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 smooth and
Statement
The real exponential function is , and for every , In particular .
Facts & Assumptions
Given: The exponential power series.
A real power series may be differentiated termwise inside its radius (Inside its radius a real power series may be differentiated term by term, and the differentiated series has the same radius), and its sum is smooth there (A power-series sum is infinitely differentiable inside its radius and satisfies at its centre).
The radius is infinite, , and the canonical embedding preserves products and sends positive naturals to nonzero reals (The exponential series converges absolutely for every real argument, The factorial and the falling factorial , defined by recursion in , The canonical natural of a field, Canonical naturals are positive and strictly increasing).
Proof
Termwise differentiation gives .
Reindex and cancel using the factorial recurrence. The series becomes .
Smoothness follows from [L1] and the infinite radius; iterating step 2.1 gives every higher derivative.
Depends on
The exponential series converges absolutely for every real argument
Inside its radius a real power series may be differentiated term by term, and the differentiated series has the same radius
A power-series sum is infinitely differentiable inside its radius and satisfies at its centre
The factorial and the falling factorial , defined by recursion in
The canonical natural of a field
Canonical naturals are positive and strictly increasing