File size: 3,267 Bytes
e63eb7f | 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 | 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
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