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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 | import OAI.Probability.MatroidProphet.ProductSplit
import OAI.Probability.MatroidProphet.Thinning
namespace OAI
namespace MatroidProphet
open Set Finset
variable {α : Type*} [Fintype α] [DecidableEq α]
theorem conditionally_independent_thinning_rank
(M : Matroid α) (hE : M.E = Set.univ) (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1)
(V G : Finset α) (hG : G ⊆ V) (S : Set α) (hS : S ⊆ (G : Set α))
(P : Finset α → Set α)
(hP : ∀ A B : Finset α, B ⊆ G → P (A ∪ B) = P A) :
t * bitsExpectation (fun _ => t) V (fun T => (conditionalRank M S (P T) : ℝ)) ≤
bitsExpectation (fun _ => t) V
(fun T => (conditionalRank M (S ∩ (T : Set α)) (P T) : ℝ)) := by
classical
have heq : bitsExpectation (fun _ => t) V (fun T => (conditionalRank M S (P T) : ℝ)) =
bitsExpectation (fun _ => t) (V \ G) (fun A => (conditionalRank M S (P A) : ℝ)) := by
rw [bitsExpectation_split (fun _ => t) V G hG]
apply bitsExpectation_congr
intro A hA
calc
_ = bitsExpectation (fun _ => t) G (fun _ => (conditionalRank M S (P A) : ℝ)) := by
apply bitsExpectation_congr
intro B hB
rw [hP A B hB]
_ = _ := bitsExpectation_const _ _ _
rw [heq, ← bitsExpectation_mul_left, bitsExpectation_split (fun _ => t) V G hG]
apply bitsExpectation_mono _ (fun _ => ht0) (fun _ => ht1)
intro A hA
have hdis : S ∩ (A : Set α) = ∅ := by
apply Set.eq_empty_iff_forall_notMem.mpr
rintro e ⟨heS, heA⟩
exact (Finset.mem_sdiff.mp (hA heA)).2 (hS heS)
have hb := independent_thinning_rank M hE G S.toFinset
(fun e he => hS (by simpa only [Set.mem_toFinset] using he)) (P A) t ht0 ht1
calc
_ ≤ bitsExpectation (fun _ => t) G
(fun B => (conditionalRank M (S ∩ (B : Set α)) (P A) : ℝ)) := by simpa only [Finset.coe_inter, Set.coe_toFinset] using hb
_ = _ := by
apply bitsExpectation_congr
intro B hB
rw [hP A B hB]
have hset : S ∩ ((A ∪ B : Finset α) : Set α) = S ∩ (B : Set α) := by
rw [Finset.coe_union, Set.inter_union_distrib_left, hdis, Set.empty_union]
rw [hset]
end MatroidProphet
end OAI
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