| import OAI.Combinatorics.CliqueFree.Numerical |
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| namespace OAI |
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| noncomputable section |
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| open scoped BigOperators |
| open Finset |
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| attribute [local instance] Classical.propDecidable |
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| namespace CliqueFreeIndependence.WeightedGraph |
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| universe u v |
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| variable {V : Type u} [Fintype V] |
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| def mass (w : V → ℝ) (A : Finset V) : ℝ := ∑ v ∈ A, w v |
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| noncomputable def neighbors (G : SimpleGraph V) (v : V) : Finset V := by |
| classical |
| exact univ.filter (G.Adj v) |
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| @[simp] lemma mem_neighbors (G : SimpleGraph V) (u v : V) : |
| v ∈ neighbors G u ↔ G.Adj u v := by |
| classical |
| simp [neighbors] |
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| noncomputable def crossMass (G : SimpleGraph V) (w : V → ℝ) (A B : Finset V) : ℝ := by |
| classical |
| exact ∑ u ∈ A, ∑ v ∈ B, if G.Adj u v then w u * w v else 0 |
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| / |
| noncomputable def edgeMass (G : SimpleGraph V) (w : V → ℝ) : ℝ := |
| crossMass G w univ univ / 2 |
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| noncomputable def neighborMass (G : SimpleGraph V) (w : V → ℝ) (u : V) : ℝ := |
| mass w (neighbors G u) |
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| noncomputable def commonMass (G : SimpleGraph V) (w : V → ℝ) (u v : V) : ℝ := by |
| classical |
| exact mass w (neighbors G u ∩ neighbors G v) |
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| / |
| noncomputable def triangleAt (G : SimpleGraph V) (w : V → ℝ) (u : V) : ℝ := |
| ∑ v ∈ neighbors G u, w v * commonMass G w u v |
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| / |
| noncomputable def triangleMass (G : SimpleGraph V) (w : V → ℝ) : ℝ := |
| (∑ u, w u * triangleAt G w u) / 6 |
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| / |
| noncomputable def growth (x s : ℝ) : ℝ := s * (x + max 0 (Real.log (1 / s))) |
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| / |
| def CrossBound (G : SimpleGraph V) (w : V → ℝ) (C : ℝ) : Prop := |
| ∀ (x : ℝ), 1 ≤ x → ∀ (A B : Finset V), |
| mass w A ≤ Real.exp x → mass w B ≤ Real.exp x → |
| crossMass G w A B ≤ C * growth x (mass w A) |
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| omit [Fintype V] in |
| lemma mass_nonneg {w : V → ℝ} (hw : ∀ v, 0 ≤ w v) (A : Finset V) : |
| 0 ≤ mass w A := sum_nonneg fun v _ ↦ hw v |
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| omit [Fintype V] in |
| lemma mass_mono {w : V → ℝ} (hw : ∀ v, 0 ≤ w v) {A B : Finset V} (hAB : A ⊆ B) : |
| mass w A ≤ mass w B := sum_le_sum_of_subset_of_nonneg hAB (fun v _ _ ↦ hw v) |
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| omit [Fintype V] in |
| lemma crossMass_symm (G : SimpleGraph V) (w : V → ℝ) (A B : Finset V) : |
| crossMass G w A B = crossMass G w B A := by |
| classical |
| unfold crossMass |
| rw [sum_comm] |
| simp_rw [G.adj_comm, mul_comm] |
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| omit [Fintype V] in |
| lemma crossMass_nonneg (G : SimpleGraph V) {w : V → ℝ} (hw : ∀ v, 0 ≤ w v) |
| (A B : Finset V) : 0 ≤ crossMass G w A B := by |
| classical |
| exact sum_nonneg fun u _ ↦ sum_nonneg fun v _ ↦ by |
| split_ifs |
| · exact mul_nonneg (hw u) (hw v) |
| · exact le_rfl |
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| omit [Fintype V] in |
| lemma crossMass_le_product (G : SimpleGraph V) {w : V → ℝ} (hw : ∀ v, 0 ≤ w v) |
| (A B : Finset V) : crossMass G w A B ≤ mass w A * mass w B := by |
| classical |
| unfold crossMass mass |
| rw [sum_mul_sum] |
| exact sum_le_sum fun u _ ↦ sum_le_sum fun v _ ↦ by |
| split_ifs |
| · exact le_rfl |
| · exact mul_nonneg (hw u) (hw v) |
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| lemma crossMass_factor (G : SimpleGraph V) (w : V → ℝ) (A B : Finset V) : |
| crossMass G w A B = ∑ u ∈ A, w u * mass w (B ∩ neighbors G u) := by |
| classical |
| unfold crossMass mass |
| apply sum_congr rfl |
| intro u _ |
| rw [← filter_mem_eq_inter] |
| simp only [sum_filter, mem_neighbors, mul_sum, mul_ite, mul_zero] |
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| lemma commonMass_comm (G : SimpleGraph V) (w : V → ℝ) (u v : V) : |
| commonMass G w u v = commonMass G w v u := by |
| classical |
| simp [commonMass, inter_comm] |
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| lemma commonMass_nonneg (G : SimpleGraph V) {w : V → ℝ} (hw : ∀ v, 0 ≤ w v) (u v : V) : |
| 0 ≤ commonMass G w u v := mass_nonneg hw _ |
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| lemma neighborMass_nonneg (G : SimpleGraph V) {w : V → ℝ} (hw : ∀ v, 0 ≤ w v) (u : V) : |
| 0 ≤ neighborMass G w u := mass_nonneg hw _ |
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| lemma commonMass_le_neighborMass (G : SimpleGraph V) {w : V → ℝ} (hw : ∀ v, 0 ≤ w v) |
| (u v : V) : commonMass G w u v ≤ neighborMass G w u := by |
| classical |
| exact mass_mono hw inter_subset_left |
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| lemma edgeMass_nonneg (G : SimpleGraph V) {w : V → ℝ} (hw : ∀ v, 0 ≤ w v) : |
| 0 ≤ edgeMass G w := div_nonneg (crossMass_nonneg G hw _ _) (by norm_num) |
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| lemma triangleAt_nonneg (G : SimpleGraph V) {w : V → ℝ} (hw : ∀ v, 0 ≤ w v) (u : V) : |
| 0 ≤ triangleAt G w u := |
| sum_nonneg fun v _ ↦ mul_nonneg (hw v) (commonMass_nonneg G hw u v) |
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| lemma triangleMass_nonneg (G : SimpleGraph V) {w : V → ℝ} (hw : ∀ v, 0 ≤ w v) : |
| 0 ≤ triangleMass G w := div_nonneg |
| (sum_nonneg fun u _ ↦ mul_nonneg (hw u) (triangleAt_nonneg G hw u)) (by norm_num) |
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| lemma edgeMass_eq_sum (G : SimpleGraph V) (w : V → ℝ) : |
| 2 * edgeMass G w = ∑ u, w u * neighborMass G w u := by |
| classical |
| rw [edgeMass, crossMass_factor] |
| simp only [univ_inter, neighborMass] |
| ring |
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| lemma triangleMass_eq_sum (G : SimpleGraph V) (w : V → ℝ) : |
| 6 * triangleMass G w = ∑ u, w u * triangleAt G w u := by |
| unfold triangleMass |
| ring |
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| lemma triangleAt_le (G : SimpleGraph V) {w : V → ℝ} (hw : ∀ v, 0 ≤ w v) |
| {L : ℝ} (hL : ∀ u, neighborMass G w u ≤ L) (u : V) : |
| triangleAt G w u ≤ L * neighborMass G w u := by |
| calc |
| _ ≤ ∑ v ∈ neighbors G u, w v * L := sum_le_sum fun v _ ↦ |
| mul_le_mul_of_nonneg_left ((commonMass_le_neighborMass G hw u v).trans (hL u)) (hw v) |
| _ = _ := by rw [← sum_mul]; exact mul_comm _ _ |
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| lemma triangleMass_le_of_neighborMass (G : SimpleGraph V) {w : V → ℝ} |
| (hw : ∀ v, 0 ≤ w v) {L : ℝ} (hL : ∀ u, neighborMass G w u ≤ L) : |
| triangleMass G w ≤ L / 3 * edgeMass G w := by |
| have h : 6 * triangleMass G w ≤ L * (2 * edgeMass G w) := by |
| rw [triangleMass_eq_sum, edgeMass_eq_sum, mul_sum] |
| exact sum_le_sum fun u _ ↦ by |
| have := mul_le_mul_of_nonneg_left (triangleAt_le G hw hL u) (hw u) |
| nlinarith |
| linarith |
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| lemma neighborMass_ge_of_commonMass (G : SimpleGraph V) {w : V → ℝ} |
| (hw : ∀ v, 0 ≤ w v) {τ : ℝ} (hc : ∀ u v, G.Adj u v → τ ≤ commonMass G w u v) |
| {u : V} (hu : (neighbors G u).Nonempty) : τ ≤ neighborMass G w u := by |
| rcases hu with ⟨v, hv⟩ |
| exact (hc u v (mem_neighbors G u v |>.1 hv)).trans (commonMass_le_neighborMass G hw u v) |
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| lemma triangleAt_ge_of_commonMass (G : SimpleGraph V) {w : V → ℝ} |
| (hw : ∀ v, 0 ≤ w v) {τ : ℝ} (hc : ∀ u v, G.Adj u v → τ ≤ commonMass G w u v) |
| (u : V) : τ * neighborMass G w u ≤ triangleAt G w u := by |
| calc |
| _ = ∑ v ∈ neighbors G u, w v * τ := by rw [← sum_mul]; exact mul_comm _ _ |
| _ ≤ _ := sum_le_sum fun v hv ↦ mul_le_mul_of_nonneg_left |
| (hc u v (mem_neighbors G u v |>.1 hv)) (hw v) |
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| lemma commonMass_eq_sum (G : SimpleGraph V) (w : V → ℝ) (u v : V) : |
| commonMass G w u v = ∑ z ∈ neighbors G u, if G.Adj v z then w z else 0 := by |
| classical |
| unfold commonMass mass |
| rw [← filter_mem_eq_inter, sum_filter] |
| simp only [mem_neighbors] |
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| lemma crossMass_all_right (G : SimpleGraph V) (w : V → ℝ) (A : Finset V) : |
| crossMass G w A univ = ∑ u ∈ A, w u * neighborMass G w u := by |
| classical |
| rw [crossMass_factor] |
| simp [neighborMass] |
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| lemma crossMass_all_left (G : SimpleGraph V) (w : V → ℝ) (A : Finset V) : |
| crossMass G w univ A = ∑ u ∈ A, w u * neighborMass G w u := by |
| rw [crossMass_symm, crossMass_all_right] |
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| lemma crossMass_neighbors (G : SimpleGraph V) (w : V → ℝ) |
| (A : Finset V) (v : V) : |
| crossMass G w A (neighbors G v) = ∑ u ∈ A, w u * commonMass G w u v := by |
| rw [crossMass_factor] |
| simp only [commonMass, inter_comm] |
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| end CliqueFreeIndependence.WeightedGraph |
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| end |
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| end OAI |
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