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.

Addition is a well-defined total operation

Statement

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

Facts & Assumptions

Given: the natural numbers $N$ with successor $σ$ .

[L1]

The recursion theorem: for a set $A$ , an element $a \in A$ , and a function $f : A \to A$ there is a unique $g : N \to A$ with $g (0) = a$ and $g (σ (n)) = f (g (n))$ (The recursion theorem).

[L2]

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

Proof

technique · direct

1.1L1L2

Fix $m \in N$ ; since $(N, 0, σ)$ is a Peano system [L2], apply the recursion theorem with $A = N$ , $a = m$ , and $f = σ$ : there is a unique function $s_{m} : N \to N$ with $s_{m} (0) = m$ and $s_{m} (σ (n)) = σ (s_{m} (n))$ .

2.1step 1.1

Define $m + n : = s_{m} (n)$ ; since $s_{m}$ is a total function on $N$ for every $m \in N$ , the assignment $(m, n) \mapsto m + n$ is a well-defined function $N \times N \to N$ .

3.1step 1.1step 2.1∎

The recursion equations become $m + 0 = s_{m} (0) = m$ and $m + σ (n) = s_{m} (σ (n)) = σ (s_{m} (n)) = σ (m + n)$ , the defining clauses of addition.

Addition is a well-defined total operation

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources