File size: 2,621 Bytes
e5a97c6 | 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 69 70 71 72 73 74 75 76 | import Mathlib.Data.Nat.Log
namespace OAI
/-! # Computing a rejection range width and its cached modulus
A doubling cursor starts at one and stops at the first power of two at
least the requested range size. The width and modulus are computed together.
-/
namespace LowSpaceSubsetSum.WordSeedModulusExecution
structure Result where
width : ℕ
modulus : ℕ
work : ℕ
def loop (D : ℕ) : ℕ → ℕ → ℕ → ℕ → Result
| 0, width, modulus, work => ⟨width,modulus,work+1⟩
| fuel+1, width, modulus, work =>
if modulus < D then loop D fuel (width+1) (2*modulus) (work+5)
else ⟨width,modulus,work+1⟩
theorem loop_value (D fuel width modulus work : ℕ)
(hm : modulus = 2^width) (hw : width ≤ Nat.clog 2 D)
(hf : Nat.clog 2 D ≤ width+fuel) :
loop D fuel width modulus work =
⟨Nat.clog 2 D,2^Nat.clog 2 D,work+5*(Nat.clog 2 D-width)+1⟩ := by
induction fuel generalizing width modulus work with
| zero =>
have he : width = Nat.clog 2 D := by omega
subst width
subst modulus
simp only [loop,Nat.sub_self,Nat.mul_zero,Nat.add_zero]
| succ fuel ih =>
change (if modulus < D then loop D fuel (width+1) (2*modulus) (work+5)
else ⟨width,modulus,work+1⟩) = _
split
next h =>
have hlt : width < Nat.clog 2 D := by
exact (Nat.lt_clog_iff_pow_lt (by decide : 1 < 2)).mpr (hm ▸ h)
have hm' : 2*modulus = 2^(width+1) := by rw [hm,Nat.pow_succ]; omega
rw [ih _ _ _ hm' (by omega) (by omega)]
congr 1
omega
next h =>
have hge : Nat.clog 2 D ≤ width := by
apply Nat.clog_le_of_le_pow
omega
have he : width = Nat.clog 2 D := by omega
subst width
subst modulus
simp only [Nat.sub_self,Nat.mul_zero,Nat.add_zero]
def run (D : ℕ) : Result := loop D D 0 1 0
theorem run_value (D : ℕ) : run D = ⟨Nat.clog 2 D,2^Nat.clog 2 D,5*Nat.clog 2 D+1⟩ := by
have hf : Nat.clog 2 D ≤ D := Nat.clog_le_of_le_pow (Nat.lt_two_pow_self.le)
simpa only [run,Nat.pow_zero,Nat.zero_add,Nat.sub_zero] using
loop_value D D 0 1 0 (by simp) (Nat.zero_le _) (by omega)
theorem run_width (D : ℕ) : (run D).width = Nat.clog 2 D := by rw [run_value]
theorem run_modulus (D : ℕ) : (run D).modulus = 2^Nat.clog 2 D := by rw [run_value]
theorem run_work (D : ℕ) : (run D).work = 5*Nat.clog 2 D+1 := by rw [run_value]
theorem run_modulus_fits (D w : ℕ) (hw : Nat.clog 2 D < w) :
(run D).modulus < 2^w := by
rw [run_modulus]
exact Nat.pow_lt_pow_right (by decide : 1 < 2) hw
end LowSpaceSubsetSum.WordSeedModulusExecution
end OAI
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