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.

A value outside the image can still be regular

Example

For the smooth map $f : R \to R$ , $f (x) = e^{x}$ , the value $0$ is a regular value even though $f^{- 1} (0) = \emptyset$ .

Facts & Assumptions

Given: The map $f (x) = e^{x}$ .

[F1]

The definition of regular value allows the fibre to be empty (Regular and critical points and values).

[L1]

The derivative of $e^{x}$ is $e^{x}$ (The exponential function is smooth and $(\exp)^{'} = \exp$ ), and the exponential maps $R$ bijectively onto $(0, \infty)$ (The exponential is a continuous bijection from $R$ onto $(0, \infty)$ ).

Verification

technique · direct

1.1

The image of $f$ is $(0, \infty)$ , so $f^{- 1} (0) = \emptyset$ .

given

2.1

By [F1], regularity of $0$ asks whether every point of the empty fibre is regular, which is vacuously true. The derivative fact [L1] is compatible with that conclusion because the map has no critical point over $0$ .

F1L1step 1.1

3.1

Hence $0$ is a regular value although it is not attained.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

Nigel Hitchin, Differentiable Manifolds, Theorem 3.3 context (standard reference, not scraped)