| import OAI.Combinatorics.ExpectationThreshold.SelectorDelayFeasibility |
|
|
| namespace OAI |
|
|
| /-! |
| # Minimal moves reconstruct inside the heavy sets |
|
|
| The family member in this theorem is any member witnessing feasibility of the |
| selected endpoint. It need not be the member restricting the changed set in |
| the minimization. This permits one witness to be used for an entire profile. |
| -/ |
|
|
| noncomputable section |
|
|
| namespace TalagrandThreshold.Selector |
|
|
| variable {α : Type*} [Fintype α] [DecidableEq α] |
|
|
| / |
| common feasibility witness. Otherwise its actual color can be delayed by one, |
| preserving that witness and contradicting minimal total movement. -/ |
| theorem selected_movedRow_subset_heavySet {F : Family α} (mu : MassFamily F) |
| {s : ℕ} (a z : Coloring α s) (hselected : Selected mu a z) |
| (H : ↥F) (hfeasible : FeasibleFor mu a z H) (i : Fin s) (ε : ℝ) |
| (hε : Admissible (mu.weight H) (prefixSet z i) |
| (crossings a z i : ℝ) (dscale i) ε) : |
| movedRow a z i ⊆ heavySet (mu.weight H) ε := by |
| classical |
| intro x hxrow |
| obtain ⟨hx, hmove⟩ := (Finset.mem_filter.mp hxrow).2 |
| apply mem_heavySet.mpr |
| by_contra hheavy |
| have hweight : mu.weight H x ≤ ε := le_of_not_gt hheavy |
| obtain ⟨H₀, hbounded, hminimal⟩ := hselected |
| have hbounded' : BoundedFeasible mu a H₀ (delayColor z x i) := by |
| refine ⟨⟨earlier_delayColor a z x i hx hmove hbounded.1.1, ?_⟩, ?_⟩ |
| · exact ⟨H, feasibleFor_delayColor mu a z H hfeasible x i hx hmove ε hε hweight⟩ |
| · intro y hy |
| exact hbounded.2 (changes_delayColor_subset a z x i hx hy) |
| have hmin := hminimal (delayColor z x i) hbounded' |
| have hdec := movement_delayColor_add_one a z x i hx hmove |
| have hstrict : movement a (delayColor z x i) < movement a z := |
| lt_of_lt_of_eq (Nat.lt_succ_self _) hdec |
| exact (not_lt_of_ge hmin) hstrict |
|
|
| / |
| theorem selected_movedRow_subset_filter {F : Family α} (mu : MassFamily F) |
| {s : ℕ} (a z : Coloring α s) (hselected : Selected mu a z) |
| (H : ↥F) (hfeasible : FeasibleFor mu a z H) (i : Fin s) (ε : ℝ) |
| (hε : Admissible (mu.weight H) (prefixSet z i) |
| (crossings a z i : ℝ) (dscale i) ε) : |
| movedRow a z i ⊆ Finset.univ.filter (fun x => ε < mu.weight H x) := by |
| simpa only [heavySet] using |
| selected_movedRow_subset_heavySet mu a z hselected H hfeasible i ε hε |
|
|
| end TalagrandThreshold.Selector |
|
|
| end |
|
|
| end OAI |
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|
