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.

Order is compatible with addition

For all $m, n, k \in N$ : $m \leq n ⟺ m + k \leq n + k$ ; and $m < n ⟺ m + k < n + k$ (Order on the natural numbers).

Given: The order $m \leq n ⟺ \exists j (m + j = n)$ (Order on the natural numbers) and addition with $m + 0 = m$ , $m + σ (n) = σ (m + n)$ (Addition of natural numbers).

[L1]

[L2]

Cancellation: $m + k = n + k \Rightarrow m = n$ (Addition is cancellative).

technique · direct

1.1givenL1

Forward: if $m \leq n$ , say $m + j = n$ , then $(m + k) + j = (m + j) + k = n + k$ , so $m + k \leq n + k$ .

1.2givenL1L2

Backward: if $m + k \leq n + k$ , say $(m + k) + j = n + k$ , then $(m + j) + k = n + k$ , so $m + j = n$ by cancellation, giving $m \leq n$ .

2.1step 1.1step 1.2L2∎

The strict forms follow: $m < n$ means $m \leq n$ and $m \neq n$ , while $m + k = n + k ⟺ m = n$ by cancellation, so $m < n ⟺ m + k < n + k$ .

Depends on