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 principal inverse tangent
Definition
By Tangent is a continuous strictly increasing bijection from onto , tangent restricts to a continuous strictly increasing bijection
Its inverse is the principal inverse tangent
Thus for every real , while precisely for in the displayed principal interval. The inverse is continuous and strictly increasing by Continuous inverse theorem: a continuous injective on an interval is a bijection onto the order-convex set , and the inverse is continuous and strictly monotone in the same sense as .
Depends on
- Tangent is a continuous strictly increasing bijection from $(-\pi/2,\pi/2)$ onto $\mathbb R$
- Continuous inverse theorem: a continuous injective $f$ on an interval $I$ is a bijection onto the order-convex set $f[I]$, and the inverse $g : f[I] \to I$ is continuous and strictly monotone in the same sense as $f$
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 69 results over 22 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- NIST Digital Library of Mathematical Functions, §4.23 Inverse Trigonometric Functions (standard reference, not scraped)