ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Discrete, continuous, and mixed distribution functions

Example

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Three standard distribution functions illustrate three different atom patterns:

$F_{d} (x) = {\begin{cases} 0, & x < 0, \\ 1 / 2, & 0 \leq x < 1, \\ 1, & x \geq 1, \end{cases}$

$F_{c} (x) = {\begin{cases} 0, & x < 0, \\ x, & 0 \leq x \leq 1, \\ 1, & x \geq 1, \end{cases}$

$F_{m} (x) = {\begin{cases} 0, & x < 0, \\ \frac{1}{2} + \frac{x}{2}, & 0 \leq x < 1, \\ 1, & x \geq 1. \end{cases}$

The first law is purely atomic, the second has no atoms on $R$ , and the third is mixed: it has an atom at $0$ and a continuous part on $(0, 1)$ .

Facts & Assumptions

Given: The Axiom of Countable Choice and the three displayed functions.

[L1]

Assuming the Axiom of Countable Choice, distribution functions determine probability laws (Probability laws correspond to distribution functions).

[L2]

Atoms are positive point masses of the law, while continuity points are points where the distribution function is continuous (Atoms and continuity points of a law).

Verification

technique · direct

1.1

Each displayed function is nondecreasing, right-continuous, tends to $0$ at $- \infty$ , and tends to $1$ at $+ \infty$ . Therefore [L1] gives a probability law for each of them.

L1given

2.1

The jumps identify the atoms. For $F_{d}$ , the jumps are $1 / 2$ at $0$ and $1 / 2$ at $1$ , so the law is purely discrete. For $F_{c}$ there are no jumps, hence no atoms. For $F_{m}$ there is a jump of size $1 / 2$ at $0$ and no jump on $(0, 1]$ , so the law is mixed.

L1L2step 1.1

3.1

Thus the three distribution functions realize discrete, continuous, and mixed behavior without any implication that a density must exist in general.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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

J. R. Norris, Probability and Measure, Section 2.3 (standard reference, not scraped)