| import OAI.MeasureTheory.DyadicAvoidance.GridStabilityFin |
| import OAI.MeasureTheory.DyadicAvoidance.FiniteTableModel |
|
|
| namespace OAI |
|
|
| universe u_ι |
|
|
| noncomputable section |
| namespace Problem310.FiniteTableModel |
|
|
| / |
| theorem selectorAddress_eq_of_grid_stability {ι : Type u_ι} {a b : ι → ℕ} |
| {M d : ℕ} (bS : Node M d → Fin M → ℕ) {x t : ℝ} |
| (hx : x ∈ gridStableCenters a b) (e : ι) (P : Node M d) (i : Fin M) |
| {n : ℕ} (hresolution : bS P i ≤ b e) (hindex : a e ≤ n) |
| (ht : t ∈ Set.Icc (1 : ℝ) 2) : |
| selectorAddress bS P i (x + t * ((2 : ℝ)⁻¹) ^ n) = selectorAddress bS P i x := by |
| unfold selectorAddress |
| rw [gridStableCenters_preserves_fin hx e hresolution hindex ht] |
|
|
| / |
| selector evaluations at a stable center. -/ |
| theorem selectorValue_eq_of_grid_stability {ι : Type u_ι} {a b : ι → ℕ} |
| {M d : ℕ} (bS : Node M d → Fin M → ℕ) (ω : SelectorTable bS) {x t : ℝ} |
| (hx : x ∈ gridStableCenters a b) (e : ι) (P : Node M d) (i : Fin M) |
| {n : ℕ} (hresolution : bS P i ≤ b e) (hindex : a e ≤ n) |
| (ht : t ∈ Set.Icc (1 : ℝ) 2) : |
| selectorValue bS ω P.val i (x + t * ((2 : ℝ)⁻¹) ^ n) = |
| selectorValue bS ω P.val i x := by |
| rw [selectorValue_valid, selectorValue_valid, |
| selectorAddress_eq_of_grid_stability bS hx e P i hresolution hindex ht] |
|
|
| end Problem310.FiniteTableModel |
|
|
| end |
|
|
| end OAI |
|
|
