Section bte_depth.
Opaque max.
(* Well-foundness *)
Fact bte_depth_eq s t : s ≈ t -> ⌞s⌟ = ⌞t⌟.
Fact btm_depth s t : s ∈ t -> ⌞s⌟ < ⌞t⌟.
(* bt is well-founded for ∈ *)

Theorem btm_wf : well_founded (fun s t => s ∈ t).