| 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] |
| (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 |
|
|
