/-
Copyright 2026 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
https://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/
module
public import Mathlib.Algebra.Order.Archimedean.Real.Basic
public import Mathlib.Algebra.Order.Round@[expose] public sectionThe distance from a real number to the nearest integer.
noncomputable def distToNearestInt (x : ℝ) : ℝ := |x - round x|The distance to the nearest integer is nonnegative.
@[simp]
theorem distToNearestInt_nonneg (x : ℝ) : 0 ≤ distToNearestInt x := abs_nonneg _