File size: 3,110 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 | import OAI.Combinatorics.MatchingPSD.NumericalBounds
namespace OAI
namespace PerfectMatchingPSD
open MatchingSmoothing MatchingInterpolation Matrix Filter Asymptotics
open scoped BigOperators MatrixOrder
lemma main_nat (p : ℕ) : ∃ n₀ : ℕ, 4 ≤ n₀ ∧
∀ n : ℕ, n₀ ≤ n → Even n → (n:ℝ)^p < (psdRank n:ℝ) := by
classical
obtain ⟨d,H,hd,D,hD,hdecode⟩ := conditional_cell_lower_bound (10*(p+1))
obtain ⟨Te,hTe⟩ := eventually_atTop.mp (eventually_pow_lt_half_exp (6*p))
let T := 2+H+1024*(d+1)+Fintype.card (SampleIndex d)+⌈8*(D+1)⌉₊+Te
refine ⟨vertexSize d T+4, by omega, ?_⟩
intro n hn hneven
obtain ⟨t,ht,hleft,hright⟩ := bracket_vertexSize d T n (by omega)
dsimp only [T] at ht
have ht2 : 2 ≤ t := by omega
have ht1 : 1 ≤ t := by omega
have hHt : H ≤ t := by omega
have hdt : 1024*(d+1) ≤ t := by omega
have hTt : Fintype.card (SampleIndex d) ≤ t := by omega
have hDt : 8*(D+1) ≤ (t:ℝ) :=
(Nat.le_ceil (8*(D+1))).trans (by exact_mod_cast (show ⌈8*(D+1)⌉₊ ≤ t by omega))
have htet : Te ≤ t := by omega
have hn6 : n ≤ t^6 := hright.le.trans (vertexSize_succ_le_pow6 d t ht1 hdt)
have hl : 1 ≤ t^4 := one_le_pow₀ ht1
have hs : 1 ≤ blockSize d t := hl.trans (blockSize_ge d t)
have hH : H ≤ 3*blockSize d t+1 := by
have hts : t ≤ blockSize d t := (Nat.le_pow (by decide : 0 < 4)).trans (blockSize_ge d t)
omega
by_contra hnot
have hrn : (psdRank n:ℝ) ≤ (n:ℝ)^p := le_of_not_gt hnot
obtain ⟨k,⟨e⟩⟩ := exists_padded_global_equiv hneven hleft
obtain ⟨Q,B,hQ,hB,htrQ,htrB,hpair⟩ := global_factors_of_factorization
hneven (by omega : 1 ≤ n) (hasFactorization_psdRank n) e (standardPairing (H := Fin k))
have hsel : (t:ℝ)*(psdRank n)^4*(Fintype.card (SampleIndex d))*
((2*(3*(blockSize d t:ℝ)+1))^(2*(10*(p+1))))⁻¹ ≤
(1/((psdRank n:ℝ)*((n:ℝ)+1)*(2*(t^4:ℕ))))^2 := by
simpa only [Nat.cast_pow] using numerical_selection p d t (psdRank n) n
(Fintype.card (SampleIndex d)) ht2 (psdRank_pos n) hn6 hTt hrn
have hsmall : 16*(t:ℝ)^2*((t:ℝ)*D+1) ≤ 2*(t^4:ℕ) := by
simpa only [Nat.cast_pow] using numerical_smallness D t ht1 hDt
have hlower : (2:ℝ)^t/2 ≤ (psdRank n:ℝ) :=
hdecode (blockSize d t) t (t^4) (psdRank n) hH hs ht1 (blockSize_grid d t) hl
(psdRank_pos n) ((n:ℝ)+1) (by positivity) Q B hQ hB htrQ htrB hpair hsel hsmall
have hn6r : (n:ℝ) ≤ (t:ℝ)^6 := by exact_mod_cast hn6
have hpow : (n:ℝ)^p ≤ (t:ℝ)^(6*p) := by
simpa only [pow_mul] using pow_le_pow_left₀ (Nat.cast_nonneg n) hn6r p
exact (not_lt_of_ge hlower) ((hrn.trans hpow).trans_lt (hTe t htet))
theorem main : MainClaim := by
intro C _hC
obtain ⟨n₀,hn₀,hmain⟩ := main_nat ⌈C⌉₊
refine ⟨n₀,hn₀,?_⟩
intro n hn heven
have hn1 : (1:ℝ) ≤ n := by exact_mod_cast (show 1 ≤ n by omega)
have hpow := Real.rpow_le_rpow_of_exponent_le hn1 (Nat.le_ceil C)
rw [Real.rpow_natCast] at hpow
exact hpow.trans_lt (hmain n hn heven)
end PerfectMatchingPSD
end OAI
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