| import OAI.Combinatorics.EditApproximation.Queries.QueryInitialSupport |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryLocalRepresentativeWork |
|
|
| namespace OAI |
|
|
| /-! A scale guard bounds the number of grid points even when the center |
| radius is as large as the input. These bounds include construction of |
| all points, before subsequent eligibility filtering. -/ |
|
|
| namespace EditApproximation |
|
|
| theorem queryGridSpacing_count (P b radius K : ℕ) (hP : 0 < P) (hr : radius ≤ K * b) : |
| 2 * radius / seedGridSpacing b P + 2 ≤ 4 * K * P + 2 := by |
| have hg : 0 < seedGridSpacing b P := by unfold seedGridSpacing; omega |
| have hq : b / P ≤ seedGridSpacing b P := le_max_right _ _ |
| have hdiv := Nat.lt_mul_div_succ b hP |
| have hb : b ≤ 2 * P * seedGridSpacing b P := by nlinarith only [hq, hdiv, hg] |
| have hn : 2 * radius ≤ 4 * K * P * seedGridSpacing b P := by |
| nlinarith only [hr, Nat.mul_le_mul_left (2 * K) hb] |
| have hfloor := Nat.div_mul_le_self (2 * radius) (seedGridSpacing b P) |
| nlinarith only [hfloor, hn, hg] |
|
|
| def queryGridPointsWithWork {n : ℕ} (P b radius : ℕ) (q : TargetInterval n) : |
| List (TargetInterval n) × ℕ := |
| let spacing := seedGridSpacingWithWork b P |
| let points := gridStateBoxWithWork n (bitWordValue spacing.1) radius q.lo q.hi |
| (points.1, spacing.2 + points.2 + 1) |
|
|
| def queryGridPointsBitBudget (P K B : ℕ) : ℕ := |
| 100 * (B + 2) ^ 2 + gridStateBoxBitBudget (4 * K * P + 2) (B + 1) + 1 |
|
|
| theorem queryGridPointsWithWork_bounds {n : ℕ} (P b radius K B : ℕ) (q : TargetInterval n) |
| (hP : 0 < P) (hPb : Nat.size P ≤ B) (hb : Nat.size b ≤ B) |
| (hr : Nat.size radius ≤ B) (hn : Nat.size n ≤ B) (hcount : radius ≤ K * b) : |
| (queryGridPointsWithWork P b radius q).1.length ≤ (4 * K * P + 2) ^ 2 ∧ |
| (queryGridPointsWithWork P b radius q).2 ≤ queryGridPointsBitBudget P K B := by |
| let spacing := seedGridSpacingWithWork b P |
| have hs := seedGridSpacingWithWork_bounds b P B hb hPb |
| have hp := seedGridSpacingWithWork_pos b P hP |
| have hsp : Nat.size (bitWordValue spacing.1) ≤ B + 1 := |
| (Nat.size_le.mpr (bitWordValue_lt_pow_length spacing.1)).trans hs.1 |
| have hc := queryGridSpacing_count P b radius K hP hcount |
| have hv : bitWordValue spacing.1 = seedGridSpacing b P := seedGridSpacingWithWork_value b P hP |
| have hgc := gridStateBoxWithWork_cost n (bitWordValue spacing.1) radius q.lo q.hi (B + 1) |
| hp (by omega) hsp (by omega) |
| ((Nat.size_le_size (q.ordered.trans q.valid)).trans (by omega)) |
| ((Nat.size_le_size q.valid).trans (by omega)) |
| rw [hv] at hgc |
| have hgc' := hgc.trans (gridStateBoxBitBudget_mono_count _ _ (B + 1) hc) |
| have hlength : (gridStateBoxWithWork n (bitWordValue spacing.1) radius q.lo q.hi).1.length ≤ |
| (4 * K * P + 2) ^ 2 := by |
| rw [gridStateBoxWithWork_value _ _ _ _ _ hp, hv] |
| exact (localTargetStates_length_le (by unfold seedGridSpacing; omega) q).trans |
| (Nat.pow_le_pow_left hc 2) |
| have hcost : spacing.2 + |
| (gridStateBoxWithWork n (bitWordValue spacing.1) radius q.lo q.hi).2 + 1 ≤ |
| queryGridPointsBitBudget P K B := by |
| rw [hv] |
| exact Nat.add_le_add_right (Nat.add_le_add hs.2 hgc') 1 |
| simpa only [queryGridPointsWithWork, spacing] using And.intro hlength hcost |
|
|
| end EditApproximation |
|
|
| end OAI |
|
|
