instNonempty — Mathlib

English

There is a subsingleton structure at every level under the well-founded order LT.

Русский

Подструктура на каждом уровне в рамках добре основанного порядка LT — одиночна.

LaTeX

$$$\\text{instSubsingletonOfWellFoundedLT}(j)$$$

Lean4

instance (j : J) : Nonempty (d.Extension val₀ j) := by
  induction j using SuccOrder.limitRecOn with
  | isMin i hi =>
    obtain rfl : i = ⊥ := by simpa using hi
    exact ⟨zero d val₀⟩
  | succ i hi hi' => exact ⟨hi'.some.succ hi⟩
  | isSuccLimit i hi hi' => exact ⟨limit i hi (fun l hl ↦ (hi' l hl).some)⟩

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