f l x) =
|
||||
koszulSign le1 q l • ofListM f (List.insertionSort le1 l) x := by
|
||||
rw [koszulOrder_ofListM, koszulOrder_ofList, map_smul]
|
||||
rfl
|
||||
|
||||
end
|
||||
end Wick
|
||||
|
|
|
|||
|
|
@ -27,15 +27,15 @@ namespace Wick
|
|||
noncomputable section
|
||||
|
||||
class SuperCommuteCenterMap {A : Type} [Semiring A] [Algebra ℂ A]
|
||||
(f : FreeAlgebra ℂ I →ₐ[ℂ] A) : Prop where
|
||||
prop : ∀ i j, f (superCommute q (FreeAlgebra.ι ℂ i) (FreeAlgebra.ι ℂ j)) ∈ Subalgebra.center ℂ A
|
||||
|
||||
(q : I → Fin 2) (F : FreeAlgebra ℂ I →ₐ[ℂ] A) : Prop where
|
||||
prop : ∀ i j, F (superCommute q (FreeAlgebra.ι ℂ i) (FreeAlgebra.ι ℂ j)) ∈ Subalgebra.center ℂ A
|
||||
dif_grade : ∀ i j, q i ≠ q j → F (superCommute q (FreeAlgebra.ι ℂ i) (FreeAlgebra.ι ℂ j)) = 0
|
||||
namespace SuperCommuteCenterMap
|
||||
|
||||
variable {I : Type} {A : Type} [Semiring A] [Algebra ℂ A]
|
||||
(f : FreeAlgebra ℂ I →ₐ[ℂ] A) [SuperCommuteCenterMap f]
|
||||
(f : FreeAlgebra ℂ I →ₐ[ℂ] A) (q : I → Fin 2) [SuperCommuteCenterMap q f]
|
||||
|
||||
lemma ofList_fst (q : I → Fin 2) (i j : I) :
|
||||
lemma ofList_fst (i j : I) :
|
||||
f (superCommute q (ofList [i] xa) (FreeAlgebra.ι ℂ j)) ∈ Subalgebra.center ℂ A := by
|
||||
have h1 : f (superCommute q (ofList [i] xa) (FreeAlgebra.ι ℂ j)) =
|
||||
xa • f (superCommute q (FreeAlgebra.ι ℂ i) (FreeAlgebra.ι ℂ j)) := by
|
||||
|
|
@ -51,7 +51,7 @@ lemma ofList_fst (q : I → Fin 2) (i j : I) :
|
|||
lemma ofList_freeAlgebraMap {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
||||
(q : I → Fin 2) (c : (Σ i, f i)) (x : ℂ)
|
||||
{A : Type} [Semiring A] [Algebra ℂ A] (F : FreeAlgebra ℂ (Σ i, f i) →ₐ[ℂ] A)
|
||||
[SuperCommuteCenterMap F] (b : I) :
|
||||
[SuperCommuteCenterMap (fun i => q i.1) F] (b : I) :
|
||||
F ((superCommute fun i => q i.fst) (ofList [c] x) ((freeAlgebraMap f) (FreeAlgebra.ι ℂ b)))
|
||||
∈ Subalgebra.center ℂ A := by
|
||||
rw [freeAlgebraMap_ι]
|
||||
|
|
@ -64,7 +64,7 @@ end SuperCommuteCenterMap
|
|||
|
||||
lemma superCommuteTake_superCommuteCenterMap {I : Type} (q : I → Fin 2) (lb : List I) (xa xb : ℂ) (n : ℕ)
|
||||
(hn : n < lb.length) {A : Type} [Semiring A] [Algebra ℂ A] (f : FreeAlgebra ℂ I →ₐ[ℂ] A)
|
||||
[SuperCommuteCenterMap f] (i : I) :
|
||||
[SuperCommuteCenterMap q f] (i : I) :
|
||||
f (superCommuteTake q [i] lb xa xb n hn) =
|
||||
f (superCommute q (ofList [i] xa) (FreeAlgebra.ι ℂ (lb.get ⟨n, hn⟩)))
|
||||
* (superCommuteCoef q [i] (List.take n lb) •
|
||||
|
|
@ -83,7 +83,7 @@ lemma superCommuteTakeM_F {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
|||
(q : I → Fin 2) (c : (Σ i, f i)) (r : List I) (x y : ℂ) (n : ℕ)
|
||||
(hn : n < r.length)
|
||||
{A : Type} [Semiring A] [Algebra ℂ A] (F : FreeAlgebra ℂ (Σ i, f i) →ₐ[ℂ] A)
|
||||
[SuperCommuteCenterMap F] :
|
||||
[SuperCommuteCenterMap (fun i => q i.1) F] :
|
||||
F (superCommuteTakeM q [c] r x y n hn) = superCommuteCoefM q [c] (List.take n r) •
|
||||
(F (superCommute (fun i => q i.1) (ofList [c] x) (freeAlgebraMap f (FreeAlgebra.ι ℂ (r.get ⟨n, hn⟩))))
|
||||
* F (ofListM f (List.eraseIdx r n) y)) := by
|
||||
|
|
@ -105,104 +105,102 @@ lemma superCommuteTakeM_F {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
|||
|
||||
lemma superCommute_koszulOrder_le_ofList {I : Type}
|
||||
(q : I → Fin 2) (r : List I) (x : ℂ)
|
||||
(le1 :I → I → Prop) [DecidableRel le1]
|
||||
(le1 :I → I → Prop) [DecidableRel le1] [IsTotal I le1] [IsTrans I le1]
|
||||
(i : I)
|
||||
{A : Type} [Semiring A] [Algebra ℂ A]
|
||||
(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap F] :
|
||||
(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap q F] :
|
||||
F ((superCommute q (FreeAlgebra.ι ℂ i) (koszulOrder le1 q (ofList r x)))) =
|
||||
∑ n : Fin r.length, (superCommuteCoef q [r.get n] (r.take n)) •
|
||||
(F (((superCommute q) (ofList [i] 1)) (FreeAlgebra.ι ℂ (r.get n))) *
|
||||
F ((koszulOrder le1 q) (ofList (r.eraseIdx ↑n) x))) := by
|
||||
rw [koszulOrder_ofList]
|
||||
rw [map_smul, map_smul, ← ofList_singleton]
|
||||
rw [superCommute_ofList_sum]
|
||||
rw [map_sum]
|
||||
rw [← (HepLean.List.insertionSortEquiv le1 r).sum_comp]
|
||||
rw [koszulOrder_ofList, map_smul, map_smul, ← ofList_singleton, superCommute_ofList_sum]
|
||||
rw [map_sum, ← (HepLean.List.insertionSortEquiv le1 r).sum_comp]
|
||||
conv_lhs =>
|
||||
rhs
|
||||
rhs
|
||||
enter [2, 2]
|
||||
intro n
|
||||
rw [superCommuteTake_superCommuteCenterMap]
|
||||
lhs
|
||||
rhs
|
||||
rhs
|
||||
rhs
|
||||
enter [1, 2, 2, 2]
|
||||
change ((List.insertionSort le1 r).get ∘ (HepLean.List.insertionSortEquiv le1 r)) n
|
||||
rw [HepLean.List.insertionSort_get_comp_insertionSortEquiv]
|
||||
have hListErase (n : Fin r.length ) : (List.insertionSort le1 r).eraseIdx ↑((HepLean.List.insertionSortEquiv le1 r) n)
|
||||
= List.insertionSort le1 (r.eraseIdx n) := by sorry
|
||||
conv_lhs =>
|
||||
rhs
|
||||
rhs
|
||||
enter [2, 2]
|
||||
intro n
|
||||
rw [hListErase]
|
||||
rw [HepLean.List.eraseIdx_insertionSort_fin le1 r n]
|
||||
rw [ofList_insertionSort_eq_koszulOrder le1 q]
|
||||
rw [Finset.smul_sum]
|
||||
conv_lhs =>
|
||||
rhs
|
||||
intro n
|
||||
rw [map_smul]
|
||||
rw [smul_smul]
|
||||
rw [Algebra.mul_smul_comm]
|
||||
rw [smul_smul]
|
||||
rw [map_smul, smul_smul, Algebra.mul_smul_comm, smul_smul]
|
||||
congr
|
||||
funext n
|
||||
congr 1
|
||||
trans superCommuteCoefLE q le1 r i n
|
||||
· rw [superCommuteCoefLE]
|
||||
rw [mul_assoc]
|
||||
exact superCommuteCoefLE_eq_get q le1 r i n
|
||||
by_cases hq : q i ≠ q (r.get n)
|
||||
· have hn := SuperCommuteCenterMap.dif_grade (q := q) (F := F) i (r.get n) hq
|
||||
conv_lhs =>
|
||||
enter [2, 1]
|
||||
rw [ofList_singleton, hn]
|
||||
conv_rhs =>
|
||||
enter [2, 1]
|
||||
rw [ofList_singleton, hn]
|
||||
simp
|
||||
· congr 1
|
||||
trans superCommuteCoefLE q le1 r i n
|
||||
· rw [superCommuteCoefLE, mul_assoc]
|
||||
refine superCommuteCoefLE_eq_get q le1 r i n ?_
|
||||
simpa using hq
|
||||
|
||||
lemma koszulOrder_of_le_all_ofList {I : Type}
|
||||
(q : I → Fin 2) (r : List I) (x : ℂ) (le1 : I → I → Prop) [DecidableRel le1]
|
||||
(i : I) (hi : ∀ j, le1 j i)
|
||||
{A : Type} [Semiring A] [Algebra ℂ A]
|
||||
(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap F] :
|
||||
(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap q F] :
|
||||
F (koszulOrder le1 q (ofList r x * FreeAlgebra.ι ℂ i))
|
||||
= superCommuteCoef q [i] r • F (koszulOrder le1 q (FreeAlgebra.ι ℂ i * ofList r x)) := by
|
||||
conv_lhs =>
|
||||
rhs
|
||||
rhs
|
||||
enter [2, 2]
|
||||
rw [← ofList_singleton]
|
||||
rw [ofListM_ofList_superCommute' q]
|
||||
rw [map_sub]
|
||||
rw [sub_eq_add_neg]
|
||||
rw [map_add]
|
||||
conv_lhs =>
|
||||
rhs
|
||||
rhs
|
||||
enter [2, 2]
|
||||
rw [map_smul]
|
||||
rw [← neg_smul]
|
||||
rw [map_smul, map_smul, map_smul]
|
||||
|
||||
sorry
|
||||
|
||||
lemma le_all_mul_koszulOrder_ofList {I : Type}
|
||||
(q : I → Fin 2) (r : List I) (x : ℂ) (le1 : I → I→ Prop) [DecidableRel le1]
|
||||
(i : I) (hi : ∀ (j : I), le1 j i)
|
||||
{A : Type} [Semiring A] [Algebra ℂ A]
|
||||
(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap F] :
|
||||
(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap q F] :
|
||||
F (FreeAlgebra.ι ℂ i * koszulOrder le1 q (ofList r x)) =
|
||||
F ((koszulOrder le1 q) (FreeAlgebra.ι ℂ i * ofList r x)) +
|
||||
F (((superCommute q) (ofList [i] 1)) ((koszulOrder le1 q) (ofList r x))) := by
|
||||
rw [koszulOrder_ofList, Algebra.mul_smul_comm, map_smul, ← ofList_singleton,
|
||||
ofList_ofList_superCommute q, map_add, smul_add, ← map_smul]
|
||||
conv_lhs =>
|
||||
lhs
|
||||
rhs
|
||||
enter [1, 2]
|
||||
rw [← Algebra.smul_mul_assoc, smul_smul, mul_comm, ← smul_smul, ← koszulOrder_ofList,
|
||||
Algebra.smul_mul_assoc, ofList_singleton]
|
||||
rw [koszulOrder_mul_ge, map_smul]
|
||||
congr
|
||||
· rw [koszulOrder_of_le_all_ofList]
|
||||
sorry
|
||||
sorry
|
||||
rw [superCommuteCoef_perm_snd q [i] (List.insertionSort le1 r) r
|
||||
(List.perm_insertionSort le1 r)]
|
||||
rw [smul_smul]
|
||||
rw [superCommuteCoef_mul_self]
|
||||
simp [ofList_singleton]
|
||||
exact fun j => hi j
|
||||
· rw [map_smul, map_smul]
|
||||
· exact fun j => hi j
|
||||
|
||||
def superCommuteCenterOrder {I : Type}
|
||||
(q : I → Fin 2) (r : List I) (i : I)
|
||||
{A : Type} [Semiring A] [Algebra ℂ A]
|
||||
(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap F]
|
||||
(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap q F]
|
||||
(n : Option (Fin r.length)) : A :=
|
||||
match n with
|
||||
| none => 1
|
||||
|
|
@ -212,7 +210,7 @@ def superCommuteCenterOrder {I : Type}
|
|||
lemma superCommuteCenterOrder_none {I : Type}
|
||||
(q : I → Fin 2) (r : List I) (i : I)
|
||||
{A : Type} [Semiring A] [Algebra ℂ A]
|
||||
(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap F] :
|
||||
(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap q F] :
|
||||
superCommuteCenterOrder q r i F none = 1 := by
|
||||
simp [superCommuteCenterOrder]
|
||||
|
||||
|
|
@ -220,9 +218,10 @@ open HepLean.List
|
|||
|
||||
lemma le_all_mul_koszulOrder_ofList_expand {I : Type}
|
||||
(q : I → Fin 2) (r : List I) (x : ℂ) (le1 : I → I→ Prop) [DecidableRel le1]
|
||||
[IsTotal I le1] [IsTrans I le1]
|
||||
(i : I) (hi : ∀ (j : I), le1 j i)
|
||||
{A : Type} [Semiring A] [Algebra ℂ A]
|
||||
(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap F] :
|
||||
(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap q F] :
|
||||
F (FreeAlgebra.ι ℂ i * koszulOrder le1 q (ofList r x)) =
|
||||
∑ n, superCommuteCenterOrder q r i F n * F ((koszulOrder le1 q) (ofList (optionEraseZ r i n) x)) := by
|
||||
rw [le_all_mul_koszulOrder_ofList]
|
||||
|
|
@ -242,9 +241,10 @@ lemma le_all_mul_koszulOrder_ofList_expand {I : Type}
|
|||
|
||||
lemma le_all_mul_koszulOrder_ofListM_expand {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
|
||||
(q : I → Fin 2) (r : List I) (x : ℂ) (le1 : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le1]
|
||||
[IsTotal (Σ i, f i) le1] [IsTrans (Σ i, f i) le1]
|
||||
(i : (Σ i, f i)) (hi : ∀ (j : (Σ i, f i)), le1 j i)
|
||||
{A : Type} [Semiring A] [Algebra ℂ A]
|
||||
(F : FreeAlgebra ℂ (Σ i, f i) →ₐ A) [SuperCommuteCenterMap F] :
|
||||
(F : FreeAlgebra ℂ (Σ i, f i) →ₐ A) [SuperCommuteCenterMap (fun i => q i.1) F] :
|
||||
F (ofList [i] 1 * koszulOrder le1 (fun i => q i.1) (ofListM f r x)) =
|
||||
F ((koszulOrder le1 fun i => q i.fst) (ofList [i] 1 * ofListM f r x)) +
|
||||
∑ n : (Fin r.length), superCommuteCoef q [r.get n] (List.take (↑n) r) •
|
||||
|
|
@ -318,19 +318,19 @@ lemma le_all_mul_koszulOrder_ofListM_expand {I : Type} {f : I → Type} [∀ i,
|
|||
rhs
|
||||
intro a0
|
||||
rw [← Finset.mul_sum]
|
||||
have hl (a0 : f (r0 :: r)[↑n]) (a : (i : Fin r.length) → f (r0 :: r)[↑(n.succAbove i)]):
|
||||
have hl (n : Fin (r0 :: r).length) (a0 : f (r0 :: r)[↑n]) (a : (i : Fin r.length) → f (r0 :: r)[↑(n.succAbove i)]):
|
||||
(ofList (optionEraseZ (liftM f (r0 :: r) ((liftMCongrEquiv f r0 r n).symm (a0, a))) i (some (Fin.cast (by simp ) n))) x)
|
||||
= ofList ((liftM f ((r0 :: r).eraseIdx ↑n) ((listMEraseEquiv q n).symm a))) x := by
|
||||
simp only [optionEraseZ, List.get_eq_getElem, List.length_cons, Fin.coe_cast]
|
||||
simp [liftMCongrEquiv]
|
||||
congr
|
||||
sorry
|
||||
conv_lhs =>
|
||||
rhs
|
||||
intro a0
|
||||
rhs
|
||||
rhs
|
||||
enter [2, 2]
|
||||
intro a
|
||||
erw [hl a0 a]
|
||||
erw [hl n a0 a]
|
||||
rw [← Finset.sum_mul]
|
||||
conv_lhs =>
|
||||
lhs
|
||||
|
|
|
|||
|
|
@ -87,6 +87,10 @@ def koszulSign {I : Type} (r : I → I → Prop) [DecidableRel r] (q : I → Fin
|
|||
| a :: l => koszulSignInsert r q a l * koszulSign r q l
|
||||
|
||||
def natTestQ : ℕ → Fin 2 := fun n => if n % 2 = 0 then 0 else 1
|
||||
def natTest3 : ℕ × ℕ × ℕ → Fin 2 := fun ⟨a, b, c⟩ => if a % 2 = 0 then 0 else 1
|
||||
|
||||
#eval List.insertionSort (fun i j => i.2 ≤ j.2) [(1, 1, 0), (1, 0, 3)]
|
||||
#eval koszulSign (fun i j => i.2 ≤ j.2) natTest3 [ (0, 0, 2), (1, 1, 0), (1, 1, 3)]
|
||||
|
||||
lemma koszulSign_mul_self {I : Type} (r : I → I → Prop) [DecidableRel r] (q : I → Fin 2)
|
||||
(l : List I) : koszulSign r q l * koszulSign r q l = 1 := by
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue