ExampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-generatedaudited 2026-09-22

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.

Cylinder covers generate the Frechet tails in the Raisonnier filter

Example

For fixed $n$ , enumerate the finitely many length- $n$ binary strings $s$ and use their cylinders $[s]$ as a countable cover of $L [x] \cap 2^{ω}$ . Any two distinct reals in one cylinder first differ at a coordinate at least $n$ . Hence $ω ∖ n$ belongs to $F (x)$ , concretely demonstrating that $F (x)$ extends the Fréchet filter.

Verification

Given: A real $x$ , a natural number $n$ , and the Raisonnier family $F (x)$ of the definition item.

[F1] Rapid filters and the Raisonnier family: cylinders, the first-difference function and the defining cover criterion for F(x).

1.1

Let $s_{0}, \dots, s_{2^{n} - 1}$ enumerate all binary strings of length $n$ in the canonical order and put $F_{i} = [s_{i}]$ for $i < 2^{n}$ , padded by empty sets for $i \geq 2^{n}$ . Every real in $L [x] \cap 2^{ω}$ extends exactly one of the listed strings, so $L [x] \cap 2^{ω} \subseteq ⋃_{i} F_{i}$ ; this is a countable cover of the required kind.

2.1

If $u \neq v$ both lie in one cylinder $[s]$ with $∣ s ∣ = n$ , then $u$ and $v$ agree on all coordinates below $n$ . Their first differing coordinate is therefore at least $n$ , so the prefix length defined by [F1] satisfies $h (u, v) \geq n + 1$ , and in particular $⋃_{i} H (F_{i}) \subseteq {k : k \geq n} = ω ∖ n$ .

F1step 1.1

3.1

Therefore $ω ∖ n \in F (x)$ by the defining cover criterion, for every $n < ω$ , so $F (x)$ contains the Fréchet filter.

F1step 2.1

3.2

The case $n = 0$ is included: the unique length- $0$ string has cylinder $2^{ω}$ , every pair of distinct reals in it has first differing prefix length at least $1$ , and the cover is the single set $2^{ω}$ padded by empty sets, giving $ω = ω ∖ 0 \in F (x)$ .

step 2.1

4.1

The steps above exhibit the cofinite tails as members of $F (x)$ through explicit cylinder covers, which is the claim.

step 3.1step 3.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hiromi Ishii, Regularity Properties and Inaccessible Cardinals (standard reference, not scraped)