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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 | 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
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