| import OAI.Combinatorics.ExpectationThreshold.Basic |
| import Mathlib.Order.ConditionallyCompleteLattice.Basic |
|
|
| namespace OAI |
|
|
| /-! |
| # Elementary bounds for the actual expectation thresholds |
|
|
| When the empty set is absent from the family, zero is feasible for both |
| thresholds. Their feasible sets are bounded by one, and indicator covers |
| give the integral-to-fractional comparison. |
| -/ |
|
|
| namespace TalagrandThreshold |
|
|
| noncomputable section |
|
|
| variable {α : Type*} [Fintype α] [DecidableEq α] |
|
|
| omit [Fintype α] [DecidableEq α] in |
| theorem integralFeasible_subset_Icc (F : Family α) : |
| integralFeasible F ⊆ Set.Icc (0 : ℝ) 1 := |
| fun _ hp => ⟨hp.1, hp.2.1⟩ |
|
|
| omit [DecidableEq α] in |
| theorem fractionalFeasible_subset_Icc (F : Family α) : |
| fractionalFeasible F ⊆ Set.Icc (0 : ℝ) 1 := |
| fun _ hp => ⟨hp.1, hp.2.1⟩ |
|
|
| omit [Fintype α] [DecidableEq α] in |
| theorem integralFeasible_bddAbove (F : Family α) : BddAbove (integralFeasible F) := |
| ⟨1, fun _ hp => hp.2.1⟩ |
|
|
| omit [DecidableEq α] in |
| theorem fractionalFeasible_bddAbove (F : Family α) : BddAbove (fractionalFeasible F) := |
| ⟨1, fun _ hp => hp.2.1⟩ |
|
|
| omit [Fintype α] [DecidableEq α] in |
| theorem zero_mem_integralFeasible (F : Family α) (hF : (∅ : Finset α) ∉ F) : |
| (0 : ℝ) ∈ integralFeasible F := |
| ⟨le_rfl, zero_le_one, small_zero F hF⟩ |
|
|
| theorem zero_mem_fractionalFeasible (F : Family α) (hF : (∅ : Finset α) ∉ F) : |
| (0 : ℝ) ∈ fractionalFeasible F := |
| ⟨le_rfl, zero_le_one, fractionalSmall_zero F hF⟩ |
|
|
| omit [Fintype α] [DecidableEq α] in |
| theorem integralFeasible_nonempty (F : Family α) (hF : (∅ : Finset α) ∉ F) : |
| (integralFeasible F).Nonempty := |
| ⟨0, zero_mem_integralFeasible F hF⟩ |
|
|
| theorem fractionalFeasible_nonempty (F : Family α) (hF : (∅ : Finset α) ∉ F) : |
| (fractionalFeasible F).Nonempty := |
| ⟨0, zero_mem_fractionalFeasible F hF⟩ |
|
|
| omit [Fintype α] [DecidableEq α] in |
| /-- Integral feasibility is preserved by lowering the nonnegative density. -/ |
| theorem integralFeasible_mono {F : Family α} {p r : ℝ} |
| (hr : r ∈ integralFeasible F) (hp : 0 ≤ p) (hpr : p ≤ r) : |
| p ∈ integralFeasible F := |
| ⟨hp, hpr.trans hr.2.1, hr.2.2.mono hp hpr⟩ |
|
|
| omit [DecidableEq α] in |
| /-- Fractional feasibility is preserved by lowering the nonnegative density. -/ |
| theorem fractionalFeasible_mono {F : Family α} {p r : ℝ} |
| (hr : r ∈ fractionalFeasible F) (hp : 0 ≤ p) (hpr : p ≤ r) : |
| p ∈ fractionalFeasible F := |
| ⟨hp, hpr.trans hr.2.1, hr.2.2.mono hp hpr⟩ |
|
|
| theorem integralFeasible_subset_fractionalFeasible (F : Family α) : |
| integralFeasible F ⊆ fractionalFeasible F := |
| fun _ hp => ⟨hp.1, hp.2.1, hp.2.2.fractionalSmall⟩ |
|
|
| omit [Fintype α] [DecidableEq α] in |
| theorem le_q_of_mem {F : Family α} {p : ℝ} (hp : p ∈ integralFeasible F) : |
| p ≤ q F := |
| le_csSup (integralFeasible_bddAbove F) hp |
|
|
| omit [DecidableEq α] in |
| theorem le_qf_of_mem {F : Family α} {p : ℝ} (hp : p ∈ fractionalFeasible F) : |
| p ≤ qf F := |
| le_csSup (fractionalFeasible_bddAbove F) hp |
|
|
| omit [Fintype α] [DecidableEq α] in |
| theorem le_q_of_small {F : Family α} {p : ℝ} |
| (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (hp : Small p F) : p ≤ q F := |
| le_q_of_mem ⟨hp0, hp1, hp⟩ |
|
|
| omit [DecidableEq α] in |
| theorem le_qf_of_fractionalSmall {F : Family α} {p : ℝ} |
| (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (hp : FractionalSmall p F) : p ≤ qf F := |
| le_qf_of_mem ⟨hp0, hp1, hp⟩ |
|
|
| omit [Fintype α] [DecidableEq α] in |
| theorem q_nonneg (F : Family α) (hF : (∅ : Finset α) ∉ F) : 0 ≤ q F := |
| le_q_of_mem (zero_mem_integralFeasible F hF) |
|
|
| theorem qf_nonneg (F : Family α) (hF : (∅ : Finset α) ∉ F) : 0 ≤ qf F := |
| le_qf_of_mem (zero_mem_fractionalFeasible F hF) |
|
|
| omit [Fintype α] [DecidableEq α] in |
| theorem q_le_one (F : Family α) (hF : (∅ : Finset α) ∉ F) : q F ≤ 1 := |
| csSup_le (integralFeasible_nonempty F hF) (fun _ hp => hp.2.1) |
|
|
| theorem qf_le_one (F : Family α) (hF : (∅ : Finset α) ∉ F) : qf F ≤ 1 := |
| csSup_le (fractionalFeasible_nonempty F hF) (fun _ hp => hp.2.1) |
|
|
| theorem q_le_qf (F : Family α) (hF : (∅ : Finset α) ∉ F) : q F ≤ qf F := |
| csSup_le_csSup (fractionalFeasible_bddAbove F) (integralFeasible_nonempty F hF) |
| (integralFeasible_subset_fractionalFeasible F) |
|
|
| end |
| end TalagrandThreshold |
|
|
| end OAI |
|
|
