| import Mathlib |
| import OAI.Computability.VertexCover.Reduction.HonestSelectionCompatible |
| import OAI.Computability.VertexCover.Reduction.NormAttainedSupport |
|
|
| namespace OAI |
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| namespace VertexCover.LabelCover |
| namespace Query |
|
|
| noncomputable abbrev localLabelLinearOrder {Φ : LabelCover} {d : ℕ} |
| (i : Φ.Query d) : LinearOrder i.LocalLabel := |
| LinearOrder.lift' |
| (fun a : i.LocalLabel => toLex (toLex a.1.1, toLex a.1.2)) |
| (fun _ _ h => Subtype.ext h) |
|
|
| noncomputable def labels {Φ : LabelCover} {d : ℕ} (i : Φ.Query d) : List i.LocalLabel := by |
| letI : LinearOrder i.LocalLabel := localLabelLinearOrder i |
| exact (Finset.univ : Finset i.LocalLabel).sort (localLabelLinearOrder i).le |
|
|
| @[simp] theorem mem_labels {Φ : LabelCover} {d : ℕ} (i : Φ.Query d) (a : i.LocalLabel) : |
| a ∈ i.labels := by |
| simp [labels] |
|
|
| @[simp] theorem labels_length {Φ : LabelCover} {d : ℕ} (i : Φ.Query d) : |
| i.labels.length = Fintype.card i.LocalLabel := by |
| simp [labels] |
|
|
| noncomputable def slot {Φ : LabelCover} {d : ℕ} (i : Φ.Query d) (a : i.LocalLabel) : |
| Fin ((max Φ.qU Φ.qV)^(d - 1)) := |
| ⟨i.labels.idxOf a, lt_of_lt_of_le |
| (by simpa only [labels_length] using List.idxOf_lt_length_iff.mpr (i.mem_labels a)) |
| i.localLabel_card_le⟩ |
|
|
| theorem slot_injective {Φ : LabelCover} {d : ℕ} (i : Φ.Query d) : |
| Function.Injective i.slot := by |
| intro a b h |
| exact (List.idxOf_inj (i.mem_labels a)).mp (congrArg Fin.val h) |
|
|
| end Query |
|
|
| abbrev WeightDimension (Φ : LabelCover) (d : ℕ) := (max Φ.qU Φ.qV)^(d - 1) |
|
|
| noncomputable def increment (Φ : LabelCover) {d : ℕ} (i : Φ.Query d) |
| (s : Fin (Φ.WeightDimension d) → ℝ) : Φ.Coordinate d → ℝ := |
| fun p => (Pi.single (M := fun j : Φ.Query d => j.LocalLabel → ℝ) |
| i (fun a => s (i.slot a)) p.1) p.2 |
|
|
| @[simp] theorem increment_same (Φ : LabelCover) {d : ℕ} (i : Φ.Query d) |
| (s : Fin (Φ.WeightDimension d) → ℝ) (a : i.LocalLabel) : |
| Φ.increment i s ⟨i, a⟩ = s (i.slot a) := by |
| simp [increment] |
|
|
| theorem increment_other (Φ : LabelCover) {d : ℕ} (i : Φ.Query d) |
| (s : Fin (Φ.WeightDimension d) → ℝ) (p : Φ.Coordinate d) (h : p.1 ≠ i) : |
| Φ.increment i s p = 0 := by |
| simp [increment, Pi.single_eq_of_ne h] |
|
|
| theorem increment_add (Φ : LabelCover) {d : ℕ} (i : Φ.Query d) |
| (s t : Fin (Φ.WeightDimension d) → ℝ) : |
| Φ.increment i (s + t) = Φ.increment i s + Φ.increment i t := by |
| funext p |
| rcases p with ⟨j, a⟩ |
| by_cases h : j = i |
| · subst j |
| simp |
| · simp [Φ.increment_other i _ ⟨j, a⟩ h] |
|
|
| theorem increment_mul (Φ : LabelCover) {d : ℕ} (i : Φ.Query d) (a : ℝ) |
| (s : Fin (Φ.WeightDimension d) → ℝ) : |
| Φ.increment i (fun k => a * s k) = fun p => a * Φ.increment i s p := by |
| funext p |
| rcases p with ⟨j, b⟩ |
| by_cases h : j = i |
| · subst j |
| simp |
| · simp [Φ.increment_other i _ ⟨j, b⟩ h] |
|
|
| theorem increment_norm_le (Φ : LabelCover) {d : ℕ} (i : Φ.Query d) |
| (s : Fin (Φ.WeightDimension d) → ℝ) {B : ℝ} (hB : 0 ≤ B) |
| (hs : ∀ k, |s k| ≤ B) : Φ.compatibilityNorm (Φ.increment i s) ≤ B := by |
| classical |
| apply (Φ.norm_le_iff _ _).mpr |
| intro I hI |
| by_cases hex : ∃ p ∈ I, p.1 = i |
| · obtain ⟨p, hp, hip⟩ := hex |
| have heq : (∑ q ∈ I, Φ.increment i s q) = Φ.increment i s p := by |
| apply Finset.sum_eq_single p |
| · intro q hq hqp |
| apply Φ.increment_other |
| intro hqi |
| exact hqp (hI.1 q hq p hp (hqi.trans hip.symm)) |
| · intro hn |
| exact (hn hp).elim |
| rw [heq] |
| rcases p with ⟨j, a⟩ |
| change j = i at hip |
| subst j |
| simpa using hs (i.slot a) |
| · have hz : ∀ p ∈ I, Φ.increment i s p = 0 := by |
| intro p hp |
| exact Φ.increment_other i s p (fun h => hex ⟨p, hp, h⟩) |
| simpa [Finset.sum_eq_zero hz] using hB |
|
|
| end VertexCover.LabelCover |
|
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|
| 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 |
|
|
