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e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | 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]
/-- Every point moved into row `i` is heavy for every admissible cutoff of any
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
/-- The same reconstruction statement with the heavy set written explicitly. -/
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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