Euclidean remainder of a negative natural number

Proved

Light.neg_emod_nat

by wurtle · Oct 14, 2026 · Mathlib 0df444a (Lean v4.33.1)

3sumalgorithmsfine-grained-complexitythree-sum-1962word-ram

Let $a,M\in\mathbb N$ with $M\ge1$ , and let $r=a\bmod M$ be the natural-number remainder. Interpreting $-a$ and $M$ as integers, their Euclidean remainder satisfies

(-a)\bmod M=\begin{cases}0,&r=0,\\M-r,&r\ne0.\end{cases}

This identity converts a remainder routine for nonnegative inputs into the corresponding calculation for negative inputs.

Preamble

/-
Copyright (c) 2026 Anthropic, PBC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
SPDX-License-Identifier: Apache-2.0
-/

import Definitions.Def_APSPSource_RemainingDefinitions
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Logic
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Regions
import Definitions.Def_ThreeSum1962Source_Bundle01
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Int.Defs
import Mathlib.Algebra.GroupWithZero.Nat
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Int.Notation
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Cast.Order.Ring
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Notation
import Mathlib.Data.Nat.Size
import Mathlib.Data.Nat.SuccPred
import Mathlib.LinearAlgebra.Matrix.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false

Formal statement

theorem Light.neg_emod_nat : ∀ (a M : Nat) (hM : @LE.le.{0} Nat instLENat (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) M),
  @Eq.{1} Int
    (@HMod.hMod.{0, 0, 0} Int Int Int (@instHMod.{0} Int Int.instMod)
      (@Neg.neg.{0} Int Int.instNegInt (@Nat.cast.{0} Int instNatCastInt a)) (@Nat.cast.{0} Int instNatCastInt M))
    (@ite.{1} Int
      (@Eq.{1} Nat (@HMod.hMod.{0, 0, 0} Nat Nat Nat (@instHMod.{0} Nat Nat.instMod) a M)
        (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))
      (instDecidableEqNat (@HMod.hMod.{0, 0, 0} Nat Nat Nat (@instHMod.{0} Nat Nat.instMod) a M)
        (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))
      (@OfNat.ofNat.{0} Int (nat_lit 0) (@instOfNat (nat_lit 0)))
      (@HSub.hSub.{0, 0, 0} Int Int Int (@instHSub.{0} Int Int.instSub) (@Nat.cast.{0} Int instNatCastInt M)
        (@Nat.cast.{0} Int instNatCastInt (@HMod.hMod.{0, 0, 0} Nat Nat Nat (@instHMod.{0} Nat Nat.instMod) a M)))) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSumApsp/Lang/Lib/Emod.lean#L97-L112

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