| import OAI.Computability.WeisfeilerLeman.RAMBounds |
|
|
| namespace OAI |
|
|
| namespace WLTime |
| open OneTape.Symbolic OneTape.Symbolic.FullWriter |
| open scoped Classical |
| theorem ram_integer_lower (k : ℕ) (hk : 100000000 ≤ k) (c : Convention) (p : InputClass) (A : RAM) |
| (hA : Decides (.ram A) c k p) : |
| ∃ n₀,∀ n ≥ n₀,n^(k/1000) < worstTime (.ram A) p n := by |
| let s := k/16 |
| let d := 3*s+6 |
| let q := k-d-conventionShift c |
| let U := Universal.machine s |
| let C := coefficient U d q (d+q+2) |
| have hshift : 1 ≤ conventionShift c ∧ conventionShift c ≤ 2 := by cases c <;> decide |
| have hs : 1 ≤ s := by dsimp [s]; omega |
| have hdim : k=d+q+conventionShift c := by dsimp [d,q,s]; omega |
| have hq : d+3 ≤ q := by dsimp [q,d,s]; omega |
| have ha : 2*max 2400 (16*(k/1000)+45)+3 ≤ s := by |
| rcases max_cases 2400 (16*(k/1000)+45) with ⟨he,_⟩ | ⟨he,_⟩ <;> rw [he] <;> dsimp [s] <;> omega |
| have hCpos : 1 ≤ C := coefficient_pos U d q |
| obtain ⟨mfit,hfit⟩ := RAMWriter.designation_small A U d q |
| obtain ⟨mword,hword⟩ := RAMWriter.word_cost_small A |
| let m₀ := mfit+384*C^3+128+C+(body U d q).charge+RAMWriter.consumerFactor U d q A+ |
| mword+(8*((Universal.boundedProgram s).scratch+Fintype.card Universal.Reg+1)*1200^2)+18*s |
| refine ⟨baseOrder U d q m₀,?_⟩ |
| intro n hn |
| let m := Numeric.chooseOrder C n |
| have hchoice := Numeric.chooseOrder_bounds C n m₀ hCpos hn |
| have hm : m₀ ≤ m := hchoice.1 |
| have hbase : baseOrder U d q m ≤ n := hchoice.2.1 |
| have hmn : n ≤ m^8 := Numeric.order_polynomial C n m hCpos (by dsimp [m₀] at hm; omega) hchoice.2.2 |
| apply lt_of_not_ge |
| exact ram_pointwise s d q k (k/1000) n m c p A hA hdim |
| (by dsimp [m₀] at hm; omega) hs rfl hq ha hbase hmn |
| (hfit m (by dsimp [m₀] at hm; omega) n hmn) |
| (by change C ≤ m; dsimp only [m₀] at hm; omega) |
| (by change (body U d q).charge ≤ m; dsimp only [m₀] at hm; omega) |
| (by change RAMWriter.consumerFactor U d q A ≤ m; dsimp only [m₀] at hm; omega) |
| (hword m (by dsimp [m₀] at hm; omega) n hmn) |
| (by dsimp [m₀] at hm; omega) |
| (by dsimp [m₀] at hm; omega) |
|
|
| end WLTime |
| namespace WLTime |
| theorem ram_real_lower (k : ℕ) (hk : 100000000 ≤ k) (c : Convention) (p : InputClass) (A : RAM) |
| (hA : Decides (.ram A) c k p) : |
| ∃ n₀ : ℕ,∀ n : ℕ, n ≥ n₀ → (n : ℝ)^((1/2000 : ℝ)*(k : ℝ)) ≤ (worstTime (.ram A) p n : ℝ) := by |
| obtain ⟨n₀,hn₀⟩ := ram_integer_lower k hk c p A hA |
| refine ⟨n₀+1,?_⟩ |
| intro n hn |
| have hint := hn₀ n (by omega) |
| have hexp : (1/2000 : ℝ)*(k : ℝ) ≤ (k/1000 : ℕ) := by |
| have h : k ≤ 2000*(k/1000) := by omega |
| have hr : (k : ℝ) ≤ 2000*((k/1000 : ℕ) : ℝ) := by exact_mod_cast h |
| linarith |
| calc |
| _ ≤ (n : ℝ)^((k/1000 : ℕ) : ℝ) := Real.rpow_le_rpow_of_exponent_le (by exact_mod_cast (show 1 ≤ n by omega)) hexp |
| _ = (n : ℝ)^(k/1000) := Real.rpow_natCast _ _ |
| _ ≤ (worstTime (.ram A) p n : ℝ) := by exact_mod_cast hint.le |
| end WLTime |
|
|
| theorem WLTime.unconditional_time_lower_bound : WLTime.MainClaim := by |
| refine ⟨1/2000, by norm_num, 100000000, ?_⟩ |
| intro k hk convention p A hA |
| cases A with |
| | turing M => exact WLTime.tm_real_lower k hk convention p M hA |
| | ram M => exact WLTime.ram_real_lower k hk convention p M hA |
|
|
| end OAI |
|
|
