openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /CharacterSamplingBlocks.lean
| 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 | |