| import Mathlib | |
| import OAI.Combinatorics.MatchingPSD.AffineFactorization | |
| import OAI.Combinatorics.MatchingPSD.Main | |
| namespace OAI | |
| namespace PerfectMatchingPSD | |
| open MatchingSmoothing Finset Matrix | |
| open scoped BigOperators | |
| theorem affine_lift_lower_bound : | |
| ∀ C : ℝ, 0 < C → ∃ n₀ : ℕ, 4 ≤ n₀ ∧ | |
| ∀ n : ℕ, n₀ ≤ n → Even n → ∀ r : ℕ, | |
| HasAffineLift n r → (n : ℝ) ^ C < (r : ℝ) := by | |
| intro C hC | |
| obtain ⟨n₀, hn₀, hmain⟩ := main C hC | |
| refine ⟨n₀, hn₀, ?_⟩ | |
| intro n hn he r h | |
| have hn4 : 4 ≤ n := hn₀.trans hn | |
| have hf := affineLift_hasFactorization hn4 he h | |
| have hr := factorization_order_pos (by omega) he hf | |
| exact (hmain n hn he).trans_le (by exact_mod_cast psdRank_le hr hf) | |
| end PerfectMatchingPSD | |
| end OAI | |