isLUB_sSup' — Mathlib

English

If BddAbove t and s ⊆ t, then sSup s ≤ sSup t.

Русский

Если t ограничено сверху и s ⊆ t, то sSup s ≤ sSup t.

LaTeX

$$$BddAbove(t) \land s \subseteq t \Rightarrow sSup(s) \le sSup(t)$$$

Lean4

/-- The `sSup` of a non-empty set is its least upper bound for a conditionally
complete lattice with a top. -/
theorem isLUB_sSup' {β : Type*} [ConditionallyCompleteLattice β] {s : Set (WithTop β)} (hs : s.Nonempty) :
    IsLUB s (sSup s) := by
  classical
  constructor
  · change ite _ _ _ ∈ _
    split_ifs with h₁ h₂
    · intro _ _
      exact le_top
    · rintro (⟨⟩ | a) ha
      · contradiction
      apply coe_le_coe.2
      exact le_csSup h₂ ha
    · intro _ _
      exact le_top
  · change ite _ _ _ ∈ _
    split_ifs with h₁ h₂
    · rintro (⟨⟩ | a) ha
      · exact le_rfl
      · exact False.elim (not_top_le_coe a (ha h₁))
    · rintro (⟨⟩ | b) hb
      · exact le_top
      refine coe_le_coe.2 (csSup_le ?_ ?_)
      · rcases hs with ⟨⟨⟩ | b, hb⟩
        · exact absurd hb h₁
        · exact ⟨b, hb⟩
      · intro a ha
        exact coe_le_coe.1 (hb ha)
    · rintro (⟨⟩ | b) hb
      · exact le_rfl
      · exfalso
        apply h₂
        use b
        intro a ha
        exact coe_le_coe.1 (hb ha)

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