openAI-math-dataset / lean /OAI /Computability /VertexCover /Reduction /HonestLargeSumIndependent.lean
| import Mathlib | |
| import OAI.Computability.VertexCover.Reduction.IncrementNorm | |
| namespace OAI | |
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| namespace VertexCover.LabelCover | |
| def gridRational (d : ℕ) (v : Fin (2*d+1)) : ℚ := | |
| 2 * ((v.val : ℚ) - d) / (2*d+1) | |
| abbrev Vertex (Φ : LabelCover) (d : ℕ) := | |
| Φ.Seeds d × (Fin d → Fin (Φ.WeightDimension d) → Fin (2*d+1)) | |
| noncomputable def sumVector (Φ : LabelCover) {d : ℕ} (v : Φ.Vertex d) : | |
| Φ.Coordinate d → ℝ := | |
| ∑ j, Φ.increment (Φ.query v.1 j) (fun k => (gridRational d (v.2 j k) : ℝ)) | |
| noncomputable def graph (Φ : LabelCover) (d : ℕ) (t : ℝ) : SimpleGraph (Φ.Vertex d) where | |
| Adj v w := v ≠ w ∧ Φ.compatibilityNorm (Φ.sumVector v + Φ.sumVector w) ≤ 2*t | |
| symm := ⟨by intro v w h; exact ⟨h.1.symm, by simpa only [add_comm] using h.2⟩⟩ | |
| loopless := ⟨by intro v h; exact h.1 rfl⟩ | |
| theorem low_norm_adjacent (Φ : LabelCover) {d : ℕ} {t : ℝ} {v w : Φ.Vertex d} | |
| (hne : v ≠ w) (hv : Φ.compatibilityNorm (Φ.sumVector v) ≤ t) | |
| (hw : Φ.compatibilityNorm (Φ.sumVector w) ≤ t) : (Φ.graph d t).Adj v w := by | |
| refine ⟨hne, (Φ.norm_add_le _ _).trans ?_⟩ | |
| linarith | |
| theorem low_norm_subsingleton (Φ : LabelCover) {d : ℕ} {t : ℝ} | |
| (A : Set (Φ.Vertex d)) (hA : (Φ.graph d t).IsIndepSet A) : | |
| {v ∈ A | Φ.compatibilityNorm (Φ.sumVector v) ≤ t}.Subsingleton := by | |
| intro v hv w hw | |
| by_contra hne | |
| exact ((SimpleGraph.isIndepSet_iff _).mp hA hv.1 hw.1 hne) | |
| (Φ.low_norm_adjacent hne hv.2 hw.2) | |
| theorem honestSelection_form_increment (Φ : LabelCover) {d : ℕ} | |
| (A : Φ.Labeling) (hA : ∀ c, Φ.Satisfies A c) (i : Φ.Query d) | |
| (s : Fin (Φ.WeightDimension d) → ℝ) : | |
| ∑ p ∈ Φ.honestSelection A hA d, Φ.increment i s p = | |
| s (i.slot (Φ.localOfSatisfying A hA i)) := by | |
| classical | |
| unfold honestSelection | |
| rw [Finset.sum_image] | |
| · have hz : ∀ j : Φ.Query d, Φ.increment i s ⟨j, Φ.localOfSatisfying A hA j⟩ = | |
| if j = i then s (i.slot (Φ.localOfSatisfying A hA i)) else 0 := by | |
| intro j | |
| split_ifs with h | |
| · subst j; exact Φ.increment_same _ _ _ | |
| · exact Φ.increment_other _ _ _ h | |
| simp only [hz] | |
| simp | |
| · intro i _ j _ h | |
| exact congrArg Sigma.fst h | |
| noncomputable def honestSum (Φ : LabelCover) {d : ℕ} (A : Φ.Labeling) | |
| (hA : ∀ c, Φ.Satisfies A c) (v : Φ.Vertex d) : ℝ := | |
| ∑ j, (gridRational d (v.2 j ((Φ.query v.1 j).slot | |
| (Φ.localOfSatisfying A hA (Φ.query v.1 j)))) : ℝ) | |
| theorem honestSelection_form_sumVector (Φ : LabelCover) {d : ℕ} | |
| (A : Φ.Labeling) (hA : ∀ c, Φ.Satisfies A c) (v : Φ.Vertex d) : | |
| ∑ p ∈ Φ.honestSelection A hA d, Φ.sumVector v p = Φ.honestSum A hA v := by | |
| classical | |
| simp only [sumVector, Finset.sum_apply] | |
| rw [Finset.sum_comm] | |
| simp only [honestSelection_form_increment, honestSum] | |
| theorem honest_large_sum_independent (Φ : LabelCover) {d : ℕ} (t : ℝ) | |
| (A : Φ.Labeling) (hA : ∀ c, Φ.Satisfies A c) : | |
| (Φ.graph d t).IsIndepSet {v | t < Φ.honestSum A hA v} := by | |
| rw [SimpleGraph.isIndepSet_iff] | |
| intro v hv w hw _ hadj | |
| have hform := Φ.form_le_norm (Φ.sumVector v + Φ.sumVector w) | |
| (Φ.honestSelection A hA d) (Φ.honestSelection_compatible A hA d) | |
| simp only [Pi.add_apply, Finset.sum_add_distrib, honestSelection_form_sumVector] at hform | |
| have hsum := (le_abs_self (Φ.honestSum A hA v + Φ.honestSum A hA w)).trans | |
| (hform.trans hadj.2) | |
| change t < Φ.honestSum A hA v at hv | |
| change t < Φ.honestSum A hA w at hw | |
| linarith | |
| end VertexCover.LabelCover | |
| end | |
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| end | |
| end OAI | |