| import OAI.NumberTheory.Ostmann.Tree.CycleCut |
| import OAI.NumberTheory.Ostmann.Tree.DensityHorizontal |
|
|
| namespace OAI |
|
|
| namespace Ostmann.Tree |
| noncomputable section |
| open scoped BigOperators |
| open Ostmann.Arithmetic.ResidueHaar |
| open Ostmann.Arithmetic.GroupHaarImage |
|
|
| local instance fieldDecidableEq {F : Type*} [Field F] : DecidableEq F := Classical.decEq F |
|
|
| local instance subgroupFintype {G : Type*} [Group G] [Fintype G] (S : Subgroup G) : |
| Fintype S := Fintype.ofFinite _ |
| local instance fiberFintype {A B : Type*} [Fintype A] (f : A → B) (y : B) : |
| Fintype {x : A |
|
|
| def fiberAverage {A B : Type*} [Fintype A] (f : A → B) (F : A → ℂ) (y : B) : ℂ := |
| average (fun x : {x : A // f x=y} => F x.val) |
|
|
| def kernelFiberEquiv {A B : Type*} [Group A] [Group B] |
| (f : A →* B) (hf : Function.Surjective f) (y : B) : f.ker ≃ {x : A |
| toFun k := ⟨Classical.choose (hf y)*k.val,by |
| rw [map_mul,Classical.choose_spec (hf y),show f k.val=1 from k.property,mul_one]⟩ |
| invFun x := ⟨(Classical.choose (hf y))⁻¹*x.val,by |
| change f ((Classical.choose (hf y))⁻¹*x.val)=1 |
| rw [map_mul,map_inv,Classical.choose_spec (hf y),x.property,inv_mul_cancel]⟩ |
| left_inv k := by apply Subtype.ext; simp |
| right_inv x := by apply Subtype.ext; simp |
|
|
| theorem average_product_nested {I J : Type*} [Fintype I] [Fintype J] |
| (F : I × J → ℂ) : average F=average (fun i => average (fun j => F (i,j))) := by |
| simp only [average,Fintype.card_prod,Nat.cast_mul,mul_inv_rev,Fintype.sum_prod_type, |
| ← Finset.mul_sum] |
| ring |
|
|
| theorem average_fiberAverage {A B : Type*} [Group A] [Group B] [Fintype A] [Fintype B] |
| (f : A →* B) (hf : Function.Surjective f) (F : A → ℂ) : |
| average F=average (fiberAverage f F) := by |
| have he := average_equiv (homProductEquiv f hf).symm F |
| calc |
| average F = average (fun z : B × f.ker => F (Classical.choose (hf z.1)*z.2.val)) := he.symm |
| _ = average (fun y : B => average (fun k : f.ker => F (Classical.choose (hf y)*k.val))) := |
| average_product_nested _ |
| _ = average (fiberAverage f F) := by |
| congr 1 |
| funext y |
| exact average_equiv (kernelFiberEquiv f hf y) (fun x => F x.val) |
|
|
| theorem cut_fiber_average {F : Type*} [Field F] [Fintype F] {d k : ℕ} |
| (C : Density.Cut d k) (W : (Leaves d → Fˣ) → ℂ) : |
| average W = average (fiberAverage C.project W) := |
| average_fiberAverage (C.projectHom) C.project_surjective W |
|
|
| end |
| end Ostmann.Tree |
|
|
| end OAI |
|
|
