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import OAI.Computability.VertexCover.Reduction.SeparatorStarOscillation
import OAI.Computability.VertexCover.Analysis.SeparatorRoundingErrorAe
namespace OAI
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namespace VertexCover
theorem finiteMean_sub {α : Type*} [Fintype α] (f g : α → ℝ) :
finiteMean (fun a => f a - g a) = finiteMean f - finiteMean g := by
simp only [finiteMean, Finset.sum_sub_distrib, sub_div]
theorem finiteMean_abs_le {α : Type*} [Fintype α] (f : α → ℝ) :
|finiteMean f| ≤ finiteMean (fun a => |f a|) := by
unfold finiteMean
rw [abs_div, abs_of_nonneg (show (0 : ℝ) ≤ Fintype.card α from Nat.cast_nonneg _)]
exact div_le_div_of_nonneg_right (Finset.abs_sum_le_sum_abs _ _) (Nat.cast_nonneg _)
end VertexCover
namespace VertexCover.LabelCover
open MeasureTheory
def StarKey (Φ : LabelCover) (d : ℕ) :=
PositionPair d → Option (Fin Φ.M) × Option (Fin Φ.u) × Option (Fin Φ.v)
noncomputable def starKey (Φ : LabelCover) {d : ℕ} (k : Fin d) (seed : Φ.Seeds d) :
Φ.StarKey d := by
classical
exact fun e => (if e.1.1 = k ∨ e.1.2 = k then none else some (seed e),
if e.1.1 = k then none else some (Φ.left (seed e)),
if e.1.2 = k then none else some (Φ.right (seed e)))
noncomputable def starFiber (Φ : LabelCover) {d : ℕ} (k : Fin d) (seed : Φ.Seeds d) :
Finset (Φ.Seeds d) := by
classical
exact Finset.univ.filter (fun seed' => Φ.starKey k seed' = Φ.starKey k seed)
theorem mem_starFiber (Φ : LabelCover) {d : ℕ} (k : Fin d) (seed seed' : Φ.Seeds d) :
seed' ∈ Φ.starFiber k seed ↔ Φ.starKey k seed' = Φ.starKey k seed := by
classical
simp [starFiber]
theorem self_mem_starFiber (Φ : LabelCover) {d : ℕ} (k : Fin d) (seed : Φ.Seeds d) :
seed ∈ Φ.starFiber k seed := (Φ.mem_starFiber k seed seed).mpr rfl
theorem starKey_eq_questions (Φ : LabelCover) {d : ℕ} (k : Fin d)
(seed seed' : Φ.Seeds d) (h : Φ.starKey k seed = Φ.starKey k seed') :
(∀ e : PositionPair d, e.1.1 ≠ k → Φ.left (seed e) = Φ.left (seed' e)) ∧
(∀ e : PositionPair d, e.1.2 ≠ k → Φ.right (seed e) = Φ.right (seed' e)) := by
constructor
· intro e he
have hh := congrArg (fun key : Φ.StarKey d => (key e).2.1) h
simpa only [starKey, ite_eq_right he, Option.some.injEq] using hh
· intro e he
have hh := congrArg (fun key : Φ.StarKey d => (key e).2.2) h
simpa only [starKey, ite_eq_right he, Option.some.injEq] using hh
noncomputable def starMean (Φ : LabelCover) {d : ℕ}
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (k : Fin d)
(seed : Φ.Seeds d) (s : Fin d → Fin (Φ.WeightDimension d) → ℝ) : ℝ :=
VertexCover.finiteMean (fun seed' : Φ.starFiber k seed =>
Φ.separator A hA (Φ.continuousSum seed'.1 s))
theorem star_residual_abs_le (Φ : LabelCover) {d : ℕ}
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (k : Fin d)
(seed : Φ.Seeds d) (s : Fin d → Fin (Φ.WeightDimension d) → ℝ)
(hs : Φ.InWeightCube s) :
|Φ.separator A hA (Φ.continuousSum seed s) - Φ.starMean A hA k seed s| ≤ 2 := by
classical
let : Nonempty (Φ.starFiber k seed) := ⟨⟨seed, Φ.self_mem_starFiber k seed⟩⟩
have heq : Φ.separator A hA (Φ.continuousSum seed s) - Φ.starMean A hA k seed s =
VertexCover.finiteMean (fun seed' : Φ.starFiber k seed =>
Φ.separator A hA (Φ.continuousSum seed s) -
Φ.separator A hA (Φ.continuousSum seed'.1 s)) := by
rw [VertexCover.finiteMean_sub, VertexCover.finiteMean_const]
rfl
rw [heq]
apply (VertexCover.finiteMean_abs_le _).trans
calc
VertexCover.finiteMean (fun seed' : Φ.starFiber k seed =>
|Φ.separator A hA (Φ.continuousSum seed s) -
Φ.separator A hA (Φ.continuousSum seed'.1 s)|) ≤
VertexCover.finiteMean (fun _ : Φ.starFiber k seed => (2 : ℝ)) := by
apply VertexCover.finiteMean_mono
intro seed'
have hkey := (Φ.mem_starFiber k seed seed'.1).mp seed'.2
obtain ⟨hL, hR⟩ := Φ.starKey_eq_questions k seed seed'.1 hkey.symm
exact Φ.separator_star_oscillation A hA seed seed'.1 s hs k hL hR
_ = 2 := VertexCover.finiteMean_const _
theorem starMean_continuous (Φ : LabelCover) {d : ℕ}
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (k : Fin d)
(seed : Φ.Seeds d) : Continuous (Φ.starMean A hA k seed) := by
classical
unfold starMean VertexCover.finiteMean
exact (continuous_finsetSum _ (fun seed' _ =>
Φ.separator_continuousSum_continuous A hA seed'.1)).div_const _
end VertexCover.LabelCover
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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