File size: 4,204 Bytes
a72760f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | 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
|