ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck pass

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.

Dyadic preimages under doubling

Example

Assume countable choice. For doubling $D (x) = {2 x}$ on the circle, $D^{- 1} [0, 1 / 2) = [0, 1 / 4) \cup [1 / 2, 3 / 4)$ , of measure $1 / 2$ . In contrast, $D [0, 1 / 2) = [0, 1)$ has measure one.

Facts & Assumptions

[F1]

Doubling preserves Lebesgue probability by inverse images. Doubling preserves Lebesgue measure.

Verification

Given: Assume countable choice. For doubling $D (x) = {2 x}$ on the circle, $D^{- 1} [0, 1 / 2) = [0, 1 / 4) \cup [1 / 2, 3 / 4)$ , of measure $1 / 2$ . In contrast, $D [0, 1 / 2) = [0, 1)$ has measure one.

1.1F1

On $[0, 1 / 2)$ , $D (x) = 2 x$ , and $0 \leq 2 x < 1 / 2$ is equivalent to $0 \leq x < 1 / 4$ . On $[1 / 2, 1)$ , $D (x) = 2 x - 1$ , and $0 \leq 2 x - 1 < 1 / 2$ is equivalent to $1 / 2 \leq x < 3 / 4$ . These two branches exhaust the circle and their solution intervals are disjoint. Each interval has length $1 / 4$ , so their union has measure $1 / 2$ , agreeing with the preservation in [F1].

2.11.1F1∎

For $x \in [0, 1 / 2)$ , the image $2 x$ ranges through every $y \in [0, 1)$ , with inverse $x = y / 2$ . Thus the forward image has measure one although the source has measure $1 / 2$ . In particular inverse-image preservation does not assert equality of forward-image measures. The included points 0 and 1/2 map to 0, and the excluded right endpoints 1/4 and 3/4 map to 1/2. Countable choice is inherited from the Lebesgue probability in [F1].

Remarks

In the interval model of the circle, filled endpoints are included and hollow endpoints are excluded. The upper pair of rows shows the two branches over $A = [0, 1 / 2)$ ; the lower pair shows the forward image of $B = [0, 1 / 2)$ . All rows use the same scale. The verification above is independent of the picture.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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 (standard reference, not scraped)