| import OAI.Combinatorics.GraphThreshold.Thresholds |
| import Mathlib.Data.Nat.Choose.Bounds |
| import Mathlib.Analysis.SpecialFunctions.Log.Base |
|
|
| namespace OAI |
|
|
| /-! |
| # Explicit constants in the threshold conclusion |
|
|
| The tree argument first gives a logarithm of an edge capacity. These elementary |
| bounds turn that quantity into the two constants stated for graphs. |
| -/ |
|
|
| noncomputable section |
|
|
| namespace LeanBlast.SecondKahnKalai |
|
|
| theorem logTwo_eq_logb (x : ℝ) : logTwo x = Real.logb 2 x := rfl |
|
|
| theorem logTwo_nonneg {x : ℝ} (hx : 1 ≤ x) : 0 ≤ logTwo x := by |
| unfold logTwo |
| exact div_nonneg (Real.log_nonneg hx) (Real.log_pos (by norm_num)).le |
|
|
| theorem logTwo_mono {x y : ℝ} (hx : 0 < x) (hxy : x ≤ y) : |
| logTwo x ≤ logTwo y := by |
| unfold logTwo |
| exact div_le_div_of_nonneg_right (Real.log_le_log hx hxy) |
| (Real.log_pos (by norm_num)).le |
|
|
| theorem edgeCount_le_sq {V : Type*} [Fintype V] |
| (n : ℕ) (H : SimpleGraph V) (hcard : Fintype.card V ≤ n) : |
| edgeCount H ≤ n ^ 2 := by |
| classical |
| rw [edgeCount_eq_card_edgeFinset] |
| calc |
| H.edgeFinset.card ≤ (Fintype.card V).choose 2 := |
| SimpleGraph.card_edgeFinset_le_card_choose_two |
| _ ≤ (Fintype.card V) ^ 2 := Nat.choose_le_pow _ _ |
| _ ≤ n ^ 2 := Nat.pow_le_pow_left hcard 2 |
|
|
| theorem one_add_logTwo_le_three_logTwo (n h : ℕ) |
| (hn : 2 ≤ n) (hh : 1 ≤ h) (hhn : h ≤ n ^ 2) : |
| 1 + logTwo (h : ℝ) ≤ 3 * logTwo (n : ℝ) := by |
| have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn |
| have hh' : (1 : ℝ) ≤ h := by exact_mod_cast hh |
| have hhn' : (h : ℝ) ≤ (n : ℝ) ^ 2 := by exact_mod_cast hhn |
| have hlog2 : 0 < Real.log 2 := Real.log_pos (by norm_num) |
| have hlogn : Real.log 2 ≤ Real.log (n : ℝ) := |
| Real.log_le_log (by norm_num) hn' |
| have hone : 1 ≤ logTwo (n : ℝ) := by |
| unfold logTwo |
| exact (le_div_iff₀ hlog2).mpr (by simpa using hlogn) |
| have hlogh : Real.log (h : ℝ) ≤ 2 * Real.log (n : ℝ) := by |
| calc |
| Real.log (h : ℝ) ≤ Real.log ((n : ℝ) ^ 2) := |
| Real.log_le_log (by linarith) hhn' |
| _ = 2 * Real.log (n : ℝ) := by rw [Real.log_pow]; norm_num |
| have htwo : logTwo (h : ℝ) ≤ 2 * logTwo (n : ℝ) := by |
| unfold logTwo |
| simpa only [mul_div_assoc] using |
| div_le_div_of_nonneg_right hlogh hlog2.le |
| linarith |
|
|
| theorem graph_one_add_logTwo_le_three_logTwo {V : Type*} [Fintype V] |
| (n : ℕ) (H : SimpleGraph V) (hn : 2 ≤ n) |
| (hcard : Fintype.card V ≤ n) (hedge : 1 ≤ edgeCount H) : |
| 1 + logTwo (edgeCount H : ℝ) ≤ 3 * logTwo (n : ℝ) := |
| one_add_logTwo_le_three_logTwo n (edgeCount H) hn hedge |
| (edgeCount_le_sq n H hcard) |
|
|
| theorem graphThresholdParameter_mem_unitInterval (q : ℝ) (h : ℕ) |
| (hq : 0 ≤ q) (hh : 1 ≤ h) : |
| min 1 (2048 * Real.exp 50 * q * (1 + logTwo (h : ℝ))) ∈ Set.Icc 0 1 := by |
| have hlog : 0 ≤ logTwo (h : ℝ) := logTwo_nonneg (by exact_mod_cast hh) |
| constructor |
| · exact le_min (by norm_num) (mul_nonneg |
| (mul_nonneg (mul_nonneg (by norm_num) (Real.exp_pos 50).le) hq) (by linarith)) |
| · exact min_le_left _ _ |
|
|
| theorem treeSamplingParameter_mem_unitInterval (q : ℝ) (m : ℕ) |
| (hq : 0 ≤ q) (hm : 0 < m) : |
| min 1 (16 * Real.exp 50 * (128 * q) * |
| (1 + Real.logb 2 (m : ℝ))) ∈ Set.Icc 0 1 := by |
| have hfactor : 16 * Real.exp 50 * (128 * q) = 2048 * Real.exp 50 * q := by ring |
| rw [hfactor, ← logTwo_eq_logb] |
| exact graphThresholdParameter_mem_unitInterval q m hq hm |
|
|
| theorem treeSamplingParameter_le_graphParameter (q : ℝ) (m h : ℕ) |
| (hq : 0 ≤ q) (hm : 0 < m) (hmh : m ≤ h) : |
| min 1 (16 * Real.exp 50 * (128 * q) * (1 + Real.logb 2 (m : ℝ))) ≤ |
| min 1 (2048 * Real.exp 50 * q * (1 + logTwo (h : ℝ))) := by |
| apply min_le_min_left |
| have hlog : logTwo (m : ℝ) ≤ logTwo (h : ℝ) := |
| logTwo_mono (by exact_mod_cast hm) (by exact_mod_cast hmh) |
| calc |
| 16 * Real.exp 50 * (128 * q) * (1 + Real.logb 2 (m : ℝ)) = |
| 2048 * Real.exp 50 * q * (1 + logTwo (m : ℝ)) := by |
| rw [logTwo_eq_logb] |
| ring |
| _ ≤ 2048 * Real.exp 50 * q * (1 + logTwo (h : ℝ)) := |
| mul_le_mul_of_nonneg_left (by linarith) |
| (mul_nonneg (mul_nonneg (by norm_num) (Real.exp_pos 50).le) hq) |
|
|
| theorem graphThresholdParameter_le_vertexBound {V : Type*} [Fintype V] |
| (n : ℕ) (H : SimpleGraph V) (q : ℝ) (hn : 2 ≤ n) |
| (hcard : Fintype.card V ≤ n) (hedge : 1 ≤ edgeCount H) (hq : 0 ≤ q) : |
| min 1 (2048 * Real.exp 50 * q * (1 + logTwo (edgeCount H : ℝ))) ≤ |
| 6144 * Real.exp 50 * q * logTwo (n : ℝ) := by |
| calc |
| min 1 (2048 * Real.exp 50 * q * (1 + logTwo (edgeCount H : ℝ))) ≤ |
| 2048 * Real.exp 50 * q * (1 + logTwo (edgeCount H : ℝ)) := min_le_right _ _ |
| _ ≤ 2048 * Real.exp 50 * q * (3 * logTwo (n : ℝ)) := by |
| exact mul_le_mul_of_nonneg_left |
| (graph_one_add_logTwo_le_three_logTwo n H hn hcard hedge) |
| (mul_nonneg (mul_nonneg (by norm_num) (Real.exp_pos 50).le) hq) |
| _ = 6144 * Real.exp 50 * q * logTwo (n : ℝ) := by ring |
|
|
| end LeanBlast.SecondKahnKalai |
|
|
| end |
|
|
| end OAI |
|
|
