ExampleConstruction: AI-generatedVerification: AI-adaptedprecheck passaudited 2026-09-01

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.

A fibre product of submersions

Example

Let $F, G : R^{2} \to R$ be the projections

$F (x, y) = x, G (u, v) = u .$

Then

$R^{2} \times_{R} R^{2} = {(x, y, u, v) : x = u}$

is an embedded $3$ -dimensional submanifold of $R^{4}$ .

Facts & Assumptions

Given: The two projection maps $F$ and $G$ .

[L1]

Two smooth maps to the same target are transverse when their differential images span the target tangent space at every common value (Transverse smooth maps).

[L2]

Transverse fibre products are embedded submanifolds (Transverse fibre products are embedded submanifolds).

Verification

technique · direct

1.1

If $F (x, y) = G (u, v)$ , then both differentials are the row matrix [L1, given, algebra] $[\begin{array}{cc} 1 & 0 \end{array}]$ , so $d F_{(x, y)} (T_{(x, y)} R^{2}) + d G_{(u, v)} (T_{(u, v)} R^{2}) = R .$ Therefore [L1] gives $F ⋔ G$ .

L1givenalgebra

2.1

By [L2], the fibre product is an embedded submanifold of $R^{4}$ . [L2, step 1.1, algebra] Its defining equation $x = u$ cuts the ambient dimension down by one, so it is $3$ -dimensional.

L2step 1.1algebra

3.1

Thus the coincidence set of two coordinate projections is a concrete fibre product of submersions.

step 2.1∎

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

Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)