File size: 5,685 Bytes
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 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | import OAI.Combinatorics.ExpectationThreshold.SelectorDefinitionsLemmas
import OAI.Combinatorics.ExpectationThreshold.SelectorProfile
import OAI.Combinatorics.ExpectationThreshold.SelectorReconstruction
import OAI.Combinatorics.ExpectationThreshold.SelectorTruncation
namespace OAI
/-!
# One witness for an entire selector profile group
A nonempty group contains an actual selected move, hence an actual feasible
family member. Its admissibility tests depend on the original coloring only
through the full profile. Maximizing each cutoff for this one member therefore
produces common heavy sets for the entire group. Selected-move minimality puts
every moved row inside its corresponding common heavy set.
-/
noncomputable section
namespace TalagrandThreshold.Selector
variable {α : Type*} [Fintype α] [DecidableEq α]
/-- The same actual family member and cutoffs work for every original coloring
in a fixed final-coloring/profile group. -/
structure CommonWitness {F : Family α} (mu : MassFamily F) {s : ℕ}
(z : Coloring α s) (n : Profile s) where
member : ↥F
cutoff : Fin s → ℝ
admissible : ∀ a ∈ groupOriginals mu z n, ∀ i : Fin s,
Admissible (mu.weight member) (prefixSet z i) (crossings a z i : ℝ)
(dscale i) (cutoff i)
maximal : ∀ (i : Fin s) (u : ℝ),
Admissible (mu.weight member) (prefixSet z i) (profileCrossings n i : ℝ)
(dscale i) u → u ≤ cutoff i
card_le : ∀ i : Fin s,
(heavySet (mu.weight member) (cutoff i)).card ≤ 2 ^ (i.val + 1) * profileSize n
moved_subset : ∀ a ∈ groupOriginals mu z n, ∀ i : Fin s,
movedRow a z i ⊆ heavySet (mu.weight member) (cutoff i)
omit [Fintype α] [DecidableEq α] in
private theorem card_le_profile_of_truncation {s : ℕ}
(R : Finset α) (i : Fin s) (n : Profile s) (m : ℕ)
(hcard : (R.card : ℝ) ≤ (m : ℝ) / dscale i)
(hm : m ≤ profileSize n) :
R.card ≤ 2 ^ (i.val + 1) * profileSize n := by
have hreal : (R.card : ℝ) ≤ ((m * 2 ^ (i.val + 1) : ℕ) : ℝ) := by
simpa only [dscale, div_inv_eq_mul, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat]
using hcard
have hnat : R.card ≤ m * 2 ^ (i.val + 1) := (Nat.cast_le (α := ℝ)).mp hreal
calc
R.card ≤ m * 2 ^ (i.val + 1) := hnat
_ ≤ profileSize n * 2 ^ (i.val + 1) := Nat.mul_le_mul_right _ hm
_ = 2 ^ (i.val + 1) * profileSize n := Nat.mul_comm _ _
/-- The common witness is constructed from the actual nonempty group; no
additional feasibility, reconstruction, or cardinality certificate is assumed. -/
theorem exists_common_witness {F : Family α} (mu : MassFamily F) {s : ℕ}
(z : Coloring α s) (n : Profile s)
(hne : (groupOriginals mu z n).Nonempty) : Nonempty (CommonWitness mu z n) := by
classical
obtain ⟨a₀, ha₀⟩ := hne
have ha₀' := (mem_groupOriginals mu a₀ z n).mp ha₀
obtain ⟨H₀, hminimal⟩ := ha₀'.2.1
obtain ⟨H, hfeasible⟩ := hminimal.1.1.2
have hcross₀ (i : Fin s) : crossings a₀ z i = profileCrossings n i := by
rw [crossings_eq_profileCrossings, ha₀'.2.2]
have hcutoff (i : Fin s) : ∃ ε : ℝ,
Admissible (mu.weight H) (prefixSet z i) (profileCrossings n i : ℝ) (dscale i) ε ∧
(∀ u, Admissible (mu.weight H) (prefixSet z i) (profileCrossings n i : ℝ)
(dscale i) u → u ≤ ε) ∧
((heavySet (mu.weight H) ε).card : ℝ) ≤ (profileCrossings n i : ℝ) / dscale i := by
apply exists_maximal_truncation_div (mu.weight H) (prefixSet z i)
(profileCrossings n i : ℝ) (dscale i) (mu.nonneg H) (mu.total H)
(Nat.cast_nonneg _) (dscale_pos i) (dscale_lt_one i)
simpa only [hcross₀ i] using hfeasible i
let ε : Fin s → ℝ := fun i => Classical.choose (hcutoff i)
have hε (i : Fin s) :
Admissible (mu.weight H) (prefixSet z i) (profileCrossings n i : ℝ) (dscale i) (ε i) ∧
(∀ u, Admissible (mu.weight H) (prefixSet z i) (profileCrossings n i : ℝ)
(dscale i) u → u ≤ ε i) ∧
((heavySet (mu.weight H) (ε i)).card : ℝ) ≤
(profileCrossings n i : ℝ) / dscale i :=
Classical.choose_spec (hcutoff i)
have hadmissible : ∀ a ∈ groupOriginals mu z n, ∀ i : Fin s,
Admissible (mu.weight H) (prefixSet z i) (crossings a z i : ℝ) (dscale i) (ε i) := by
intro a ha i
have hprofile := ((mem_groupOriginals mu a z n).mp ha).2.2
rw [crossings_eq_profileCrossings, hprofile]
exact (hε i).1
refine ⟨{
member := H
cutoff := ε
admissible := hadmissible
maximal := fun i => (hε i).2.1
card_le := ?_
moved_subset := ?_
}⟩
· intro i
exact card_le_profile_of_truncation (heavySet (mu.weight H) (ε i)) i n
(profileCrossings n i) (hε i).2.2 (profileCrossings_le_profileSize n i)
· intro a ha i
have hselected := ((mem_groupOriginals mu a z n).mp ha).2.1
have hfeasible' : FeasibleFor mu a z H := fun j => ⟨ε j, hadmissible a ha j⟩
exact selected_movedRow_subset_heavySet mu a z hselected H hfeasible' i (ε i)
(hadmissible a ha i)
/-- The finite sets needed for counting are common to the whole actual group. -/
theorem exists_common_heavy_sets {F : Family α} (mu : MassFamily F) {s : ℕ}
(z : Coloring α s) (n : Profile s)
(hne : (groupOriginals mu z n).Nonempty) :
∃ R : Fin s → Finset α,
(∀ i, (R i).card ≤ 2 ^ (i.val + 1) * profileSize n) ∧
∀ a ∈ groupOriginals mu z n, ∀ i, movedRow a z i ⊆ R i := by
obtain ⟨w⟩ := exists_common_witness mu z n hne
exact ⟨fun i => heavySet (mu.weight w.member) (w.cutoff i), w.card_le, w.moved_subset⟩
end TalagrandThreshold.Selector
end
end OAI
|