| import OAI.Combinatorics.CycleDecomposition.BatchIndexing |
| |
| namespace OAI |
|
|
| section |
| open Filter Asymptotics Real |
| open scoped Topology |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Finset |
| section |
|
|
| namespace ErdosGallai |
| noncomputable section |
| open Finset |
|
|
| theorem prefix_selection (S : ℕ → ℝ) (hS : ∀ j, 0 ≤ S j) (K N : ℕ) |
| (hsupp : ∀ j, N ≤ j → S j = 0) (hne : ∃ j, 0 < S j) : |
| ∃ i, 0 < S i ∧ ∑ j ∈ range (i+K+1), S j ≤ 4*(K+1:ℝ)*S i := by |
| let A := (range N).filter (fun j => 0 < S j) |
| have hA : A.Nonempty := by |
| obtain ⟨j,hj⟩ := hne |
| exact ⟨j, by |
| simp only [A, mem_filter, mem_range] |
| exact ⟨by by_contra hn |
| let i₀ := A.min' hA |
| have hi₀A : i₀ ∈ A := A.min'_mem hA |
| have hi₀pos : 0 < S i₀ := (mem_filter.mp hi₀A).2 |
| have hstart : (∑ j ∈ range (i₀+1), S j) = S i₀ := by |
| apply sum_eq_single i₀ |
| · intro j hj hji |
| by_contra hjzero |
| have hjpos : 0 < S j := lt_of_le_of_ne (hS j) (Ne.symm hjzero) |
| have hjA : j ∈ A := by |
| simp only [A, mem_filter, mem_range] |
| refine ⟨?_, hjpos⟩ |
| have hiN := (mem_range.mp (mem_filter.mp hi₀A).1) |
| have hjle := mem_range.mp hj |
| omega |
| have hmin : i₀ ≤ j := A.min'_le j hjA |
| have hjle := mem_range.mp hj |
| omega |
| · intro hi |
| exact (hi (by simp)).elim |
| let B := A.filter (fun i => (∑ j ∈ range (i+1), S j) ≤ 2*(K+1:ℝ)*S i) |
| have hB : B.Nonempty := by |
| refine ⟨i₀, mem_filter.mpr ⟨hi₀A, ?_⟩⟩ |
| rw [hstart] |
| have hK : (0:ℝ) ≤ K := Nat.cast_nonneg K |
| nlinarith |
| let i := B.max' hB |
| have hiB : i ∈ B := B.max'_mem hB |
| have hiA : i ∈ A := (mem_filter.mp hiB).1 |
| have hipos : 0 < S i := (mem_filter.mp hiA).2 |
| have hinv : (∑ j ∈ range (i+1), S j) ≤ 2*(K+1:ℝ)*S i := (mem_filter.mp hiB).2 |
| refine ⟨i, hipos, ?_⟩ |
| by_contra hfail |
| have hfail' : 4*(K+1:ℝ)*S i < ∑ j ∈ range (i+K+1), S j := lt_of_not_ge hfail |
| have hwindow : 2*(K+1:ℝ)*S i < ∑ j ∈ Ico (i+1) (i+K+1), S j := by |
| have hsplit := Finset.sum_range_add_sum_Ico S (by omega : i+1 ≤ i+K+1) |
| linarith |
| have hwindow_nonempty : (Ico (i+1) (i+K+1)).Nonempty := by |
| by_contra hh |
| rw [Finset.not_nonempty_iff_eq_empty.mp hh] at hwindow |
| simp only [sum_empty] at hwindow |
| have hK : (0:ℝ) ≤ K := Nat.cast_nonneg K |
| nlinarith |
| obtain ⟨j,hjw,hmax⟩ := Finset.exists_max_image (Ico (i+1) (i+K+1)) S hwindow_nonempty |
| have hsum : (∑ l ∈ Ico (i+1) (i+K+1), S l) ≤ (K:ℝ)*S j := by |
| calc |
| _ ≤ ∑ _l ∈ Ico (i+1) (i+K+1), S j := sum_le_sum hmax |
| _ = _ := by simp [] |
| have hdouble : 2*S i < S j := by |
| have hK : (0:ℝ) ≤ K := Nat.cast_nonneg K |
| nlinarith |
| have hjpos : 0 < S j := by linarith |
| have hjA : j ∈ A := by |
| simp only [A, mem_filter, mem_range] |
| exact ⟨by by_contra hjN; rw [hsupp j (by omega)] at hjpos; linarith, hjpos⟩ |
| have hjbounds := Finset.mem_Ico.mp hjw |
| have hpartial : (∑ l ∈ Ico (i+1) (j+1), S l) ≤ (K:ℝ)*S j := by |
| calc |
| _ ≤ ∑ l ∈ Ico (i+1) (i+K+1), S l := by |
| apply sum_le_sum_of_subset_of_nonneg |
| · intro l hl |
| simp only [mem_Ico] at hl ⊢ |
| omega |
| · intro l _ _; exact hS l |
| _ ≤ _ := hsum |
| have hjinv : (∑ l ∈ range (j+1), S l) ≤ 2*(K+1:ℝ)*S j := by |
| have hsplit := Finset.sum_range_add_sum_Ico S (by omega : i+1 ≤ j+1) |
| have hK : (0:ℝ) ≤ K := Nat.cast_nonneg K |
| nlinarith |
| have hjB : j ∈ B := mem_filter.mpr ⟨hjA,hjinv⟩ |
| have hji : j ≤ i := B.le_max' j hjB |
| omega |
|
|
| end |
| end ErdosGallai |
|
|
| end |
|
|
| namespace ErdosGallai |
|
|
| def MainStatement : Prop := |
| ∃ C : ℝ, 0 < C ∧ ∀ (n : ℕ) (G : SimpleGraph (Fin n)), |
| ∃ k : ℕ, EdgeDecomposition G k ∧ (k : ℝ) ≤ C * n |
| end ErdosGallai |
| end |
| end |
|
|
| end OAI |
|
|
