cases' abs_cases x with hx hx
a b x : ℝ
hx : (a ≤ x ∨ b ≤ x) ∧ (x ≤ a ∨ x ≤ b)
this : |x| ≤ max |a| |b|
⊢ |↑⌊x⌋| ≤ max |a| |b| + 1
case inl
a b x : ℝ
hx✝ : (a ≤ x ∨ b ≤ x) ∧ (x ≤ a ∨ x ≤ b)
this : |x| ≤ max |a| |b|
hx : |x| = x ∧ 0 ≤ x
⊢ |↑⌊x⌋| ≤ max |a| |b| + 1

case inr
a b x : ℝ
hx✝ : (a ≤ x ∨ b ≤ x) ∧ (x ≤ a ∨ x ≤ b)
this : |x| ≤ max |a| |b|
hx : |x| = -x ∧ x < 0
⊢ |↑⌊x⌋| ≤ max |a| |b| + 1