CorollaryStatement: AI-adaptedProof: 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.

Multiplication is a well-defined total operation

Statement

For each $m \in N$ there is a unique function $p_{m} : N \to N$ with $p_{m} (0) = 0$ and $p_{m} (σ (n)) = p_{m} (n) + m$ . Writing $m \cdot n : = p_{m} (n)$ , multiplication $\cdot : N \times N \to N$ is a well-defined total binary operation satisfying $m \cdot 0 = 0$ and $m \cdot σ (n) = m \cdot n + m$ .

Facts & Assumptions

Given: $N$ with successor $σ$ and addition $+$ (Addition of natural numbers).

[L1]

The recursion theorem (The recursion theorem).

[L2]

For fixed $m$ , the map ${a d d}_{m} : x \mapsto x + m$ is a function $N \to N$ (Addition of natural numbers).

[L3]

$(N, 0, σ)$ is a Peano system (The von Neumann naturals form a Peano system).

Proof

technique · direct

1.1

Fix $m \in N$ ; the map ${a d d}_{m} : N \to N$ , ${a d d}_{m} (x) = x + m$ , is a well-defined function.

2.1

Since $(N, 0, σ)$ is a Peano system [L3], apply the recursion theorem with $A = N$ , $a = 0$ , $f = {a d d}_{m}$ : there is a unique $p_{m} : N \to N$ with $p_{m} (0) = 0$ and $p_{m} (σ (n)) = {a d d}_{m} (p_{m} (n)) = p_{m} (n) + m$ .

L1L3step 1.1

3.1

Define $m \cdot n : = p_{m} (n)$ ; as $p_{m}$ is total for each $m \in N$ , the assignment $(m, n) \mapsto m \cdot n$ is a well-defined function $N \times N \to N$ with $m \cdot 0 = 0$ and $m \cdot σ (n) = m \cdot n + m$ .

step 2.1∎

Depends on

Used by

Multiplication of natural numbers Definition

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

T. Tao, Analysis I, 3rd ed., §2.1-2.3 (Peano axioms, recursion, arithmetic) (standard reference, not scraped)
Peano axioms (Wikipedia) (standard reference, not scraped)
B. Kaya, MATH 320 Set Theory (METU lecture notes) (standard reference, not scraped)
W. Aitken, MATH 378 Ch. 1: The Peano Axioms (CSU San Marcos) (standard reference, not scraped)