openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Word /WordExecutionAsymptotics.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Search.PreparedAsymptotics | |
| namespace OAI | |
| /-! # Resource envelopes including the executed word generators -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum.WordExecutionAsymptotics | |
| variable {n : ℕ} | |
| theorem counter_limit (I : Instance n) (hn : 10000 ≤ n) | |
| (hscale : Real.logb 2 (inputScale I : ℝ) ≤ (n : ℝ)/1000000000) : | |
| inputScale I^624*2^((n+1)/2) < 2^n := by | |
| have hu : (0 : ℝ) < inputScale I := by | |
| exact_mod_cast (show 0 < inputScale I by unfold inputScale; omega) | |
| have he : (((n+1)/2 : ℕ) : ℝ) ≤ (1/2 : ℝ)*n+1 := by | |
| have h : 2*((n+1)/2) ≤ n+1 := by omega | |
| have h' : (2 : ℝ)*((n+1)/2 : ℕ) ≤ n+1 := by exact_mod_cast h | |
| linarith only [h'] | |
| have h := guarded_polynomial_exponential hn hu hscale 624 | |
| (a := (1/2 : ℝ)) (by norm_num) | |
| have hm := mul_le_mul_of_nonneg_left | |
| (Real.rpow_le_rpow_of_exponent_le (by norm_num : (1 : ℝ) ≤ 2) he) | |
| (show 0 ≤ (inputScale I : ℝ)^624 by positivity) | |
| have hh : ((inputScale I^624*2^((n+1)/2) : ℕ) : ℝ) < (2 : ℝ)^n := by | |
| simpa only [Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,Real.rpow_natCast] using hm.trans_lt h | |
| exact_mod_cast hh | |
| theorem time (c : ℕ) (hc : 0 < c) (n : ℕ) (hn : 2 ≤ n) | |
| (I : Instance n) (hI : inputBits I ≤ n^c) : | |
| ((inputScale I^624*2^((n+1)/2) : ℕ) : ℝ) ≤ | |
| ((3 : ℝ)^624*2)*(n : ℝ)^(624*c)*(2 : ℝ)^((1/2 : ℝ)*n) := by | |
| have hu : (inputScale I : ℝ) ≤ 3*(n : ℝ)^c := by | |
| exact_mod_cast polynomial_bit_bound hn hc hI | |
| have hp := pow_le_pow_left₀ (by positivity : (0 : ℝ) ≤ inputScale I) hu 624 | |
| have hhalf : (((n+1)/2 : ℕ) : ℝ) ≤ (1/2 : ℝ)*n+1 := by | |
| have h : 2*((n+1)/2) ≤ n+1 := by omega | |
| have hh : (2 : ℝ)*((n+1)/2 : ℕ) ≤ n+1 := by exact_mod_cast h | |
| linarith only [hh] | |
| have he := Real.rpow_le_rpow_of_exponent_le (by norm_num : (1 : ℝ) ≤ 2) hhalf | |
| simp only [Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat] | |
| apply (mul_le_mul hp (by simpa only [Real.rpow_natCast] using he) | |
| (by positivity) (by positivity)).trans | |
| rw [mul_pow,←pow_mul,Nat.mul_comm c 624, | |
| Real.rpow_add (by norm_num : (0 : ℝ) < 2),Real.rpow_one] | |
| exact le_of_eq (by ac_rfl) | |
| end LowSpaceSubsetSum.WordExecutionAsymptotics | |
| end | |
| end OAI | |