Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-29
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.

Boundary tracking for the sine biholomorphism of the upper half-strip

Example

Let

S:={ z∈C:−π/2<Re⁡z<π/2, Im⁡z>0 }.

The boundary components of S map under sin⁡ to the real axis:

sin⁡(π/2+iy)=cosh⁡y,sin⁡(−π/2+iy)=−cosh⁡y,sin⁡x∈(−1,1) for x∈(−π/2,π/2).

Thus the sine biholomorphism of The sine map biholomorphically sends an upper half-strip onto the upper half-plane carries the whole boundary of S onto R.

Facts & Assumptions

Given: The upper half-strip S above.

[F1]

The sine map biholomorphically sends S onto the upper half-plane (The sine map biholomorphically sends an upper half-strip onto the upper half-plane).

Verification

1.1givenalgebra

For y>0, sin⁡(π/2+iy)=sin⁡(π/2)cosh⁡y+icos⁡(π/2)sinh⁡y=cosh⁡y>0, and similarly sin⁡(−π/2+iy)=−cosh⁡y<0.

2.1F1step 1.1algebra∎

For real x∈(−π/2,π/2) one has sin⁡x∈(−1,1)⊂R; together with step 1.1, every boundary component of S maps into R, and [F1] identifies the interior image as H.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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