distribLattice_of_sSup_eq_top — Mathlib

English

A lattice generated by Boolean generators is distributive.

Русский

Латекс, порожденный булевыми генераторами, является распределенным.

LaTeX

$$$ \text{distribLattice } α \quad\text{(under the given hypotheses)} $$$

Lean4

/-- A lattice generated by Boolean generators is a distributive lattice. -/
def distribLattice_of_sSup_eq_top (hS : BooleanGenerators S) (h : sSup S = ⊤) : DistribLattice α where
  le_sup_inf a b
    c := by
    obtain ⟨Ta, hTa, rfl⟩ := hS.atomistic a (h ▸ le_top)
    obtain ⟨Tb, hTb, rfl⟩ := hS.atomistic b (h ▸ le_top)
    obtain ⟨Tc, hTc, rfl⟩ := hS.atomistic c (h ▸ le_top)
    apply le_of_eq
    rw [← sSup_union, ← sSup_union, ← hS.sSup_inter hTb hTc, ← hS.sSup_inter, ← sSup_union]
    on_goal 1 => congr 1; ext
    all_goals
      simp only [Set.union_subset_iff, Set.mem_inter_iff, Set.mem_union]
      tauto

Похожие утверждения