| import OAI.Combinatorics.ContingencyTables.Dense.DenseBooleanStep |
| import OAI.Combinatorics.ContingencyTables.Transport.BoundedOffsetLaw |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| namespace ContingencyTables.GridBoundary |
|
|
| theorem denseStepBits_upper (d : ℕ) : denseStepBits d ≤ 16 * d ^ 5 := by |
| have hl : Nat.clog 2 (8 * d) ≤ 8 * d := |
| Nat.clog_le_of_le_pow (Nat.le_of_lt Nat.lt_two_pow_self) |
| have hd : d ≤ d ^ 5 := Nat.le_self_pow (by decide) d |
| unfold denseStepBits |
| omega |
|
|
| theorem denseOffsetBits_upper (w C b h d : ℕ) (hw : w ≤ C) |
| (hb : Nat.clog 2 (C + 2) ≤ b) : |
| FairBits.offsetBitLength w h d ≤ b + 1 + 4 * (h + d) := by |
| have hl := (Nat.clog_mono_right 2 (by omega : w + 1 ≤ C + 2)).trans hb |
| unfold FairBits.offsetBitLength |
| omega |
|
|
| def denseRandomBitBudget (d b h : ℕ) : ℕ := |
| (d ^ 4 * (h + 1)) * |
| ((d ^ 50 * (h + 1) ^ 2) * denseStepBits d + d * (b + 1 + 4 * (h + d))) |
|
|
| /-- The fixed declared number of dense random bits is polynomial in the |
| dimension parameter, binary margin length, and requested precision. -/ |
| theorem denseRandomBitBudget_polynomial (d b h : ℕ) : |
| denseRandomBitBudget d b h ≤ 25 * (d + b + h + 1) ^ 62 := by |
| let S := d + b + h + 1 |
| have hd : d ≤ S := by omega |
| have hh : h + 1 ≤ S := by omega |
| have hS : 1 ≤ S := by omega |
| have hQ : denseStepBits d ≤ 16 * S ^ 5 := |
| (denseStepBits_upper d).trans (Nat.mul_le_mul_left 16 (Nat.pow_le_pow_left hd 5)) |
| have hH : d ^ 50 * (h + 1) ^ 2 ≤ S ^ 52 := by |
| calc |
| _ ≤ S ^ 50 * S ^ 2 := Nat.mul_le_mul (Nat.pow_le_pow_left hd 50) (Nat.pow_le_pow_left hh 2) |
| _ = _ := (pow_add S 50 2).symm |
| have hwalk : (d ^ 50 * (h + 1) ^ 2) * denseStepBits d ≤ 16 * S ^ 57 := by |
| calc |
| _ ≤ S ^ 52 * (16 * S ^ 5) := Nat.mul_le_mul hH hQ |
| _ = _ := by rw [mul_left_comm, ← pow_add] |
| have hr : b + 1 + 4 * (h + d) ≤ 9 * S := by omega |
| have hoffset : d * (b + 1 + 4 * (h + d)) ≤ 9 * S ^ 57 := by |
| calc |
| _ ≤ S * (9 * S) := Nat.mul_le_mul hd hr |
| _ = 9 * S ^ 2 := by rw [mul_left_comm, pow_two] |
| _ ≤ _ := Nat.mul_le_mul_left 9 (Nat.pow_le_pow_right hS (by decide : 2 ≤ 57)) |
| have hJ : d ^ 4 * (h + 1) ≤ S ^ 5 := by |
| calc |
| _ ≤ S ^ 4 * S := Nat.mul_le_mul (Nat.pow_le_pow_left hd 4) hh |
| _ = _ := (pow_succ S 4).symm |
| unfold denseRandomBitBudget |
| calc |
| _ ≤ S ^ 5 * (25 * S ^ 57) := Nat.mul_le_mul hJ (by omega) |
| _ = _ := by rw [mul_left_comm, ← pow_add] |
|
|
| end ContingencyTables.GridBoundary |
|
|
| end OAI |
|
|
