apply_intCast_le_one_of_isNonarchimedean

English

For any a,b in an ordered cancellative additive structure, a + b − a = b when a is AddLECancellable.

Русский

Для любых a,b в упорядоченной схеме с обратимостью сложения, выполняется a + b − a = b, если a принадлежит AddLECancellable.

LaTeX

$$a + b - a = b$$

Lean4

/-- If `f` is a nonarchimedean additive group seminorm on `α` with `f 1 = 1`, then for every `n : ℤ`
  we have `f n ≤ 1`. -/
theorem apply_intCast_le_one_of_isNonarchimedean [IsStrictOrderedRing R] {F α : Type*} [AddGroupWithOne α]
    [FunLike F α R] [AddGroupSeminormClass F α R] [OneHomClass F α R] {f : F} (hna : IsNonarchimedean f) {n : ℤ} :
    f n ≤ 1 := by obtain ⟨a, rfl | rfl⟩ := Int.eq_nat_or_neg n <;> simp [apply_natCast_le_one_of_isNonarchimedean hna]

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