divMod_spec — Mathlib

English

For a,b,c nonzero with a ≤ c, one has c = a + spanSingleton(c.divMod b a)·b; i.e., the division with remainder exists as a single principal correction.

Русский

Для ненулевых a,b,c с a ≤ c имеем c = a + spanSingleton(c.divMod b a)·b; т.е. разложение по частям через 'остаток'.

LaTeX

$$$c = a + \operatorname{spanSingleton}(c.divMod\, b\, a) \cdot b$ при условиях $a\le c$, $a\neq 0$, $b\neq 0$.$$

Lean4

theorem divMod_spec {a b c : FractionalIdeal R⁰ K} (hac : a ≤ c) (ha : a ≠ 0) (hb : b ≠ 0) :
    a + spanSingleton R⁰ (c.divMod b a) * b = c :=
  by
  rw [divMod, dif_pos ⟨hac, ha, hb⟩]
  exact (IsDedekindDomain.exists_add_spanSingleton_mul_eq hac ha hb).choose_spec

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