import Spa.Lattice

/-!

# Unit Lattice

This file provides a proof that in addition to being a lattice,
`PUnit` is a `Spa.FiniteHeightLattice`. This is fairly trivial result,
but the unit is used as a placeholder in various contexts (e.g.,
as a base case for the iterated product `Spa/Lattice/IterProd.lean`). -/

namespace Spa

/-- Since a singleton type's preorder has no nonempty `<` chains,
    they are vacuously bounded by any minimum height. -/
lemma 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
  toLattice := inferInstance
  longestChain := RelSeries.singleton _ PUnit.unit
  chains_bounded := boundedChains_of_subsingleton PUnit 0

end Spa