Alphabeta Math
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 ∂[a,b]={b}−{a}.

Facts & Assumptions

Given: Real numbers a<b, with [a,b] oriented by the positive tangent ∂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
1.1

The vectors +∂x at b and −∂x at a point out of the interval.

givenalgebra
2.1

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

givenL1L2step 1.1algebra∎

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