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.

Addition is commutative

Statement

For all $m, n \in N$ : $m + n = n + m$ .

Facts & Assumptions

Given: Addition (Addition of natural numbers).

[L1]

$0 + k = k$ for all $k$ (Left identity for addition).

[L2]

Left successor law: $σ (a) + k = σ (a + k)$ for all $a, k$ (Left successor law for addition).

[L3]

Induction principle (The principle of mathematical induction).

Proof

technique · induction

1.1baseL1

Base $n = 0$ : $m + 0 = m$ and $0 + m = m$ by [L1], so $m + 0 = 0 + m$ .

1.2ih

Inductive hypothesis: $m + n = n + m$ .

2.1step 1.2givenL2

Step: $m + σ (n) = σ (m + n) = σ (n + m)$ by the recursion and the hypothesis; and the left successor law [L2] gives $σ (n) + m = σ (n + m)$ , so $m + σ (n) = σ (n + m) = σ (n) + m$ .

3.1step 1.1step 2.1discharge-induction∎

By induction [L3], $m + n = n + m$ for all $n$ , hence for all $m, n \in N$ .

Addition is commutative

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources