| import Mathlib.Data.Finset.Powerset | |
| import Mathlib.Algebra.Order.Monoid.Unbundled.Pow | |
| namespace OAI | |
| namespace MatroidProphet | |
| namespace Pivots | |
| variable {α : Type*} [DecidableEq α] | |
| lemma card_family_with_fixed_core (family : Finset (Finset α)) | |
| (core extra : Finset α) | |
| (hcore : ∀ J ∈ family, core ⊆ J) | |
| (hcover : ∀ J ∈ family, J ⊆ core ∪ extra) : | |
| family.card ≤ 2 ^ extra.card := by | |
| rw [← Finset.card_powerset extra] | |
| apply Finset.card_le_card_of_injOn (fun J => J \ core) | |
| · intro J hJ | |
| apply Finset.mem_powerset.mpr | |
| intro x hx | |
| have hcov := hcover J hJ (Finset.mem_sdiff.mp hx).1 | |
| exact (Finset.mem_union.mp hcov).resolve_left (Finset.mem_sdiff.mp hx).2 | |
| · intro J hJ K hK heq | |
| apply Finset.ext | |
| intro x | |
| by_cases hx : x ∈ core | |
| · exact iff_of_true (hcore J hJ hx) (hcore K hK hx) | |
| · have hmem := congrArg (fun L : Finset α => x ∈ L) heq | |
| simpa [Finset.mem_sdiff, hx] using hmem | |
| lemma card_retained_family (family : Finset (Finset α)) | |
| (old movable omittable : Finset α) (s : ℕ) | |
| (hm : movable.card ≤ s) (ho : omittable.card ≤ s) | |
| (hcore : ∀ J ∈ family, old \ omittable ⊆ J) | |
| (hcover : ∀ J ∈ family, J ⊆ old ∪ movable) : | |
| family.card ≤ 4 ^ s := by | |
| have h := card_family_with_fixed_core family (old \ omittable) | |
| (omittable ∪ movable) hcore (by | |
| intro J hJ x hx | |
| rcases Finset.mem_union.mp (hcover J hJ hx) with hxold | hxmov | |
| · by_cases hxo : x ∈ omittable | |
| · exact Finset.mem_union_right _ (Finset.mem_union_left _ hxo) | |
| · exact Finset.mem_union_left _ (Finset.mem_sdiff.mpr ⟨hxold, hxo⟩) | |
| · exact Finset.mem_union_right _ (Finset.mem_union_right _ hxmov)) | |
| have hcard : (omittable ∪ movable).card ≤ 2 * s := | |
| (Finset.card_union_le _ _).trans (by omega) | |
| calc | |
| family.card ≤ 2 ^ (omittable ∪ movable).card := h | |
| _ ≤ 2 ^ (2 * s) := Nat.pow_le_pow_right (by decide) hcard | |
| _ = 4 ^ s := by rw [pow_mul]; rfl | |
| end Pivots | |
| end MatroidProphet | |
| end OAI | |