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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 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | import Mathlib
namespace OAI
section
section
noncomputable section
namespace HyperbolicObstruction.PlanarRecursion
attribute [local instance] Classical.propDecidable
variable {V : Type*}
/-- A median BFS level, with both strict exterior sides at most half.
This applies to any specified vertex subset, not just the full graph. -/
theorem median_level (U : Finset V) (depth : V → ℕ) :
∃ m : ℕ, 2*(U.filter (fun v => depth v < m)).card ≤ U.card ∧
2*(U.filter (fun v => m < depth v)).card ≤ U.card := by
have hex : ∃ m : ℕ, U.card ≤ 2*(U.filter (fun v => depth v ≤ m)).card := by
refine ⟨U.sup depth,?_⟩
have he : U.filter (fun v => depth v ≤ U.sup depth) = U :=
Finset.filter_eq_self.mpr (fun v hv => Finset.le_sup hv)
rw [he]; omega
let m := Nat.find hex
have hm : U.card ≤ 2*(U.filter (fun v => depth v ≤ m)).card := Nat.find_spec hex
refine ⟨m,?_,?_⟩
· by_cases hz : m = 0
· simp [hz]
· have hmin := Nat.find_min hex (show m-1 < m by omega)
have he : U.filter (fun v => depth v ≤ m-1) = U.filter (fun v => depth v < m) := by
ext v
simp only [Finset.mem_filter]
have hmpos : 0 < m := Nat.pos_of_ne_zero hz
rw [Nat.le_sub_one_iff_lt hmpos]
rw [he] at hmin
omega
· have hh := U.card_filter_add_card_filter_not (fun v => depth v ≤ m)
simp only [not_le] at hh
omega
lemma sparse_level_in_interval (U : Finset V) (depth : V → ℤ) (l h : ℤ) (hlh : l ≤ h) :
∃ t ∈ Finset.Icc l h, (Finset.Icc l h).card * (U.filter (fun v => depth v = t)).card ≤ U.card := by
classical
obtain ⟨t,ht,hmin⟩ := Finset.exists_min_image (Finset.Icc l h)
(fun t => (U.filter (fun v => depth v = t)).card) ⟨l,Finset.mem_Icc.mpr ⟨le_rfl,hlh⟩⟩
refine ⟨t,ht,?_⟩
calc
_ = ∑ s ∈ Finset.Icc l h, (U.filter (fun v => depth v = t)).card := by simp
_ ≤ ∑ s ∈ Finset.Icc l h, (U.filter (fun v => depth v = s)).card :=
Finset.sum_le_sum (fun s hs => hmin s hs)
_ = (U.filter (fun v => depth v ∈ Finset.Icc l h)).card :=
Finset.sum_card_fiberwise_eq_card_filter U _ depth
_ ≤ U.card := Finset.card_filter_le _ _
/-- Two sparse integer levels bracketing an arbitrary median. Negative
levels are permitted (their fibers are empty for graph distances), avoiding
an endpoint exception at the root. -/
theorem two_sparse_levels (U : Finset V) (depth : V → ℤ) (m : ℤ) (a : ℕ) :
∃ L H : ℤ, L ≤ m ∧ m ≤ H ∧ H-L ≤ 2*a ∧
(a+1)*(U.filter (fun v => depth v = L)).card ≤ U.card ∧
(a+1)*(U.filter (fun v => depth v = H)).card ≤ U.card := by
have ha : (0 : ℤ) ≤ a := by positivity
obtain ⟨L,hL,hLc⟩ := sparse_level_in_interval U depth (m-a) m (by omega)
obtain ⟨H,hH,hHc⟩ := sparse_level_in_interval U depth m (m+a) (by omega)
have hcardL : (Finset.Icc (m-(a : ℤ)) m).card = a+1 := by rw [Int.card_Icc]; omega
have hcardH : (Finset.Icc m (m+(a : ℤ))).card = a+1 := by rw [Int.card_Icc]; omega
rw [hcardL] at hLc
rw [hcardH] at hHc
have hLd := Finset.mem_Icc.mp hL
have hHd := Finset.mem_Icc.mp hH
exact ⟨L,H,hLd.2,hHd.1,by omega,hLc,hHc⟩
end HyperbolicObstruction.PlanarRecursion
end
end
section
end
end
end OAI
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