LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (openai/gpt-5.4)audited 2026-07-25

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.

Left identity for addition

For all $n \in N$ : $0 + n = n$ . The right identity $n + 0 = n$ is definitional, so together with this law $0$ is a two-sided identity for addition.

Given: Addition with $m + 0 = m$ and $m + σ (n) = σ (m + n)$ (Addition of natural numbers).

[L1]

technique · induction

1.1base

Base $n = 0$ : $0 + 0 = 0$ by the clause $m + 0 = m$ at $m = 0$ .

1.2ih

Inductive hypothesis: $0 + n = n$ .

2.1step 1.2given

Step: $0 + σ (n) = σ (0 + n) = σ (n)$ , using $m + σ (n) = σ (m + n)$ at $m = 0$ , then the hypothesis.

3.1step 1.1step 2.1discharge-induction∎

By induction [L1], $0 + n = n$ for all $n \in N$ .

Depends on