English
natAbs is strictly monotone on the nonnegative reals: on Ici 0, natAbs is strictly increasing.
Русский
abs(a) строго возрастает на неотрицательных числах: на интервале [0, ∞) natAbs возрастает строго.
LaTeX
$$$\operatorname{StrictMonoOn} (\operatorname{natAbs}) (\mathrm{I}\!\mathrm{ci}\,0)$$$
Lean4
theorem nonneg_or_nonpos_of_mul_nonneg {a b : ℤ} : 0 ≤ a * b → 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 :=
by
intro h
by_cases ha : 0 ≤ a <;> by_cases hb : 0 ≤ b
· exact .inl ⟨ha, hb⟩
· refine .inr ⟨?_, le_of_not_ge hb⟩
obtain _ | _ := Int.mul_eq_zero.mp <| Int.le_antisymm (Int.mul_nonpos_of_nonneg_of_nonpos ha <| le_of_not_ge hb) h
all_goals cutsat
· refine .inr ⟨le_of_not_ge ha, ?_⟩
obtain _ | _ := Int.mul_eq_zero.mp <| Int.le_antisymm (Int.mul_nonpos_of_nonpos_of_nonneg (le_of_not_ge ha) hb) h
all_goals cutsat
· exact .inr ⟨le_of_not_ge ha, le_of_not_ge hb⟩
