File size: 3,618 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 | import OAI.Combinatorics.EditApproximation.Queries.QueryRepresentativeCost
import OAI.Combinatorics.EditApproximation.Queries.QueryRepresentativeScales
import OAI.Combinatorics.EditApproximation.Queries.QueryWideBounds
namespace OAI
namespace EditApproximation.BinaryFraction
def queryWideInputsBitBudget (M P count B : ℕ) : ℕ :=
let G := (40 * P + 2) ^ 2
let C := wideChildGridBitBudget P (B + 1)
count * (M * (C + 4 * G + 1 + G + 3) + M * G + 3) +
(count * (M * G) + 1) ^ 2 * (24 * B + 12) + M + 2
theorem queryWideInputsWithWork_dyadic_bounds {nx ny : ℕ}
(M N P count B : ℕ) (parent : TargetInterval nx) (hP : 0 < P)
(representatives : List (ℕ × TargetInterval ny))
(hl : representatives.length ≤ count)
(hs : ∀ br ∈ representatives, br.1 ∈ dyadicScales N)
(hN : Nat.size N ≤ B) (hnx : Nat.size nx ≤ B) (hny : Nat.size ny ≤ B)
(hM : Nat.size M ≤ B) (hPb : Nat.size P ≤ B) :
(queryWideInputsWithWork M parent P representatives).1.length ≤ count * (M * (40 * P + 2) ^ 2) ∧
(queryWideInputsWithWork M parent P representatives).2 ≤ queryWideInputsBitBudget M P count B := by
have hb := queryWideInputsWithWork_bounds M parent P representatives ((40 * P + 2) ^ 2)
(wideChildGridBitBudget P (B + 1)) B hM hny
(fun br _ i => by rw [wideChildGridWithWork_value _ _ _ _ _ hP]; exact wideChildStateList_length_le parent br.2 hP i)
(fun br hbr i => wideChildGridWithWork_cost parent br.2 br.1 P (B + 1) i hP
(by omega) (by omega) (by omega)
((dyadicScales_size_bound N br.1 (hs br hbr)).trans (by omega)) (by omega))
constructor
· exact hb.1.trans (Nat.mul_le_mul_right _ hl)
· apply hb.2.trans
unfold queryWideInputsBitBudget
dsimp only
gcongr
def queryBandInputsBitBudget (M P F exponent B R : ℕ) : ℕ :=
queryEarlierRepresentativesBitBudget P F exponent B R +
queryWideInputsBitBudget M P (queryEarlierRepresentativesLength P F exponent B) B + 1
theorem queryBandInputsWithWork_bounds {nx ny : ℕ} (M N P F exponent B R : ℕ)
(parent : TargetInterval nx) (hP : 0 < P) (a : BinaryFraction)
(initial : TargetInterval ny → BinaryFraction × ℕ) (q : TargetInterval ny)
(hB : 8 ≤ B) (hN : Nat.size N ≤ B) (hF : Nat.size F + 2 ≤ B) (hFpos : 1 ≤ F)
(hPb : Nat.size P ≤ B) (hnx : Nat.size nx ≤ B) (hny : Nat.size ny ≤ B)
(hM : Nat.size M ≤ B) (he : exponent + 1 ≤ B) (ha : a.width ≤ B)
(hi : ∀ r, (initial r).1.width ≤ B) (hR : ∀ r, (initial r).2 ≤ R)
(hw : 0 ≤ (initial q).1.value) (hPd : ∃ e : ℕ, P = 2 ^ e) :
(queryBandInputsWithWork M parent N P F exponent hP a initial q).1.length ≤
queryEarlierRepresentativesLength P F exponent B * (M * (40 * P + 2) ^ 2) ∧
(queryBandInputsWithWork M parent N P F exponent hP a initial q).2 ≤
queryBandInputsBitBudget M P F exponent B R := by
have hr := queryEarlierRepresentativesWithWork_bounds N P F exponent B R hP a initial q
hB hN hF hFpos hPb hny he ha hi hR hw hPd
have hwide := queryWideInputsWithWork_dyadic_bounds M N P
(queryEarlierRepresentativesLength P F exponent B) B parent hP
(queryEarlierRepresentativesWithWork N P F exponent hP a initial q).1 hr.1
(fun br hb => queryEarlierRepresentativesWithWork_scale N P F exponent hP a initial q R hR br hb)
hN hnx hny hM hPb
constructor
· simpa only [queryBandInputsWithWork] using hwide.1
· simpa only [queryBandInputsWithWork, queryBandInputsBitBudget] using
Nat.add_le_add_right (Nat.add_le_add hr.2 hwide.2) 1
end EditApproximation.BinaryFraction
end OAI
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