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97c3405 | 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 | import OAI.Combinatorics.ContingencyTables.Dense.DenseBooleanStep
import OAI.Combinatorics.ContingencyTables.Transport.BoundedOffsetLaw
namespace OAI
/-! # Polynomial bounds for the manuscript's bounded dense random-bit budget -/
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
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