wellFoundedGT — Mathlib

English

The identity morphism on a finite bounded distributive lattice X fixes every element.

Русский

Идентичный морфизм на FinBddDistLat X фиксирует каждый элемент.

LaTeX

$$id_X(x) = x$$

Lean4

theorem wellFoundedGT (h : IsSupClosedCompact α) : WellFoundedGT α where
  wf := by
    refine RelEmbedding.wellFounded_iff_isEmpty.mpr ⟨fun a => ?_⟩
    suffices sSup (Set.range a) ∈ Set.range a
      by
      obtain ⟨n, hn⟩ := Set.mem_range.mp this
      have h' : sSup (Set.range a) < a (n + 1) := by
        change _ > _
        simp [← hn, a.map_rel_iff]
      apply lt_irrefl (a (n + 1))
      apply lt_of_le_of_lt _ h'
      apply le_sSup
      apply Set.mem_range_self
    apply h (Set.range a)
    · use a 37
      apply Set.mem_range_self
    · rintro x ⟨m, hm⟩ y ⟨n, hn⟩
      use m ⊔ n
      rw [← hm, ← hn]
      apply RelHomClass.map_sup a

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