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dd14c26 | 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 | import OAI.Probability.MatroidProphet.Algorithm.Statistics
namespace OAI
namespace MatroidProphet.MainAlgorithm
open Set Finset
variable {n : ℕ}
lemma groupMask_function_eq_inter (M : Matroid (Fin n)) (d : MainMasks n)
(w : Fin n → Option ℤ) (mask : Finset (Fin n)) :
groupMask M d w mask = fun j => (mask : Set (Fin n)) ∩ (pathGroups M d w j : Set (Fin n)) := by
funext j
exact groupMask_eq_inter M d w mask j
lemma listedZ_update_T (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (h : ℕ) (T : Finset (Fin n)) :
listedZ M hE {d with T := T} w h =
nominalRankStatistic M hE (2^100) (groupMask M d w d.D) (groupMask M d w d.C)
(fun j => (T : Set (Fin n)) ∩ (pathGroups M d w j : Set (Fin n))) h (pathGroups M d w h)
((d.H ∪ d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) (boolParity d.odd) := by
unfold listedZ
change nominalRankStatistic M hE (2^100) (groupMask M d w d.D) (groupMask M d w d.C)
(groupMask M d w T) h (groupMask M d w univ h)
((d.H ∪ d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) (boolParity d.odd) = _
rw [groupMask_function_eq_inter M d w T, ← pathGroups_coe]
lemma listedLambda_update_T (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (h : ℕ) (T : Finset (Fin n)) :
listedLambda M hE {d with T := T} w h =
finalRankStatistic M hE (2^100) (groupMask M d w d.D) (groupMask M d w d.C)
(fun j => (T : Set (Fin n)) ∩ (pathGroups M d w j : Set (Fin n))) h (groups M d w).length
(pathGroups M d w h) ((d.H ∪ d.D ∪ d.C : Finset (Fin n)) : Set (Fin n))
(T : Set (Fin n)) (boolParity d.odd) := by
unfold listedLambda
change finalRankStatistic M hE (2^100) (groupMask M d w d.D) (groupMask M d w d.C)
(groupMask M d w T) h (groups M d w).length (groupMask M d w univ h)
((d.H ∪ d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) (T : Set (Fin n)) (boolParity d.odd) = _
rw [groupMask_function_eq_inter M d w T, ← pathGroups_coe]
theorem listed_expected_nominal_to_final (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (h : ℕ) (hh : h < (groups M d w).length)
(t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) :
densityThreshold * t * bitsExpectation (fun _ : Fin n => t) univ (fun T =>
(listedZ M hE {d with T := T} w h : ℝ)) ≤
densityThreshold * bitsExpectation (fun _ : Fin n => t) univ (fun T =>
(listedLambda M hE {d with T := T} w h : ℝ)) +
t * ((2^100) * (groupMask M d w d.C h).ncard +
∑ j ∈ range ((groups M d w).length-(h+1)),
((2^100) * (groupMask M d w d.C (h+1+j)).ncard +
(groupMask M d w d.D (h+1+j)).ncard) : ℕ) := by
have hc := expected_nominal_to_final M hE (2^100) (by positivity)
(groupMask M d w d.D) (groupMask M d w d.C) (pathGroups M d w)
(pathGroups_disjoint M d w) h ((groups M d w).length-(h+1))
((d.H ∪ d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) (boolParity d.odd) t ht0 ht1
have hlen : h+1+((groups M d w).length-(h+1)) = (groups M d w).length := by omega
rw [hlen] at hc
simp only [listedZ_update_T, listedLambda_update_T]
convert hc using 1 <;> norm_num [densityThreshold]
end MatroidProphet.MainAlgorithm
end OAI
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