File size: 3,130 Bytes
e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | 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
end OAI
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