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import OAI.Computability.VertexCover.Reduction.NormAttainedSupport
import OAI.Computability.VertexCover.Reduction.PairSeedFiberCard
namespace OAI
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namespace VertexCover.LabelCover
def ListHit (Φ : LabelCover) {d : ℕ} (L : Φ.PrivateLists d)
(seed : Φ.Seeds d) (I : Finset (Φ.Coordinate d)) (j : Fin d) : Prop :=
∃ a ∈ L (Φ.query seed j), (⟨Φ.query seed j, a⟩ : Φ.Coordinate d) ∈ I
noncomputable def hitPositions (Φ : LabelCover) {d : ℕ} (L : Φ.PrivateLists d)
(seed : Φ.Seeds d) (I : Finset (Φ.Coordinate d)) (J : Finset (Fin d)) : Finset (Fin d) := by
classical
exact J.filter (Φ.ListHit L seed I)
noncomputable def pairEvents (Φ : LabelCover) {d : ℕ} (L : Φ.PrivateLists d)
(seed : Φ.Seeds d) (J : Finset (Fin d)) : Finset (PositionPair d) := by
classical
exact Finset.univ.filter (fun e => e.1.1 ∈ J ∧ e.1.2 ∈ J ∧ Φ.PairSeedHit L seed e)
theorem pairSeedHit_of_hits (Φ : LabelCover) {d : ℕ} (L : Φ.PrivateLists d)
(seed : Φ.Seeds d) {I : Finset (Φ.Coordinate d)} (hI : Φ.Compatible I)
(e : PositionPair d) (hj : Φ.ListHit L seed I e.1.1)
(hk : Φ.ListHit L seed I e.1.2) : Φ.PairSeedHit L seed e := by
classical
obtain ⟨a, haL, haI⟩ := hj
obtain ⟨b, hbL, hbI⟩ := hk
refine ⟨a, haL, b, hbL, Φ.compatible_mono hI ?_⟩
intro p hp
simp only [Finset.mem_insert, Finset.mem_singleton] at hp
rcases hp with rfl | rfl
· exact haI
· exact hbI
theorem hitPositions_card_le (Φ : LabelCover) {d : ℕ} (L : Φ.PrivateLists d)
(seed : Φ.Seeds d) (I : Finset (Φ.Coordinate d)) (hI : Φ.Compatible I)
(J : Finset (Fin d)) :
(Φ.hitPositions L seed I J).card ≤ 1 + (Φ.pairEvents L seed J).card := by
classical
let A := Φ.hitPositions L seed I J
by_cases hA : A.Nonempty
· let j := A.min' hA
have hj : j ∈ A := Finset.min'_mem _ _
have keep : ∀ k : ↥(A.erase j), j < k.1 := by
intro k
have hk := Finset.mem_erase.mp k.2
exact lt_of_le_of_ne (Finset.min'_le A k.1 hk.2) (Ne.symm hk.1)
let inject : ↥(A.erase j) → ↥(Φ.pairEvents L seed J) := fun k =>
⟨⟨(j, k.1), keep k⟩, by
have hmj := Finset.mem_filter.mp hj
have hmk := Finset.mem_filter.mp (Finset.mem_erase.mp k.2).2
exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, hmj.1, hmk.1,
Φ.pairSeedHit_of_hits L seed hI _ hmj.2 hmk.2⟩⟩
have hinj : Function.Injective inject := by
intro k l h
apply Subtype.ext
exact congrArg (fun e : ↥(Φ.pairEvents L seed J) => e.1.1.2) h
have hcard := Fintype.card_le_of_injective inject hinj
simp only [Fintype.card_coe] at hcard
have herase := Finset.card_erase_add_one hj
change A.card ≤ _
omega
· have he : A = ∅ := Finset.not_nonempty_iff_eq_empty.mp hA
change A.card ≤ _
simp [he]
noncomputable def listHitMaximum (Φ : LabelCover) {d : ℕ} (L : Φ.PrivateLists d)
(seed : Φ.Seeds d) (J : Finset (Fin d)) : ℕ :=
(Φ.compatibleSelections d).sup (fun I => (Φ.hitPositions L seed I J).card)
theorem listHitMaximum_le (Φ : LabelCover) {d : ℕ} (L : Φ.PrivateLists d)
(seed : Φ.Seeds d) (J : Finset (Fin d)) :
Φ.listHitMaximum L seed J ≤ 1 + (Φ.pairEvents L seed J).card := by
classical
apply Finset.sup_le
intro I hI
exact Φ.hitPositions_card_le L seed I ((Φ.mem_compatibleSelections I).mp hI) J
end VertexCover.LabelCover
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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