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f1f7953 | 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 | import OAI.Analysis.IrrationalRotation.Sampling.CharacterSamplingTrial
import OAI.Analysis.IrrationalRotation.Rotation.CircleParameterBlocks
namespace OAI
noncomputable section
open MeasureTheory
open scoped BigOperators
namespace Invariant233.CircleParameterData
theorem prod_block_eq_fin {A : Type*} [CommMonoid A]
(p : CircleParameterData) (j : ℕ) (hj : 1 ≤ j) (f : ℕ → A) :
(∏ i ∈ p.block j, f i) =
∏ i : Fin (p.m j), f (p.blockEnd (j - 1) + i.val + 1) := by
classical
have hend : p.blockEnd j = p.blockEnd (j - 1) + p.m j := by
simpa only [Nat.sub_add_cancel hj] using p.blockEnd_succ (j - 1)
symm
apply Finset.prod_bij (fun i _ => p.blockEnd (j - 1) + i.val + 1)
· intro i hi
rw [mem_block_iff, hend]
have h := i.isLt
omega
· intro i hi k hk heq
apply Fin.ext
omega
· intro i hi
obtain ⟨hlo, hhi⟩ := (p.mem_block_iff j i).mp hi
rw [hend] at hhi
refine ⟨⟨i - p.blockEnd (j - 1) - 1, by omega⟩, Finset.mem_univ _, ?_⟩
simp only
omega
· intro i hi
rfl
theorem samplingAverage_eq_characterAverage (p : CircleParameterData)
(θ : ℝ) (j r : ℕ) (hj : 1 ≤ j) (endpoint : Bool) (x : Circle) :
p.samplingAverage θ j r endpoint x =
CharacterSampling.average (p.m j) (p.samples j r) endpoint
(fun i : Fin (p.m j) => p.alpha θ (p.blockEnd (j - 1) + i.val + 1))
(fun i : Fin (p.m j) =>
(p.q (p.blockEnd (j - 1) + i.val + 1) : ℤ) • x) := by
unfold samplingAverage CharacterSampling.average
rw [one_div]
congr 1
apply Finset.sum_congr rfl
intro n hn
rw [prod_block_eq_fin p j hj]
apply Finset.prod_congr rfl
intro i hi
rw [CharacterSampling.kernel_eq_parameter_orientedKernel]
rfl
end Invariant233.CircleParameterData
end
end OAI
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