ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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 Borel representative from a random Boolean value

Example

Trace $φ (x, a)$ through the random-algebra Boolean value at the generic-real name.

Facts & Assumptions

Given: $a \in N$ , the random-real name $\dot{r}$ , and the homogeneous tail forcing $R_{\dot{r}}$ . In the random-forcing language let $ψ (\dot{r})$ say that the top condition of $R_{\dot{r}}$ forces $φ (\dot{r}, a)$ , and put $b = ∥ ψ (\dot{r}) ∥$ .

[F1]

Homogeneous truth about a generic real has Borel representatives: $b$ has an $N$ -coded Borel representative agreeing with truth on $N$ -random reals.

Verification

1.1F1

Choose Borel $B \in N$ representing $b$ modulo null. If $x$ is $N$ -random, its ultrafilter on the measure algebra contains $b$ exactly when $x \in B$ ; the random-forcing truth lemma gives $N [x] ⊨ ψ (x) ⟺ x \in B$ .

2.1F1step 1.1∎

Homogeneity makes the $R_{x}$ -Boolean value of $φ (x, a)$ either $0$ or $1$ . Consequently the actual tail-generic extension satisfies $φ (x, a)$ exactly when the top of $R_{x}$ forces it, that is, exactly when $N [x] ⊨ ψ (x)$ . Step 1.1 therefore gives $V [G] ⊨ φ (x, a) ⟺ x \in B$ . For a nongeneric $x$ no equivalence is asserted; those exceptions are removed only after placing them in the coded null set. This is a tail-forcing calculation, not upward absoluteness of arbitrary $φ$ .

Depends on

Homogeneous truth about a generic real has Borel representatives

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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.