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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 64 65 66 67 68 69 70 71 72 | import OAI.Probability.MatroidProphet.Density.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.BigOperators.Ring.Finset
namespace OAI
namespace MatroidProphet
open Set
variable {α : Type*} [Fintype α]
lemma natRank_union_le_add_ncard (M : Matroid α) (X Z : Set α) :
natRank M (X ∪ Z) ≤ natRank M X + Z.ncard := by
have hz : Z.encard ≠ ⊤ := (Set.toFinite Z).encard_lt_top.ne
have h := ENat.toNat_le_toNat (M.eRk_union_le_eRk_add_encard X Z)
(WithTop.add_ne_top.2 ⟨rank_ne_top M X, hz⟩)
simpa only [ENat.toNat_add (rank_ne_top M X) hz, natRank, Set.ncard_def] using h
lemma densityExpansion_residual_le_generators (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D P Z : Set α) :
let Q := densityExpansion M hE κ D P
(D ∩ (M.closure (Q ∪ Z) \ Q)).ncard ≤ κ * Z.ncard := by
dsimp only
have hd := densityExpansion_residual M hE κ D P Z
have hr := Nat.mul_le_mul_left κ
(natRank_union_le_add_ncard M (densityExpansion M hE κ D P) Z)
rw [Nat.mul_add] at hr
dsimp only at hd
omega
lemma safe_layer_le_witness_residual (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D F K Z A S : Set α)
(hS : S ⊆ (D ∩ A) \ densityExpansion M hE κ D (F ∪ K)) :
S.ncard ≤ κ * Z.ncard +
(A \ M.closure (densityExpansion M hE κ D F ∪ K ∪ Z)).ncard := by
let Q := densityExpansion M hE κ D (F ∪ K)
let R := A \ M.closure (densityExpansion M hE κ D F ∪ K ∪ Z)
have hX : densityExpansion M hE κ D F ⊆ Q :=
densityExpansion_mono M hE κ D subset_union_left
have hK : K ⊆ Q := subset_union_right.trans
(densityExpansion_extensive M hE κ D (F ∪ K))
have hcl : M.closure (densityExpansion M hE κ D F ∪ K ∪ Z) ⊆
M.closure (Q ∪ Z) :=
M.closure_subset_closure (union_subset_union (union_subset hX hK) Subset.rfl)
have hcharge : S \ R ⊆ D ∩ (M.closure (Q ∪ Z) \ Q) := by
intro e he
have hs := hS he.1
refine ⟨hs.1.1, hcl ?_, hs.2⟩
by_contra hn
exact he.2 ⟨hs.1.2, hn⟩
have hcount := (Set.ncard_mono hcharge).trans
(densityExpansion_residual_le_generators M hE κ D (F ∪ K) Z)
exact (Set.ncard_le_ncard_sdiff_add_ncard S R).trans (Nat.add_le_add_right hcount _)
lemma safe_sum_le_witness_residual {ι : Type*} (L : Finset ι)
(M : Matroid α) (hE : M.E = univ) (κ : ℕ) (D : Set α)
(F K Z A S : ι → Set α)
(hS : ∀ b ∈ L, S b ⊆ (D ∩ A b) \ densityExpansion M hE κ D (F b ∪ K b)) :
∑ b ∈ L, (S b).ncard ≤ κ * (∑ b ∈ L, (Z b).ncard) +
∑ b ∈ L, (A b \ M.closure (densityExpansion M hE κ D (F b) ∪ K b ∪ Z b)).ncard := by
calc
_ ≤ ∑ b ∈ L, (κ * (Z b).ncard +
(A b \ M.closure (densityExpansion M hE κ D (F b) ∪ K b ∪ Z b)).ncard) :=
Finset.sum_le_sum fun b hb => safe_layer_le_witness_residual M hE κ D
(F b) (K b) (Z b) (A b) (S b) (hS b hb)
_ = _ := by rw [Finset.sum_add_distrib, Finset.mul_sum]
end MatroidProphet
end OAI
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