openAI-math-dataset / lean /OAI /Computability /VertexCover /Reduction /DecodingUpperThreeHalves.lean
| import Mathlib | |
| import OAI.Computability.VertexCover.Reduction.HighVectorsSeparated | |
| import OAI.Computability.VertexCover.Reduction.GridBudget | |
| namespace OAI | |
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| namespace VertexCover.Parameters | |
| theorem pair_budget (m : ℕ) (hm : 4 ≤ m) : | |
| (((h m).choose 2 : ℕ) : ℝ)*(ell m : ℝ)^2*σ m = ((h m : ℝ)-1)/(2*(h m : ℝ)) := by | |
| have hpos := positive m (by omega : 0 < m) | |
| have hh : (h m : ℝ) ≠ 0 := by exact_mod_cast (ne_of_gt hpos.2.2.2.1) | |
| have hl : (ell m : ℝ) ≠ 0 := by exact_mod_cast (ne_of_gt hpos.2.2.1) | |
| rw [Nat.cast_choose_two, σ_eq] | |
| field_simp | |
| theorem pair_budget_lt (m : ℕ) (hm : 4 ≤ m) : | |
| (((h m).choose 2 : ℕ) : ℝ)*(ell m : ℝ)^2*σ m < (1:ℝ)/2 := by | |
| rw [pair_budget m hm] | |
| have hh : (0:ℝ) < h m := by exact_mod_cast (lt_of_lt_of_le (by decide : 0<2) (dimensions m hm).2.1) | |
| apply (div_lt_iff₀ (by positivity : (0:ℝ) < 2*(h m : ℝ))).mpr | |
| linarith | |
| theorem σ_between (m : ℕ) (hm : 4 ≤ m) : 0 < σ m ∧ σ m < 1 := by | |
| refine ⟨(positive m (by omega)).2.2.2.2.2, ?_⟩ | |
| have hm1 : (1:ℝ) ≤ m := by exact_mod_cast (show 1 ≤ m by omega) | |
| have hp : (1:ℝ) ≤ (m:ℝ)^12 := one_le_pow₀ hm1 | |
| unfold σ | |
| apply (div_lt_one (by positivity)).mpr | |
| norm_num only [Nat.reducePow] | |
| nlinarith | |
| theorem sqrt_d_gt (m : ℕ) (hm : 4 ≤ m) : 4*(m:ℝ) < Real.sqrt (d m) := by | |
| rw [sqrt_d] | |
| have hm1 : (1:ℝ) ≤ m := by exact_mod_cast (show 1 ≤ m by omega) | |
| have hp : (m:ℝ)^1 ≤ (m:ℝ)^5 := pow_le_pow_right₀ hm1 (by decide) | |
| simp only [pow_one] at hp | |
| norm_num only [Nat.reducePow] | |
| nlinarith | |
| theorem gap_ratio (m : ℕ) (hm : 0 < m) : | |
| (1-1/(m:ℝ))/(1/2+1/(m:ℝ)) = 2-6/((m:ℝ)+2) := by | |
| have hp : (m:ℝ) ≠ 0 := by positivity | |
| have hm2 : (m:ℝ)+2 ≠ 0 := by positivity | |
| have hh : (1:ℝ)/2+1/(m:ℝ) ≠ 0 := by positivity | |
| field_simp | |
| ring | |
| end VertexCover.Parameters | |
| namespace VertexCover.LabelCover | |
| theorem decoding_upper_lt_three_halves (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m) | |
| (hval : Φ.value ≤ Parameters.σ m) (L : Φ.PrivateLists (Parameters.d m)) | |
| (hlen : ∀ i, (L i).length ≤ Parameters.ell m) | |
| (J : Finset (Fin (Parameters.d m))) (hJ : J.card = Parameters.h m) | |
| (frozen : Φ.Seeds (Parameters.d m)) : | |
| finiteMean (fun hidden : Φ.HiddenSeeds J => | |
| (Φ.listHitMaximum L (Φ.spliceSeeds J frozen hidden) J : ℝ)) < 3/2 := by | |
| have hu := Φ.decoding_upper_bound hval L hlen J frozen | |
| rw [hJ] at hu | |
| have hp := Parameters.pair_budget_lt m hm | |
| linarith | |
| end VertexCover.LabelCover | |
| end | |
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| end | |
| end OAI | |