Alphabeta MathHow 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 .