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 cancellative

Statement

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

Facts & Assumptions

Given: Addition (Addition of natural numbers).

[L1]

$σ$ is injective (P2) (The von Neumann naturals form a Peano system).

[L2]

Induction principle (The principle of mathematical induction).

Proof

technique · induction

1.1base

Base $k = 0$ : if $m + 0 = n + 0$ then $m = n$ , since $m + 0 = m$ and $n + 0 = n$ .

1.2ih

Inductive hypothesis: for all $m, n$ , $m + k = n + k \Rightarrow m = n$ .

2.1step 1.2L1

Step: suppose $m + σ (k) = n + σ (k)$ ; then $σ (m + k) = σ (n + k)$ by $m + σ (j) = σ (m + j)$ , so $m + k = n + k$ by injectivity [L1], hence $m = n$ by the hypothesis.

3.1step 1.1step 2.1discharge-induction∎

By induction [L2], cancellation holds for all $k$ , hence for all $m, n, k \in N$ .

Addition is cancellative

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources