lean_workbook_plus_19664

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

An automorphism of $(\mathbb{Z},+)$ is a bijective mapping $f\colon \mathbb{Z} \to \mathbb{Z}$ such that $f(m+n) = f(m)+f(n)$ for all $m,n \in \mathbb{Z}$ (whence $f(0)=0$ ). Consider what the value $u$ of $f(1)$ could be, since then $f(n) = nu$ for any $n \in \mathbb{Z}$. It should be clear that it can only be $u\in \{-1,1\}$, so the (only) two automorphisms are $f(n)=n$ for all $n \in \mathbb{Z}$, and $f(n)=-n$ for all $n \in \mathbb{Z}$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_19664 : ∀ f : ℤ → ℤ, (∀ m n : ℤ, f (m + n) = f m + f n) → (∀ m : ℤ, f m = m) ∨ ∀ m : ℤ, f m = -m   :=  by sorry

The source bijectivity assumption for an automorphism was omitted, admitting the zero homomorphism.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Require f:ℤ→ℤ additive and bijective, and prove f=id or f=−id; verify both candidates are automorphisms.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.