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 exponential function is strictly increasing
Statement
The exponential function is continuous and strictly increasing on .
Facts & Assumptions
Given: The exponential function.
Its derivative equals itself (The exponential function is smooth and ) and it is everywhere positive (The exponential is positive and satisfies ).
The mean value theorem applies to a continuous function on a closed interval and converts a positive interior derivative into strict increase (The mean value theorem, as the case of Cauchy's: for continuous on with and differentiable on there is with ). A power-series sum is continuous at every point strictly inside its convergence interval, and the exponential series has infinite radius (The sum of a real power series is continuous at every point strictly inside its interval of convergence, The exponential series converges absolutely for every real argument).
Proof
If , the mean value theorem gives for some .
Both factors on the right are positive, so . Continuity is the cited power-series conclusion.
Depends on
The exponential series converges absolutely for every real argument
The exponential function is smooth and
The exponential is positive and satisfies
The mean value theorem, as the case of Cauchy's: for continuous on with and differentiable on there is with
The sum of a real power series is continuous at every point strictly inside its interval of convergence