ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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.

The boundary of an oriented interval

Example

With the standard orientation on $[a, b]$ , the induced orientation of its boundary is $\partial [a, b] = {b} - {a}$ .

Facts & Assumptions

Given: Real numbers $a < b$ , with $[a, b]$ oriented by the positive tangent $\partial_{x}$ .

[L1]

Boundary orientation uses the outward-normal-first rule (Induced boundary orientation).

[L2]

The orientation obtained from that rule does not depend on the chosen outward vector (Boundary orientation is independent of the outward vector field).

Verification

technique · direct

1.1givenalgebra

The vectors $+ \partial_{x}$ at $b$ and $- \partial_{x}$ at $a$ point out of the interval.

2.1givenL1L2step 1.1algebra∎

At $b$ , the outward vector $+ \partial_{x}$ is positive, so [L1] gives the point $b$ the positive orientation. At $a$ , the outward vector $- \partial_{x}$ is negative, so [L1] gives $a$ the negative orientation; [L2] makes these conclusions independent of the particular outward vectors. Hence $\partial [a, b] = {b} - {a}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)