DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-16

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.

Simple continued fractions, convergents, and the integer-coordinate coding of $N^{N}$

Definition

Define a bijection $z : N \to Z$ by $z (2 k) = k$ and $z (2 k + 1) = - (k + 1)$ ; the division algorithm makes these two cases exhaustive (Division with remainder in $Z$ : for $a \in Z$ and $b > 0$ there are unique $q, r \in Z$ with $a = q b + r$ and $0 \leq r < b$ ). For $x \in N^{N}$ put $a_{0} = z (x_{0})$ and $a_{n} = x_{n} + 1$ for $n \geq 1$ . Its finite simple continued fractions $[a_{0}; \dots, a_{n}]$ are defined by recursion on the length, evaluated in $Q$ (The rationals as equivalence classes of pairs of integers, Arithmetic on the rationals): $[a_{n}] : = a_{n}, [a_{k}; a_{k + 1}, \dots, a_{n}] : = a_{k} + \frac{1}{[a_{k + 1}; \dots, a_{n}]} (k < n) .$ The recursion never divides by zero, because $a_{j} \geq 1$ for $j \geq 1$ makes every tail value $[a_{k + 1}; \dots, a_{n}]$ at least $1$ . A finite prefix determines the cylinder of all codes extending it. Infinite continued-fraction values are established, rather than assumed, in Infinite simple continued fractions parametrise the irrational real numbers.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 81 results over 18 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

David Marker, Descriptive Set Theory, §§1–2 (standard reference, not scraped)
Michael Kunzinger, General Topology, §§11.3–11.4 (standard reference, not scraped)
MFF General Topology course summary, §4.3 (standard reference, not scraped)
Jesse Peterson, Real Analysis, §§3.6–3.7 (standard reference, not scraped)

Simple continued fractions, convergents, and the integer-coordinate coding of NNNN