ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 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.

Coordinate inclusions are immersions and coordinate projections are submersions

Example

For $m \leq n$ , the coordinate inclusion

$j : R^{m} \to R^{n}, j (u) = (u, 0)$

is an immersion. For $n \leq m$ , the coordinate projection

$π : R^{m} \to R^{n}, π (u, v) = u$

is a submersion.

Facts & Assumptions

Given: The two displayed coordinate maps.

[F1]

A smooth map is an immersion exactly when its differential is injective at every point, and is a submersion exactly when its differential is surjective at every point (Immersions, submersions, and constant-rank maps).

[L1]

For a totally differentiable Euclidean map, the differential is computed by the Jacobian matrix (A total derivative computes every directional derivative, and its matrix is the Jacobian).

Verification

technique · direct

1.1

The coordinate inclusion $j$ is linear, so directly from the definition of the total derivative its differential at every point is $j$ itself. Its Jacobian is the injective block matrix $[\begin{array}{c} I_{m} \\ 0 \end{array}]$ , in agreement with [L1]. Hence $j$ is an immersion by [F1].

F1L1givenalgebra

1.2

The projection $π$ is linear, so its differential at every point is $π$ itself. Its Jacobian is the surjective block matrix $[\begin{array}{cc} I_{n} & 0 \end{array}]$ , in agreement with [L1]. Hence $π$ is a submersion by [F1].

F1L1givenalgebra

2.1

This verifies the example.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, Immersions and Submersions (standard reference, not scraped)