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Copyright 2026 The Formal Conjectures Authors.
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module
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Algebra.Group.Pointwise.Set.Basic
public import Mathlib.Order.Filter.AtTopBot.Basic@[expose] public sectionAdditive complements
Two sets of naturals are additive complements when their sumset contains every large integer.
open Filteropen scoped Pointwise
A and B are additive complements if A + B contains all large integers.
def IsAdditiveComplement (A B : Set ℕ) : Prop :=
∀ᶠ n : ℕ in atTop, n ∈ A + Btheorem IsAdditiveComplement.symm {A B : Set ℕ} (h : IsAdditiveComplement A B) :
IsAdditiveComplement B A := A:Set ℕB:Set ℕh:IsAdditiveComplement A B⊢ IsAdditiveComplement B A
filter_upwards [h] with n A:Set ℕB:Set ℕh:IsAdditiveComplement A Bn:ℕhn:n ∈ A + B⊢ n ∈ B + A using A:Set ℕB:Set ℕh:IsAdditiveComplement A Bn:ℕhn:n ∈ A + B⊢ n ∈ B + A rwa [A:Set ℕB:Set ℕh:IsAdditiveComplement A Bn:ℕhn:n ∈ A + B⊢ n ∈ A + BA:Set ℕB:Set ℕh:IsAdditiveComplement A Bn:ℕhn:n ∈ A + B⊢ n ∈ A + B