| 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 | |