ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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.

Base-three cylinders and their preimages

Example

Assume countable choice. For $D_{3} (x) = {3 x}$ , one has $D_{3}^{- 1} [0, 1 / 3) = [0, 1 / 9) \cup [1 / 3, 4 / 9) \cup [2 / 3, 7 / 9)$ , of measure $1 / 3$ . With $I_{n, k} = [k / 3^{n}, (k + 1) / 3^{n})$ , $D_{3} (I_{2, 3 a + c}) = I_{1, c}$ for $a, c \in {0, 1, 2}$ : applying $D_{3}$ deletes the first ternary digit.

Facts & Assumptions

[F1]

On the base-three branch $I_{1, a}$ the map is $3 x - a$ , and the intervals use half-open endpoints. Integer-base maps and b-adic circle intervals.

[F2]

Integer-base maps preserve Lebesgue probability under countable choice. Integer-base circle maps preserve Lebesgue measure.

Verification

Given: Assume countable choice. For $D_{3} (x) = {3 x}$ , one has $D_{3}^{- 1} [0, 1 / 3) = [0, 1 / 9) \cup [1 / 3, 4 / 9) \cup [2 / 3, 7 / 9)$ , of measure $1 / 3$ . With $I_{n, k} = [k / 3^{n}, (k + 1) / 3^{n})$ , $D_{3} (I_{2, 3 a + c}) = I_{1, c}$ for $a, c \in {0, 1, 2}$ : applying $D_{3}$ deletes the first ternary digit.

1.1

For $x \in [a / 3, (a + 1) / 3)$ with $a = 0, 1, 2$ , [F1] gives $D_{3} x = 3 x - a$ . The condition $0 \leq 3 x - a < 1 / 3$ is equivalent to $a / 3 \leq x < a / 3 + 1 / 9$ . Substitution of the three values of $a$ gives exactly the three stated disjoint intervals. Their lengths add to $3 (1 / 9) = 1 / 3$ , as required by [F2].

F1F2

2.1

The interval $I_{2, 3 a + c} = [a / 3 + c / 9, a / 3 + (c + 1) / 9)$ lies in branch $a$ . Its affine image under $3 x - a$ is $[c / 3, (c + 1) / 3) = I_{1, c}$ . Surjectivity onto that interval is explicit: for $y \in I_{1, c}$ use $x = (y + a) / 3$ , which lies in $I_{2, 3 a + c}$ . The included left and excluded right endpoints are preserved by this increasing affine map, including $a = c = 2$ where the right endpoint is the excluded point 1. No ambiguous choice of ternary expansion is required. Countable choice enters only in the Lebesgue measure statement of [F2].

1.1F1F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

E–W Example 2.4 pp.14–15, base-three specialization; MT-22 binding example amendment (standard reference, not scraped)