ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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.

Transverse and tangent intersections of plane curves

Example

In $R^{2}$ , the line $L_{1} = {y = x}$ meets the parabola $P = {y = x^{2}}$ transversely at $(0, 0)$ and $(1, 1)$ , while the line $L_{0} = {y = 0}$ is tangent to $P$ at $(0, 0)$ and is not transverse there.

Facts & Assumptions

Given: The three embedded curves $L_{1}$ , $L_{0}$ , and $P$ in $R^{2}$ .

[F1]

Two embedded submanifolds are transverse when their tangent spaces span the ambient tangent space (Transverse embedded submanifolds).

[L1]

Transverse intersections have the expected codimension (Transverse embedded submanifolds intersect in the expected codimension).

Verification

technique · direct

1.1F1givenalgebra

The tangent lines to $L_{1}$ and $P$ are spanned by $(1, 1)$ and $(1, 2 x)$ , respectively. At $x = 0$ and $x = 1$ these are distinct, so their spans add to $R^{2}$ . Thus $L_{1} ⋔ P$ at both intersection points by [F1].

1.2F1L1givenalgebra

The tangent line to $L_{0}$ is spanned by $(1, 0)$ , and the tangent line to $P$ at $(0, 0)$ is also spanned by $(1, 0)$ . Their sum is only one-dimensional, so [F1] fails there. This is the tangent situation warned about by [L1].

2.1step 1.1step 1.2∎

Therefore the parabola exhibits both transverse and tangent intersections in the plane.

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)