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.
Taylor polynomials and their remainders
Definition
Let , let have derivatives through order at , and let be the canonical embedding. The Taylor polynomial of degree at most about is The Taylor remainder is .
The factorials are natural numbers as in The factorial and the falling factorial , defined by recursion in and enter real arithmetic only through (The canonical natural of a field); they are nonzero (Canonical naturals are positive and strictly increasing). The sum and powers are those of Finite sums and finite products, by recursion, Laws of finite sums and finite products, and Integer powers . For , is the constant .
Depends on
- Higher derivatives and the classes $C^k$ and $C^\infty$
- The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$
- Finite sums and finite products, by recursion
- Laws of finite sums and finite products
- Integer powers $a^m$
- The canonical natural $\iota(n) = n \cdot 1_F$ of a field
- Canonical naturals are positive and strictly increasing
Used by
- The multivariable Taylor polynomial in multi-index notation Definition
- The Taylor polynomial of (1-x)⁻¹ at 0 has the exact geometric remainder xⁿ⁺¹/(1-x) Example
- Taylor polynomials match the prescribed derivatives at the centre Lemma
- Peano's form: the normalized Taylor remainder tends to zero Theorem
- Taylor's Schlömilch–Roche remainder formula Theorem
- The first nonzero higher derivative classifies a stationary point Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 72 results over 20 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
- MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)
- J. Lebl, Basic Analysis I, Taylor's theorem and related calculus (standard reference, not scraped)