| import OAI.Combinatorics.MatchingPSD.Parameters |
| |
| namespace OAI |
| |
| namespace PerfectMatchingPSD |
| open MatchingSmoothing MatchingInterpolation Matrix |
| open scoped BigOperators MatrixOrder |
| |
| lemma blockSize_ge (d t : ℕ) : t^4 ≤ blockSize d t := by |
| dsimp [blockSize] |
| simpa only [one_mul] using Nat.mul_le_mul_right (t^4) (show 1 ≤ 4*(d+1) by omega) |
| |
| lemma blockSize_grid (d t : ℕ) : 2*t^4*d ≤ blockSize d t := by |
| dsimp [blockSize] |
| calc |
| 2*t^4*d = (2*d)*t^4 := by ring |
| _ ≤ 4*(d+1)*t^4 := Nat.mul_le_mul_right _ (by omega) |
| |
| lemma selection_of_cleared {q r T m L b : ℝ} (hb : 0 < b) (hc : 0 < r*m*L) |
| (hnum : q*r^6*T*m^2*L^2 ≤ b) : |
| q*r^4*T*b⁻¹ ≤ (1/(r*m*L))^2 := by |
| rw [← div_eq_mul_inv, div_pow, one_pow] |
| apply (div_le_div_iff₀ hb (sq_pos_of_pos hc)).mpr |
| calc |
| _ = q*r^6*T*m^2*L^2 := by ring |
| _ ≤ b := hnum |
| _ = _ := by ring |
| |
| lemma numerical_selection (p d t r n T : ℕ) (ht : 2 ≤ t) (hr : 0 < r) |
| (hn : n ≤ t^6) (hT : T ≤ t) (hrn : (r:ℝ) ≤ (n:ℝ)^p) : |
| (t:ℝ)*r^4*T*((2*(3*(blockSize d t:ℝ)+1))^(2*(10*(p+1))))⁻¹ ≤ |
| (1/((r:ℝ)*((n:ℝ)+1)*(2*(t:ℝ)^4)))^2 := by |
| have ht2 : (2:ℝ) ≤ t := by exact_mod_cast ht |
| have ht1 : (1:ℝ) ≤ t := by linarith |
| have ht0 : (0:ℝ) ≤ t := Nat.cast_nonneg t |
| have hr0 : (0:ℝ) ≤ r := Nat.cast_nonneg r |
| have hn6 : (n:ℝ) ≤ (t:ℝ)^6 := by exact_mod_cast hn |
| have hrp : (r:ℝ) ≤ (t:ℝ)^(6*p) := by |
| simpa only [pow_mul] using hrn.trans (pow_le_pow_left₀ (Nat.cast_nonneg n) hn6 p) |
| have hm : (n:ℝ)+1 ≤ (t:ℝ)^7 := by |
| have h6 : (1:ℝ) ≤ (t:ℝ)^6 := one_le_pow₀ ht1 |
| calc |
| _ ≤ (t:ℝ)^6+(t:ℝ)^6 := add_le_add hn6 h6 |
| _ = 2*(t:ℝ)^6 := by ring |
| _ ≤ (t:ℝ)*(t:ℝ)^6 := mul_le_mul_of_nonneg_right ht2 (pow_nonneg ht0 _) |
| _ = _ := by ring |
| have hL : 2*(t:ℝ)^4 ≤ (t:ℝ)^5 := by |
| calc |
| _ ≤ (t:ℝ)*(t:ℝ)^4 := mul_le_mul_of_nonneg_right ht2 (pow_nonneg ht0 _) |
| _ = _ := by ring |
| have hTr : (T:ℝ) ≤ t := by exact_mod_cast hT |
| have hbase : (t:ℝ)^4 ≤ 2*(3*(blockSize d t:ℝ)+1) := by |
| have hs : (t:ℝ)^4 ≤ (blockSize d t:ℝ) := by exact_mod_cast blockSize_ge d t |
| have hs0 : (0:ℝ) ≤ (blockSize d t:ℝ) := Nat.cast_nonneg _ |
| linarith |
| have hb : 0 < (2*(3*(blockSize d t:ℝ)+1))^(2*(10*(p+1))) := by positivity |
| apply selection_of_cleared hb (by positivity) |
| calc |
| _ ≤ (t:ℝ)*((t:ℝ)^(6*p))^6*(t:ℝ)*((t:ℝ)^7)^2*((t:ℝ)^5)^2 := by gcongr |
| _ = (t:ℝ)^(36*p+26) := by |
| have he : 36*p+26 = 1+(6*p)*6+1+7*2+5*2 := by omega |
| rw [he] |
| simp only [pow_add, pow_mul, pow_one] |
| _ ≤ (t:ℝ)^(8*(10*(p+1))) := pow_le_pow_right₀ ht1 (by omega) |
| _ = ((t:ℝ)^4)^(2*(10*(p+1))) := by rw [← pow_mul]; congr 1; omega |
| _ ≤ _ := pow_le_pow_left₀ (pow_nonneg ht0 _) hbase _ |
| |
| lemma numerical_smallness (D : ℝ) (t : ℕ) (ht : 1 ≤ t) (hD : 8*(D+1) ≤ (t:ℝ)) : |
| 16*(t:ℝ)^2*((t:ℝ)*D+1) ≤ 2*(t:ℝ)^4 := by |
| have ht1 : (1:ℝ) ≤ t := by exact_mod_cast ht |
| have ht0 : (0:ℝ) ≤ t := Nat.cast_nonneg t |
| calc |
| _ ≤ 16*(t:ℝ)^2*((t:ℝ)*(D+1)) := by gcongr; linarith |
| _ = 2*(t:ℝ)^3*(8*(D+1)) := by ring |
| _ ≤ 2*(t:ℝ)^3*(t:ℝ) := mul_le_mul_of_nonneg_left hD (by positivity) |
| _ = _ := by ring |
| |
| end PerfectMatchingPSD |
| |
| end OAI |
| |
