LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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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.

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Equality near a point is an equivalence relation on local smooth functions

Statement

Equality on some neighbourhood of a fixed point $p$ is an equivalence relation on smooth real-valued functions defined near $p$ .

Facts & Assumptions

Given: Smooth functions defined on open neighbourhoods of a fixed point $p$ .

[F1]

The germ relation declares $(U, f)$ and $(V, g)$ equivalent when $f$ and $g$ agree on some open neighbourhood of $p$ inside $U \cap V$ (The germ of a smooth function at a point).

Proof

technique · direct

1.1F1given

Reflexivity holds because each function agrees with itself on its whole domain, and symmetry holds because equality of functions is symmetric.

1.2F1given

If $(U, f)$ agrees with $(V, g)$ on a neighbourhood $W_{1}$ of $p$ and $(V, g)$ agrees with $(Z, h)$ on a neighbourhood $W_{2}$ of $p$ , then all three agree on the neighbourhood $W_{1} \cap W_{2}$ of $p$ , so transitivity holds.

2.1step 1.1step 1.2∎

Therefore the relation is reflexive, symmetric, and transitive, hence an equivalence relation.

Depends on

The germ of a smooth function at a point

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by The germ of a smooth function at a point.

Dependency tree · two levels

2 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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)