File size: 4,026 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 66 67 68 69 70 | import OAI.Combinatorics.EditApproximation.Queries.QueryGroupLabelCost
import OAI.Combinatorics.EditApproximation.Queries.QueryRoundedStateCost
import OAI.Combinatorics.EditApproximation.Queries.QueryRoundedCounts
namespace OAI
namespace EditApproximation.BinaryFraction
def queryGroupInputsLength (M exponent : ℕ) : ℕ :=
M * ((24 * 2 ^ exponent * M + 2) * (4 * 2 ^ exponent + 4))
def queryGroupInputsBitBudget (M exponent B R : ℕ) : ℕ :=
let L := (24 * 2 ^ exponent * M + 2) * (4 * 2 ^ exponent + 4)
queryGroupLabelsBitBudget M B R +
M * (queryRoundedStatesBitBudget (24 * 2 ^ exponent * M + 2) (4 * 2 ^ exponent + 4) exponent B +
5 * L + 4) + 1
theorem queryGroupInputsWithWork_bounds {nx ny M : ℕ} (parent : TargetInterval nx)
(b exponent Q B R : ℕ) (cell : ℕ × ℕ) (h : BinaryFraction)
(draw : Fin (groupSampleCount M Q b h.value) → Fin M × ℕ)
(hB : 4 ≤ B) (hMpos : 0 < M) (hnx : Nat.size nx ≤ B) (hny : Nat.size ny ≤ B)
(hM : Nat.size M + 2 ≤ B) (hQ : Nat.size Q + 2 ≤ B) (hb : Nat.size b + 2 ≤ B)
(hc : Nat.size cell.1 ≤ B) (he : exponent + 1 ≤ B) (hh : h.width ≤ B)
(hhv : 0 ≤ h.value) (hfloor : (b : ℚ) / M ≤ h.value) (hR : ∀ i, (draw i).2 ≤ R) :
(queryGroupInputsWithWork (ny := ny) parent b exponent Q cell h draw).1.length ≤
queryGroupInputsLength M exponent ∧
(queryGroupInputsWithWork (ny := ny) parent b exponent Q cell h draw).2 ≤
queryGroupInputsBitBudget M exponent B R := by
let labels := queryGroupLabelsWithWork M Q b h draw
let L := (24 * 2 ^ exponent * M + 2) * (4 * 2 ^ exponent + 4)
let C := queryRoundedStatesBitBudget (24 * 2 ^ exponent * M + 2) (4 * 2 ^ exponent + 4) exponent B
let run := fun i : Fin M =>
let states := queryGroupRoundedStatesWithWork (ny := ny) parent b exponent cell i h
let tagged := arithmeticMapWithWork (fun r => ((i, r), 1)) states.1
(tagged.1, states.2 + tagged.2 + 1)
have hl := queryGroupLabelsWithWork_bounds M Q b B R h draw hM hQ hb hh hR
have hcounts := groupRoundSpacing_endpoint_counts M b exponent h.value hMpos hhv hfloor
have hs (i : Fin M) := queryGroupRoundedStatesWithWork_bounds (ny := ny) parent b exponent B
(24 * 2 ^ exponent * M + 2) (4 * 2 ^ exponent + 4) cell i h hB hnx hny (by omega)
(by omega) hc he hh hcounts.1 hcounts.2
have hrun (i : Fin M) : (run i).1.length ≤ L ∧ (run i).2 ≤ C + 4 * L + 1 := by
have hm := (arithmeticMapWithWork_cost (fun r => ((i, r), 1))
(queryGroupRoundedStatesWithWork (ny := ny) parent b exponent cell i h).1 1
(fun _ _ => le_rfl)).trans (Nat.mul_le_mul_right _ (hs i).1)
constructor
· simpa only [run, arithmeticMapWithWork_value, List.length_map] using (hs i).1
· change (queryGroupRoundedStatesWithWork (ny := ny) parent b exponent cell i h).2 +
(arithmeticMapWithWork (fun r => ((i, r), 1))
(queryGroupRoundedStatesWithWork (ny := ny) parent b exponent cell i h).1).2 + 1 ≤ _
have hsc : (queryGroupRoundedStatesWithWork (ny := ny) parent b exponent cell i h).2 ≤ C := (hs i).2
change _ ≤ L * 4 at hm
rw [Nat.mul_comm L 4] at hm
omega
have hlen := (arithmeticFlatMapWithWork_length run labels.1 L
(fun i _ => (hrun i).1)).trans (Nat.mul_le_mul_right _ hl.1)
have hcost := (arithmeticFlatMapWithWork_cost run labels.1 (C + 4 * L + 1) L
(fun i _ => (hrun i).2) (fun i _ => (hrun i).1)).trans (Nat.mul_le_mul_right _ hl.1)
rw [show C + 4 * L + 1 + L + 3 = C + 5 * L + 4 by omega] at hcost
have hinputCost : labels.2 + (arithmeticFlatMapWithWork run labels.1).2 + 1 ≤
queryGroupInputsBitBudget M exponent B R := by
have hlc : labels.2 ≤ queryGroupLabelsBitBudget M B R := hl.2
simpa only [queryGroupInputsBitBudget, C, L] using
Nat.add_le_add_right (Nat.add_le_add hlc hcost) 1
simpa only [queryGroupInputsWithWork, labels, run, queryGroupInputsLength, L] using
And.intro hlen hinputCost
end EditApproximation.BinaryFraction
end OAI
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