| import OAI.Combinatorics.MatchingPSD.NumericalBounds |
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| namespace OAI |
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| namespace PerfectMatchingPSD |
| open MatchingSmoothing MatchingInterpolation Matrix Filter Asymptotics |
| open scoped BigOperators MatrixOrder |
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| 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)) |
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| 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) |
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| end PerfectMatchingPSD |
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| end OAI |
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