| import OAI.Combinatorics.EditApproximation.Coarse.CoarsePrecision |
| import OAI.Combinatorics.EditApproximation.Coarse.CoarseGlobalSeed |
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|
| namespace OAI |
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| namespace EditApproximation |
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| / |
| base in its denominator. This includes the literal sampled child label. -/ |
| theorem coarseChildAllowance_grid (B q L j : ℕ) (allowance : ℚ) (draw : Fin B) |
| (hB : 0 < B) (hq : 0 < q) (hL : 0 < L) |
| (ha : allowance.den ∣ (L * B * q) ^ j) : |
| ((1 / (L : ℚ)) * coarseChildThreshold B q allowance draw).den ∣ |
| (L * B * q) ^ (j + 1) := by |
| have ht : (coarseChildThreshold B q allowance draw).den ∣ |
| (L * B * q) ^ j * (B * q) := by |
| simpa only [coarseChildThreshold, Nat.cast_mul] using |
| coarse_rat_div_nat_den_dvd (allowance * ((draw.val + 1 : ℕ) : ℚ)) (B * q) ((L * B * q) ^ j) |
| (Nat.mul_pos hB hq) (coarse_rat_mul_nat_den_dvd allowance (draw.val + 1) _ ha) |
| have heq : (1 / (L : ℚ)) * coarseChildThreshold B q allowance draw = |
| coarseChildThreshold B q allowance draw / L := by ring |
| rw [heq] |
| convert coarse_rat_div_nat_den_dvd _ L _ hL ht using 1 |
| rw [pow_succ] |
| ring |
|
|
| / |
| its root; it is unnecessary to normalize any array sum by a gcd. -/ |
| theorem coarseDescendant_grid {B d : ℕ} {α : Type*} [DecidableEq α] |
| (k q L j : ℕ) (frame : List (Option α)) (tree : CoarseSourceTree B α d) |
| (draws : CoarseThresholdDraws B d) (allowance : ℚ) |
| (hB : 0 < B) (hq : 0 < q) (hL : 0 < L) |
| (ha : allowance.den ∣ (L * B * q) ^ j) (start : ℕ) (shift : Fin (2 * k + 1)) : |
| (coarseThresholdEstimate k q (1 / L) frame tree draws allowance start shift).den ∣ |
| coarseCommonDenominator 1 q L B (j + d) := by |
| have h := coarseThresholdEstimate_den_dvd k q L frame tree draws allowance |
| ((L * B * q) ^ j) hB hq hL (pow_pos (by positivity) _) ha start shift |
| convert h using 1 |
| simp only [coarseCommonDenominator, one_mul, pow_add] |
| ring |
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|
| / |
| theorem coarseDescendant_magnitude {B d : ℕ} {α : Type*} [DecidableEq α] |
| (k q L D n : ℕ) (frame : List (Option α)) (tree : CoarseSourceTree B α d) |
| (draws : CoarseThresholdDraws B d) (allowance : ℚ) |
| (hB : 2 ≤ B) (hq : 0 < q) (hL : 0 < L) (hd : d ≤ D) |
| (hn : tree.word.length ≤ n) (ha : 0 ≤ allowance) (hak : allowance ≤ k) |
| (start : ℕ) (shift : Fin (2 * k + 1)) : |
| |coarseThresholdEstimate k q (1 / L) frame tree draws allowance start shift| ≤ |
| ((n + B * k * B ^ D : ℕ) : ℚ) := by |
| have hqQ : (1 : ℚ) ≤ q := by exact_mod_cast hq |
| have hLQ : (1 : ℚ) ≤ L := by exact_mod_cast hL |
| have hLpos : (0 : ℚ) < L := by exact_mod_cast hL |
| have hnode : coarseInternalNodes B d ≤ B ^ D := |
| (Nat.le_of_succ_le (coarseInternalNodes_add_one_le_pow B d hB)).trans |
| (Nat.pow_le_pow_right (by omega) hd) |
| have hm := coarseThresholdEstimate_magnitude k q (1 / L) frame tree draws allowance hqQ |
| (by positivity) ((div_le_one hLpos).mpr hLQ) ha start shift |
| rw [abs_of_nonneg (coarseThresholdEstimate_nonneg k q (1 / L) frame tree draws allowance start shift |
| (by positivity) ha)] |
| apply hm.trans |
| have hs : (B : ℚ) * allowance / q ≤ B * k := by |
| exact (div_le_self (by positivity) hqQ).trans |
| (mul_le_mul_of_nonneg_left hak (by positivity)) |
| have hh := mul_le_mul hs (by exact_mod_cast hnode : (coarseInternalNodes B d : ℚ) ≤ B ^ D) |
| (by positivity) (by positivity : (0 : ℚ) ≤ B * k) |
| push_cast |
| exact add_le_add (by exact_mod_cast hn) hh |
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|
| end EditApproximation |
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| end OAI |
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