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9b6e014 | 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 | import Mathlib
import OAI.Probability.BinarySweep.Processes.PointMassOne
import OAI.Probability.BinarySweep.Analytic.FiniteAverageGap
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps
def complexLineLaw (d : ℕ) (z : ℂ) (g : Equiv.Perm (Slot d)) : ℂ :=
(1-z)*(uniformLaw (Equiv.Perm (Slot d)) g:ℂ)+z*(binaryLaw d g:ℂ)
lemma complexLineLaw_real (d : ℕ) (z : ℝ) (g : Equiv.Perm (Slot d)) :
complexLineLaw d (z:ℂ) g=(lineLaw d z g:ℂ) := by
simp [complexLineLaw,lineLaw]
lemma complexLine_average {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
[FiniteDimensional ℂ V] (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Slot d)) V)
(z : ℂ) : complexAverage ρ (complexLineLaw d z)=
complexAverage ρ (fun g => (uniformLaw (Equiv.Perm (Slot d)) g:ℂ))+
z • (complexAverage ρ (fun g => (binaryLaw d g:ℂ))-
complexAverage ρ (fun g => (uniformLaw (Equiv.Perm (Slot d)) g:ℂ))) := by
ext v
simp only [complexAverage_apply,complexLineLaw,add_smul,mul_smul,
Finset.sum_add_distrib,←Finset.smul_sum,add_apply,
smul_apply,sub_apply]
module
theorem binary_line_disk (d : ℕ) : ∃ R : ℝ, 1<R ∧
∀ (V : Type*) [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V],
∀ (ρ : Representation ℂ (Equiv.Perm (Slot d)) V),
(∀g v, ‖ρ g v‖=‖v‖) → ∀z : ℂ, ‖z‖≤R →
‖complexAverage ρ (complexLineLaw d z)‖ ≤ 1 := by
have he : ∀ᶠ n in Filter.atTop,
totalVariation (sweepLaw d n) (uniformLaw (Equiv.Perm (Slot d)))<1/4 :=
(fixed_dimension_sweep_convergence d).eventually (gt_mem_nhds (by norm_num))
obtain ⟨N,hN⟩ := Filter.eventually_atTop.mp he
have ht := (hN (max N 1) (le_max_left _ _)).le
have hp : IsProbability (binaryLaw d) :=
⟨finiteLaw_nonneg _,finiteLaw_sum _⟩
have hp1 : 0<binaryLaw d 1 := by
rw [binaryLaw,←binarySweep_zero d]
exact finiteLaw_pos_of_mem _ _
obtain ⟨R,hR,h⟩ := finite_average_disk hp hp1 (by omega : 0 < max N 1) ht
refine ⟨R,hR,?_⟩
intro V _ _ _ ρ hρ z hz
rw [complexLine_average]
exact h V ρ hρ z hz
theorem binary_lines_disk (m : ℕ) : ∃ R : ℝ, 1<R ∧ ∀ d, d ≤ m →
∀ (V : Type*) [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V],
∀ (ρ : Representation ℂ (Equiv.Perm (Slot d)) V),
(∀g v, ‖ρ g v‖=‖v‖) → ∀z : ℂ, ‖z‖≤R →
‖complexAverage ρ (complexLineLaw d z)‖ ≤ 1 := by
induction m with
| zero =>
obtain ⟨R,hR,h⟩ := binary_line_disk 0
refine ⟨R,hR,?_⟩
intro d hd
have : d=0 := by omega
subst d
exact h
| succ m ih =>
obtain ⟨R,hR,h⟩ := ih
obtain ⟨S,hS,k⟩ := binary_line_disk (m+1)
refine ⟨min R S,lt_min hR hS,?_⟩
intro d hd V _ _ _ ρ hρ z hz
by_cases he : d=m+1
· subst d
exact k V ρ hρ z (hz.trans (min_le_right _ _))
· exact h d (by omega) V ρ hρ z (hz.trans (min_le_left _ _))
end BinaryCoordinateSweeps
end
end OAI
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