insert_Ioc_right_eq_Ioc_succ_of_not_isMax

English

If a ≤ b and b is not maximal, then inserting succ b into Ioc a b yields Ioc a (succ b).

Русский

Если a ≤ b и b не максимален, вставка succ b в Ioc a b даёт Ioc a (succ b).

LaTeX

$$$\\text{insert }(\\mathrm{Order.succ}\\,b) (\\text{Ioc } a b) = \\text{Ioc } a (\\mathrm{Order.succ}\\,b)$$$

Lean4

theorem insert_Ioc_right_eq_Ioc_succ_of_not_isMax (h : a ≤ b) (hb : ¬IsMax b) :
    insert (succ b) (Ioc a b) = Ioc a (succ b) := by ext x;
  simp +contextual [or_and_left, le_succ_iff_eq_or_le, lt_succ_of_le_of_not_isMax h hb]

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