import Spa.Lattice

namespace Spa

theorem boundedChains_of_subsingleton (α : Type*) [Preorder α] [Subsingleton α]
    (n : ℕ) : BoundedChains α n := fun c => by
  by_contra hc
  push_neg at hc
  exact (c.step ⟨0, by omega⟩).ne (Subsingleton.elim _ _)

instance : FiniteHeightLattice PUnit where
  bot := PUnit.unit
  top := PUnit.unit
  height := 0
  longestChain := { series := RelSeries.singleton _ PUnit.unit, head_series := refl _, last_series := refl _, length_series := refl _ }
  chains_bounded := boundedChains_of_subsingleton PUnit 0

end Spa