LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-08-02 (claude-opus-5)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Every nonzero natural number is a successor

For every $n \in N$ with $n \neq 0$ , there exists $m \in N$ with $n = σ (m)$ . Equivalently, every natural number is either $0$ or a successor.

Given: $N$ with distinguished element $0$ and successor $σ$ . Let $S = {n \in N : n = 0 or \exists m \in N (n = σ (m))}$ .

[L1]

technique · induction

1.1base

Base case: $0 \in S$ since $0 = 0$ .

1.2ih

Inductive hypothesis: $n \in S$ .

2.1step 1.2

$σ (n)$ is the successor of $n$ , so $σ (n) \in S$ (it satisfies $\exists m, σ (n) = σ (m)$ with $m = n$ ).

3.1step 1.1step 2.1discharge-induction∎

By induction [L1], $S = N$ , so every $n \in N$ is $0$ or a successor; if $n \neq 0$ then $n = σ (m)$ for some $m$ .

Depends on