| import Mathlib |
| 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 |
| |
