CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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 immersion is locally an embedding

Statement

Let $F : M^{m} \to N^{n}$ be a smooth immersion and let $p \in M$ . Then some neighbourhood $U$ of $p$ is sent homeomorphically onto an embedded $m$ -dimensional submanifold of $N$ , and $F ∣_{U}$ is still an immersion.

Facts & Assumptions

Given: A smooth immersion $F : M^{m} \to N^{n}$ and a point $p \in M$ .

[L1]

Near $p$ , suitable coordinates identify $F$ with the coordinate inclusion $u \mapsto (u, 0)$ (Local normal form for immersions).

Proof

technique · direct

1.1L1given

By [L1], after shrinking about $p$ and $F (p)$ there are charts in which $F$ becomes $u \mapsto (u, 0)$ .

2.1step 1.1

The coordinate inclusion is injective, is a homeomorphism onto the slice $R^{m} \times {0}$ with the subspace topology, and that slice is an embedded submanifold of the ambient Euclidean space. Pulling this description back through the charts gives the stated neighbourhood $U$ .

3.1step 1.1step 2.1∎

The restricted map remains an immersion because it is locally identified with the coordinate inclusion.

Depends on

Local normal form for immersions

Every immersion is locally an embedding

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources