LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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.

Every finite power of an at most countable set is at most countable

If $C$ is at most countable and $m \in N$ , then the finite product $C^{m} = \prod_{i < m} C$ is at most countable.

Given: An at most countable set $C$ and a natural number $m$ .

[L1]

The product of two at most countable sets is at most countable (A product of two at most countable sets is at most countable).

[L2]

[L3]

technique · induction

1.1baseL2L3

For $m = 0$ , $C^{0}$ is a one-point set by [L2], hence at most countable by [L3].

1.2ih

Assume $C^{m}$ is at most countable.

2.1step 1.2L1

The product $C^{m + 1}$ is canonically $C^{m} \times C$ , so it is at most countable by [L1].

3.1basestep 1.2step 2.1L3discharge-induction∎

By induction, $C^{m}$ is at most countable for every $m \in N$ .

Depends on