openAI-math-dataset / lean /OAI /Computability /VertexCover /Reduction /SeparatorContinuousSumOdd.lean
| import Mathlib | |
| import OAI.Computability.VertexCover.Reduction.HighVectorsSeparated | |
| namespace OAI | |
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| namespace VertexCover.LabelCover | |
| theorem norm_sum_le (Φ : LabelCover) {d : ℕ} {ι : Type*} | |
| (T : Finset ι) (f : ι → Φ.Coordinate d → ℝ) : | |
| Φ.compatibilityNorm (∑ i ∈ T, f i) ≤ ∑ i ∈ T, Φ.compatibilityNorm (f i) := by | |
| classical | |
| induction T using Finset.induction_on with | |
| | empty => | |
| simp only [Finset.sum_empty] | |
| exact (Φ.norm_zero d).le | |
| | @insert i T hi ih => | |
| simp only [Finset.sum_insert hi] | |
| exact (Φ.norm_add_le _ _).trans (add_le_add_right ih _) | |
| theorem increment_sub (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 | |
| have hn : Φ.increment i (-t) = -Φ.increment i t := by | |
| ext p | |
| change Φ.increment i (fun k => -t k) p = -Φ.increment i t p | |
| have h := congrFun (Φ.increment_mul i (-1) t) p | |
| simpa only [neg_one_mul] using h | |
| rw [sub_eq_add_neg, Φ.increment_add, hn, sub_eq_add_neg] | |
| noncomputable def continuousSum (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) | |
| (s : Fin d → Fin (Φ.WeightDimension d) → ℝ) : Φ.Coordinate d → ℝ := | |
| ∑ j, Φ.increment (Φ.query seed j) (s j) | |
| theorem sumVector_eq_continuousSum (Φ : LabelCover) {d : ℕ} (v : Φ.Vertex d) : | |
| Φ.sumVector v = Φ.continuousSum v.1 (fun j k => (gridRational d (v.2 j k) : ℝ)) := rfl | |
| theorem continuousSum_neg (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) | |
| (s : Fin d → Fin (Φ.WeightDimension d) → ℝ) : | |
| Φ.continuousSum seed (-s) = -Φ.continuousSum seed s := by | |
| unfold continuousSum | |
| rw [← Finset.sum_neg_distrib] | |
| apply Finset.sum_congr rfl | |
| intro j _ | |
| ext p | |
| change Φ.increment (Φ.query seed j) (fun k => -s j k) p = | |
| -Φ.increment (Φ.query seed j) (s j) p | |
| have h := congrFun (Φ.increment_mul (Φ.query seed j) (-1) (s j)) p | |
| simpa only [neg_one_mul] using h | |
| theorem continuousSum_sub_same_seed (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) | |
| (s t : Fin d → Fin (Φ.WeightDimension d) → ℝ) : | |
| Φ.continuousSum seed s - Φ.continuousSum seed t = Φ.continuousSum seed (s-t) := by | |
| simp only [continuousSum, ← Finset.sum_sub_distrib, Pi.sub_apply, increment_sub] | |
| theorem continuousSum_close (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) | |
| (s t : Fin d → Fin (Φ.WeightDimension d) → ℝ) {b : ℝ} (hb : 0 ≤ b) | |
| (hclose : ∀ j k, |s j k - t j k| ≤ b) : | |
| Φ.compatibilityNorm (Φ.continuousSum seed s - Φ.continuousSum seed t) ≤ d*b := by | |
| rw [Φ.continuousSum_sub_same_seed] | |
| apply (Φ.norm_sum_le _ _).trans | |
| calc | |
| (∑ j, Φ.compatibilityNorm (Φ.increment (Φ.query seed j) ((s-t) j))) ≤ | |
| ∑ _j : Fin d, b := by | |
| apply Finset.sum_le_sum | |
| intro j _ | |
| exact Φ.increment_norm_le _ _ hb (hclose j) | |
| _ = d*b := by simp | |
| def InWeightCube (Φ : LabelCover) {d : ℕ} | |
| (s : Fin d → Fin (Φ.WeightDimension d) → ℝ) : Prop := | |
| ∀ j k, |s j k| ≤ 1 | |
| theorem continuousSum_change_blocks (Φ : LabelCover) {d : ℕ} | |
| (seed seed' : Φ.Seeds d) (s t : Fin d → Fin (Φ.WeightDimension d) → ℝ) | |
| (hs : Φ.InWeightCube s) (ht : Φ.InWeightCube t) (B : Finset (Fin d)) | |
| (heq : ∀ j ∉ B, Φ.query seed j = Φ.query seed' j ∧ s j = t j) : | |
| Φ.compatibilityNorm (Φ.continuousSum seed s - Φ.continuousSum seed' t) ≤ 2*B.card := by | |
| classical | |
| unfold continuousSum | |
| rw [← Finset.sum_sub_distrib] | |
| apply (Φ.norm_sum_le _ _).trans | |
| calc | |
| (∑ j, Φ.compatibilityNorm | |
| (Φ.increment (Φ.query seed j) (s j) - Φ.increment (Φ.query seed' j) (t j))) ≤ | |
| ∑ j : Fin d, if j ∈ B then (2 : ℝ) else 0 := by | |
| apply Finset.sum_le_sum | |
| intro j _ | |
| by_cases hj : j ∈ B | |
| · rw [ite_eq_left hj] | |
| have hb := Φ.norm_sub_triangle (Φ.increment (Φ.query seed j) (s j)) 0 | |
| (Φ.increment (Φ.query seed' j) (t j)) | |
| have hh1 := Φ.increment_norm_le (Φ.query seed j) (s j) (by norm_num : (0:ℝ) ≤ 1) (hs j) | |
| have hh2 := Φ.increment_norm_le (Φ.query seed' j) (t j) (by norm_num : (0:ℝ) ≤ 1) (ht j) | |
| simp only [sub_zero, zero_sub, Φ.norm_neg] at hb | |
| linarith | |
| · obtain ⟨hq, hw⟩ := heq j hj | |
| rw [hq, hw, sub_self, ite_eq_right hj] | |
| exact (Φ.norm_zero d).le | |
| _ = 2*B.card := by simp [Finset.sum_ite_mem, mul_comm] | |
| theorem separator_continuousSum_odd (Φ : LabelCover) {d : ℕ} | |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (seed : Φ.Seeds d) | |
| (s : Fin d → Fin (Φ.WeightDimension d) → ℝ) : | |
| Φ.separator A hA (Φ.continuousSum seed (-s)) = | |
| -Φ.separator A hA (Φ.continuousSum seed s) := by | |
| rw [Φ.continuousSum_neg, Φ.separator_odd] | |
| end VertexCover.LabelCover | |
| end | |
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| end | |
| end OAI | |