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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 | import OAI.Probability.MatroidProphet.Density.Birth
import OAI.Probability.MatroidProphet.LayerRank
namespace OAI
namespace MatroidProphet
open Set
variable {α : Type*} [Fintype α]
omit [Fintype α] in
lemma natRank_eq_ncard_of_indep (M : Matroid α) {I : Set α} (hI : M.Indep I) :
natRank M I = I.ncard := by
simp only [natRank, hI.eRk_eq_encard, Set.ncard_def]
lemma baseline_density_rank (M : Matroid α) (hE : M.E = Set.univ) (κ : ℕ) (D : Set α) :
κ * natRank M (densityExpansion M hE κ D (M.closure ∅)) ≤ D.ncard := by
let Q := densityExpansion M hE κ D (M.closure ∅)
have hmax := (densityExpansion_spec M hE κ D (M.closure ∅)).1.2.2
(M.closure ∅) (Matroid.isFlat_closure (M := M) ∅) Set.Subset.rfl
have hcount := Set.ncard_mono (show D ∩ Q ⊆ D from inter_subset_left)
have hnn : (0 : ℤ) ≤ (D ∩ M.closure ∅).ncard := by exact_mod_cast Nat.zero_le _
have hcount' : ((D ∩ Q).ncard : ℤ) ≤ D.ncard := by exact_mod_cast hcount
have hmax' : (κ : ℤ) * natRank M Q ≤ D.ncard := by
dsimp only [densityObjective] at hmax
rw [natRank_closure, natRank_empty] at hmax
change _ ≤ (D ∩ Q).ncard - (κ : ℤ) * natRank M Q at hmax
omega
exact_mod_cast hmax'
lemma baseline_independent_count (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D I : Set α) (hI : M.Indep I) :
κ * (I ∩ densityExpansion M hE κ D (M.closure ∅)).ncard ≤ D.ncard := by
have hi := natRank_eq_ncard_of_indep M
(hI.subset (show I ∩ densityExpansion M hE κ D (M.closure ∅) ⊆ I from inter_subset_left))
have hr := natRank_mono M (show I ∩ densityExpansion M hE κ D (M.closure ∅) ⊆
densityExpansion M hE κ D (M.closure ∅) from inter_subset_right)
rw [hi] at hr
exact (Nat.mul_le_mul_left κ hr).trans (baseline_density_rank M hE κ D)
lemma baseline_birth_iff (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (d : α) (hd : d ∉ M.closure ∅) :
nominalBirth M hE κ D C h d = activation h ↔
d ∈ densityExpansion M hE κ (D h) (M.closure ∅) := by
rw [← nominalPath_baseline M hE κ D C h,
mem_nominal_iff_birth_le M hE κ D C h d hd]
exact ⟨fun he => he.le, fun he => le_antisymm he (nominalBirth_bounds M hE κ D C h d).1⟩
end MatroidProphet
end OAI
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