/-
Port of `Lattice/Unit.agda`.

The lattice structure itself (`_⊔_`, `_⊓_`, all semilattice/lattice laws) is
lifted into mathlib: `PUnit.instLinearOrder` provides `Lattice PUnit`.
What remains is the fixed-height structure: the unit lattice has height 0.
-/
import Spa.Lattice

namespace Spa

/-- Chains in a subsingleton order are bounded by any `n` (Agda: the `bounded`
field of `Lattice/Unit.agda`'s `fixedHeight`, generalized). -/
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 _ _)

/-- Agda: `Lattice/Unit.agda`'s `fixedHeight`/`isFiniteHeightLattice`. -/
instance : FiniteHeightLattice PUnit where
  height := 0
  fixedHeight :=
    { bot := PUnit.unit
      top := PUnit.unit
      longestChain := RelSeries.singleton _ PUnit.unit
      head_longestChain := rfl
      last_longestChain := rfl
      length_longestChain := rfl
      bounded := boundedChains_of_subsingleton PUnit 0 }

end Spa
