| import OAI.NumberTheory.Ostmann.Tree.CoupledHaar |
| import OAI.NumberTheory.Ostmann.Tree.CycleProducts |
|
|
| namespace OAI |
|
|
| namespace Ostmann.Tree |
| noncomputable section |
| open scoped BigOperators ComplexConjugate |
| open Ostmann.Arithmetic.ResidueHaar |
|
|
| lemma average_fintype_congr {I : Type*} (i j : Fintype I) (f : I → ℂ) : |
| @average I i f = @average I j f := by |
| cases Subsingleton.elim i j |
| rfl |
| lemma real_average_fintype_congr {I : Type*} (i j : Fintype I) (f : I → ℝ) : |
| @Density.average I i f = @Density.average I j f := by |
| cases Subsingleton.elim i j |
| rfl |
|
|
| lemma average_mul_const {I : Type*} [Fintype I] (f : I → ℂ) (c : ℂ) : |
| average (fun i => f i*c)=average f*c := by |
| simp only [average,← Finset.sum_mul] |
| ring |
| lemma average_const {I : Type*} [Fintype I] [Nonempty I] (c : ℂ) : |
| average (fun _ : I => c)=c := by |
| have hn : (Fintype.card I:ℂ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero |
| simp only [average,Finset.sum_const,Finset.card_univ,nsmul_eq_mul] |
| exact inv_mul_cancel_left₀ hn c |
| lemma average_comm {I J : Type*} [Fintype I] [Fintype J] (f : I → J → ℂ) : |
| average (fun i => average (f i)) = average (fun j => average (fun i => f i j)) := by |
| simp only [average,← Finset.mul_sum] |
| rw [Finset.sum_comm] |
| ring |
|
|
| def BalancedSelection.projection {E V W U : Type*} |
| [DecidableEq E] [DecidableEq V] [DecidableEq W] [CommGroup U] [Fintype U] |
| {left : E → V} {right : E → W} (c : BalancedSelection left right) |
| (F : (E → U) → ℂ) (M : E → U) : ℂ := average (fun z : U => F (c.act z M)) |
|
|
| theorem cycle_projection_correlation {E E' V W U : Type*} |
| [Fintype E] [Fintype E'] [Fintype U] [CommGroup U] |
| [DecidableEq E] [DecidableEq E'] [DecidableEq V] [DecidableEq W] |
| {left : E → V} {right : E → W} (other : E' → V) |
| (c : BalancedSelection left right) (F : (E → U) → ℂ) (G : (E' → U) → ℂ) : |
| average (fun x : coupledGroup (labelProducts left) (labelProducts other) => |
| c.projection F x.val.1*conj (G x.val.2)) = |
| average (fun x : coupledGroup (labelProducts left) (labelProducts other) => |
| F x.val.1*conj (G x.val.2)) := by |
| classical |
| let S := coupledGroup (labelProducts (U:=U) left) (labelProducts other) |
| calc |
| _ = average (fun x : S => average (fun z : U => F (c.act z x.val.1)*conj (G x.val.2))) := by |
| congr 1 |
| funext x |
| exact (average_mul_const _ _).symm |
| _ = average (fun z : U => average (fun x : S => F (c.act z x.val.1)*conj (G x.val.2))) := |
| average_comm _ |
| _ = average (fun _z : U => average (fun x : S => F x.val.1*conj (G x.val.2))) := by |
| congr 1 |
| funext z |
| have h := coupledGroup.average_change_first (labelProducts left) (labelProducts other) |
| (c.actEquiv z) (c.labelProducts_act z) F G |
| simp only [average_fintype_congr _ (inferInstance : Fintype S)] at h |
| change average (fun x : S => F (c.act z x.val.1)*conj (G x.val.2)) = _ at h |
| exact h |
| _ = _ := average_const _ |
|
|
| theorem cycle_projection_reduction {E E' V W U : Type*} |
| [Fintype E] [Fintype E'] [Fintype U] [CommGroup U] |
| [DecidableEq E] [DecidableEq E'] [DecidableEq V] [DecidableEq W] |
| {left : E → V} {right : E → W} (other : E' → V) |
| (hl : Function.Surjective left) (ho : Function.Surjective other) |
| (c : BalancedSelection left right) (F : (E → U) → ℂ) (G : (E' → U) → ℂ) : |
| ‖average (fun x : coupledGroup (labelProducts left) (labelProducts other) => |
| F x.val.1*conj (G x.val.2))‖^2 ≤ |
| Density.average (fun M => ‖c.projection F M‖^2)*Density.average (fun N => ‖G N‖^2) := by |
| classical |
| let S := coupledGroup (labelProducts (U:=U) left) (labelProducts other) |
| have hi := cycle_projection_correlation other c F G |
| have h := coupled_cauchy_sq (labelProducts left) (labelProducts other) |
| (labelProducts_surjective left hl) (labelProducts_surjective other ho) (c.projection F) G |
| simp only [average_fintype_congr _ (inferInstance : Fintype S)] at hi h |
| simp only [real_average_fintype_congr _ (inferInstance : Fintype (E → U)), |
| real_average_fintype_congr _ (inferInstance : Fintype (E' → U))] at h |
| rw [hi] at h |
| exact h |
|
|
| end |
| end Ostmann.Tree |
|
|
| end OAI |
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|
