| import OAI.Combinatorics.ExpectationThreshold.ThresholdCompactness |
| import OAI.Combinatorics.ExpectationThreshold.ThresholdBounds |
|
|
| namespace OAI |
|
|
| /-! |
| # Attainment of the integral and fractional thresholds |
|
|
| There are finitely many integral covers, so integral feasibility is closed |
| in the density interval. Fractional feasibility is a compact projection. |
| The actual suprema therefore belong to their respective feasible sets as |
| soon as those sets contain density zero. |
| -/ |
|
|
| namespace TalagrandThreshold |
|
|
| open scoped BigOperators |
|
|
| variable {α : Type*} [Fintype α] [DecidableEq α] |
|
|
| omit [Fintype α] [DecidableEq α] in |
| theorem continuous_cost (G : Family α) : Continuous (fun p : ℝ => cost p G) := by |
| unfold cost |
| exact continuous_finsetSum G (fun S _ => continuous_pow S.card) |
|
|
| omit [DecidableEq α] in |
| / |
| theorem isClosed_small (F : Family α) : IsClosed {p : ℝ | Small p F} := by |
| classical |
| have hset : {p : ℝ | Small p F} = |
| ⋃ G : Family α, {p : ℝ | Covers G F ∧ cost p G ≤ 1 / 2} := by |
| ext p |
| simp only [Set.mem_ofPred_eq, Set.mem_iUnion, Small] |
| rw [hset] |
| apply isClosed_iUnion_of_finite |
| intro G |
| by_cases hG : Covers G F |
| · simpa only [hG, true_and] using |
| (isClosed_le (continuous_cost G) (continuous_const : |
| Continuous (fun _ : ℝ => (1 / 2 : ℝ)))) |
| · simpa only [hG, false_and, Set.ofPred_false] using |
| (isClosed_empty : IsClosed (∅ : Set ℝ)) |
|
|
| omit [DecidableEq α] in |
| theorem isCompact_integralFeasible (F : Family α) : IsCompact (integralFeasible F) := by |
| have hset : integralFeasible F = Set.Icc (0 : ℝ) 1 ∩ {p : ℝ | Small p F} := by |
| ext p |
| constructor |
| · intro hp |
| exact ⟨⟨hp.1, hp.2.1⟩, hp.2.2⟩ |
| · intro hp |
| exact ⟨hp.1.1, hp.1.2, hp.2⟩ |
| rw [hset] |
| exact (isCompact_Icc : IsCompact (Set.Icc (0 : ℝ) 1)).inter_right (isClosed_small F) |
|
|
| omit [DecidableEq α] in |
| / |
| theorem q_mem_integralFeasible (F : Family α) (hF : ∅ ∉ F) : |
| q F ∈ integralFeasible F := by |
| exact (isCompact_integralFeasible F).sSup_mem (integralFeasible_nonempty F hF) |
|
|
| / |
| theorem qf_mem_fractionalFeasible (F : Family α) (hF : ∅ ∉ F) : |
| qf F ∈ fractionalFeasible F := by |
| exact (isCompact_fractionalFeasible F).sSup_mem (fractionalFeasible_nonempty F hF) |
|
|
| omit [DecidableEq α] in |
| theorem small_q (F : Family α) (hF : ∅ ∉ F) : Small (q F) F := |
| (mem_integralFeasible.mp (q_mem_integralFeasible F hF)).2.2 |
|
|
| theorem fractionalSmall_qf (F : Family α) (hF : ∅ ∉ F) : FractionalSmall (qf F) F := |
| (mem_fractionalFeasible.mp (qf_mem_fractionalFeasible F hF)).2.2 |
|
|
| end TalagrandThreshold |
|
|
| end OAI |
|
|
