129 lines
4.8 KiB
Text
129 lines
4.8 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.PerturbationTheory.FieldStatistics.ExchangeSign
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/-!
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# Field statistics of a finite set.
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-/
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namespace FieldStatistic
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variable {𝓕 : Type}
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/-- The field statistic associated with a map `f : Fin n → 𝓕` (usually `.get` of a list)
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and a finite set of elements of `Fin n`. -/
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def ofFinset {n : ℕ} (q : 𝓕 → FieldStatistic) (f : Fin n → 𝓕) (a : Finset (Fin n)) :
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FieldStatistic :=
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ofList q ((a.sort (· ≤ ·)).map f)
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@[simp]
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lemma ofFinset_empty (q : 𝓕 → FieldStatistic) (f : Fin n → 𝓕) :
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ofFinset q f ∅ = 1 := by
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simp only [ofFinset, Finset.sort_empty, List.map_nil, ofList_empty]
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rfl
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lemma ofFinset_singleton {n : ℕ} (q : 𝓕 → FieldStatistic) (f : Fin n → 𝓕) (i : Fin n) :
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ofFinset q f {i} = q (f i) := by
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simp [ofFinset]
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lemma ofFinset_finset_map {n m : ℕ}
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(q : 𝓕 → FieldStatistic) (i : Fin m → Fin n) (hi : Function.Injective i)
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(f : Fin n → 𝓕) (a : Finset (Fin m)) :
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ofFinset q (f ∘ i) a = ofFinset q f (a.map ⟨i, hi⟩) := by
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simp only [ofFinset]
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apply FieldStatistic.ofList_perm
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rw [← List.map_map]
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refine List.Perm.map f ?_
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apply List.perm_of_nodup_nodup_toFinset_eq
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· refine (List.nodup_map_iff_inj_on ?_).mpr ?_
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exact Finset.sort_nodup (fun x1 x2 => x1 ≤ x2) a
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simp only [Finset.mem_sort]
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intro x hx y hy
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exact fun a => hi a
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· exact Finset.sort_nodup (fun x1 x2 => x1 ≤ x2) (Finset.map { toFun := i, inj' := hi } a)
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· ext a
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simp
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lemma ofFinset_insert (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a : Finset (Fin φs.length))
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(i : Fin φs.length) (h : i ∉ a) :
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ofFinset q φs.get (Insert.insert i a) = (q φs[i]) * ofFinset q φs.get a := by
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simp only [ofFinset, instCommGroup, Fin.getElem_fin]
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rw [← ofList_cons_eq_mul]
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have h1 : (φs[↑i] :: List.map φs.get (Finset.sort (fun x1 x2 => x1 ≤ x2) a))
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= List.map φs.get (i :: Finset.sort (fun x1 x2 => x1 ≤ x2) a) := by
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simp
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erw [h1]
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apply ofList_perm
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refine List.Perm.map φs.get ?_
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refine (List.perm_ext_iff_of_nodup ?_ ?_).mpr ?_
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· exact Finset.sort_nodup (fun x1 x2 => x1 ≤ x2) (Insert.insert i a)
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· simp only [List.nodup_cons, Finset.mem_sort, Finset.sort_nodup, and_true]
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exact h
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intro a
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simp
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lemma ofFinset_erase (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a : Finset (Fin φs.length))
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(i : Fin φs.length) (h : i ∈ a) :
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ofFinset q φs.get (a.erase i) = (q φs[i]) * ofFinset q φs.get a := by
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have ha : a = Insert.insert i (a.erase i) := by
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exact Eq.symm (Finset.insert_erase h)
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conv_rhs => rw [ha]
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rw [ofFinset_insert]
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rw [← mul_assoc]
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simp only [instCommGroup, Fin.getElem_fin, mul_self, one_mul]
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simp
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lemma ofFinset_eq_prod (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a : Finset (Fin φs.length)) :
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ofFinset q φs.get a = ∏ (i : Fin φs.length), if i ∈ a then (q φs[i]) else 1 := by
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rw [ofFinset]
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rw [ofList_map_eq_finset_prod]
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congr
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funext i
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simp only [Finset.mem_sort, Fin.getElem_fin]
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exact Finset.sort_nodup (fun x1 x2 => x1 ≤ x2) a
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lemma ofFinset_union (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a b : Finset (Fin φs.length)) :
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ofFinset q φs.get a * ofFinset q φs.get b = ofFinset q φs.get ((a ∪ b) \ (a ∩ b)) := by
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rw [ofFinset_eq_prod, ofFinset_eq_prod, ofFinset_eq_prod]
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rw [← Finset.prod_mul_distrib]
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congr
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funext x
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simp only [instCommGroup, Fin.getElem_fin, mul_ite, ite_mul, mul_self, one_mul, mul_one,
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Finset.mem_sdiff, Finset.mem_union, Finset.mem_inter, not_and]
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split
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· rename_i h
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simp [h]
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· rename_i h
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simp [h]
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lemma ofFinset_union_disjoint (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a b : Finset (Fin φs.length))
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(h : Disjoint a b) :
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ofFinset q φs.get a * ofFinset q φs.get b = ofFinset q φs.get (a ∪ b) := by
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rw [ofFinset_union, Finset.disjoint_iff_inter_eq_empty.mp h]
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simp
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lemma ofFinset_filter_mul_neg (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a : Finset (Fin φs.length))
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(p : Fin φs.length → Prop) [DecidablePred p] :
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ofFinset q φs.get (Finset.filter p a) *
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ofFinset q φs.get (Finset.filter (fun i => ¬ p i) a) = ofFinset q φs.get a := by
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rw [ofFinset_union_disjoint]
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congr
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exact Finset.filter_union_filter_neg_eq p a
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exact Finset.disjoint_filter_filter_neg a a p
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lemma ofFinset_filter (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a : Finset (Fin φs.length))
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(p : Fin φs.length → Prop) [DecidablePred p] :
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ofFinset q φs.get (Finset.filter p a) = ofFinset q φs.get (Finset.filter (fun i => ¬ p i) a) *
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ofFinset q φs.get a := by
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rw [← ofFinset_filter_mul_neg q φs a p]
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conv_rhs =>
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rhs
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rw [mul_comm]
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rw [← mul_assoc]
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simp
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end FieldStatistic
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