ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-02

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.

Complex sine is unbounded on the imaginary axis

Example

For real $t$ , $\sin (i t) = i \sinh t$ , so complex sine is unbounded on the imaginary axis. The conventions and prerequisite facts used below are recorded in The exponential formulas, real restrictions, and trigonometric-hyperbolic dictionary over $C$ , The exponential tends to $+ \infty$ at $+ \infty$ and to $0$ at $- \infty$ .

Facts & Assumptions

Given: A real parameter $t$ .

Verification

1.1

The dictionary gives $\sin (i t) = i \sinh t$ .

given

2.1

Since $\sinh t = (e^{t} - e^{- t}) / 2$ is unbounded as $t \to \infty$ , so is its modulus.

given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)