File size: 2,053 Bytes
7d9acbd | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 | import Mathlib
namespace OAI
section
section
noncomputable section
namespace HyperbolicObstruction.PlanarRecursion
attribute [local instance] Classical.propDecidable
variable {I : Type*} [Fintype I] [DecidableEq I]
/-- 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⟩
end HyperbolicObstruction.PlanarRecursion
end
end
section
end
end
end OAI
|