Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27
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.

A simple function and its canonical representation

Example

On the Borel measurable space (R,B(R)), the function

s(x):=2 1[0,1)(x)−1[1,2)(x)

is simple. Its distinct values are 2, −1, and 0, with level sets

[0,1),[1,2),R∖[0,2).

So its canonical representation is

s=2 1[0,1)−1[1,2)+0⋅1R∖[0,2).

Facts & Assumptions

Given: The Borel measurable space (R,B(R)) and the function s(x)=2 1[0,1)(x)−1[1,2)(x).

[L1]

A measurable real-valued function with finite range is simple, and its canonical representation is the sum over its level sets. (A simple function and its canonical representation)

Verification

technique · direct
1.1given

The function s takes only the three values 2, −1, and 0, and the [given] corresponding level sets are exactly the three Borel sets displayed above. Thus s is measurable.

2.1step 1.1L1∎

Therefore [L1] identifies s as a simple function and the displayed sum as [step 1.1, L1] its canonical representation.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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