| import Mathlib |
| import OAI.Computability.VertexCover.Reduction.CompatibleSingleton |
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| namespace OAI |
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| namespace VertexCover.LabelCover |
| namespace Decoding |
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|
| variable (Φ : LabelCover) {d : ℕ} |
| variable (iU : Fin Φ.u → Φ.Query d) (iV : Fin Φ.v → Φ.Query d) |
| variable (pU : ∀ x, x ∈ (iU x).scopeU) (pV : ∀ y, y ∈ (iV y).scopeV) |
| variable (LU : ∀ x, List (iU x).LocalLabel) (LV : ∀ y, List (iV y).LocalLabel) |
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| def leftAt (ρ : ℕ) (x : Fin Φ.u) : Fin Φ.qU := |
| if h : ρ < (LU x).length then ((LU x)[ρ]).1.1 ⟨x, pU x⟩ else ⟨0, Φ.qU_pos⟩ |
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| def rightAt (τ : ℕ) (y : Fin Φ.v) : Fin Φ.qV := |
| if h : τ < (LV y).length then ((LV y)[τ]).1.2 ⟨y, pV y⟩ else ⟨0, Φ.qV_pos⟩ |
|
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| def PairHit (c : Fin Φ.M) : Prop := |
| ∃ labU ∈ LU (Φ.left c), ∃ labV ∈ LV (Φ.right c), |
| Φ.Compatible {⟨iU (Φ.left c), labU⟩, ⟨iV (Φ.right c), labV⟩} |
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| noncomputable def pairHits : Finset (Fin Φ.M) := by |
| classical |
| exact Finset.univ.filter (PairHit Φ iU iV LU LV) |
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| theorem pairHit_of_selection (c : Fin Φ.M) (I : Finset (Φ.Coordinate d)) |
| (hI : Φ.Compatible I) |
| (labU : (iU (Φ.left c)).LocalLabel) (hlabU : labU ∈ LU (Φ.left c)) |
| (labV : (iV (Φ.right c)).LocalLabel) (hlabV : labV ∈ LV (Φ.right c)) |
| (hlabUI : ⟨iU (Φ.left c), labU⟩ ∈ I) (hlabVI : ⟨iV (Φ.right c), labV⟩ ∈ I) : |
| PairHit Φ iU iV LU LV c := by |
| classical |
| refine ⟨labU, hlabU, labV, hlabV, Φ.compatible_mono hI ?_⟩ |
| intro p hp |
| simp only [Finset.mem_insert, Finset.mem_singleton] at hp |
| rcases hp with rfl | rfl |
| · exact hlabUI |
| · exact hlabVI |
|
|
| theorem pairHit_decodes {ell : ℕ} |
| (hLU : ∀ x, (LU x).length ≤ ell) (hLV : ∀ y, (LV y).length ≤ ell) |
| (c : Fin Φ.M) (hc : PairHit Φ iU iV LU LV c) : |
| ∃ ρ : Fin ell, ∃ τ : Fin ell, |
| Φ.Satisfies (leftAt Φ iU pU LU ρ, rightAt Φ iV pV LV τ) c := by |
| classical |
| obtain ⟨labU, hlabU, labV, hlabV, hcompat⟩ := hc |
| obtain ⟨ρ, hρ, eρ⟩ := List.mem_iff_getElem.mp hlabU |
| obtain ⟨τ, hτ, eτ⟩ := List.mem_iff_getElem.mp hlabV |
| refine ⟨⟨ρ, lt_of_lt_of_le hρ (hLU _)⟩, ⟨τ, lt_of_lt_of_le hτ (hLV _)⟩, ?_⟩ |
| have hs := hcompat.2.2.2 ⟨iU (Φ.left c), labU⟩ (by simp) |
| ⟨iV (Φ.right c), labV⟩ (by simp) c (pU _) (pV _) |
| simpa only [Satisfies, leftAt, rightAt, dite_eq_left hρ, dite_eq_left hτ, eρ, eτ] using hs |
|
|
| include pU pV |
|
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| theorem pairHits_card_le {ell : ℕ} {σ : ℝ} |
| (hval : Φ.value ≤ σ) |
| (hLU : ∀ x, (LU x).length ≤ ell) (hLV : ∀ y, (LV y).length ≤ ell) : |
| ((pairHits Φ iU iV LU LV).card : ℝ) ≤ (ell : ℝ)^2 * σ * Φ.M := by |
| classical |
| let good : (Fin ell × Fin ell) → Finset (Fin Φ.M) := fun a => |
| Finset.univ.filter (Φ.Satisfies (leftAt Φ iU pU LU a.1, rightAt Φ iV pV LV a.2)) |
| have hs : pairHits Φ iU iV LU LV ⊆ Finset.univ.biUnion good := by |
| intro c hc |
| have hh : PairHit Φ iU iV LU LV c := (Finset.mem_filter.mp hc).2 |
| obtain ⟨ρ, τ, ha⟩ := pairHit_decodes Φ iU iV pU pV LU LV hLU hLV c hh |
| exact Finset.mem_biUnion.mpr ⟨(ρ, τ), Finset.mem_univ _, Finset.mem_filter.mpr |
| ⟨Finset.mem_univ _, ha⟩⟩ |
| have hcard := (Finset.card_le_card hs).trans Finset.card_biUnion_le |
| calc |
| ((pairHits Φ iU iV LU LV).card : ℝ) ≤ |
| ((∑ a : Fin ell × Fin ell, (good a).card : ℕ) : ℝ) := Nat.cast_le.mpr hcard |
| _ = ∑ a : Fin ell × Fin ell, ((good a).card : ℝ) := by simp |
| _ ≤ ∑ _a : Fin ell × Fin ell, σ * Φ.M := by |
| apply Finset.sum_le_sum |
| intro a _ |
| exact Φ.count_le_of_value_le hval _ |
| _ = (ell : ℝ)^2 * σ * Φ.M := by |
| simp [Fintype.card_prod, pow_two] |
| ring |
|
|
| theorem pairHits_probability_le {ell : ℕ} {σ : ℝ} |
| (hval : Φ.value ≤ σ) |
| (hLU : ∀ x, (LU x).length ≤ ell) (hLV : ∀ y, (LV y).length ≤ ell) : |
| ((pairHits Φ iU iV LU LV).card : ℝ) / Φ.M ≤ (ell : ℝ)^2 * σ := by |
| apply (div_le_iff₀ (Nat.cast_pos.mpr Φ.M_pos)).mpr |
| exact pairHits_card_le Φ iU iV pU pV LU LV hval hLU hLV |
|
|
| end Decoding |
| end VertexCover.LabelCover |
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| end |
| end |
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| end |
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| end OAI |
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