| import Mathlib |
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| namespace OAI |
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| section |
| section |
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| noncomputable section |
| namespace HyperbolicObstruction.PlanarRecursion |
| attribute [local instance] Classical.propDecidable |
| variable {I : Type*} [Fintype I] [DecidableEq I] |
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| /-- Group blocks of size at most half into two groups of size at most two |
| thirds. The total may be less than n because removed vertices are absent. -/ |
| theorem balanced_grouping (w : I → ℕ) (n : ℕ) (htotal : ∑ i, w i ≤ n) |
| (hblock : ∀ i, 2*w i ≤ n) : |
| ∃ A : Finset I, 3*(∑ i ∈ A, w i) ≤ 2*n ∧ 3*(∑ i ∈ Aᶜ, w i) ≤ 2*n := by |
| classical |
| by_cases hsmall : 3*(∑ i, w i) ≤ 2*n |
| · exact ⟨∅,by simp,by simpa using hsmall⟩ |
| by_cases hlarge : ∃ i, n ≤ 3*w i |
| · obtain ⟨i,hi⟩ := hlarge |
| refine ⟨{i},?_,?_⟩ |
| · simp only [Finset.sum_singleton] |
| have hh := hblock i |
| omega |
| · have hs := Finset.sum_add_sum_compl ({i} : Finset I) w |
| simp only [Finset.sum_singleton] at hs |
| omega |
| push Not at hlarge |
| let P := (Finset.univ : Finset I).powerset.filter (fun A => n ≤ 3*(∑ i ∈ A, w i)) |
| have hP : P.Nonempty := ⟨Finset.univ,Finset.mem_filter.mpr ⟨Finset.mem_powerset.mpr (Finset.Subset.refl _),by omega⟩⟩ |
| obtain ⟨A,hAP,hmin⟩ := Finset.exists_min_image P Finset.card hP |
| have hAn := (Finset.mem_filter.mp hAP).2 |
| have hAne : A.Nonempty := by |
| by_contra h |
| have he := Finset.not_nonempty_iff_eq_empty.mp h |
| simp only [he,Finset.sum_empty,mul_zero] at hAn |
| omega |
| obtain ⟨a,ha⟩ := hAne |
| have hremove : 3*(∑ i ∈ A.erase a, w i) < n := by |
| by_contra h |
| have hm := hmin (A.erase a) (Finset.mem_filter.mpr |
| ⟨Finset.mem_powerset.mpr (Finset.subset_univ _),by omega⟩) |
| have hcard := Finset.card_erase_lt_of_mem ha |
| omega |
| have hs : w a + ∑ i ∈ A.erase a, w i = ∑ i ∈ A, w i := Finset.add_sum_erase A w ha |
| have ha' := hlarge a |
| have hcompl := Finset.sum_add_sum_compl A w |
| exact ⟨A,by omega,by omega⟩ |
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| end HyperbolicObstruction.PlanarRecursion |
| end |
| end |
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| section |
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| end |
| end |
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| end OAI |
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