| import OAI.Combinatorics.MatchingPSD.Padding |
|
|
| namespace OAI |
|
|
| namespace PerfectMatchingPSD |
| open MatchingSmoothing MatchingInterpolation Matrix Filter Asymptotics |
| open scoped BigOperators MatrixOrder |
|
|
| def blockSize (d t : ℕ) : ℕ := 4*(d+1)*t^4 |
| def vertexSize (d t : ℕ) : ℕ := 2*blockSize d t*(t+2)+2 |
|
|
| lemma index_le_vertexSize (d t : ℕ) : t ≤ vertexSize d t := by |
| have ht : t ≤ t^4 := Nat.le_pow (by decide) |
| have hc : 1 ≤ 8*(d+1)*(t+2) := by |
| have : 0 < 8*(d+1)*(t+2) := by positivity |
| omega |
| calc |
| t ≤ t^4 := ht |
| _ ≤ (8*(d+1)*(t+2))*t^4 := by simpa only [one_mul] using Nat.mul_le_mul_right (t^4) hc |
| _ ≤ vertexSize d t := by dsimp [vertexSize, blockSize]; nlinarith |
|
|
| lemma vertexSize_succ_bound (d t : ℕ) (ht : 1 ≤ t) : |
| vertexSize d (t+1) ≤ 1024*(d+1)*t^5 := by |
| have hp : 1 ≤ (d+1)*t^5 := by |
| have : 0 < (d+1)*t^5 := by positivity |
| omega |
| calc |
| _ = 8*(d+1)*(t+1)^4*(t+3)+2 := by dsimp [vertexSize, blockSize]; ring |
| _ ≤ 8*(d+1)*(2*t)^4*(4*t)+2 := by gcongr <;> omega |
| _ = 512*(d+1)*t^5+2 := by ring |
| _ ≤ _ := by nlinarith |
|
|
| lemma vertexSize_succ_le_pow6 (d t : ℕ) (ht : 1 ≤ t) (hd : 1024*(d+1) ≤ t) : |
| vertexSize d (t+1) ≤ t^6 := by |
| calc |
| _ ≤ 1024*(d+1)*t^5 := vertexSize_succ_bound d t ht |
| _ ≤ t*t^5 := Nat.mul_le_mul_right _ hd |
| _ = _ := by ring |
|
|
| lemma bracket_vertexSize (d T n : ℕ) (hn : vertexSize d T ≤ n) : |
| ∃ t : ℕ, T ≤ t ∧ vertexSize d t ≤ n ∧ n < vertexSize d (t+1) := by |
| classical |
| let P := fun t => vertexSize d t ≤ n |
| let t := Nat.findGreatest P n |
| have hT : T ≤ n := (index_le_vertexSize d T).trans hn |
| refine ⟨t, Nat.le_findGreatest (P := P) hT hn, Nat.findGreatest_spec (P := P) hT hn, ?_⟩ |
| by_contra h |
| have hnext : P (t+1) := le_of_not_gt h |
| have htn : t+1 ≤ n := (index_le_vertexSize d (t+1)).trans hnext |
| exact (Nat.findGreatest_eq_iff.mp (show Nat.findGreatest P n=t from rfl)).2.2 |
| (Nat.lt_succ_self t) htn hnext |
|
|
| lemma exists_padded_global_equiv {n s q : ℕ} (hn : Even n) (hle : 2*s*(q+2)+2 ≤ n) : |
| ∃ k : ℕ, Nonempty ((GlobalVertices s q ⊕ (Fin k × Bool)) ≃ Fin n) := by |
| have hg : Even (2*s*(q+2)+2) := ⟨s*(q+2)+1, by ring⟩ |
| have hd : Even (n-(2*s*(q+2)+2)) := (Nat.even_sub hle).mpr ⟨fun _ => hg, fun _ => hn⟩ |
| obtain ⟨k,hk⟩ := hd |
| refine ⟨k, ⟨Fintype.equivOfCardEq ?_⟩⟩ |
| rw [Fintype.card_sum, globalVertices_card] |
| simp only [Fintype.card_prod, Fintype.card_fin, Fintype.card_bool] |
| omega |
|
|
| lemma eventually_pow_lt_half_exp (p : ℕ) : |
| ∀ᶠ t : ℕ in atTop, (t:ℝ)^p < (2:ℝ)^t/2 := by |
| have ho := isLittleO_pow_const_const_pow_of_one_lt (R := ℝ) p (by norm_num : 1 < (2:ℝ)) |
| have hb := ho.bound (by norm_num : (0:ℝ) < 1/4) |
| filter_upwards [hb] with t ht |
| have ht' : (t:ℝ)^p ≤ (1/4)*(2:ℝ)^t := by |
| simpa only [Real.norm_eq_abs, abs_of_nonneg (show (0:ℝ) ≤ (t:ℝ)^p by positivity), |
| abs_of_pos (pow_pos (by norm_num : (0:ℝ)<2) _)] using ht |
| calc |
| _ ≤ (1/4)*(2:ℝ)^t := ht' |
| _ < _ := by nlinarith [pow_pos (by norm_num : (0:ℝ)<2) t] |
|
|
| end PerfectMatchingPSD |
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|
| end OAI |
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