| import OAI.NumberTheory.TwoPoint.Bounds.ActiveStateExpansion | |
| namespace OAI | |
| /-! The complete low-degree state family fits the scalar-comparison budget. -/ | |
| namespace TwoPointCorrelations | |
| lemma active_state_family_cost (L : ℝ) (n M R : ℕ) | |
| (hL : 4800 ≤ L) (hn : (n : ℝ) ≤ Real.exp L) | |
| (hM : (M : ℝ) ≤ 400 * Real.log L) (hR : (R : ℝ) ≤ 4 * L) : | |
| (((M + 1) * (n + 1) ^ M : ℕ) : ℝ) ^ R ≤ Real.exp (L ^ 4) := by | |
| have hLp : 0 ≤ L := by linarith | |
| have hlog : 0 ≤ Real.log L := Real.log_nonneg (by linarith) | |
| have hexp : (2 : ℝ) ≤ Real.exp L := by linarith [Real.add_one_le_exp L] | |
| have hn' : (n : ℝ) + 1 ≤ Real.exp (2 * L) := by | |
| have he : Real.exp (2 * L) = Real.exp L * Real.exp L := by | |
| rw [show 2 * L = L + L by ring, Real.exp_add] | |
| rw [he] | |
| nlinarith | |
| have hM' : (M : ℝ) + 1 ≤ Real.exp (400 * Real.log L) := by | |
| linarith [Real.add_one_le_exp (400 * Real.log L)] | |
| have hbase : (((M + 1) * (n + 1) ^ M : ℕ) : ℝ) ≤ | |
| Real.exp (1200 * L * Real.log L) := by | |
| push_cast | |
| calc | |
| _ ≤ Real.exp (400 * Real.log L) * (Real.exp (2 * L)) ^ M := | |
| mul_le_mul hM' (pow_le_pow_left₀ (by positivity) hn' M) (by positivity) (by positivity) | |
| _ = Real.exp (400 * Real.log L + M * (2 * L)) := by | |
| rw [← Real.exp_nat_mul, ← Real.exp_add] | |
| _ ≤ _ := by | |
| apply Real.exp_le_exp.mpr | |
| have hm := mul_le_mul_of_nonneg_right hM (show 0 ≤ 2 * L by positivity) | |
| have hl := mul_le_mul_of_nonneg_right (show 1 ≤ L by linarith) hlog | |
| nlinarith | |
| have hlogL : Real.log L ≤ L := | |
| (Real.log_le_sub_one_of_pos (by linarith)).trans (by linarith) | |
| calc | |
| _ ≤ (Real.exp (1200 * L * Real.log L)) ^ R := | |
| pow_le_pow_left₀ (by positivity) hbase R | |
| _ = Real.exp (R * (1200 * L * Real.log L)) := by rw [Real.exp_nat_mul] | |
| _ ≤ Real.exp (4800 * L ^ 3) := by | |
| apply Real.exp_le_exp.mpr | |
| have hr := mul_le_mul_of_nonneg_right hR | |
| (show 0 ≤ 1200 * L * Real.log L by positivity) | |
| have hl := mul_le_mul_of_nonneg_left hlogL (show 0 ≤ 4800 * L ^ 2 by positivity) | |
| nlinarith | |
| _ ≤ _ := by | |
| apply Real.exp_le_exp.mpr | |
| nlinarith [mul_nonneg (show 0 ≤ L - 4800 by linarith) (pow_nonneg hLp 3)] | |
| end TwoPointCorrelations | |
| end OAI | |