import Mathlib
import OAI.RepresentationTheory.Saxl.Main
import OAI.RepresentationTheory.UniversalSquare.Support.UnionRowChecker

namespace OAI

/-! Split Union Row Checker. -/

section

namespace UniversalTensorSquare

lemma checkRegionPrefixes_step (P : List ℕ → Bool) (k n upper : ℕ)
    (regions : List PrefixRegion) (pref : List ℕ)
    (hc : ∀ a, max 1 (n/(k+1)) ≤ a → a ≤ min n upper → n ≤ (k+1)*a →
      checkRegionPrefixes P k (n-a) a
        (nextRegions a (liveRegions (k+1) n upper regions)) (a::pref) = true) :
    checkRegionPrefixes P (k+1) n upper regions pref = true := by
  rw [checkRegionPrefixes]
  split_ifs with hnone
  · rfl
  · apply List.all_eq_true.mpr
    intro a ha
    have ha' := List.mem_range'_1.mp ha
    split_ifs with hn
    · apply hc a ha'.1 ?_ hn
      omega
    · rfl

end UniversalTensorSquare
end

end OAI