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.

No natural number equals its own successor

For every $n \in N$ , $n \neq σ (n)$ .

Given: $N$ with distinguished element $0$ and successor $σ$ .

[L1]

$σ (n) \neq 0$ for all $n$ (P1) and $σ$ is injective (P2) (The von Neumann naturals form a Peano system).

[L2]

technique · induction

1.1baseL1

Base case $n = 0$ : $σ (0) \neq 0$ by P1, so $0 \neq σ (0)$ .

1.2ih

Inductive hypothesis: $n \neq σ (n)$ .

2.1step 1.2L1

If $σ (n) = σ (σ (n))$ then injectivity of $σ$ gives $n = σ (n)$ , contradicting the hypothesis; hence $σ (n) \neq σ (σ (n))$ .

3.1step 1.1step 2.1discharge-induction∎

By induction [L2], $n \neq σ (n)$ for all $n \in N$ .

Depends on